P R EFACE

This wo rk outlines for students of the third and fourth high-school y ears a more a dvanced and more thorough course in a pplied business ma thematics than the ordinary first- ea u y r co rse in elementary commercial arithmetic . The attempt has b een made to construct a practical course which will contain all the essential mathematica l knowledge

e e a b s ess a ee e h as em ee ma a e r quir d in u in c r r , it er ploy , n g r , or employer . The fa ct that the field has been covered in this text both more intensively andmore comprehensively than it has yet bee e e the text and the a e a ct that the n cov r d in o r s, dd d f ’ material gathere d together has stood the test of six years e xperience in the tea ching of large and varie d classes of the

th ea a t h h s h seem s f e t a a t four y r in ci y ig c ool , u fici n w rr n for it s publication . The work is a da pted no t only for use in the classroom but also as a reference manual for those a ctively engaged in

Th s it b e u a a t a e business life . u will fo nd pr c ic l guid for young employ ee s who wish through private study to master “ the fundamental mathematics involved in running a busi ” h ab at s ms st at e exam es ha ts ness . T e t ul ion , for , illu r iv pl , c r ,

ands m e es are all a ab e logarithmic applications, i pl rul , pplic l to the financial and other mathematical problems which

e e th s s e a b s business presents . Lack of knowl dg of i id of u i

s ab t to out its mathemat s o te es ts nes , or in ili y work ic , f n r ul in h apha zard guessing where a ccurate and careful calcula tions are re quired . iv PRE FACE

The m aterial has been submitte d to the criticism of many prominent business m en and spe cialists in the commercial

e m h m a ab e s est s and t ms ha fi ld , fro w o v lu l ugg ion cri icis ve bee M a b s ess b at s ha e b n received . ny u in pu lic ion v een drawn upon for illustrative material of various kinds : such “ b ” ’ s as Th . S . t t t n ook e U Sta is ical A las, Finne y a dBrown s

” ’ “ M e s ess th et M e an od rn Bu in Ari m ic, oor dMiner s Co n ” ’ “ se s ess r thmet Van Tu l s C mm h ci Bu in A i ic , y o ercial Arit ” ’ ” met t s G a h Meth s ese t a ic, Brin on r p ic od for Pr n ing F cts, ’ “ ” ’ e E eme ta Ma a Stat st s an K Bowl y s l n ry nu l of i ic , d ester s ” C at t andAuditin ands h e orpor ion Accoun ing g , uc p riodi

’ a s as The Annalzst ss e b The New York Ti mes an c l , i u d y , d

s The e ema s to a e e h he Sy tem. r r in cknowl dg t e lpful sug t h es . . D . s e e e m s P . a h g ion r c iv d fro J J Hopkin , , Princip l of t e D h h n Wm . . s S a d a T bbetts e L ickin on Hig c ool , Fr nk i , H ad

the C mme a De a tme t the same s h as of o rci l p r n of c ool , Well as the t a ea the ma s t b a cri ic l r ding of nu crip y H mlet P .

Collins .

W th hate e a e a th s ha a te is e a e i w v r c r work of i c r c r pr p r d , it is ea e that e s are ab e to ee r liz d rror li l cr p in undiscovered . The authors will be grateful to receive andglad to a ckno wl edge any corrections o r criticisms the reader may care to ff o er .

EDW AR D I . EDGE RTON

TH L WALLAC E R . BAR O OME W CONTENTS

CH A PTE R

I . Sales and Profits Statistics

I I . Profits Base d on Sales P a - I I I . y roll Calcul ations

IV. I nterest D V . epreciation

I a VI . nsur nce

VI I . E x ha e c ng .

VI II . Taxes

I te es a IX. n r t on B nk Accounts

X. Building and Loan Associations

a h a e ese tat XI . Gr p ic l R pr n ion

I h M th s and Che s XI . S ort e od ck

a es m e and We hte XII I . Aver g , Si pl ig d

The ess s XI V. Progr ion

a thms XV. Log ri

a at s o f a thm XVI . Commerci l Applic ion Log ri s

h R e XVII . T e Slide ul

m ate N mbe s XVI I I . Deno in u r

a t a Meas eme ts XIX. Pr c ic l ur n APPE NDIX- Tables and Formulas FORMS

FORM PAGE

1 . e e iati Cha t sh R ate e e at C m te D pr c on r . owing of D pr ci ion o pu d According to the Straight Line Method 5° E xpress Money Order

3. Sight Draft Bank Draft ’ Bank ers Bill of Exchange Lette r of Credit ’ Travelers Che ck Circle Chart Showing Distribution o fI ncome Re ctangle Chart A Variation of the Straight-Line Graph Curve Graph Conl parative Curves P eriod Chart Composite Chart Showing R elation Between I ncome and Outgo Chart Showing Component P arts Correlative andCumulative Curves Map Chart Fre quency Curves Showing Changes 1n Costs The Slide Rule BUSINESS MATHEMATICS

CHAPTER I

SALES AND PR OFITS STATI STICS

1 U e — . s of Comparative R e cords Every business is carried

th The th s on for e purpose of selling somethi ng at a profit . ing

s ma be the s th tai e h esa e ma old y good of e re l r, w ol l r, or nu

a t re the se n a ba an f c u r, or rvices of a advertising concern , nk,

No matte s a e m a b t at . in ur nc co p ny , or pu lic u ility corpor ion r h t th m w a e kind of organization or enterprise, co parative figures of its sales and profits pla y an important part in its management . The figures should be compiled regularly andtabulated to show increases or decreases covering corresponding periods of

h ta a ma be ma e to sh the t e time . T e bul tions y d ow r nd of

es s sa esme sales or profits by departments, lin of good , or l n , or

The the y may be worked out for the business as a whole . computations involved seldom require more than the use of simple arithmetic and percentage calculations.

e P e enta e ures —I n a t to sh 2. Comparativ rc g Fig ddi ion ow

a t t es it is a a ta e s to ing sales and profits by qu n i i , dv n g ou

a e es and sh reduce t hese quantities to percent g figur , ow ih

I n e e ta es the creases or decreases in this way. finding p rc n g

ma e the ast e ma e proper adjustment should be d in l d ci l figur , up or down as the case requires . 1 2 BUSINESS MATHEMATICS

The t ta sa a e th are I llustrati ve Exampl e . o l les for giv n mon

r a s a . of which sales amounting to a e credited to s le m n A Find,

e t to two e a a es the er e t t ta sa es ma e . corr c d cim l pl c , p c n of o l l d by A SOLUTI ON : 12 7 . 3 8 , or Adj usted to two de cimals the last figure is

— h ab t m t com a e R e c ds . T e t a s s 3. Tabul ated S l s or ul ion o

r th se e the a ee andm th mouly used a e o cov ring d ily , w kly, on ly

C m ma b e e sales by departments or by salesmen . olu ns y rul d to show increases or decreases both by quantities and per

he m at e es to ate the ave a e centages or t cu ul iv figur d , or r g Th mb figures reduced to percentages . e possible co inations h are numerous andare determined by t e kind of business .

De artme t — Th e the 4 . Month ly Sales by p n s e figur s in tabulation shown below represent the monthly department

The t ta s at the t ea h sales totals for the year . o l foo of c column ll m give the annual sa les of a depart ents . Supply th e missing totals at th e foot of each column andin the las t column .

MONTH LY R E CORD OF SA LE S B Y DE PAR TME NTS

2 D E . E E . T 4 DE . TO L MON TH S D P T P 3 D P T. PT 5 TA SALES AND PROFITS STATISTICS 3

D — 5. aily Recordof Sale s by Departments The figures in the following tabulation give the daily sales in each depart ment and are designed to show total daily andweekly sales b th b e a tme ts and the b h o y d p r n for usiness as a whole. I n t e ’ last li ne is shown the percentage Of each day s sales to the

a t ta Of sa es and the gr nd o l l , in last column the percentage of ’ each department s weekly sales to the grand total of sales is shown .

S the e uire dt t upply r q o al s , andcompute the percentages in each ase a be c s sh own low .

C OMPARATIVE DAI LY R ECORD OF SALE S B Y DE PARTMENTS

i Week beg nning .

W E D . TH R S . MON . TU E s . DE P T. U

1 .

3.

To tal .

— al es b Sa esme . The tab l at be is 6. Month ly S y l n u ion low designed to furnish the monthly total sales by salesmen ; the ’ are ea h sa esma s eek sa figures entered thereon c l n w ly les. ’ l men s t t s This give s a comparison of sa es o al . ndm nth t ta s and m te Compute the weekly a o ly o l , co pu the ma ’s sal es to the t ta m th percentage of each sal e s n o l on ly sales correct to the neare st tenth per cent. 4 BUSINESS MATHEMATICS

COMPARATIVE MONTH LY R E CORD o r SALE S

' SALE SM AN S NUMB E R

er Cen ease Decrea - h 7 . P t of Incr or se T e tabulation be low gives the daily departmental sales figures andis designed h th e ta e ease to s ow e perc n g of incr or decrease in each case . Compute th e percentages corre ct to one de cimal place of per cent andindicate a decre ase by an asterisk or by

redink .

COMPAR ATI VE SALES FOR CORRE S PONDING DAYS OF Two YE ARS

S LE S Wan. A . I NCR E ASE DE CRE AS E DE C. 4 , [ 9 1 9 SALES AND PROFITS STATISTICS 5

— 8 . P er Cent of Average I t is sometimes desirable to com

a e the eek m th a tme t p r w ly or on ly s les of a clerk or depar n , with the average weekly or monthly figures as the case may

be .

In the following table compute the total and average sales and the per cent of th e sale s of e ach clerk to the

average figure s .

MONTH LY SA LE S o r A NUMB E R OF CLE R KS

' C LE R K S NUM BE R SALE S P E R CE NT OF AVE R AGE

c c c c c c c

o o o o o o o o o o o o o o o o

To tal

Avera ge

— ed— P er Cent I n s me es b s ess 9 . Sale s R eturn o lin of u in Th it is desirable to keep a close watch on goods returned . is

can be effectively done by means of the percentage figures

shown in the two following tables. de ar tmental sal es returned Compu te the per cent of th e p , e er cent of all sal es returned to t tal to total sales , andth p , o sale s . 6 BUSINESS MATHE MATI CS

SALE S AND R ETURNE D G OODS R Y DE P AR TME NTS

Year ending

R w as 333?

To ta l

1 - — 0. Average Net Sale s per Ch eck Compute th e to tal and ne t sale s and th e average ne t sal e s per ch e ck for e ach clerk .

I ND I V ID UAL DA I LY SALE S SH E E T

Se t 13 G s ate Ma 11 19 c ion , Dry ood D , y ,

R E TUR NE D AVE R A G E NE T NE T SALE S C H E C KS G o o ns SA LE S P E R C H EC K

3

To ta l SALES AND PROFITS STATISTICS 7

11 Tab — . ulations fo r O ther Comparative Purposes Tabu latio ns similar to those given may be used to compare sales

m m th to m th and m ea t nd a s the fro on on fro y r o year, a l o month of one year with the same month in preceding

Sh ow th e month ly increas e or decrease as th e case may re quir e andcompute th e percentage of increase or decrease of sal e s u n th e e d ri g pr sent year .

COMPA RI SON OF SALE S B Y COR RE S PONDING MONTH S

Sa es a Doe l m n , John

SALE S SALE S OF O F M ONTH I NC R E AS E DE C R E AS E LAS T YE AR TH I S YE A R 1NCR E ASE DE CR E ASE

0 0 0 0 0 0 0

' R rd S a es — B mea s of this m 12. Cumul ative eco of l y n for ' we may have no t only a very complete record of each de ’ a st m partment s sa les by any particular ro d or ore sales an th but a s a m a at e e the from month to mon , l o co p r iv r cord of h m e m ths an ea total sales for two or t ree or or on of y y r , or n a comparison of these totals for a y previous year since this particular salesman has been connected with the business. 8 BUSINESS MATHE MATICS

Compute th e cumul ative sal es to th e end of April and sh ow th e average at th e foot of each column .

’ SALE SMAN s CUMULATI VE R E CORD OF SALE S B Y DE PARTME NTS

i ms . W a Salesman , H ill

DE E R AR Y RI L PT. F E U AP

3 546 . 3

Average .

G ain — I n th 13. Computation of Loss or figuring e profits hethe b e a tme ts the b s ess as of a business, w r y d p r n or for u in

he e ct the st the s s m the a whole, t d du ion of co of good old , fro

net sa es es the a the e t me goods sold or l , giv g in for p riod of i

To eterm h h covered by the figures . d ine t e cost of t e goods so ld it is necessary to deduct the cOst of the goods unsold at the endof the month or year from the purchases made during

ss m that the r h the same period . A u ing re a e no goods on and ,

e n e t at the be the e the i . e ., no op ni g inv n ory ginning of p riod , computation would be as follows : C Goods sold (sales) 7 Purchases . 8 end ea Cost of goods unsold (inventory of y r) .

’ Cost of goods sold .

Gain

Gain SALES AND PROFITS STATI STI CS 9

WRI TTEN EXERCI SES

1 . G s s the ea ood old during y r . Original cost of the goods Cost of goods unsold (inventory end) the a and h Find g in t e gain per cent .

2 . Goods sold during the year Original cost of the goods Cost of goods

Find the gain andthe gain per cent .

3.

P urchases . I nventory at endof year

Find the loss andthe loss per cent .

t n e i I e t e s — ss m 14 . Accoun i g for Op n ng nv n ori A u ing

h the a the ma b m opening inventory , t en g in for period y e co

G e the sa es e t es and as sh b e . puted own low iv n l , inv n ori , h s s th s a ma be e b an the cost of t e good old , i pl n y follow d y y business to arrive at gain or loss .

Il lustrative Example . Sales

Opening inventory .

Purchases of year . .

Total Closing inventory

Difference

WRI TTEN EXER CI SES

ha a a 1 1920 me ha t had s , , 1 . A rc n good on nd J nu ry s to the a t d Dun ng that year he purchased good moun of an H is e t at the end sold goods to the amount of inv n ory of ha that year showed goods on nd

Find his gain or loss for the year . 10 BUSINESS MATHEMATICS

2. I e tor at th b i th a nv n y e eg nning of e ye r . I nventory at the endof the ye ar P urchases during the year

Sales during the year . th Find e gai n or loss.

Sales during the year Beginning inventory E ndinventory

Find the gain or loss .

Find the gain or loss pe r ce nt .

a Re s and Stateme ts — I 15. Comp rative cord n ncreases or

a tme ts sa esme s decreases in profits of dep r n , l n, lines of good ,

ses and sa es are somet mes m a e as st at expen , l , i co p r d illu r ed

e ta es s a e the ai e m a s . below . P rcen g u u lly giv f r r co p ri on

WRITTEN EXERCISES

m ete the re r s to sh w th e e ui edt ta s and e e ta . 1 . Co pl co d o r q r o l p rc n ges

’ SALE SM AN S RE coRD o r COM PAR ATI VE SALE S R Y DE P ARTME NTS

Sa es a . l m n , A Jones

FI RS T YE AR SE COND YE AR

Sal es Profits P rofits

32

To ta l .

12 BUSINESS MATHEMATI CS

P R OFI T AND Loss STATE ME NT — Sales Le ss R eturned Go ods Les: Disco unt and Freight Allo wa nce 7

Cost o f Go o ds So ld

Add I nvento r a nuar 1 y . J y

Less I nventor De y. cember 31 .

Add: Purchase Cash Disco unt I nterest R eceived Miscellaneo us I ncome

Less Selling E xpense Admin—istratio n E xpense . Taxes State and Federal I nterest Paid

R eserve for Bad Debts Miscellaneo us E xpense

100% CHAPTER II

PROFITS BASED ON SALES

1 - — 6. M ethods of Marking up Goods I n the marking of

' goods bought for resale the percentage of profit may be added to the st the s h h is h co price of good , w ic t eir invoice price plus freight and cartage ; or the percentage of profit may be com

ute h h h i hei t p don t e sales, w ic s t r cos price plus the expense

Th b s h of carrying on the business . e u iness man w o adds a certain per cent of profit to the cost price of the goods rarely

h m h t he is a t a mak e a se he knows ow uc profi c u lly ing, b c u rarely knows how much the overhead expense of carrying on hi n hi h i . If sta a ar a d s t e business s , for in nce, s s les e expenses for the year are it is apparent that each $1 2 2 to se . Th o ma e 0 worth of goods costs $ 0 ll erefore, t k % clear profit on his goods he must first add20% to the cost

ss o st h h 20 price to give him their gro c price, and t en t e %

the sa es e t the st to bu profit required . Thus l r presen co y andsell .

Ex e ses — e hea ex ses are th se in 17. Overhead p n Ov r d pen o

— s h as e t taxes sa ar es ht curred in doing business uc r n , , l i , lig

a e te e h e a e t s sta e e e and heat, insur nc , l p on , dv r i ing, po g , d pr These ex e ses m st be ta e t s e a ciation, etc. p n u k n in o con id r

- h x e es s a tion when marking up goods . Over ead e p ns u u lly

s a s n m eX eri have a fairly constant ratio to gro s s le , a dfro p

Th s er ence the merchant determines what this ratio is. i p centage plus the percentage of profit decided upon is deducted — — fro m 100% representing the selling price to determine the 13 14 BUSINESS MATHEMATICS percentage which the cost of goods plus freight bears to the se lling price . The cost andfreight figured in dollars divided b th s e e ta Thi s y i p rc n ge will give the selli ng price in dollars . allows for the computation of the overhead andgain as a per

‘ e ta e th c n g of e sales .

I st at e Ex a t i 2 llu r iv ample . An r icle s invoice d at $ 3 overhead charges t to 12 a sa es andthe a is sa es e t is 1 . i moun % of l g in 8 % of l ; fr igh $ F nd, th e selling price .

SOLUTI ON : 12% 8 % 20% 100 % 20 % 80 % 23 1 24 the t $ $ $ , cos 24 80 30 the se e $ % $ , lling pric

CH E CK : 20 30 6 the e hea ha e % of $ $ , ov r d c rg s 30 6 24 the st $ $ $ , co

Ove rhe ad on S al es Net P rofit on Sales G ross P rofit o n Sal es 100 % or S elling P ric e G ross P rofit Cost of S al es Invoice P ri ce Freight Cost Cost Cost of S al es S elling P rice

ORAL EXER CI SES

1 . If an a t e st 50 is s 60 at is the a er e t r icl co ing $ old for $ , wh g in p c n on the cost? What is the gain per cent on the selling price ?

2 . t 2 nd w s s What is the r An article cos $ 5a a old for $30 . pe cent of gain on the cost? On the selling price?

a t e is s 60 . I f th a th 3. An r icl old for $ e g in on e selling price is what is the gain? What is the cost? What is the gain per cent on the cost?

4 . man s an a t e 8 and a e the se li A old r icl for $ m d on l ng price . What was the per cent of profit on the cost? hi h t 5. e a t s an a t e 15e ts s 12 A m rch n old r icl for c n , w c co him cents . Wha t per cent profit is that on the cost? On the selling price? d th a it t l 6 . I f I an a t e 1 an e o se buy r icl for $ , n m rk l for how much must it be reduced to sell at cost? What per cent is this on the marked price? On the cost? PROFI TS BASE D ON SALES 15

7 . If an a t e st 1 and i ta e r icl co $ , you sell t for what percen g of t a profi do you m ke, minus overhead? I f 8 . overhead expense is what will an article that cost $1 and you sell for figure as profit?

WRITTE N EXER CI SES

1 . Co mplete the followi ng form .

o r R E DUCTI ON To NUMBE R MAR KE D PR I CE PR ODUCE Co s,

r e t the se l e the l wi 2. Find the gain pe c n on l ing pric of fol o ng

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

l s The e hea ha es are and 3. A man sel s good for ov r d c rg h i e the 5. t e e the profits The freight is $12 Find invoic pr c of goods. 16 BUSINESS MATHE MATICS

4. An automobile is invoiced at The freight charges are $50 . If we a 15 e ea and 15 t at s be the llow % for ov rh d % for profi , wh hould selling price of the automobile? e 5. An invoice of merchandise amoun ts to If the overh ad ha es are a h 5find the se li c rg the gain nd t e freight $12 , l ng pri ce . ’ 6 . The e st a a s a t is 30 the e hea a es are invoic co of l dy co $ , ov r d ch rg

Fi the se in e . the profit is andthe freight is $2 . nd ll g pric

7 . 2 h in to the A merchant marked goods at 0 % above t e cost . Ow g fact that these goods did no t sell well he re duced them 20 % and th t his claimed that he was selling the m o ut at cost . Find e amoun of e I f h had e e t e he rror in per cent . e r duc d h m how much would have lost? f s t e o as to a 25 an 8 . I a man buys some articles andmark h m s g in d the e es the t st e to m e the at er e t st n r duc m o co in ord r ov m , wh p c n mu he reduce them on the marked price?

9 . The e st an a t e was The e ht is 2 5the invoic co of r icl fr ig $ , overhead charges are 20 and the profit is compute d at Find h t e selling price .

10 . eta e ta e t 40 h h he a s to sel at a t A r il r buys bl s a $ , w ic m rk l profi 4 a t s s ess he e es to eta of 0 % on the cost . On ccoun of low bu in d cid r il At at e h e l t ? them at 25% less on the marked price . wh pric does e s l hem Does he gain or lose andho w much? What per cent is this on the selling price? i 1 i The e t 50 . If we 1 . An automobile s invoiced at fr igh s $ a 10 e ea and20 t hat sh be the se llow % for ov rh d, % for profi , w ould lling price of the automobil e ? C 12. omplete the following form

COST G A I N

— P er Cent of C st The tab es e 18 . Adding o l giv n on the “ ” remaining pages of thi s chapter are short-cuts for qui ckly calculating profits and selling prices . PROFITS BASED ON SALES 17

The following table shows the percentage of cost which must be added to effect a given percentage of profit on sales .

To MAK E PR OFI T TO MAKE PR OFI T ON SALE S O N SALE S

— utin Ne t P ro fits The tabl e sh s the 19 . Comp g following ow he the er e t ex e se to per cent of the net profit w n p c n of p n - sales andthe per cent of mark up are known . If the cost of doing business figured on sales is represented

h tab e be and the ma -u the in the to p line of t e l low, rk p on the st m to goods is one of the p ercentages shown in fir colu n

the e e ta e net t be at the the left, p rc n g of profi will found junc tion of the line and column .

2 18 BUSINESS MATHEMATICS

w « R w 0 m N m 0 A 2 m

3 m « E 2 2 N 8

' v-i oc g Q Q

“ x 3 N m : 2 ” S m

“ « “ S m m o 3 H 2 u m n

a m « 5N n o S 3 h A “ m“ N a m

m A r S 8 m 2 : m m m. m

0 « “ a m S 2 : a N 8 S

20 BUSI NESS MATHEMATI CS

SOL TI ON : a e the es 5 the e i th 19 h is U T k figur in column on lin w , whic

6 h e . 7 t e selling pric

The percentage of cost of do ing business and profit are figured on selling price

— 21 . P er Cent of Profit o n S ales The following table shows the per cents of profit made on the selling price when the per cents shown in the first column ar e added to cost of the goods sold .

TABLE F OR COMPUTING P ROFIT

added to cost profit on selling price ( 1 H 7% U l ‘ N ( 1

I l %% PROFITS BASED ON SALES 21

WRI TTEN EXER CI SE S

l . A man buys an article for $125andwishes to make 20% profit on sales . How much profit shall he compute on the cost? m 2. A erchant buys an article for a ce rtain sum Of money andwishes to ma it to se so that he can ma e the st hat er ent rk ll k on co . W p c pro fit is this equivalent to when compute d on the selling price?

8 . me ha t ma s an a t l e to se 200 the e mak in 33 A rc n rk r ic ll for $ , r by g } o n the . the cost Findthe cost . What is the equivalent per cent on selling price? 4 If . your co st of doing bus iness is 12 % of gross sales andyou mark a ine at 50 ab e st i l % ov co , what s your net profit on sales? If m ess 5. you ark a line 75% above cost andyour cost of doing busin is 16 ss sa es hat is net r t sal es? % of your gro l , w your p ofi on is 6. If goods sold amount to andyour cost of doing b usiness nd h th h hat is the a you ave marked e line at 40% above t e cost, w net profit on the sales? What is the cost?

7 . If bu an a t e 12 andthe e t is 5and es re you y r icl for $ , fr igh $ 7 , you d i n t andit t 1 t s ss hat is the to make a et profi of cos s 8% o do bu ine , w selling price? r e net t ti e h h sts 8 . If you desi e to mak a profi of 8 % on an ar cl w ic co n it s s 1 to b s ess at s b the se i e a d co t 6% do u in , wh hould e ll ng pric of that article? i e an a t e is 100 the net t is cost 9. If the sell ng pric of r icl $ , profi th s that a of doing business is find e co t of rticle . s an a t e 100 andma es 26 the sel in 10. If a man se ll r icl for $ k % on l g r es he ma e the st? price, what pe cent do k on co CHAPTE R I I I

PAY-R OLL CALCULATI O NS

2 M h — ma a t 2 . e t ods of Wage P ayme nt . Every nuf c uring business and many mercantile concerns maintain a pay

m Th f h i o ee roll depart ent . e duty o t is de partme nt s t k p ’ the employees time re cords or re cords of the quantity of

an at th en h he e to work done , d, e d of t e wee k or ot r p riod , compute the wages or salary earne d by each employee .

he e is a b the h da ee a W r work p id for y our , y, or w k , or ccord

to the mbe e es ma e he m t the a ing nu r of pi c d , t co pu ing of p y roll may b e a simple arithmetical problem of multiplication . Where scales of wa ges vary with the efficiency of ea ch

ma and he e i s e a e work n , w r good work or quick work r w rd d by the pa yment of a premium or bonus in a ddition to the

a h e e - ate a e h b em ma e regul r ourly or pi c r w g , t e pro l y involv Th intricate fra ctional and percentage calculations . e method of computing the premium or bonus may be based upon the number of pieces produced within a given time or upon the amount of time saved in the performance of a

given operation .

Effi e a m — 23. ci ncy P y ent Systems The more complicated methods of wage payment are met with in the highly o r ganiz edmechanical industries where the modern method of “ ” mana gement known as scientific management is in

l D f e t t e creasing y employ ed . i f ren yp s of industry often adopt different methods ; andthe efficiency expert who has 22 PAY- ROLL CALCULATIONS 23 been responsible for the introduction of a special system of wage payment usually gives it his name to distinguish it h from ot ers.

a - a — 24 . D y R te System Wages which are b ased on time

e are s a m n h h n work d u u lly co puted at a ourly rate, wit o e and o ne- half times the regular ra te for overtime and ho liday

and s m me e h and work, o eti s doubl time for oliday work h Sundays . T e pra ctice with respect to overtime wages

t ff The t me varies wi h di erent manufacturing concerns . i ’ clock is generally used to record ea ch employee s time , and the time cards serve as a basis for computing the wages of the employees .

WRI TTEN EXERCISES

i se ti a a - the e a i da is 1 . I n the follow ng c on of p y roll, r gul r work ng y

If man s e t a 8 hr . an s e assume d to be 8 hr . a work mor h n on y ingl h i a t e and a h a e t e a th s e a s he day, e s p id im lf for ov r im , l ough on om d y h may work less tha n the numbe r of hours in t e standard day . M ak e th e required computations to S how th e wages due to each employee .

H O UR S P E R DAY

NAME M T W T F S 24 BUSINESS MATHEMATI CS

- 2 . R e a a e the re e e te the ata. ul p y roll lik p c ding model , n r following d and he a a is 8 find t mount of wages due to each employee . A full d y hr andt e and a al is a im h f p id for overtime .

' No EM LOY E E S N M E TE . P A H OURLY RA 1 Henry Jones h 2 Wm . Jo nson 30 Bel 3 Chas . l 32

4 . T ha A . Wig m 35

R . a t 5 G . M r in 34

rin the Wee e Oct . 13 t me a s e e t e h Du g k nding , i c rd w r urn d in by t e ema sh the e h s e ea h e ee b for n, owing numb r of our work d by c mploy to e a s follows : 1 0 , 3, 8 ; 4, 2 ; , 8 ; 3, 9 ; We es a 1 8 2 9 3 8 4 8 57 dn d y , , ; , ; , ; ,

; 2,

; 2, 7 ; 2 Sat a . 8 3 8 4 56 urd y , ; , ; , 8 ; ,

The cashier has already advance d the following s ums : to

5No . 5 No . 3, 3 ; ,

Wo rkman No . 1 . 8 hr .

— I k S tem . 25. P i ecewor ys nstead of basing the wage rate

t me it ma be base the a t t upon i , y d upon qu n i y produced . The principle of all straight piecework systems is that the employee is likely to work harder and produce more if he is paid in proportion to hi s production tha n he would if paid

- by the day or hour . A pay roll designed to re cord piece work wage pa yments is similar to the o ne use d under the

te stem ex e t that hourly ra sy , c p ing no provision need be

e t me as e t me is e e a made for ov r i , ov r i g n r lly paid for at the same rate as regular time .

WRITTEN EXERCI SES

l ete e f l win a -r l b eterminin 1 . Comp th ol o g p y o l y d g the amount due to each employee . PAY-ROLL CALCULATIONS 25

PI E CE RK P AY- LL E E K G E EMBE R 27 192 WO RO FOR. W ENDIN D C ,

NUM BE R PR OD UCED

N M E L A A TI ON No . T W T F S O T

14 17 1512 89 23 26 26 25 32 30 34 32 42 4540 24 26 2527 20

— a a a e 26:Th e Difi erential Rate . Under this pl n c r ful estimate is made of the number of pieces each employee can

n ea h em ee i ex e te to e produce in a day, a d c ploy s p c d produc

f he es h he sta a the standard number . I produc less t an t nd rd ,

If he es m e tha the he re ceives l ess per pie ce . produc or n

h e es m e tha the sta a ate . standard , e rec iv or n nd rd r This system is based upon the idea that the expense of

ma h am hethe the l ht e t e etc . e s t e s e heat, ig , r n , pow r, , r in w r employees produce a small amount of product or a large

as the m t the t these amount . By incre ing a oun of produc ion, expenses are distributed over a larger quantity of manu

n the st ma ea h a t e i s the eb facturedgoo ds, a d co of king c r icl r y a ef e te b eas the t t is de creased . The s ving f c d y incr ing ou pu divide d between the owner of the fa ctory andthe workmen who by their skill and industry increase the output . On he em ees who b a sma the other hand, t ploy y producing ll

r a t e are a ess . quantity increase the cost pe r icl , p id l

— tin the D ffe e tial Rate . The b em 27 . Compu g i r n pro l of dete rmi ni ng the standard number of pieces which shall con ’ The stitute a fair day s work i s Often a difficult matter . employer is naturally desirous of basing the rate on the 26 BUSI NESS MATHE MATICS amount of pieces produced or work done by his most effi cient em The h se who are ployees . less efficient employees or t o no t temperamentally fast workers may be penalized if the rate is placed at so high a figure that they earn less tha n the

t Us a a c m m a e b a s andard rate . u lly o pro ise is m d y pl cing the rate at such a figure that all industrious workmen can easily earn the standard rate and only the ineffi cient or lazy fail to earn it .

If x that I llustrative E xample . a manufacturer found by e perience the a e a e a his a t e 10 a t es e ’ da v r g workm n in f c ory could produc r icl p l. y, andt at 35er a t e be a the he ht t e t e h ¢ p r icl could p id for work , mig h n ou lin the following differential rate :

I E S No . O F P CE PR OD UC E D RATE P E R PI EC E 8 9 34 10 (standard number andrate) 35 11 36 12 37 13 38

s es 8 e es he e e es 8 If William produc pi c , r c iv X I f Hartman 13 13

WR I TTEN EXERCI SES

’ out a a - an s i e a e e e 1 . Work p y roll bl k how ng ch mploy s production he a du t e a d a and wages , using t d ily pro c ion r cord n t bl e of rates give n b elow .

DAI LY P R ODUCTI ON NAME

28 BUSINESS MATHEMATICS

WR ITTEN EXER CI SES

1 . five B . the aid a e o f th e y of th e above table , compute th e total w g s e at s op r or b elow .

PR ODUCTI ON AND SOL UTI ON OF TH E EXAMPLE

OP E R ATOR B ONUS P E R TOTAL H OURLY TOTAL S . I H WAG E R E Q N . OUR S A T No . H O UR R ATE WAGE S

2. Us n th e s tab e en ab e c m te the t tal e i g bonu l giv ov , o pu o arnings of O the following six typewrite r perators .

R W G E RA TE ON S TOT L E R N N G I N . H O S I S SQ. U A B U A A

— -R wan emi m R ate U e th s meth 29 . H al sey o Pr u nd r i od the workman is paid a premium which is generally from

- - th m h one half to o ne third the value of e ti e saved . T e time saved is computed by setting a standard task to be done

a t me andthe ffe e e bet ee the sta within a cert in i , di r nc w n nd ardtime and the actual time (where the a ctual time is less than the standard time) constitutes the time saved on the task . PAY-ROLL CALCULATI ONS 29

I If th l a e str h llustrative Example . e standard time for cur ing doz n o ic h i n 5 es the tas eat e s s 3hr . a da at i 0 f r , worker whose hourly r e s ¢ do k in

h 2 hr . 2 hr . he sa es 1 hr . Th h s 1 t , v , or erefore e receive $ for e work H e plus a pre mium of 25¢ this being half the value of the hour saved . still has o ne hour left in which he can do other work for which he should s b : re ceive at least so that his total payment for 3hr . work hould e

EXPL AN ATI ON

er . 2 hr . 1 doz . (a ctual time ) at p hr 50 hr .

sa e 1 hr . v d

Total

- —B h s a an n Efi cienc Wa e S stem. t 30. Emers o y g y y i pl n

efficiency bonus is paid to those workers who maintain or

The b s a e is a exceed a given rate of production . onu dd d

a e at the e a h ate perce ntage of the wages e rn d r gul r ourly r , and the percentage i s calculated by dividing the standard time by the actual number of hours taken to do the work . effi e he e e es b s If the worker reaches 70% ci ncy, r c iv a onu of If! on every dollar of wages ; if 55! on the dollar ; if 90% 10¢ on the dollar ; if 25g! on the dollar ; and so on progressively . The method of calculating e fii ciency is :

If A Actual time in hours S Standard time in hours E Efficiency per cent Then E S/A

Ill ustrative E xample .

Standard time is pieces per hr.

100 in hr . Wage rate is 50¢ 30 BUSINESS MATHE MATI CS

If the e a es 30 e es the st t e l be : 30 work r m k pi c , andard im wou d 2 039 hr .

I f th e er e a t a t me the 30 hr . the e c u l i on pieces were , ffici ncy p cent would be : Re e to the s ta e e e an effi e e f rring bonu bl giv n b low, ci ncy of corr sponds approximate ly to a bonus per cent of The computation woul d then be as follows : (M) X (wage at r e) X .7 % (bonus fa ctor) (bonus earned . )

B ON US SC ALE

E F F I C I E NC Y E F F I CI E NCY

O 0

The efficie ncy per ce nt corresponding to pie ces hour andthe bonus in cents is as follows :

0 PI EC ES P E R H O UR BONUS I NCE NTS 50 1 per dollar of wage 60 5( f H i f M 75 25

WRI TTEN EXE RCI SES

ete th e l n ta at fr th e n a 1 . Compl fo lowi g bul ion om followi g d ta i Standard time s 100 pieces in 4 hr .

u u u ’ 25 p er hl .

1 . 4 in 0 hr .

Wage rate is 40¢ per hr . PAY - ROLL CALCULATIONS 31

TI M E BONUS EA R N E D No . OF E F F I AM T . NAM E P I E C E S C I E c EA R NE D AT M DE R E G R TE A A ct ua l Standard A Amo unt

Tab ate th e a e- e ti e e ees t th e sa e num 2 . ul bov m n on d mploy , wi h m e sam e a tua me th e s b er of pie ces of work and th c l ti , from figure following

Standard time is 100 pieces in 5hr .

20 per hr .

1 in hr .

Wage rate is 50¢ per hr .

— a C mm ss as s The sa a es 31 . S alar ie s on o i ion B i l ri of employees are usually fixed in their amount and no extra

ex e t to th s st m is pay is given for overtime . An c p ion i cu o sometimes made in the case of salesmen whose salaries are frequently paid on what i s known as a commission basis ;

is a e ta e e ta e the sa es ea h em ee is that , c r in p rc n g of l of c ploy

paid to him instead of a regular salary or a small salary

with a mm ss may be paid and supplemented co i ion on sales made .

WRI TTEN EXER CISE

a es a e ta firm are ete m e th e 1 . Th e w g of c r in d r in d on following com mission basis

e a t e t 1 10 Ou goods sold in D p r m n , % U ( l ( l U ( f 2 2, 1 % 2 3, 1 % 4 5 , 1 % 32 BUSINESS MATHEMATICS

S LE DE T. 4 SM N E T. 2 DE T. 3 A A D P 1 DE PT. P P

P lan a form for th e above Whi ch will show

’ (a) E ach salesman s commission in ea ch depa rtment . T t (b) o al commissions paid to each salesman .

(c) Total commi ssions paid for ea ch department .

(d) Total commissions paid to all salesmen . (c) Total sales in ea ch department

(f) Total sales of each salesman .

(g) Total sales in all depa rtments .

-R — h ma e s 32. P ay oll S lips It is t e pra ctice of ny conc rn

“ to ha out the ee a e e e es e a h em ee nd w kly w g in nv lop , c ploy receiving an e nvelope containing the exact amount of his

n h s se a h is e a e as be . pay. I t i ca , coin s eet pr p r d low

WRI TTEN EXE RCI SE

a s a e b s and c ns re uired to 1 . Find the total w ge nd th total of ill oi q

k . pay th e wage s shown b elow . Che ck your wor

COI N SH E E T

B I LLS NAME WAGE S 310 3532 31

o nes 1 E . J

ester . 2 H . J etts 3 C . B 4 artl ett J. B

5R . Dibbl e

To ta l PAYR- OLL CALCULATIONS 33

— 3. a Th ta e to the 3 Currency Memor ndum. is is used to k bank to show the paying te ller j ust what kinds of mone y are re quire d andhow many Of ea ch kind are required to pay the

- weekly pay roll .

CE ASE NATI ON A L B ANK CURRE NCY ME MOR ANDUM New o 1 192 Y rk, April , — Depositor John Do e

D O LLA R S CE NTS

5Bills 31

[0

50 100 Coin :

WRI TTEN EXE R CI SE

e e a fo r the exe se 32 . 1 . Make a curr ncy m mor ndum rci in CHAPTE R I V

INTEREST

— 34 . Natur e o f Inte re st I nterest is pa yment for the use

The te st to b e a a e ta u of money . in re p id for c r in s m de pends upon the time for which the sum is loane d as well as h on the rate per cent charged . T e principal is the sum

Th a and t a e . e e est a e t the ar lo n d princip l in r , dd d oge r , e

calle d the amount . Legal interest is the interest determined a ccording to a rate fixed by law . If no rate is stated in a business tran sac

Th xa t the e a ate is se . e e t te est at a ion , l g l r u d c ing of in r

h he ate tha that xe b l aw is as an i ig r r n fi d y known usury , d s l punishable by aw . I nterest rates are limited to 6% in all states with the

ex e t s : 5 I s s a a an h following c p ion % in llinoi , Loui i n , d Mic i

an 7 Ca a G e a I ah Neb a a th g ; in liforni , orgi , d o , r sk , Sou

a a a nd S th Da ta 8 ab am C rolin , ou ko ; % in Al a, Alaska,

a a M ta a Utah and W m 12 Color do , Florid , on n , , yo ing ; % in Nevada . Lending mone y is generally beneficial to the lender be cause it allows him to re ceive a return for the mone y earned by

I t a s he s th b him the s the ast . e or o r in p l o lp orrower , be

a e he i s e ab e to et a a e et m hi c us n l d g l rg r r urn fro s work .

ORAL EXERCI SES

n s to e a e him t 1 . If a ma borrow n bl o manufacture a new h t is the sum a e hi h he m st a t article, w a c ll d w c u p y for he money? 34

36 BUSINESS MATHE MATI CS

a interest for 1 yr . t 5% 3i

te es t 3 and3mo . or in r for yr .

WRI TTE N EXER CI SE S

Find the interest on 2 6 . 1 . 3 40 for 2 yr . mo at

2 . 5 4 8 . t . 337 for yr mo a t 3. for 9 mo . a

4. for 3yr . 6 mo . at

h M eth s C m t n — 37 . S ort od of o pu i g Interest The follow ing principles andmethods of computing interest are short

' cuts fo r calculating interest when the rate is

To findth e interest at 6% for

6 da . nt 0 3 3 a e s to th e e t in th e i poi pl c l f pr ncipal . 60 U H M 2 U U N I f U ( f 600 1 no

WR ITTE N EXER CI SES

the t ta a t te est at 6 the 1 . Find o l moun of in r % on following

for 60 da .

da . int . 60 e b 2 an 90 Find for divid y da dd. 6 and ti b 42 mul ply y 7 . 40 60 divide by 3and subtract 60 i e b 12 an 65 div d y d add. 24 873 80 70

th tota amo t te est at 6 2 . Find e l un of in r % on INTE REST 37

for 45da 15 25 21 int . 6 da . e 2 n . Find for , divid by , a dmultiply by 7

33 i 3 da . a nd 0 3 d . i t . 55 F nd a n andadd. 60 a . and int . and a t . Find d 5da . subtr c

3. i the t ta amo unt of i te est F nd o l . n r at 6 % on : 72 for da . 84 48 6 9 10 7

4 . Find the total interest at 6% on 3 for

’ ‘hn h a t te est at 6 : 5. I dt e to tal moun of in r % on 3 2 . 7 . . 15da mo . 1 3 for 7 mo mo X mo . 2 2 . 1 a . 60 da . . 1 yr . mo 0 d mo

6 mo . 20 da .

2 a . 1 yr . 6 mo . 1 d

a . 8 mo . 8 d

Rule : P ointing o fi two places in any principal gives rates oth er than 6% as follows

At 1% for 1 yr . 1 0 da At 2% 6 mo . or 8 At 3% 4 120 At 4% 3 90 At 80 38 BUSINESS MATHE MATICS

ORAL EXE RCI SE S

7‘ 1 . At 5% for how many da M N M ( f At 8 %

? i h f' Tth t - a e t-o ff bta e 8 . How s t e numbe r of days o e wo pl c poin o in d

WRITTEN EXER CI SES

total interest on the follo wing :

t 1 for 1 yr . a %

2 %

t te e t the 2 . Find amoun of in r s on following

3 800 for 40 da . at 9 % 800 20 48 8 4a 15 60

To find th e‘ intere st at

add $2 to th e interest at ‘ ‘ u (4 l. f

. 6%

1 . 7‘ 2.

v e 6 te est 6 a d lt b h t? 4 . di id in r by , n mu iply y w a INTEREST 39

Rul e : To find th e inte rest at

ate d 6 inte e st 6 and l b t at ate . Any r , ivide. % r by , mu tiply y h r

OR AL EXE R CISES

To find

deduct from 6% interest .

6% divide 6% interest by what

Wha t is the interest on

SOLUTI ON :

' 324 60 da . int . a t 6 % 324 60 da . int . at

320 60 316 60

324 60 da . int . at 6 % 4 60 ( J ( I U 1 %

WRI TTEN EXER CIS E S

$1 far 6 da , at 7 % 12 5 _ % 3 15 8% 30 9 % 40 BUSINESS MATHE MATICS

Ru To th e nte est le : Interchanging principal and tim e . find i r on 60 s o fi di th e nte est 53 600 $ 0 for 53da . at change thi t n ng i r on $ for da ., at

WRI TTE N EXE R CI SE

1 . Find total interest on 3 600 for 60 ( l 200

660

s Rule : To findaccurate intere t. e l e s First : Findordinary inte rest by abov princip . ‘ Second : Deduct 7 s of the ordina ry intere st from itself .

t te est at 6 60 da . Example . Find the accura e in r on % for

SOLUTI ON

d a t int . at 6 e est 373 60 da . % (or in ry in r ) 1 373 73

int . 6 a u ate te est 372 60 da . % ( cc r in r )

MISCELLANE OUS WRI TTEN EXE R CI SES

h t ta te est 1 . Find t e o l in r on 3600 for 550 780 800 720

the t ta te est 2. Find o l in r on

3 640 for 60 da . at 60 600 60 10 % I NTEREST 41

1 44 3 0 for 60 da . at 60 u 60 ( I u

i th 3. F nd e total interest on

for 60 da . at 4 “ 800 60 4m,

800 60 2m, 800 60 9 %

4 I f the a t ta . ordin ry in erest on a certain sum of money for a cer in t e is a t a e ate at is the a ate te e t the same im giv n r , wh ccur in r s for time andthe sa me rate ?

38 . To i th e T m R e st . F nd i e from Princ ipal , ate , andInter — It is sometimes necessary to find the time at which a certain sum of mone y at a gi ven rate will produce a certain

Th s i m t b fin the ih amount of interest . i s co pu ed y ding terest on the given amount of money forthe given rate for

ne ea an h t e b o y r , d dividing t e amoun of interest requir d y

1 Th e t he e the amount for yr . e r sul will be t numb r of years anda decimal or fraction of a year .

l I strat e E xam e 1 . If the te est 600 is 108 at a a llu iv pl in r on 3 3 nnu ly ,

find the time it w as on interest .

SOL UTI ON

0 3600 at 6 0 for 1 yr . 336 3108 336 3

Exa e 2 . If 4 0 at 6 e s 84 te est Illustrative mpl 3 0 % yi ld 3 in r , find time .

SOL UTI ON

3400 at 6% for 1 yr . 384 324 42 BUSINESS MATHEMATI CS

WRI TTEN EXER CISES

1 I f 2 . 3500 yields 31 0 at find the time .

2. I n what time will it take to produce 3210 at

3. How long will it take to produce 3260 at

— i 39 . To FindI nte re st Betwe en Certai n Dates I t s often necessary to find the time between certain d ates and then

Th i a find the interest for that amount Of time . is s ccom plishedby '

(a ) Subtracting the dates as illustrated b elow .

h t 4 at 6 I llustrative E xample . Find t e in erest on 3 00 % from Jul y

12 1 1 to n . 1 2 . , 9 9, Ja 3, 9 1

SOLUTI ON

m . n EX L N TI ON : S t a t . In so 1 o a d add its P A A ub r c doing , borrow ,

h . h e t e ta 12 . e i a e t 30 da . to t e 3da t e e o 33 21 qu v l n in minu nd , h n k fr m d Next ta e 7 0 but e to t s we st 1 . an k from , in ord r do hi , mu borrow yr

addits e a e t 12 . to the 0 t e ta e 7 12 . n a s t a t quiv l n mo , h n k from Fi lly ub r c T 19 19 from 1920 . hen find the interest for the given time at the given rate on 3400 .

(b) Finding the exact number of days from o ne date to the next date .

Mar . 1 1 1 Il s at e Exam e . the e a s 5 lu tr iv pl Find numb r of d y from , 9 9,

to e 26 1919 . Jun ,

SOLUTI ON

Total

44 BUSINESS MATHE MATI CS

I st at e Exa 3 llu r iv mple . Find the compound interest on for

r . at y compounded annually .

SOL UTI ON

’ 5 50 1st ea s te est % of 3 , y r in r

350 new principa l 2ndyr . ’ 5% of 2nd year s interest

new principal 3rd yr . ’ 5% Of 3rd year s interest

a mount end of 3rdyr . compound interest

WRI TTEN EXER CI SES

1 . he at 6 n e Find t compound interest on for 4 yr . % compou d d annually . 2 h t . Find t e compound in erest on for 4 yr . at 4% compounded semiannually . h hi 14th 3. A boy as deposited in a savings bank for him on s f th 4 hat th a . I e a a s te est se a a bir d y b nk p y % compound in r , mi nnu lly , w amount will he have on his 21st birthday if there are no other deposits or withdrawals?

4 the f e e e et ee the s e te est 6 . . Find di r nc b w n impl in r on for yr at and the compound interest on the sa me a mount for the same If time at the same rate if the interest is compounded semiannually . compounded quarterly .

th te est 2 . at 4 e 5. Find e compound in r on for yr % compound d quarterly . 2 is e s te a sa s a h a s 4 6 . I f 3 00 d po i d in ving b nk w ich p y % compound 2 it t se a an . 1 1 1 at a inte rest compounded miannu lly , on J , 9 , wh will moun t 1 1924? o July ,

at is the a t 975 3 . at com 7 . Wh compound moun on 3 for yr pounded se miannually?

I te e st Tab e — If a e s has m h 41 . Compound n r l p r on uc

at to he sh compound interest calcul ion work do , ould resort

I t is m h eas e s m e and e tha to the table . uc i r , i pl r , quick r n the method explained in 40 . I NTE REST 45

This table sh ows th e amount of $1 compounde d annually at th e difi erent rate s .

Ye ars

the te est 5 Illustrative E xample . Find compound in r on for

yr . at

a u SOLUTI ON : 31 compounded annually at 6 % for 5yr . mo nts to

133 226 as s the a e ta e . 3 8 , hown by bov bl

400 . X compound interest

h t is e se a a ta e the at NOTE : I f t e interes compound d mi nnu lly, k r e for twice the time . t ta e the ate 4 t I f the interest is compounded quar erly, k r for imes the time

x e the om o te est Illustrative E ampl . Find c p und in r on for 5

te est m e sem a a . yr . at in r co pound d i nnu lly 46 BUSINESS MATHEMATICS

SOLUTI ON : of 6%

2 t mes 5 . i yr 10 yr .

a nt mou of $1 compounded at 3% for 10 yr . is 313439 16 . X 31343916 compound interest

WRITTEN EXER CI SES

inte st the m e 6 . at 1 . Find co pound r on for yr compounded

hat sum i a t 9 r . if 2 . To w w ll moun in y inves ted at inte rest compo un ded semiannually? hat is the m te est a an 500 at 3. W co pound in r on lo of $ com pounde d quarterly for 5yr . ? sum b i este at 4 om i t 4 . What must e nv d % c pound n erest to amount the i te est is n ? to $800 in 10 yr . if n r compou ded annually

um m st be e s te Jan . 1 1921 s that a 5. What s o J n. 1 u d po i d on , , on , 1 w th te est at 5 m e a a l the a t 193 , i in r % co pound d nnu l y, moun will be

What is the a e a l o- ea e e t i e s a e e 6 . v lu of y r ndowm n l f in ur nc pr mium a at 4 te est of if pl ced % compound in r , compounde d semi

a l at the end the 10 r . ? annu l y , of y If e a e a ai e asse s a rie th 7 . th v r ge d ly numb r of p nger c r d on e I nter borough subways and elevate d lines of New York was in ndth e a e i ease er a n is h w ma n 1920, a e av r g ncr p n um o ny passe gers ate ? must be providedfor 25yr . l r

u De sits at C m ndInte e t — 42. Ann al po o pou r s The follow ing table is very useful when o ne has to find the amount of $1 de posited annually at compound interest for any number I t is e e t of years up to 25inclusive . p rf c ly obvious that such an example would be an endless task without a table of this

I t a t a use be me e a a nature . s pr c ic l will co v ry pp rent with h the written exercises whic follow it. INTEREST 47

This table shows the amount o f $1 deposite d annually at cornpo undinterest for any numb e r of years to 25inclusive .

Ye ars

th a Il lustrative Example . Find e mount of $10. deposited annuall y

f r 10 . a a a i 4 o yr in a s vings b nk p y ng % compoundinte rest .

SOL TI ON : I n the hea e and s te 10 r U column d d down oppo i y . , we find that 31 under the stated conditions will amount to 312 486351 ; th 10 t to 10 12 486351 en 3 will amoun X 3 , or

WRI TTEN EXE R CI SES

1 man 28 . a e has his e s e b . A yr of g lif in ur d for y taking o ut a, e e t i h e a s a a l r 20 yr . ndowm n pol cy, for which p y nnu l y pe he If at the expiration of the 2oth yr . receives the face val ue of the find the ain to the s a e m a if m e is rth policy, g in ur nc co p ny on y wo 4%

See a e tab . compound interest to them. ( bov le ) 2 f the s e in Exe se 1 had e at the a e . I in ur d rci di d g of 37, would the st n h insurance company have gained or lo , a d ow much? 48 BUSI NESS MATH E MATICS

3. man sta ts a sa s a a t his 16th th a A young r ving b nk ccoun on , bir d y e t I s 30 . f he e s t a t n t he is by d po i ing 3 d po its 330 e very 6 mo . here f er u il

25. a e hat a t he a e t hi t the a a s yr of g , w moun will h v o s credi , if b nk p y 4 % inte rest compounded semiannually?

4 ~ . What amount of money deposited in a savings bank paying 4é % a a ? nnu lly will amount to in 20 yr .

i s — A s k i um m 43. S nking Fund in ing fund s a s of one y set aside at regular periods for the purpose of pa ying o ff a n ex st a t ate ebte ess Of e a a a e i ing or n icip d ind dn , or r pl cing v lu

hi h sa ea b t exh o r w c will di pp r y deprecia ion , austion , termination . The pa yment Of a public or a corporation debt and the

a e ta b ate repl cing of c r in pu lic , corpor , or private values due to de preciation or other causes a re ofte n m ade easier by

est a ta um s re gularly inv ing cer in s in ome form of security . The interest and principal from these investments from

a to ea m a s h h it is a ha y e r y r for inking fund , w ic , pl nned , s ll a ccumulate to an amount needed to redeem the debt when h i u e a e t e a ue he t sa . itfalls d e , or r pl c v l w n di ppears

Exa e at sets as e a a Il lustrative mpl . A corpor ion id nnu lly o ut of I profits of the pre ceding yea r for 20 yr . f this amount is in t te est e a a the vested a compound in r , compound d nnu lly , find amount at the endof the 20th yr .

SOL UTI ON : Amount of 31 depos ited annually for 20 yr . at

332 783137 .

Amount Of deposite d a nnua lly for 20 yr . x 332 783137

R efer to above table .

WR I TTEN EXER CI SE S

h e i ea ea 10 . a e ta m a set asi e 1 . At t e b g nning of ch y r for yr c r in co p ny d s ea as a s If o ut Of the profits of the previou y r inking fund . this at 4 te est e a al sum was invested % compound in r , compound d nnu ly, what did it amount to at the endof the l oth year? INTEREST 49

2 . Jan . 1 1910 a e ta t e nd to a , , c r in ci y borrow d a agreed p y

the sa e Ja n . 1 1920 . W at sum s a e n te m on , h hould h ve b en i ves d on 1 1910 an a h s ee a Jan . d e e 10 . a 5 , , c ucc ding y r for yr in bonds p ying %

" te t e a all compound in res , compound d nnu y , in order to pay the loan when it became due ? W at sum st a t set as e and est a a t l a 3. h mu ci y id inv nnu lly o bui d n school building costing if it is to be paid for in 20 yr . a dthe city receives on the money thus set aside? W at sum st a a e t a set as e to eet the 4 . h mu l rg prin ing comp ny id m

t a t ess t e e at 15r . it st cos of prin ing pr , hrough d pr ci ion, in y , if co andthe money is worth 4 % compounded annually? CH APTE R V

DEP RECIATION

De — i h Natur e o f ec ati n . De at s t e ss 44 . pr i o preci ion lo or expense incurred in business through decline in the value of

hi m a h property . W le repairs y be made to prolong t e useful

s a b a ma h e s e ate the t me nes of uilding or c in , oon r or l r i comes when the property is either worn o ut or it is good business eco nomy to repla ce it .

m h sta e st o ut 10 A ac ine, for in nc , co ing is worn in

at the end h h t me he the ma h e is e a e yr . , of w ic i , w n c in r pl c d, there will have been a loss of due to depreciation . Unless a portion of the depreciation is charge d to profits as an a a ex e se the e t e ss be ha e nnu l p n , n ir lo will c rg d

h t h The a t e b s ess against t e profi s of t e last year . pr c ic in u in is to spread this loss over the life of the property by charg

h h h These ing off part of t e loss to t e operations of eac year . charges are called depreciation charges .

M e th ds C m ut De ec ati n — Th 45. o of o p ing pr i o e following methods are those most commonly used to compute the de preciation charges

h - h The st a t e met . 1 . r ig lin od

fixe ate m te ea h ea h 2 . A d r , co pu d c y r on t e original value

of the property .

e eas ate m te o n the 3. A d cr ing r , co pu d original value of

the property .

te m te a ecrea in 4 . xe a s a A fi d r , co pu d on d g v lue . 50

52 BUSINE SS MATH EMATI CS

ea h Th - h c year . is is calle d the straight line method or t e

xe t meth b h m a e a es are fi d propor ion od , e cause if t e re ind r v lu plotted on the vertical lines (see Form 1) and the y ears on the h ta e h h h b th orizon l lin , t en t e rem ainder value is s own y e

b e st a ht e o liqu r ig lin .

x — 47 . Fi e d R ate Compute d on Original Value This is a

e s m e meth Th f he a v ry i pl od . e di ference b etween t origin l

a e nd h Th v lu a t e prob able scrap value is first obtained . is difference is the n divided by the number of y ears that the ma h e is est mate to ast and th s es t is a e the c in i d l , i r ul c ll d

a Th r a is the depreci tion per year . e de preciation pe ye r n

v e b the a a e the ma h e h h es di id d y origin l v lu of c in , w ic giv the rate per cent of the original value to be charged Off h ea c year .

I st ati e E xa e i t a st llu r v mpl . A printing press s purchased a co of i i andit s ex e te t at t s ess can be se 10 r . e t a e p c d h hi pr u d for y , wh n will h v f T at a a e O e e e th 10 . use a e e v lu h r for , during e yr of , d pr ci ion of

4 Thi s is an a a e e at 600 . 600 will occur . nnu l d pr ci ion of 3 3 7% The refore of the original value is charged Off each year as an e xpense .

4 De e a n R ate C m e a a e 8 . cr si g o put d on Origin l V lu of — Prope r ty I t is som etime s preferred to charge the largest

e e at the st ea a a e it amount of d pr ci ion fir y r, gr du lly r ducing

h Th s is e b e a se a eate e ea ch year t ereafter . i don c u gr r d pre ciatio n a ctually occurs during the first year than during any

m e an a t m b is s -h ea . exa e later y r For pl , u o o il econd and ’ a few m ths use a nd the e f h a fter only on , own r su fers a muc greater loss from its use during the first year than he does

I a during the second year . t will lwa ys depend upon the

t e as to hat a m t m st b e e te e ar icl w oun u d duc d ach year . DEP RECIATION 53

x m te a a n a e — Th s 49 . Fi e d R ate Co pu d on De cre si g V lu i method in a som ewhat similar way as in 48 results in a de

h Tha i the e e eas a a a e e e a . s cr ing nnu l c rg for d pr ci tion t , d pr ciatio n for the first year will amount to more than that for the se ea and that the se ea be m e cond y r , of cond y r will or h h tha t e t ea etc . n for ird y r ,

h e t is I llustrative E xample . Suppose the original value of t e prop r y and the ate e e at is 10 a ea t e e e at r of d pr ci ion % y r , h n d pr ci ion unde r this method is computed as follows :

original value 10

depre ciation first year

a e e de crease d or carrying v lu , b ginning of second year

depre ciation se cond year

a e e t ea decreased v lu , b ginning of hird y r

. 10

e a etc . depre ciation third y r ,

t The fixe d r ate is obtained by somewhat more complica ed h s n can h h s a e a t m , a d calculations, w ic u u lly involv log ri rea dily be understood by anyone having a working knowl

' I n e e a the xe ate is as sh edge of them . g n r l fi d r found own in the following : 54 BUSI NESS MATHE MATICS

I x r n n ase a t ess llustrative E ample . A p i ti g firm purch d prin ing pr for and a e a s a It was estimated that it would last 10 yr . h v cr p

the a te at . value of 3200 . Find nnual ra of depreci ion

SOLUTI O N : The followin g e quation is used R V

where V present value of the asset R residual value after 11 periods 71 number of periods T pe rce ntage of diminishing va lue to be deducted a n a ate e e at n u lly, or r of d pr ci ion

If we substitute the values of the proble m we have

1 0 3 200 T 1

. 04

1 7 221 + 7 9 . 2 7

will be the rate to be use d on the de creasing value each

year .

WRITTEN EXER CI SES

1 the a a e e at a i t 4 . Find nnu l d pr ci ion of build ng wor h if %

is charged o ff each year . 2 is a e o ff a n a e e at a a . How much ch rg d n u lly for d pr ci ion by m nu fa cturer who owns property which depreciates at the following rates?

PR O P E R TY VA L UE DE P R EC I ATI O N R A TE l 5( Facto ry bui ding 72, Machinery 75 Too ls 121 P atents 6 %

The ne a u i est mates the a a e e at a 3. ow r of b ild ng i nnu l d pr ci ion s 3% T st ‘ What is the a t of its cost . he building co moun of the annual depreciation?

The i dn is e te at 40 er th . I f the taxes a e n bu l i g r n d 3 p mon , insur nc , a d 0 er ea hat ne othe r e xpe nses amount to 38 p y r, w t income does the owne r of this property re ceive on his investment after allowing for depre ciation? DEPRE CIATI ON 55

4 I i . t s estimated that a machine costing can be sold at the

en . at r dof 8 yr . for 3500 Wh pe cent should be charged annually for

a e a t t t 5. M chin ry in fac ory cos Depre ciation is compu ed

10 % of the original val ue the l st year 8% H U H U N 2d ( 1 6% 3d 4th 3% andeach year thereafter .

What was the a t e e at ha ea h ea 5 ? moun of d pr ci ion c rged o ff c y r for yr. Wh at was the inventory value of the machinery at the begin ning of each year? a the i e t a e th a 2 ? Wh t was nv n ory v lu of e m chinery at the endof 1 yr . fl mi was e t a hin ti e 6 . A our ll quippe d wi h m c ery cos ng D pre te at i st i st ciatio n was compu d of ts co the l st yr ., 8 % of ts co the

he . the 4th r . an 2 h 2 r . 5t 3d d ea ea t e ea te . dy , % yr y , % c y r h r f r i t a the inve Fin d the a mount of deprec a ion e ch year for 7 yr . Find n

‘ h a hine t the e i tory value of t e m c ry a b g nning of each year . at e tai e t st a t 7 . Depreci ion on c r n prop r y co ing w s compu ed at h 8% of the de creased value for 4 yr . Find t e annual depre ciation and the de crease d value each year .

u a e as e a e s ess 10 . H is a hi 8 . A man f ctur r w ng g d in bu in for yr m c n ery cost andhe charged 6 % depre ciation annuall y on de creased

the a a e e at a ndthe e ease a e ea ea . values . Find nnu l d pr ci ion d cr d v lu ch y r a hi e a a t st and e e i at w as te 9 . M c n ry in f c ory co d pr c ion compu d at was th e at the 4th at 7 % on de crease d values. Wh e d preci ion during

h a e at the end the 4th r . ? . andt e e t yr , inv n ory v lu of y 0 Th s a h e was and e e at was 1 . e co t of m c in ry d pr ci ion com a What was the a t puted at on decreasing annual v lues . moun of t th en r . nd h e e a e a e d the depreciation during the 6th y , a t e r duc d v lu of as what er e that year? The depreciation during this 6th yr . w p c nt of the original cost of the machinery?

1 n e te ess to i e andB Th e a uthors acknowledge their i d b dn F nn y rown, ” t ese r ems and h th Mode rn B us iness Arithmetic, for h p obl muc of e other material in this chapter . C HAPTER VI

I NSURANCE

— ~ E e b s ess m st ta e 50 . Ne ce ssity of I nsur ance . v ry u in u k the precaution of insuring its premises and stock- in-trade a a st fire its me a a st a c e t and ma g in , work n g in c id n ; in ny

ase the e a a t e a ma a e t a n im c s lif of p r n r , n ging dir c or , or

ta t O e m s a s b e e to te t the b s ess por n ffic r , u t l o insur d pro c u in h against t e loss that his death might cause. I t is also considered wise for ea ch employer or employ ee h to insure imself against death or a ccident .

K — h s 51 . inds of I nsurance T ere are very many kind of

The be s e e insurance . first four named below will con id r d

te eta h m s s a e are : qui in d il in t is work . So e kind of in ur nc

Fire Transportation Life Keys (loss of) Fraternal Mail Accident Flood Liability P rofit (loss of) I nspection Use and occupancy Burglary War risk P late glas s R iot Steam boiler Damage Automatic sprinkler claim Furniture Casualty I ndemnity ’ Automobile Musicians fingers Live stock E arthquake Marine Title to property Hail E xpress Cyclone Health INSURANCE 57

52 — e s a in . Fire Insurance . Fir in ur nce is guaranty of

emni S h d ty for loss or damage to property by fire . uc con t a ts s a e sses b ht n met mes ss r c u u lly cov r lo y lig ning, a d so i lo a se b es and t a es I a e m an es c u d y cyclon orn do . nsur nc co p i are liable for loss or damage resulting from the use of water h em a s se ext sh the fir and m sm e . or c ic l u d in ingui ing e , fro ok The fire insurance policies of all the companies in the states New New e se t and e of York, J r y , Connecticu , P nn “ s a a are n m nd ta the New sta ylv ni , u ifor a con in York nd ” Th a t i a ard clause . is in p r s s follows : This company shall no t be liable for a larger proportion of anv loss or damage to the property described herein than the sum hereby insured bears to 80% of the actual cash value of ” Th s is said property at the time su ch loss shall happen . i

h an exam e easily understood wit pl .

e e e t i a e at For instance , if a pi c of prop r y s v lu d and is insured for and a fire and water loss is the amount paid by th e insurance company would be as follows

80% of

3 I

2of (amount pa id by the company on this loss)

I t will be observed that the c ompany pays much less than

to th s a se the . I t has the a ctual loss, owing i cl u in policy been claimed that the companies use this clause to force manufacturers andother large owners of property to insure n their property for what it is worth . I t ca be easily seen that a large plant compose d of many detached buildings is not as liable to burn up completely as a loft buildi ng situated in the city ; andconsequently the owner of the latter is keen 58 BUSINESS MATHEMATI CS to insure his bui lding for more nearly what it is a ctually

th h e h e s a wor , w il t e own r of l rge manufa cturing plants are more liable to take a chance and not insure for what they are a t a h e c u lly wort . Cons quently the insurance companies by the aid of the 80% clause are able to penali ze the large Th m plant . e insurance co pany would have had to pay a much larger amount had the owners insured the property

at the be as sh the for ginning , own in following computation

80% of $ 125 H , or % 380 000 l %of

se it is ha e ab e that th s a co Of cour , rdly conc iv l e in ur nce m pany would have insured the plant for at the begin ning but they might have insure d it for or and then the amount received for the loss would have been much larger than it was . ” These policies also contain a waiver clause whi ch is

n e the a e the b h I cas of loss, if v lu of property descri ed erein does not exceed the 80% average clause sha ll be ” a es ha e asse h h waived . Some st t v p d laws w ic require the poli cy to state definitely the amount of loss for which the

th h m a i company is liable . By is policy t e co p ny s compelled

the s no t ex ee h a h to pay a ctual lo s c ding t e f ce of t e policy .

n me states the ta s a s a e a se h h I so policy con in coin ur nc cl u , w ic spe cifies that only such a part of the loss will be paid as the

e the bea s to the a e the e t i s fac of policy r v lu of prop r y n ured .

e tha o ne m a s es the same e t ea If mor n co p ny in ur prop r y, ch company pays only its pro rata share of any loss on the property .

60 BUSI NESS MATHE MATI CS

WRI TTEN EXER CI SES

' 1 . Find the premium o n each of the followin g policies

FACE OF P OLICY RA TE OF I NS URANCE AM O UNT OF P R E MI UM

p e r 3100 $100 i % l ess 10% p er 3100 l e ss 10%

To th e m t a 55. Find A oun P idby th e I nsur er .

x If e t Il lustrative E ample . prop r y valued at is insured for at e r a n a d fire and ate a se 3% p n um, n w r c u a loss of th e a t t at be a the s a e a find moun h would p id by in ur nc . comp ny (a) Under an ordinary policy

’ (b) Under a coinsurance clause policy (0 ) Unde r the New York standard a ve rage clause policy

SOLUTI ON

(a)

(b)

ORAL EXE R CI SE S

hat a t the ss es the a a 1. W moun of lo do comp ny p y in (a) ? te the a t the ss a as a a t 2 . Sta p r of lo p id in (b) fr c ion . Write the names e in the numerator andthe d nominator .

e as E xe se 2 i th the a e . 3. Sam rci w (c) in pl c of (b) I NSURANCE 61

I st a x llu r tive E ample . A stock of merchandise is insured in Company X for in Company Y for andin Company ! for I f the damage is ho w much should each company pay?

SOLUTI ON

total amount i nsurance 8 Of i ; 3 pai d by CO . X

“7 7 6 ; 2 “Of pal d by Co . Y $40 000

° f 4 000 a1d Co . z i $ , . p by $40 000

CH E CK :

WRI TTEN EXE R CI SE S

I f a se is a e at andis s e o f its a e at 1 . hou v lu d in ur d for 3 v lu andits contents are value d at andare insured for o f their value at and fire causes a total loss Of the building and a loss of the the te ts s a e a i . on con n , find how much in ur nc comp ny w ll pay

(a) Under an ordinary policy (b) Unde r a coinsurance clause policy (0 ) Under a Ne w York standard 80% clause policy

e andits te ts a re s e C a 2 . A stor con n in ur d in omp ny A for at 5595pe r 3100 ; in Company B for at andin Company C for T is a a fir at 60¢ p er 3100 . his property d m ged by e and water to the amount of

(a) What will ea ch company pay? ’ (b) What is each company s net loss if they have held the insurance . e e is t for 8 yr . wh n mon y wor h

— ta a d Sh t-R ate Tab e Th s tab e is e 56. S nd r or l i l us d for the purpose of computing premiums for terms less than 1

the se m t the am t em r . m y , or for purpo of co pu ing oun of pr iu to be returned by the insurer (the insurance company) when

It is the poli c y is canceled by the insured . used as follows : 62 BUSINESS MATHEMATICS

Take the percentage opposite the number of days that r s is to th m t the e ate and i k run, on e premiu for 1 yr . a giv n r , this result will b e the premium to be charged in case of short

s e e a risk , or earn d in cas of c ncellation .

x e the st ns a Illustrative E ampl 1 . Find co of i uring stock of goods a a ate is for 4 mo . if the nnu l r

SOLUTI ON 8% or 1% of 3300 of 3300 350

00 50 250 a n a em m 33 3 3 , n u l pr iu 50 250 125 em m 4 % of 3 3 , pr iu for mo INSURANCE 63

E X P L AN ATI ON é of the value of 1% less than the value of 1 Of % the amount .

I st at e E x llu r iv ample 2. Merchandise valued at is insured for its a e 1 . r 2of v lu for yr at 55¢ pe 3100 . How much of the premium should be et e if the is r urn d policy canceled at the expiration of 9 mo . (3) by the insured? (b) By the insurance company?

220 a a e 3 , nnu l pr mium 3 a t 3 3, moun returned if insured cancels policy

3x :3220

55a t et e s e a s 3 , moun r urn d if in ur r c ncel policy

WRI TTE N EXE R CI SES

1 . s a e a t er a was ate Jan . An in ur nc policy for p nnum d d ,

Six t s ate it wa s a e e the s e . 1921 . mon h l r c nc l d by in ur d How much of the pre mium was re turned?

2 . e 1 1920 I t o ut a r t e at 45 Jun , , ook policy on my fu ni ur for ¢

r 100 er a . Feb . 10 1921 I a e e the . pe 3 p nnum , , c nc l d policy (a) How much of the premium should be returned to me ? (b) How much would have been returned to me had the company canceled the policy on that date ? a e at are s e at 60 er 100 er a 3. Goods v lu d in ur d ¢ p 3 p nnum for he n The is a e e the i s re at t e d 2 . 3yr . policy c nc l d by n u r of yr (a) H ow much pre mium should be returned to the insured? (b) How much would have been returned in case the policy hadbeen canceled by the insured .

‘

the st s a st s 7 . at 4 . Find co of in uring ock of good for for mo r 70¢ per 3100 pe annum . e s a e is ss e e a i se st e 5. An op n policy of in ur nc i u d on m rch nd or d in a the e o n is to be 75 er 100 er warehouse , pr mium which ¢ p 3 p year . are to be Goods which are withdr awn within 1 yr . charged the short the t ta em s a and the t ta et s the rate . Find o l pr ium p id , o l r urn on following : 64 BUSINESS MATHEMATICS

R EC E I PTS W I TH DR AW ALS PR E MI UM TO TAL R E TUR N

Amo unt Date Am o unt

Feb . 10 1920 ur s De c . 7 1920 urs , F , F

Ma . r 151920 Sil . 51 920 Sil , k Aug 2 . k Ma 51 2 W o o l e n No v . 51 2 W o e n y , 9 0 . 9 0 ol

une 8 1920 H o sier Dec . 4 1920 H o sier J , y . y

A . 7 u 1920 l o ve s N v . 251 920 Gl o ves g . G o ,

fe I sur a e — e s a e i a t a 57 . Li n nc Lif in ur nc s con r ct by which

m a s e a a me t ma a co p ny , in con id r tion of p y n s de at stated intervals by an individual (or by a company for th e in

v a a ees to a a e ta sum m e t hi h di idu l) , gr p y c r in of on y o s eirs

h to h m l h a his eat se e atta s a e ta a . t d , or i f if in c r in ge The contra ct is called a policy and the mone y paid by

em ms ar th e individual a pr emi um. Pr iu e payable either

m th a te em a ee s a a a . w kly , on ly , qu r rly , i nnu lly , or nnu lly M any policies now contain the permanent disability clause which states that the company shall waive payment Of all fut ure premiums andpay 10% of the face Of the policy annually

sab t ma e s me the s mi . during di ili y, or k o o r i lar provision

a Kinds fe I s a — 58 . Princip l of Li n ur nce Policie s The principal kinds Of policies are :

a fe E" Ordin ry li policy N Limite d life policy W E ndowment policy P Term policy

‘ P Life in come policy Q Joint life policy

I T Survivorship annuity INSURANCE 65

They differ in three important ways

1 . The number of premiums paid by the insure d

2 . The amount of each premium

3. The time when payment is made by the company

59 . C m a s Diffe e t in o p ri on of r n K ds of Policies .

N M B E R OF E R S TI ME H E N YME NT I s MADE K m” OF P o u c v U Y A W PA P RE MI UMS AR E P AID BY TH E COMPANY

Ordina r Life Durin ife At death of insured. y g l o f insure d. This perio d ma y b e sh ortened in so me c o m

pa nics if th e dividends

are allo wed to acc umu

late .

Limited Life

l o- a ment ife At dea th ns p y l of i ured.

20 At death of insured.

E ndo wment Policy 2 - wmen 2 At dea th o insur n r . e n m 0 y do t 0 yr . f edpa y e t ma de to beneficiary ; or at ex pira ti o n of 2 en m 0 yr . pa ym t ade to ins ured

if still li ving . - l o yr . endo wment 10 At dea th of insuredor at e xpiratio n

of 10 yr . 20-pa yment - d a x 30 yr . endo wme nt 20 At e th of insuredo r at e pira tio n

o f 30 yr.

At death of insuredif he dies with I h in 20 yr . f e lives be yo nd this term no payment is made

Other periods of time may be obtained in the last three above . Life income policy provides an annual payment to the in

a sta sured after ted date . 66 BUSI NESS MATHEMATICS

— h am nt the 60 . Premiums andP r emium Ra te s T e ou of

th s an e premium i s a certain amount on wor of in ur c , andde pe nds upon the age of the insured at the time of buy The e the he n the . ing t poli cy, a don kind of policy young r

s th h he st the a e . per on, e c eaper t co of insur nc The premium rates per thousand for different kinds of participating policies at different ages are shown in the f ll m o o w g table .

ANN UAL P RE MI UM ON DIF FE RE NT K I NDS OF I NS URANCE P E R

- - l O Y R . 20 Y R . E NDO WME NT E NDO W M E NT

m ums at n e . 61 . Comput io of Pr i

WRI TTE N EXER CI SES

From the preceding table find the annual premium for the following f l o r ta e man 2 di a e a 5. a 1 . An or n ry li po icy f k n by yr of ge . - t e man 20 . 20 r . e a a e . 2 . A y ndowm n policy for for yr of g - an 2 . a e t o ut a l o r . e me t 3. A m 8 yr of g ook y ndow n policy for H o h anddie d after making 6 payments . w muc less would the combined pre miums have bee n on an ordi na ry life policy? man his 26th b th a t out a 20- a me t 4 . A on ir d y ook p y n life po li cy for ate he to o ut an r i a e i and4 yr . l r ok o d n ry lif pol cy for H e rs die d afte r making 12 payments on his fi t policy . How much more did the beneficiary receive from the insurance co mpany than the insured ? i e s not to be si hadpaid premiums (Div d nd con dered . )

68 BUSINESS MATHE MATICS

ANN UA L CA SH DI v NDs AND NE T COST OF INS UR A NCE ON P O LI CI E S o r —AGE 25

- 20 YR . E NDO M E N T OLI C Y - W P 20 P AY . LI FE I SS E D I N 1893 NE T E R U u g Y A NN L R E MI M C OS T A UA P U ANN Jii ggEfil éi g

r en e a andP a d-u u an — h 63. Cash S u r d r , Lo n , i p I ns r ce T e policies of most companies have certain privileges which the insured may take advantage of after the policies have been in force for 2 or 3yr . These privileges are as follows

h m e m t e m a . 1 . Borrowing on y fro co p ny

th a ash a e e e me t . 2 . Surr nd ring policy for c p y n “ ” R e e a a -u h h states a x 3. c iving p id p policy w ic fi ed amount of insurance during the remainder of life

h the a me t mi wit out fur r p y n of pre ums.

e s ed the a e the a x 4 . B ing in ur for f c of policy for fi ed

ea an m hs number of y rs d ont . INSURANCE 69

exam e a te 20- e For pl , f r a yr . ndowment policy for ta e at the a e 33 a e ta a ha e e k n g of in c r in comp ny , s be n in forc h 10 r . t e s e n for y , in ur d ca

1 . Borrow from the company on the security of

the policy .

2 . e e th Surr nd r e policy andre ceive in cash .

3. Stop paying premiums and be insured for the re

ain er i m d of h s life for 3532 .

4 St he a m n . op t p y ent of premiums a d be insure d for

e e e 46 h h en for 10 yr . or r c iv 3 0 in cas at t e d

m he he . of 20 yr . fro t date of t policy

e S ett eme t — U h h 64 . M th ods of l n pon proof of t e deat

f e the ase an e me t at O insur d (or in c of ndow n policy , the expiration of the endowment period) the policy is to

The a s a s be paid by the company . v riou w y of settle ment are :

Payment of cash to the b eneficiary . Annual pa yment of interest during the life of the

a a nd a me t Of the a e the benefici ry , p y n f c of policy

at the death Of the beneficiary.

a e t Of e a a a sta me ts the mbe 3. P ym n qu l nnu l in l n for nu r

of years specifie d to the beneficiary .

e a a a sta me ts a e ta 4 . Pa yment of qu l nnu l in l n for c r in period (usually 20 and for as many years

e e a sha e thereafter as the b n fici ry ll liv . The

amount of each annual pa yment depends upon the age of the beneficiary at the death of the

insured .

me t to the s r the a e the 5. Pay n u vivor of f c of policy or of

an annuity. 70 BUSI NESS MATHE MATICS

— 65. a s e I f h he due the L p s t e premium is not paid w n ,

n ma states policy la pses . Most compani es a llow (a d ny m a e ake it a law that they sha ll a llow) 30 da . of gr c during whi ch time the premium m ay b e paid plus the interest on that m f he s e shes pre ium for the overdue time . I t in ur d wi ’ to a the em m a te th s a s me he m st e p y pr iu f r i 30 d y ti , u und r go another examination by a physician and if successful he may pay it plus the interest on that premium for the time overdue .

a e a u — h 66. Fr t rn l Ins rance T is is an insurance Offered by

f e e a i The tateme ts di f r nt fr ternal organ zations . following s n are answers to a questionnaire sent to offi cers or those thoroughly a cquainted with thefinancial obligations of each th of ese organizations . The arabic numbers in each group refer to the same num ber I exam e I I I e s to the e se C t of group , for pl , (3) r fer J r y i y T h ’ eac er s life insura nce organization .

Names of the organizations .

1 . T Now and hen Association . p Name omitted . e C T Jersey ity eachers . e Na e t m omi te d . w U H

s U te States I i at Se v e Be e a s s at ni d mm gr ion r ic n fici l A oci ion .

Amount paid to dependents of members at the death of

member . ea h e e the as s at from c m mb r of oci ion . 3100 (some claim 1 ea th 3 from ch member of e association . 1 000 3 , or 3500 to 31 from e ach member in good standing (average I NSURANCE 71

assess e t to eet i How m n m number I I s paid .

1 . ssess e ts h h to t e e at e e 20 da . A m n upon d h of m mb r, in whic a e e se ot e th a t a fee a e a tax p y b for cond n ic , wi ddi ion l , c ll d , i s sent . e t Weekly dues 139i . a d Assessment of 31 due by the first of the following month for ea e e h a e h ch m mb r who s di d during t e month .

4 . At a e 27 st r I ease is a e O e g co 99¢ pe month . ncr l rg for ld r

men . — 5. A e 27 29 75a th th se a a l 2 off g , ¢ mon ; mon ly , mi nnu l y ( % ) ;

annual ly (4% o ff) .

e . 6 . Assessment at death of memb r

To whom is money payable ? When payable ?

T n e ea e . I e ate 1 . o o e designated by d c s d mm di ly upon proof of

death . d no t t e to the e an sur 2 . To e a n wif , if living, if , h n childr n if y

e e i e t e to the ea est . At viv ; if no wif or ch ldr n , h n n r of kin

once upon proof of death .

T the ea est . I e ate eat . 3. o n r of kin mm di ly upon proof of d h

T ate . 4 . O person design d

T e e e t s a a e at e es ate . 5. o d p nd n (u u lly r l iv ) d ign d

To a e es ate the e e . As s as 6 . nyon d ign d by m mb r oon proof of de ath is shown (by officia l certificate or personal view of

the remains by a n officer of the society) .

there a le gal prior claim which can be put on this money? ‘ T None e xcept for unpaid dues to the association .

P None .

W None . P None .

Q None .

Q None .

Any other information you think advisable . es the ass at ta e a e s a e and t e 1 . Du in oci ion k c r of in ur nc o h r

expenses of the club house . When the sick benefit fund rea ches a stated amount an assessment is levied on

members .

No e . 2 . r ply th m the t e t at the e is a 3. A good ing fro poin of vi w h mon y p id ove r immediat ely a t a time when the dependents may 72 BUSINESS MATHEMATI CS

ee it e h i an - i e m a it n d v ry muc , wh le in old l n co p ny t eat and et generally takes from 30 to 60 da . o prove d h g

the insurance .

P roof of death is suffi cient to get the money . C m a a s a th to a s e e Of e s etc . mon p y upr m fic r , o p ny p y 0 36 for funeral e xpenses .

6 . I t a f s i el e at s a st a e a es a t 16 f ord qu ck r i f m ll co , now v r g bou 3 per year per The only salaries paid are 350 per a the s t n nnum for ecre ary a dtreasurer .

m t s e e ts . Are the rm the i ffe A oun of ick b n fi y unifo , or do y d r for iffe e t nesses a e ts andma a me e a e d r n ill or ccid n , y mb r p y mor and e e e e are t e the a e all? r c iv mor , or h y s m for l — ’ b S e e ts Of r il 5e ee n t ai 1 ee s ess . ick b n fi 3 p w k, o p d for w k ln

2 . 4 r . 3 pe wk for 13wk . $3 U H U ( l 32 31

Uniform. " S None . P a Loc l organization (lodge) dues quarterly . Lo dge a e r is s mone y for sick membe rs .

5. e . S and a e t onc in 3 mo for home e xpenses . ick ccid n

e e ts st a th a 6 er . 12 . b n fi co mon , p y 3 p wk for wk s a 750 ss th e es th e s t Al o p y 3 for lo of bo y , or bo l g or bo h

arms .

6 . m t Of a me ts the sa e all e e di n the A oun p y n m for , d p n g upon e N a e t numb r of members in good standing . o ccid n or h h s e e ts . a n a a s a affa is e ick b n fi An n u l b ll or oci l ir h ld , w ic

usually nets about 3200 which pays incidental expenses .

e t u a — an i a e 67 . Accid n I ns r nce Accide nt insur ce s insur nc

an e a in r which covers loss by a ccidents . Accide nt dh lth su a nce i s insur ance which covers loss to the insured through

hea th b h e ts ss t . accid n , lo of l , or o

S e m ta t ata the a a to om i por n d for l ym n know follow .

at C st t te s th e at — Th 68 . Wh on i u O ccup ion? e profession ,

s ess t a e em me t an at e as a bu in , r d , ploy n , or y voc ion follow d INSURANCE 73

mea s h h n of livelihood constitutes t e occupation . S ould a

e s e a e h re an the a p r on ng g in work , for i , in y o r occup tion , s h sha hi uc ll constitute s occupation .

69 . Gre atest az ard Dete m ne th H r i s e Cl assifi cation .

If th a a ha m e tha o n e pplic nt s or n e occupation , a ll occupa h h n tions must be named in t e application . T e o e involving

he eatest ha a ete m es the ass a t gr z rd d r in cl ific tion .

o ant — at n 70. Age f Applic s Applic ions will ot be a ccepted 1 e 65but h from persons under 8 nor ov r , t ose who insured before the age of 65are usually carried to the 66th birthday Disability or health insurance is issued only to male persons between 18 and 59

B i — The m st al a 71 . eneficiar es y u w ys be named in the

and m st be e s s ha n n application blank, u p r on ving a i sur ab e terest in the e th e s e a s a e ath l in lif of in ur d, wif , f er,

h br he s ste the e at e a e e e t mot er, ot r, i r , or o r r l iv , d p nd n or a

h e es no t a e to me t a b creditor . If t e insur d do c r n ion ene “ ” ficiar he can state m estate m exe t s ad y, y or y cu or , ” i ni o rs ass s h h ase the m e m strat , or ign , in w ic c on y goes into the estate and has the same legal meaning as cash in the ba nk .

e —The es s a 72 . S cape o f Polici s polici u u lly cover a cci dents sustained while residing o r traveling for business or pleasure in any part of the civilized globe ; while discharging the usua l duties pertaini ng to the occupation named in the policy ; while pursuing any ordinary form of pleasure or re creation ; and while engaged in athletic exercises usually in

n ess n dulgedin by business a dprof ional me . 74 BUSI NESS MATHE MATICS

R k — 73. Limit of is The maximum amount of death benefit andweekly indemnity most companies will carry on any o ne risk (exclusive of the double clause) is stated op

osite ea h ass at n and a t be ease p c cl ific io , c nno incr d except by

h t m the h me special aut ori y fro o ofli ce .

h e d R sk s — e r n 74 . Pro ibit i P rsons a e ot insurable who are

ea m e e to use a t h a h blind , d f , or co p ll d cru c or c ne ; w o are

e eme te eeb e- m e s b e t fi insan , d n d , f l ind d , or u j c to ts ; who have suffered from paralysis or are paralyzed ; who are in

’ ess o r s e tab e who are temperate , reckl , di r pu l ; suffering from

ha e a n e m t any bodily injury ; or v y d for i y, disease, or in

firmity.

— h 5. C e s . A e s who has st a a a 7 rippl p r on lo nd , or foot, or

h an e but is the se an ab -b the sig t of ey , o rwi le odied andac

l and h se at is no t a ceptab e risk , w o occup ion cl ssed as more

ha a m a b e s e at an hazardous t n ordin ry , y in ur d advanced rate by applying to the home offi ce ; but o ne who has lost

he ee who is b e to u a l eg above t kn , or o lig d se a crutch or

e who ha an the a esa e e ts i can , or , ving y of for id d f c , s engaged

n m e ha a s tha in a occupation or z rdou n ordinary, wil l not be acce pted .

ra of W men — A ma n 76. Insu nce o wo n will ot be accepted for weekly indemni ty unless enga ge d in a stated business or employment from which she derives a regul ar income on If which she is dependent for support . in receipt of any

me h ma be su e h such inco , s e y in r d for deat benefit and weekly indemnity (without doubling clause) at the rate

h r at but s h named for e occup ion , uc policy will not be issued for more than death benefit and $15weekly indem

76 BUSI NESS MATHE MATI CS

e b e ee . E e t me 5 em m is e s te w k y w k v ry i a g! pr iu d po i d , s meth i E a es can o ing s saved . very m a n who works for w g se e the s a ea s a f cur in ur nce protection which his m n f ord ,

th t ma i a ea wi ou k ng gr t strain on his income . The companies usually send a gents to the home e very

ee to e t the m Th is e the w k coll c pre iums . is don for con veni nce the h e but an as the a e t e of policy old r , if for y re on g n sh a to a th m h h fi ould f il c ll , e one y may b e sent to t e ome of ce h m n t e m a . One a w as ta t 1 a d of co p ny co p ny s r ed in 866, now has more than twenty million policies of this kind in

m m h n e . e be t e am h 1 a d forc Any r of f ily over t e age of yr .

h r x b h a h i hea th to t e a e 65. e t t w o s up g of y n ir d y , in good l , n n ca obtain o e of these policies .

I llustrative Example . Suppose a youn g man of 25pays 10¢ a week to H e se es a h h t a the ee the company . cur policy upon which e as o p y w kly premium e a ch wee k andthe company a gre es to pay in case of his death

th um 180 e he has a all e s 6 . e e s of $ , provid d p id pr mium for mo or mor up I f hi n t t the to the time of his de ath . s death should occur a y ime wi hin t andhe has a e a to t at t e the a l firs 6 mo . p id r gul rly up h im , comp ny wou d o ne- a the 180 e e h e e me ate a te the e e pay h lf of 3 , v n if di d im di ly f r d liv ry S e a e d n t e i e the a e t of the policy to him . om comp ni s o o r qu r p ym n of further pre mium on any industrial policy after the insure d rea ches 75 I h a es a - u s a e a e . st a e e t as yr . of g ndu ri l ndowm n , c v lu , p id p in ur nc , - e t and a t mat exte e i s an e are in paid up endowm n , u o ic nd d n ur c common this kind of insurance . CH AP TER vii

EXCH ANGE

80 D m — . o e stic Exchange E xchange is the payment of a

ebt s b ht he h d for good oug , or for some ot r purpose , wit out the se m D mes h nding of oney . o tic exc ange is such pay me t bet ee e s s a s the s n w n p r on or corpor tion in ame country .

ORAL EXER CISES

S 1 . tate some obje ctions to sending money through the mails even if the ette ste l r is regi red .

2. St i ate some obje ctions to sending t by express .

1 — 8 . M e th ods of Exch ange The paying of debts without the transmission of real mone y is effected by :

1 h . Persona l c ecks 5° Postal office money orders 3 / S Bank drafts P Express mone y orders P Telegra phic mone y orders

6 mme a a ts . Co rci l dr f

If a merchant sends his check to a manufa cturer in De

it he atte e s t it his ba and it be tro , t l r will d po i in nk, will

i The he is the et e h h credited to h s a ccount . c ck n r urn d t roug

the ma e an h h proper channels to the bank of k r , dt e merc ant h hi is charged with t at amount on s account .

a the t 25and es not ha e a If A owes B , in no r own , $ , A do v

he ma to the st-office andb a m bank account, y go po uy oney 77 78 BUSI NESS MATHE MATI CS

e h m Th he e s b ma ord r for t e a ount pa yab le to B . is s nd y il to B h h h e o u m st- th e , w ic pa ys t e d bt . Find t fro your po o e who is liable in case the mone y order and letter are lost or destroy ed .

If so shes he can to n h h A wi , go a bank a dpay t e cas for a ba a t a ab e to the m f he e n nk dr f , p y l B , for a ount o t d bt a d

an a then send it to B . B c t ke it to his b ank and get the

h ha that am i cas or ve ount credited to h s a ccount . Or A can go to the express offi ce andpay cash andbuy an ex ess m e e a ab e to andse th s pr on y ord r p y l B , nd i on to B .

r n to a te e a h o fli h n O A ca go l gr p ce , pay cas a d buy a te e a h m e e a ab e to and can e th l gr p ic on y ord r p y l B , s nd i s by hi h n telegra phic communication to B in s city . B t en ca get h h the money at t e telegrap offi ce in his city . Or if A i s a merchant to whom a debt is due from some ’ e s who es B s t ma se a mme a p r on liv in own , A y nd B co rci l draft drawn on C for the amount that A owes B .

— Th sta M e de . s is a e m e e b 82 . Po l on y Or r i gov rn nt ord r y a post- o ffice in o ne pla ce to a post-o ffi ce in some other place

I n e to to pay a stated amount to a Specified person . ord r obtain a postal mone y order a person must fill out an appli cation blank which states : h The ame and a ess t e a ee . 1 . n ddr of p y

2 Th am t to b e a . . e oun p id

The ame anda ess the o ne who b h 3. n ddr of uys t e order .

The rates charge d for postal mone y orders are :

For or less From to From to 05 08 10 12 EXCHANGE 79

Th e largest amount fo r whi ch a single postal mone y order ma b e y ssue is $100 . I f ar e s ms are to be se n i d l g r u nt, o e

m st hase a t a m n e Th u purc ddi ion l o y orders . ese orders are

to b e presented for pa yment at the po st-offi ce on which the y are a a t a b a . dr wn , or nk If th ey a re no t presented W ithin

1 . the are st a yr or if y lo , duplicate may be obtained from Washington upon proper presentation of evidence of such loss .

WR ITTE N EXE R CI SE

1 . Find of the following money orders

(a) (b) (c) (d) (e )

Ex e — 83. pr ss Money Orde r s An express mone y order

m 2 i s te s m a to a sta m e e I i (For ) qui i il r po l on y ord r . t s a written order by o ne expre ss agent to another agent to p ay a um Th stated s to a spe cified person . e largest amount of an f n express mone y order is $50 . I o e desires to send more he

E x must pur chase additiona l o rders . press mone y orders may b e indorsed a ndtransferred in a manner similar to b ank

h a s ar th same as s m drafts and c ecks . R te e e for po tal oney orders .

WR ITTE N EXE R CI SES

125 ex ess e the t ta fee the t a s e $ e . 1 . Find o l for r n f r of by pr mon y ord r t is the est w a to ex ess e e s and 2 . Wha b y buy pr mon y ord r for what would be the to tal cost of the same ? man a a a a t e e s to se a e to 3. A h ving b nk ccoun pr f r nd ch ck of he es at e t a ex ess e e sta e pay a bill ow , r h r h n pr mon y ord r or po l mon y Wh ? order . y BUSI NESS MATHE MATICS EXCHANGE 8 1

84 — . Te le graphic Mone y Orde rs Sometimes it be comes necessary to send mone y immediately for some special p ur se sa po , y o ne of the following :

To ba to mee m w nks t aturing obligations . To fire and e s a e m a es m lif in ur nc co p ni for premiums . a To t a e e s and t a r v l r r veling salesmen . e t To st e ts and s at s h sem a ud n pupil c ools, in ries, colleges,

To a a tee ha s gu r n purc se .

To a m a b s cco p ny id for contra cts .

For pa yment of bills .

h h a msh an h 8 . t e ase a stea d t eate For purc of r ilro d , ip , r

tickets . h ll ases a s . 9 . For purc of kind

h a ts and the ememb a es . 10 . For olid y gif o r r r nc

11 mem a as s anda e sa es . . For ori l occ ion nniv r ri ll 12 . a me t taxes and assessme ts and fo r a For p y n of n ,

other purposes requiring the quick remittance of mone y .

To an a e e e the 10- te e a ate I llustrative E xample . y pl c wh r word l gr m r is 60¢ o ne can send $50 anda 15-word message for

WRITTEN EXER CI SE S

h t ta a e se $30 te e a h to a a e 1 . What is t e o l ch rg for nding by l gr p pl c where the cost of a -word message is '

the st se 250 tele ra h to the sa e a e . 2 . Find co of nding $ by g p m pl c s t at he is a st a e a e a nd ee s e his 3. A ma n find h in r ng pl c n d mon y from

- H e te e a s 75. The a e a ten essa e firm at once . l gr ph for $ ch rg for word m g the st the ha e between those places is $ 72 . Find co , including c rg for the message . an a s a sta e e a the 4 . A m p y bill of by po l mon y ord r, no r bill of t e 115 a nd a te e . by e xpre ss money order, no h r bill of $ by l graph

- e is 42 i the t ta st to him. If the 10 word rat 9, find o l co

6 82 BUSINESS MATHEMATICS

I n cases like these mone y may be sent by the use of a te e a h m e Th h e s are o b l gr p on y order . e rates for suc ord r

tained in a table like the o ne following .

! TAB LE o r CHAR GE S FOR TE LE GR APH MONE Y ORDE RS

To ANY PLACE W H E R E m s l o-Wo nn TE LE G R AM RATE I s Fo x A TR ANS F E R O F

8 o r l ess

t o 8 . 78 . 84 . 90

t o

t o

t o 1 . 59

to

to 5 2 . 1

t o

t o

t o

t o

t o

t o

t o

t o

t o

t o

to

to

t o

t o

t o 6 . 13

t o

t o

t o

to

t o 7 . 15

t o

t o

t o

t o

t o

t o

a nd up a dd fo r ea ch a dditio nal one hundred do llars o r fractio n thereof t o the cha rges for

in r -wo rd m I ncl ud g s essa ge .

84 BUSI NESS MATHE MATI CS m n ent . A . H . Jo es must re ceive this bill of lading andpre sent it to the freight office in Alb any before he can obtain h t e goods . Company A takes this bill of lading to the Fi rst Nationa l Bank in Binghamton and deposits it along with a

a t s mi a to ha h dr f i l r t t s own in Form 3. The First National Bank of Binghamton sends this bill of

a and the a t o a h N l ding dr f t , s y, t e Alb any Se cond ational h a h a e th . . es B nk , w ic in turn t k s e draft to A H Jon for pay m If e t . the atte a s it a e ts it case it is a t me n l r p y (or cc p , in i

a t he h e e th a and t can dr f ) , t en r c ives e bill of l ding in urn se h Sh A. . e cure t e shoes from the freight o fiice . ould H Jon s

e se to a the t the h a Na a r fu p y draf , n t e Alb ny Second tion l

‘ B nk Th the a so informs the Binghamton bank . e latter n

t es m a who m t h the h es s me no ifi Co p ny A , us t en sell s o in o other manner .

ts it the I n a i i a me a n . . s a cce c se t s ti dr ft a d A H Jone p , ‘ i h e a to m a and the atte can draft s t en s nt b ck Co p ny A , l r take it to the bank andhave it discounted andthe proceeds

he a t the ba C m a can credited to t ir ccoun in nk , or o p ny A ask the Albany b ank to discount it and send them the pro

If a t me a t i s s te t me is e m the ceeds . i dr f di coun d , i figur d fro day it is discounted .

— a e s the C mme a D a t 1 . the 88 . Advant g of o rci l r f By above method the purchaser must pay . for the goods before he receives them . n t is ast due and a a t is se t to be 2 . I f a a ccoun p , dr f n ’ h s b a it is te e effe t e collected by the purc aser nk , of n v ry c iv in se curing the mone y .

m a the ha e e t the h me 3. Co p ny A , if y v good cr di in ir o

a n mme ate ease the s be a se the bank, c i di ly incr ir fund c u

‘ home bank will discount the draft at once for them . EXCHANGE 85

4 . s a t se th ess A . H . Jone c nno nd a check which is wor l and h m t us cause Co pany A much loss in money andtrouble .

The h se a 5. purc a r c nnot claim that he has no t received the bill of the goods .

Th hase an 6 . e purc r c not maintain that he has already sent a remittance in some other way;

WRI TTE N EXER CISE S

1 . If mme i a aft a a a te ate e e dis a co rc l dr of p y ble 90 da . f r d w r

te 15da . a te a te and the ate s t nd h coun d f r d , r of di coun were a t e t ex ha e e e the net e s o f th cos of c ng w r find proce d is draft.

C . tha R . . o 2 . Suppose t D Ford of New York were to sell A . H .

ll a s Ch a 3. s a t t 4 Wi i m of ic go bill of good moun ing o on Apr . 1 .

- h e a a 60 da . a t l Suppose t at th y dr w dr f on A . H . Wi liams through the I First National Ba nk of New York . f a Chicago bank should buy this 1 draft at 6% discount andexchange we re 7 0 what are the proceeds for

Co . ? R . D . Ford

. 4 ase an e o f s 1 f 3. On Apr 1 , you purch d invoic good of $ 00 rom H . R . T - Co . a N . . e s 30 da . a t m ate Brown of Alb ny, Y rm , dr f fro d of sale, h ss . 16 e e e a t e a t ate . 14 le On Apr , you r c iv d by m il dr f d d Apr , d i a e te it an et e t t . . anddue in 30 days . You cc p d r urn d o H R Brown e l ha e to a t s a t? Co . Wh n wi l you v p y hi dr f

ed in D me st Ex han e —The 89 . Te rms Us o ic c g maker or h h drawer of a draft is t e person w o signs it .

n who i to it The drawe e is the o e s pay .

he o ne to h m the m e is a The paye e is t w o on y p id . Pa r means that -a draft is bought exactly for its face value . Premium means that the buyer of the draft must pay E x ha e is the more than its face value . c ng n said to be at a

h s he sa th e b a s premium . T is occur w n, y, nk of St . Louis

ew a a e sum be a o we the banks of N York l rg , c use the banks

s m st e the a the t a s tat h of St . Loui u i r p y for r n por ion of t e

te est money to New York or pay in r upon it . If A lived in 86 BUSI NESS MATHEMATICS

h me h u h o a a em m be St . Louis at t at ti e wo ld ave t p y pr iu cause his draft would increase the amount that the St .

Louis b anks o we New York b anks . Dis count is a term used when the draft can be bought Th h D slightly below fa ce value . is mig t occur if in New

to e to se a a t St . s e th York wer nd dr f A , in Loui (giv n in e a e a a a h at the same t me b e a se it e bov p r gr p ) i , c u would l ssen

h w ha a the ba lance whic Ne York d gainst St . Loui s . Exch an ge on ch e ck s is a small sum usually charged by a

a a he m a the t s h bank for p ying c ck fro no r ci y , ince t e bank is put to some expense in sending the che ck b a ck and colle ct

i The m a the m e t . s s m 10 but ing on y on u v ry fro ¢ up , usu ally are no t more than A sight dr aft is o ne which must be pai d immediately when

it is presented .

im a t is a ab e a te a stat A t e dr f p y l f r ed time . I t is pre

te at e to the a ee he a e ts i t h h sen d onc dr w ; if cc p , e writes t e ” word Accepte d a cross its fa ce and signs it as we ll a s dat

h h s that he mi ses to it . T s s a it a n ing i ow pro p y , d really

makes it a promissory note .

D af t — A ba ina ma 90 . Bank s s t r nk ll own , for instance , will kee p funds in some large city bank on which it can draw che cks just as an individual can draw a check on his bank

for some other person . A b ank dr aft (Form 4) is an order drawn by o ne bank h a gainst its de posits in anot er bank .

— e ul ne ss Bank D afts A b a 91 . Us f of r nk draft is issued

h a h e a he is e b a by t e b nk w il c ck giv n y person or a firm . The cashing of the former is considered safer tha n that of the

latter . EXCHANGE 87

There is less expense in collecting a bank draft than a personal check be cause the former are drawn on banks in large cities while the latter are very often drawn on small

ba h nks in t e country . E XP LANA TI ON : The cashier of the Franklin Nation al a i D i a e . h b a e s ts ts B nk of Fr nklin s W . . Ogd n T is nk d po i

s h h h h fund wit t e C ase Nationa l B ank of New York . W en

NK L N 1 FRA I NATI ONAL BANK No . 687

R NKLI N N. Y Ma 1 192 1 F A , y ,

P a y t o th e o rder E H . Willia ms ' Twe n t y-fi ve a n d fob

To CH S E NAr i ONAL NK W . D . OGDE N Ca shier A BA . of Ne w Y o rlc

Form 4 . Bank Draft

the latter bank pa ys the amount of thi s draft it will reduce the balance of the account of the Franklin National B ank for m this a ount . h E . . W ams w o es a a ts to a h H illi , liv in Fr nklin , w n p y t is

se b a . H e u amount to T . H . Mor of Al ny accordingly b ys

H c u h a h i this draft to send to Morse . e o ld ve ad t made

e to M se but it sh ea h ‘ M se th no pa y abl or , if ould r c or wi

to ex a it M se m ht no t e t it to the ht letter pl in , or ig cr di rig

The e e W ams has it ma e a ab e to h mse man . r for illi d p y l i lf “ an a te e e it ses it ab t as s P a d, f r r c iving , indor ou follow y ” ndthe s s his W ams h e T . . M se a to t e ord r of H or , n ign ( illi )

it Th a es him the e e t b M se name to . is ssur of prop r cr di y or and also a cts as a receipt if Morse afterwards receives it a nd uses it .

M se e e e it and has it ashe at his ba the Mr . or r c iv s c d nk,

The ba Nat a Albany National Ba nk . Al ny ion l Bank col 88 BUSI NESS MATHEMATICS

e ts it C t ba l c as follows : Thi s b ank has a New York i y nk, the Chem I the e e se s the a t ical National Bank . t r for nd dr f

The Chemi a to the Chemical National B ank as a deposit . c l

Nat a h h ea se h h ion l Bank sends t e draft to t e Cl ring Hou , w ic

The atte in turn collects it from the Chase National B ank . l r b ank upon pa ying the draft charges the amount of the draft to the Franklin Nationa l Bank .

Ban D afts —The ra i Nat a a 92. Cost of k r F nkl n ion l B nk

h Th s ha e i would make a small charge for t e draft . i c rg s

The ex ha e be a b the called exchange . c ng would p id y pur 1 Th a ha e is th a m i chaser (Williams) . e usu l c rg 1 6 70 wi in mum charge of The following rates are in cents for between these

cities . — Chi cago New York — San Francisco New York — Boston New York

~ N w St . Lo uis e

At 20 e the st a a t Ill ustr ative E xampl e 1 . ¢ pr mium find co of dr f on New York in San Francisco .

SOLUTI ON : 3 x $2 0 premium cost

Ex P LANAr' I ON : 3 thous ands

I f x a e is se l at 25 s t the I llustrative E xample 2. e ch ng l ing ¢ di coun , find cost of a draft .

SOLUTI ON : 3X discount on

WR ITTEN EE RCI SE S

Using the above rates find st a a t New at C a 1 . The co of dr f on York hic go for

t a a t at San a s New . 2 . The cos of dr f Fr nci co on York

t a t New a t B st . 3. The cos of dr f on York o on EXCHANGE 89

93. To ndthe r ee s a Fi P oc d of Draft.

I st a e E llu r tiv xample 1 . Find the proceeds of a sight draft of if the colle ction andexchange is

SOLUTION 5% of

Il strat e E xam e 2 . the e s - lu iv pl Find proce d of a 60 da . commercial a t s the da it was ate at is t h m e dr f of if old y d d d co un , w en on y is worth

S L TI ON - O U $40 60 da interest . of 310 40 10 50 to ta $ $ 3 , l discount $50 proceeds

WRITTEN EXER CI SES

Findthe proceeds of the following sight drafts aft he e ti and x ha e i 1 . dr w n coll c on e c ng s i % . a t he e t andex ha is 2 . dr f w n coll c ion c nge - the ee s a 0 da . a t 3. Find proc d of 9 dr f of if sold at dis t he e is th coun , w n mon y wor the ee s a a t he e t n x 4 . Find proc d of dr f for w n coll c ion a d e change is t and ex han e is hat are the ee s a t 5. I f colle c ion c g w proc d of draf for ar he ee a s ht a t and 6 . What e t proc ds of ig dr f for if collection e xchange is i % .

ta M e e s a k D a ts and T a e A 94. Pos l on y Ord r , B n r f r d c — ceptance a (a) The following differences exist between a postal mone y order anda bank draft '

sta m e e m st be ese te to th st 1 . A po l on y ord r u pr n d e po

h i is a to s me ba h othee on w ich t dr wn , or o nk whic

- ffi ce can present it to that post o .

h an ba a t can be as e at ba . 2 . A nk dr f c d y nk 90 BUSINESS MATHEMATI CS

sta m e e i po l on y ord r s to be indorsed but once .

ba a t ma b e se an mbe nk dr f y indor d y nu r of times . postal mone y order will no t be cashed until the post master re ceives a notice of such order from the

fi c h h w e h o e w ic rot t e order .

6 . ba aft can be ashe as s as it i A nk dr c d oon s presented .

(b) A trade a ccepta nce is like an ordinary bil l of exchange except that it has a written guara ntee upon it that the in debtedness has nate a n ex ha e m origi d on c ng of erchandise . The advantage of the trade a cceptance is shown by the following example

B st b s a firm New h A firm in o on uy from in York wort of goods . Simultaneously with the shipment the selle r draws on the buye r a draft at 90 days from date or sight (a ccording to the terms of sale) andmails it e and the Th to the latte r with the invoic bill of lading . e usual form of draft is used with this additiona l clause ; the obligation of the acceptor h ase h hereof arises o ut of t e purch of goods from t e drawe r . The e a e ts the a t t a ss the a e of it e te buy r cc p dr f by wri ing cro f c , Acc p d ”

Ba B t . Pa a e at . s H e ates y bl nk, o on d andsigns this acceptance and returns the acce pte d draft to the seller in “ N w The e t is a t a e a e ta ce bec e York . docum n now r d cc p n , oming u a e B st 90 a s the a te of the a t if aw d e for p ym nt in o on d y from d dr f , dr n “ ’ a te ate but 90 da . ate a e ta e a 90 a s f r d , in from d of cc p nc , if dr wn d y " sight . If the se ller re quires the mone y repres ente d by this acceptance he may take the accepte d draft to his ba nk and the bank wil l pur chase it e e see t Th provided the names appea ring th r on m sa isfa ctory . e bank in

t r a e is t .the a e tan e w t the e e al ese e a the u n m y r d coun cc p c i h f d r r rv b nk , as the hase s andth s f obliga tion arise s o ut of purc of good , u alls under the The a t provisions of the Federal Reserve Act . m rke rate of discount is charged by the bank for its courtesy .

t P es e T a e . New Sav ay , Norber , rincipl of For ign r d York, Ronald

1919 . Press,

92 BUSINESS MATHE MATI CS

VA LUE S o r FORE I GN COI NS I N UN ITE D STATE S MONE Y

L E I N Te ams CO NTR Y VA U R ImAR xs U MO NE o r U . S . Y

Argentine R e public gold ustr a -H un a r old Grea l de reciate d A i g y . . g t y p Canada gold

rance . o d E xcha n e al ue 0 698 F g l . g v 3

G erman y . go ld Greatl y de preciated Grea t Brita in gold sterling E xchange va lue

I tal . o d E xchan e val ue y g l . g ld E xchan e a lue Ja pa n. go g v

Bi l s Ex ha e — B s ex ha e are at 98 . l of c ng ill of c ng dr f s of a person or bank in o ne country on a person or a

The bank in another country . y are divided into three classes :

’ 1 B s m . ankers bill (For

2 a b s a b one me ha t an . Commerci l ill dr wn y rc n on th o er .

D me ta b s h h are a b o ne m e ha t 3. ocu n ry ill , w ic dr wn y rc n

a the a nd ha e a b a atta he upon no r v ill of l ding c d , together with an insura nce policy covering the

goods en route .

h ar s a ss e ate a e th Bills of exc ange e u u lly i u d in duplic , c ll d e

The ar e t b ff original andthe duplicate . y e s n y di erent mails h m h S and the payment of one of t e cancels t e other . ome

i t andthe ate is a e fil times the original s sen , duplic pl c d on e

~ and sent later if needed .

— P of Exchan e Th s is the a t a a e the 99 . ar g i c u l v lu of pure metal of the monetary unit of o ne country expressed in h terms of the monetary unit of anot er country . EXCHANGE 93 94 BUSI NESS MATHE MATI CS

I x s n . llu trative E ample . O e pound ste rling £ 1 contains 1130016 gr fin e . 1 ta fin . of gold 8 con ins gr . of e gold T e e e £ 1 h is a e the ar ex an e h r for , whic c ll d p of ch g et ee the ite States and b w n Un d England .

— 100 . R ate Ex h e Th s i the ma et a e o ne of c ang . i s rk v lu in

b ex h h country of a ill of c ange of anot er country . The price paid for a bill of ex change is constantly fluctuat If th due e the th s to s and em a . e ing, lik o r ing uppl y d nd Unite d States should o we Great Britain the same amo urit

ha eat ta es the U te States the ex ha e t t Gr Bri in ow ni d , n c ng If h h would be at par . t e United States s ould o we Great

a m e tha G eat B ta es the U te States Brit in or n r ri in ow ni d , then exchange in the United States would be at a pr emium

i b a u If me a s a ndin Great Britain t would e t a dis co nt. A ric n

m h m tha he m t he will ha e are exporting uc ore n t y i por , t y v many b ills Of exchange in the form of documentary bills to

h m a b a e and s ex ee em a sell t e A eric n nk r , upply will c d d nd ,

x h b e ar . th the ha thus ca using e c ange to fall low p On e o r nd , if Great Britain is exporting to the United States much

h h ite States is ex t to G eat B ta more t an t e Un d por ing r ri in , then bills in the United States will be scarce and sell a t a premium.

R ate s Ex han e —Ex ha e 101 . Quotations of of c g c ng on Great Britain is usually quoted at the number of dollars to the pound sterling ; means that a pound bill on Lon don will cost

i te at th num a e B e m etc . s e E xchange on Fr nc , lgiu , , quo d

h ex ha e a e h a . T s te ber of cents to t e fr nc u , c ng on Fr nc quo d at means that 1 franc costs Exchange on Germany is quoted at the number of cents

Th s mea s that hase 1 ma . to 1 mark . u , n will purc rk

96 BUSI NESS MATHE MATI CS

WRITTEN EXE RCI SE S

Us the e tat s h t a h the ing for going quo ion , find t e cost of draf s of e c of i a ts st at the a at x an s at the follow ng moun , fir norm l r e of e change , d econd quoted pri ces :

1 . £200 5. marks

2. 2 6 . 40 marks 5 a s 3. 0 0 fr nc 7 . 150 guilders (Holland)

4 . 6 5 00 marks 8 . 3 0 guilders

T 9 . Do e Ne es . . e w s Lo £300 5d. John of York ow H Jon of ndon , 88 H e s a e a t t s a t to hi S se t buy for ign dr f of hi moun pay s bill . uppo hat exchange on Lo ndon is what is the cost of the draft?

e — Th 102. Le tte r of Cr dit is is a circular letter (Form 6)

e b s me ba ba e t h h issu d y o nk or nk r, in roducing t e older and ’ instructing the bank s correspondents in stated pla ces of the world to pay the holder any amount up to the face of the

letter. The holder de posits with the bank cash or securities to h the face amount of the letterof credit . T e purchaser must sign this letter at the purchasing bank in order that he may

be properly identified by their correspondents . H e a lso writes other copies of his signature which the bank forwards

to its correspondents . When the holder re quires money he presents the letter to some o ne of the banks specified (as a correspondent) to gether with a draft drawn by the holder for the amount re

If the s at es a ee he is a the sum as e quired . ign ur gr p id k d for , and such sum is indorsed on the b a ck of the letter by the

The ba ma the ast a m t ta s pa ying bank . nk king l p y en re in

n t s it to h a the letter of cre dit a d re urn t e drawee . B nks usually Charge a commi ssion of 1% for issuing a letter of

credit . EXCHANGE

F m 6 Letter o f Cred t or . i 98 BUSI NESS MATHE MATI CS

103 ’ — . Trave le r s Ch e ck This is a circular che ck (Form 7) which i s made pa y able for a stated amount in the cur rency of the foreign countries named on the face of the check .

The are s a am ts 10 20 50 100 and y u u lly in oun of $ , $ , $ , $ ,

$200 . A commission of is the customary charge .

t M e — es a 104 . Pos al on y Orde rs The following rat prev il

e sta m e e s a ab e st a for for ign po l on y ord r , if p y l in Au ri ,

e m B a Ca e C C sta R a De ma B lgiu , olivi , p olony , o ic , n rk,

E t Ge man G eat ta as k gyp , r y, r Bri in , Hondur , Hong ong,

a I ta a a be a xemb New S th Hung ry, ly , J p n , Li ri , Lu urg, ou

Wa es New ! ea a e t a ee s a R ss a l , l nd , P ru, Por ug l, Qu n l nd , u i ,

Sa a S th st a a S t e a Tasma i a The lv dor , ou Au r li , wi z rl nd , n ,

T a s aa U a and t a . r n v l , rugu y , Vic ori

FOR OR DE R S FOR OR DE R S

3 . 01 to 3 to 6

c If pa yable in any other foreign ountry .

F OR OR DE R S F OR ORDE R S Co s r

01 to to .6 From 3 . From 3 0 70 80 90

WR I TTEN EXER CI SE S

the st a sta e e $35se t to a e s 1 . Find co of po l mon y ord r for n p r on in

Canada .

100 BUSINESS MATHEMATI CS

? 2. What will it cost me to buy $250 worth of orders for a man in P aris

3. How much would it cost me to buy the following orders made pay able to myself in the following countries :

AMOUNT OF O R DE R PAYA BLE I N $25 London 50 P aris 25 Constantinople 50 C alcutta

4 . 2 0 B If 1 t A sends 0 francs to in Switzerland . franc cos s find cost of postal order .

— 105. Use C mme a B l s Ii E a of o rci l il A in ngl nd owes B , a me ha t the U te States and shes to h rc n in ni d , B wi colle ct, e may draw up a commercial bill andl et his bank colle ct it in a manner similar to the colle ction of sight or time drafts in h domestic exc ange .

mme a a B — 1 . m Mr 06 I te e t b i o f a n . J es di P y n y ll L di g . on ,

me ha t the U te States se s a b a rc n in ni d , nd ill of goods to Mr .

Mr . es h ams who es . Willi , liv in London Jon delivers t e goods to the transportation company and re ceives a bill of

H e es the s a a t n lading . also insur good g ins loss in transit a d re ceives the certificate of insurance from the insurance h r . e t e a s a b ex ha company . M Jon s n dr w ill of c nge on M r .

ams and atta hes the b a and the s e Willi , c ill of l ding in uranc

x h h certificate to this bill of e c ange . All t ese pa pers are

t th e the ba hi h b s the indorsed o e ord r of nk w c uy draft . Mr .

he es a the s he has shi If Jones t n receiv p y for good pped . the goods are lost the bank is reimbursed by the insurance

I f the b is e t b e the r company . ill uncoll c i l , goods a e taken

The U te States ba over by the bank . ni d nk indorses these

n them to a e ba the papers a d sends for ign nk , reby receiving Th credit by the latter for the amount . e foreign bank then collects the bill . EXCHANGE 101

MI SCELLANE OUS EXERCISE S

1 At 25a and 1 the a t the st o f a 25- . ¢ word % of moun , find co word a e e e New to P a s a s ex a e c bl mon y ord r from York ri for fr nc , if ch ng is quoted at (ce nts) . ex te s to a e the s ex ha e : £500 2 . An por r old brok r following bill of c ng at francs at and marks at Find the total

net ee s the e ha e e t . proc d , if brok r c rg d for coll c ion

3. W a t b e the st a L a t £220 ex ha e e h would co of ondon dr f for , c ng b ing quoted at E s e 4 e hi so n 200 to L . . A man s nds s 3 ondon How much ngli h mon y would the so n re ceive if exchange is quoted at

e ts t Ne a a a t an . 5. A Frenchman pres n o a w York b nk dr f for fr cs What money should he re ceive if exchange is quoted at CHAPTE R VI I I

TAXES

1 — 07 . Kinds of Tax M any businesses a s well as many

e s s ar e tax i p r on e r qui re d to p ay so me form of tax . A s money levied upon a person or property for the payment of

the b x pu lic e penses .

re t tax is a tax e e a e s his e t his A di c l vi d on p r on , prop r y , or If i i business . t s upon his business it usually takes the form

a i e se fee an hi i i a a l ax. of l c n , dif upon s person t s c lled pol t An indire ct tax is a tax (called a duty) on imported goods or a tax (called an internal r eve nue ) on tobacco products .

h x n x T e latter ta need o t be paid on goods e ported .

m ax rm An inco e t is a tax on the income of a person or a fi . An e xce ss profits tax is a tax upon the excess profits of a business .

The taxes are levi ed by officers called asse ss ors, or by people called in come tax colle ctors .

u Taxe s — The se taxes i to m 108 . P rpose of purpo of s eet

These se m the expenses of the government . purpo s ay be classified somewhat as follows :

the a m an 1 . Nat a taxes are to a d a he ion l p y r y n vy, t

a a es the e s andem ees e s n s l ri of offic r ploy , p n ions, a d

an the U te States e me t ex for y o r ni d gov rn n penses .

2 taxes are to a the o fli cers and em . State p y ir ployees,

t the s h s e s t es as m to suppor ir c ool , univ r i i , ylu s, and

to pay all state expenses . 102

104 BUSINESS MATHEMATICS

I st at e E x Wh h as a e ta er llu r iv ample 2. en t e tax rate is stated c r in p cent ’ Baker s property is asse ssed at the tax rate is Find his

SOLUTI ON assesse d valuation 15 a .0 tax r te 0

$75tax

t at e E xam h th a x t Illus r iv ple 3. W en e t ra e is stated as a certain num e ber of dollars on each hundr d of dollars .

’ h F n White s prope rty is assessed at t e tax is p er $100 . i d the amount of his tax .

SOL UTI ON 30 hundreds of dolla rs 30 X tax

’ a e Ex 4 t T Illustr tiv ample . Jones proper y is assessed at he i 2 r h a i x tax rate s $ 5pe Find t e mount of h s ta .

SOLUTI ON 5thousands of doll ars 5X $25 $125

ORAL EXE R CI SES , f

te ea h the tax ates t th a s 1 . Sta c of following r in wo o er w y

32 mills per $100

t its ea a at i h 2 . I f a e ta sta assesses e at e s t e c r in te prop r y i of r l v lu , wh assessed value of property worth Of a building worth Of a manufa cturing plant worth If e t a e at is assesse at its a e andthe tax 3. prop r y v lu d d é of v lu er 100 hat is the tax? rate is p $ , w

an a 2 tax the real a e his h se . What er 4 . A m p ys % on of v lu of ou p ? cent of the real value does he pay es s e t t and is taxe 2 er h re 5. Jon own prop r y wor h d $ p und d i W a s s a h se t an dollars on f; of the real value . illi m own ou wor h d l hi h a s the a is taxed, 19 mills on 4 of the rea value . W c p y l rger tax? How much does he pay? TAXES 105

WRI TTEN EXERCISES

1 C . omplete the following form

R E L L E R CTI ON OF A VA U F A R TE OF TAx M O NT O F TAx OF p Ro p E R TY VALUE ASSE S S E D A A U

n 1 Full va l ue . 004 o $ p er $ 100 p er 2 mills o n $ 1

I n a e ta t a 1 is a e all taxes a e e c r in ci y, discount of % llow d on p id b for i 10 a nd e e Mar . 1 s nt s a te Feb . 10 a Feb . ; if p id on or f r b for , di cou 1 s t is al e a te M r . 1 and e e . allow d ; if p id on or af r a b for Apr , no di coun

' is a e the st da ea h e if a o u a te . 1 low d ; p id or f r Apr , dd d on fir y of c the a nt tax ea month for the re mainder of the year . Find mou of on ch of the following in that city at per $100

ss ax a Feb . 1 . 2. A house a esse d at t p id 0

2 . A store assessed at tax paid Mar . 9

4 assesse a t tax a 25. . A building d p id July

e assesse at tax a No v . 15. 5. An apa rtment hous d p id I n New York City o ne-half of the tax on real estate andall the tax on T personal property are due a ndpayable on andafter May 1 . he othe r half of the real estate tax is due and payable on and after No v . 1 . A discount of 4% pe r annum is allowed on the second half of the real estate

h a has ee a . I te est e No v . 1 e t e st tax if p aid befor , provid d fir h lf b n p id n r at 7% per annum from May 1 is adde d to all payments of the first half n of the real estate tax and all personal taxes paid on a d after June 1 . i e to all a e ts the I nterest at 7 % per annum from No v . 1 s add d p ym n of

1 . the tax ue se cond half of the rea l estate tax on andafte r Dec . Find d to t ese e at s the : andpayable , according h r gul ion , on following

6 7 8

A t . h o use Pro perty B usine ss buil ding R ailroad p

A ssesse d valua tio n. Bo ro ugh Manhatta n Queens Bro o kl yn R ate (1920)

Date o f Pa yment

2 Se . 51920 Feb . 10 1920 u ne 151920 Ma 20 19 0 t . , First h a lf . J , y , p

2 . 151920 Feb . 10 1921 Se t . 1 1920 “ Oct 10 19 0 No v , , o nd 11 3 , p Se c .

To tal Ta x 106 BUSI NESS MATHEMATI CS

I Taxes are n anothe r city the taxes are paid in two e qua l payments . d a s due Jan . Th a es e e t an 1 of ea ch year . e first h lf becom d linqu n dr w es The se half e mes inter t at 1 per month on anda fter Mar . 1 . cond b co

1 . delinquent anddraws inte rest at l i % a month on andafte r Aug . I f th s a n i r 1 findthe t ta e t te tax is a dthe city tax s pe $ 00 , o l tax paid on the following

e 25. 10 . A building valued at both instalments paid Jun

1 F b . 2 se 1 . A dwelling valued at first instal ment paid e 0; cond

2 . instalment paid No v . 5

111 —A tab e s m a . Computation of Taxe s by a Table . l i il r to the following may b e pre pared -for any city tax rate and then used for the computation of the taxes of that city .

TAX TAB LE

(Type rate 15mills on o ne dollar)

$ 10 $100

20 . 30 200

30 . 45 300

40 . 60 400

WRI TTEN EXER CI SES

Using the above table findthe tax on the following

1 . $600

2 .

3. $80

4 .

5.

H w h Tax R ate I s Determine — 112. o t e d The assessed

’ the e t is m the asses valuation of prop r y found fro sors lists .

108 BUSI NESS MATHE MATI CS

I n e t h 2. a c r ain city t e tax rate is as follows :

x Find the total ta . t If e t is assesse at its ea a e and t s the 3. prop r y d a of r l v lu Hor on own foll owing property

Find his tota l tax . hi a x h Th T 4 . the a t s t t e state . e t . he t Find moun of for coun y ci y .

The schools . er t the t ta tax E x s 2 a es to ea s o ? 5. Wh at p cen of o l in erci e ppli ch divi i n

x — An h e a e ax 113. Inh e ritance Ta e s in rit nc t is a tax on

Th is s sse h the property of a de ceased person . is as e d in t e

mbe the states but a es f e t states greater nu r of v ri in dif er n , and a person interested in this subj ect should look up the

The ates are New e s law for e a ch sta te . following r for J r ey and New York .

T h sba e h a h ew J erse . o te N y u nd or wif , c ild , dop d c ild or i e ea es e a t the ates are : ts issu , or lin l d c nd n , r

1% from to

2 % 3% above exempt

ts b the s ste so n-in-law and a h - To paren , ro r , i r , , d ug ter i n

w the ates are : la , r TAXES 109

2% from to $

3% 4 % above Al l others 5% exempt

Preferred obligations

1 . Judgments

2 a x . Funer l e penses

M x 3. edical e penses of last sickness

h New York . If he ta e is e e b a he m t e in ri nc r c ived y f t r , o r, h a w a t h : usb nd , ife or child , dop ed c ild

Exemption to amount of 1% on amounts up to 2% on next 3% on next 4% upon all additiona l sums

h b b th s t o r If in eritance 18 receive d y ro er, is er, wife widow

n ht : of so , or husband of daug er

Exemption to the amount of $500 2% on amounts up to 3% on next 4% on the next 5% thereafter

I f inheritance is received by any person or corporation othe r tha n those above name d :

E xemption to the amount of $500 5% on amounts up to 6 % on the next 7 % on the next 8% thereafter 110 BUSI NESS MATHEMATI CS

Preferred obligations :

n a m s 1 . Funeral a d d ini tration expenses

D e e e n 2 . ebts pr f rr d under U ite d States

3. Taxes

an 4 . Judgments d de crees

a h — 114 . Feder l In eritance Tax The federal tax is

th state as a h e no t h h posed on e e w ol , on t e s ares of the se atees r es e t e th b veral leg , i r p c iv of e eneficiaries to the decedent .

a h estate is exem of e c pt from tax. The rates on the excess are as follows :

Not exceeding From

Exceedin g

WRI TTE N EXERCI SE S

i the am t he ta e tax to be ai to ea 1 . Find oun of in ri nc p d ch , the state of d the U ite State s an estate New Jerse y an n d , on of willed as ’ follows : to decedent s wife ; to each of three children ; ivi e e a l am the the a b the andtwo s balance d d d qu l y ong mo r , ro r, isters. the state and e e a he ta e tax am u 2 . What would f d r l in ri nc o nt to in New York? the amo t he ta e tax to be e te d m 3. Find un of in ri nc d duc fro each of the following amounts willed by a decedent of New Jersey

112 BUSINESS MATHE MATICS

The net income is obtained by deducting from the gross income all necessary expenses a ctually paid in carrying on the b s ess s h a e b ess ss m u in , uc s inter st paid on inde tedn ; lo fro bad ebts ha e ff x e n e e b d , if c rg d o ; ta es ; fire loss s ot cov r d y insurance or otherwise ; and a reasonable depreciation on th h h h and e a e t e e t . e a se v lu of prop r y P rson l , ou old , living expenses are not included .

I t Exam e man t his wif has a net me o f llustra ive pl . A living wi h e inco How much income tax is he re quire d to pay?

SOLUTI ON net income exe mption

taxa ble income 4 a ate . 0 norm l r

35120 income tax

- — An a x i x Su tax . t a ta s assesse 117. Super Ta or r ddi ion l d Th b on large incomes in excess of a certain amount . e ta le below shows the a dditional rates charged on the portions of net income above certain state d amounts (not deduct ing the or exemption applying to the nor mal tax) .

Exceeds but does not exceed 3 1 % r u u 4 r 2

37, 4% 5% 6% 7% 8% 9 % 10 % 11%

127, TAXES 113

Ex s 28 000 but ex ee ceed $ , c d

32 , 000 114 BUSINESS MATHEMATICS

I l strati e Exam e : es a a e man th h d e re l u v pl Jon , m rri d wi no c il r n , his ceives a sal ary of a year but has no other income . Find t tal o income tax .

SOLUTI ON

NORMAL Tax

net income exemption

taxable income (normal tax) at 4% $160 8 % 480

SU R TAx

From to at 1 % 10 2% 40 3% 60 4% 80

TOTAL TAX

$160 $480 $190 $830

Credits for determi ni ng normal tax

s m at s hi h are s 1 . Dividend fro corpor ion w c ubj ect to the

e tax upon net incom . m U ite States be t b est s. 2 . I nter fro n d Li r y ond

e e s s ha e an exem t 3. Singl p r on v p ion of married persons or heads of famil ies have exemptions of but the total exemption of both husband a nd wife shall not exceed a t a e t 200 is a e ea h 4 . An ddi ion l cr di of $ llow d c head of a

family for each dependent under 18 years of age or incapable

- t be a se me ta h s a e of self suppor c u n lly or p y ic lly d fective .

116 BUSI NESS MATHEMATICS

To DE TE RMI NE NOR MAL

Net income subje ct to tax Le ss Credits

1 . Co rporation dividends 2 . I nte rest on Liberty bonds i 3. F xed exemption for married

n t hi . 4 . O e dependen c ld

I ncome subj e ct to no rmal tax

To DE TE RMI NE SU RTAx

Net Le ss Credits A andB income from Liberty bonds

I ncome subj e ct to surtax

Accumul ated surtax at various rate s on Surtax on $400 at 15%

Total surtax

S e t to 4 ate ubj c % r , S t to 8 ate ubj ec % r ,

Normal Tax Surtax

Total tax $4 326

The interest on (A and B) might be considered the following year .

WRI TTEN EXE R CI SES

s e man has a sa a a ea and t 1 . A ingl l ry of y r no o her income , x Find his total income ta . ar e man w t h e has a earl e 2 . A m ri d i h no c ildr n y y incom of in hi s e tax . salary andno other income . Find incom s man has a ea sa a and t e 3. A ingle y rly l ry of no o h r income .

Find his total income tax . TAXES 117

4 H e has one . A married man has a yearly income of salary . e en e t h l hat i his t ta e tax? d p d n c i d . W s o l incom 5 h man . Find the net income andt e total income tax for a married th h en m the ata wi no c ildr , fro following d te Salary interest on money loaned by him in the form of a no , $500 e t b i i s e him ; r n on u ld ng own d by , H e a s hi h h e s in p y inte rest on money w ch e as borrowed ; r pair , s a e and e t e e t 1 5state and a ur nc , d pre cia ion on his r nted prop r y $ 7 ; loc l

taxes $225. the 6. Find the net income andthe to tal tax for a ma rried man from following information : Cost of goods sold during the year Gross sales Wa es e ees s ra e e t and the b s ess ex enses g of mploy , in u nc , r n , o r u in p

Lo ss m bad e ts ete m e and ha e off 895. fro d b , d r in d c rg d $ ’ H e hadborrowed on which he paid a year s interest at His store building was worth and he estimated the annual de

preciation at 2% of the value of the building .

Net Sales G ross Sales Re turned Sales Gross P rofit Net Sale s Cost of Goods Sold Cost of Goods Sold Net Sales G ross P rofit Net Profit G ross P rofit Expenses

tax an e a e as s 7 . Find the income on incom of m d up follow

I NCOME To B E R E P ORTE D B UT E XE MPT FR OM TAx

I ncome from Muni cipal bonds ’ Inte rest on 3} Libe rty s

I NCOME WH I CH MUST BE RE P ORTE D

Interest from real estate and rent ’ ’ I nte rest on Libe rty 4 s and 4%s issue d previous to the 4th The e i a s bs e the Loan . ( own r orig n lly u crib d for of 4th Loan andstill holds them) Interest on railroad bonds no t tax-free CHAPTER I X

INTEREST ON BANK ACCOUNTS

— 118 . Bank I nte re st E very business banks its money and if the amount on deposit is large or is no t re quire d for

c e t use t i The c m urr n , in erest s usually earned thereon . o puting of the amount of interest earned is often a difficult

b em the f t be u h ff n pro l , di ficul y ing in part d e to t e di ere t methods of pa ying interest used by different banks and in

a t to the a t that the am i h p r f c ount on deposit s often c anging ,

to the e e t n h h owing fr qu n deposit a dwit drawal of cas .

1 — 1 9 . Kinds of Banks Mone y may be de posited in a

sa s ba a sta sa s ba a n a co m ving nk , po l ving nk , or ordin ry

mer cial ba nk . A savings b ank is primarily for the purpose of a ccepting deposits from persons who wish to put their savings in an institution which shall guarantee them a certain rate per

e t These ba s are ha te e b the c n upon their mone y . nk c r r d y state and are under the supervision of the state banking

e a tm C mm ates nte est a b these ba s d p r ent . o on r of i r p id y nk are and Money in these banks cannot be

a o ut ex e t b a he a ab e to the e s t him dr wn , c p y c ck p y l d po i or

h h e ba s the e e e self . T e l aw allows t es nk privil g of r quiring ’ m 5o 6 a s t e th a a but th s is se m fro 1 t 0 d y no ic of wi dr w l , i ldo exe rcised . A postal savings bank is o ne conducted by the United h h sa s ma be e s te h States government , in w ic ving y d po i d w ere

ma ate te est m the e they will draw a s ll r of in r fro gov rnment . 118

120 BUSINESS MATHEMATICS

B . KS J . JAC ON

DATE I NTE R E S T WI TH DR AWALS BALANCE

0 0 0 0 0 0 0 0 0 0 0 0

Second form a ccompanying the above form

SM ALLES T QUA R TE R LY BALANC E I N TE R E S T First quarter Se cond Thi rd

. 1 b t EXP LANATI ON : The first interest te rm was from Jan 1 to Apr . ; u

a s ma e e s t t Jan . 18 he s a est a a e since Mr . J ck on d no d po i un il , t m ll b l nc on deposit during the entire interest term was and therefore no

1 . interest is added Apr .

1 to 1 . T t The seco nd interest term was from Apr . July he smalles Th balance on deposit during the entire term was $200. e interest was

$2.

l 1 to Oct . 1 the s a l The third inte rest term was from Ju y , m l est balance The terest was being $402 . in h s a l t ct . 1 to Jan. 1 t e es ba a be n The fourth term was from O , m l l nce i g The interest was

— of C m utin . Interes 121 . Oth er Meth ods o p g t Some

the te est m th b savings banks compute in r on on ly alances ,

an es and s me se some on quarterly bal c , o on miannual I NTEREST ON BANK ACCOUNTS 121 ba a e S me ba the arte ands me l nc s. o nks add interest qu rly, o a di a d t semi nnually . The following illustrations will make these principles

e x th cl ar . The e planation form should be carried along wi h t e other form .

Illust a e Exa - o m r tiv mple . Suppose that the above mentioned bank c uted the te est e th e nte e t sem p in r on quarterly balances, but add d i r s i — annually we would then have

Exp LANATION FORM :

QUA R TE R LY I NTE R E S T

First quarter Se cond Third Fourth

Then the a ccount would

B . CKS N J . JA O

DATE I NTE R E ST WI TH DR AWALS BALANCE

c c c c c c c c c

I n case the bank computed the interest on the monthly

est a te the x balances and a dded inter qu r rly, e planation ’ form of Jackson s account would appear thus : 122 BUSI NESS MATHE MATICS

B . CKSO N J . JA

1 M ON TH I NTE R E S T Q R TE R LY DI I DE ND M ONTH UA V 1 o r 1 % o r I NTE RE S T

s s s s s s s

' Th e cents i n the r in ci al ar e dro e d a nd th e arts of c ents 1n the NOTE : p p pP , p interest are al so dro pped in this w o rk .

WRI TTEN EXERCI SES

C . Ste a t to and i ts E . Jan. 1 1 . Make accoun for w r includ ng , one for each of the following methods of interest at 4 %

(a ) Interest compute d qua rterly andadded qua rterly . a e (b) I nteres t computed qua rterly and dded s miannually .

(0) I nterest computed monthly and adde d quarterly .

DATE DE POSI TS W I THDRAWALS 1920

Jan. 19

o o o o o o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o

124 BUSINESS MATHE MATICS

— 122. Postal Savings Bank s NO person can deposit less tha 51 m e tha 100 a m th these ba s can n 3 nor or n $ on in nk , nor he ha e a ba a e at an 500 ex s e v l nc y time of more than $ , clu iv

Of I t est is a h 2 er te est . e at t e ate ea in r n er llow d r of % p y r,

e a h e a that the m e for c full y r one y remains on d posit, beginning with the first da y of the month following that in which it is deposited .

e s t ma ex ha e his e t ate u A d po i or y c ng c r ific s, nder definite

t s mu t es 20 U te ta condi ion , in l ipl of $ , for ni d S tes govern ment registered or coupon bo nds pa ying but this is to be done at the beginning of the year if such bonds are a vail

h These ma b h able at t at time . y e eld in addition to the

$500 mentioned above . Money may b e withdrawn if o ne gives up to the postal

O fi e he e the e s t was ma e the sa s e t a es f c r , w r d po i d , ving c r ific t h h m for t e wit dra wal a ount .

Illustrative Example 1 . How much interest would I re ceive in a postal

I te n . sa s a e s 8 Ja 1 1920 a nd t e it Jan . I ving b nk , if d po i d $ , , wi hdr w . 1921?

SOL TI O N : The a s e is e e a se the e es n t U n w r , non , b c u mon y do o begin

to a te est t Feb . 1 1920 . dr w in r un il ,

Illustrative Example 2 . How much inte rest will I re ceive if I deposit d t a it 1 1 21? 15 Jan . 1 1920 a n 9 $ on , , wi hdr w on July , SOL UTI ON : Since the money is in the bank 1 full ye ar but not all of ’ the 2nd ea I e e e 1 ea s te est 2 15 y r, would r c iv only y r in r or % of $ $30 .

’ m tat De s t s Da B — 123. Co pu ion of po i or ily alances The following method is o ne employed by some banks to co m ’ t a e s t s a b a a e a he k a pu e d po i or d ily l nc on c c ing ccount . I t

. 1 2 will b e noted that from Oct 3 to No v . there were 2

h h the e s t had 1 th sa da . upon w ic d po i or ou nd dollars on

Th s mea s the same a e a s 2 deposit . i n in v lu thousand

N . h a s 1 da . a ov 8 t e e had bee doll r for Ag in on , r n 7 da . on I NTEREST ON BANK ACCOUNTS 125

which the deposits had no t fa llen below 2 hundred dollars .

Th s is e a e t to 14 h e 1 th sa 4 h e i quiv l n undr d or ou nd , undr d

The h a e a dollars for 1 da . a ccount may t us be c rri d long for the month and at the end of the month the aggregate

he 1 a . a b The n mber h e s is t for d m y e found . iI of undr d n

e ndthe h h er dropp d , a interest on t e aggregate at t e given p cent may be found as follows :

at 2 % 365 amount of interest to be credited to

the a ccount .

T N . R ST C M NY H . U O PA

ST ATE ME NT OF I NTE RE ST

F . E S ACCO UNT O B H . JON

BALAN C E A G G R E G A TE DATE DAY S R ATE I NTE R E S T

H n . Th u . H Th o us . u d o s und.

WR ITTEN EXER CI SE S

te est I e e e I sh e s 5 1 . How much in r would r c iv if ould d po it $2 in a i 4 . m a Jan . 1 and i t a t at ? postal sav gs b nk , w hdr w mo l er I e e e I s l t a it 1 . an 2 . How much would r c iv if hou d wi hdr w yr d 1 mo . late r? I ma e the e s ts a sta sa s a 3. k following d po i in po l ving b nk 126 BUSINESS MATHEMATICS

ne 1 1921 Ju , 5192 1 1 , Se t 1 1921 p ,

No v . 10 , 192 1

’ When would I be able to re ceive a full yea r s interest on each of these si andhow m te e st I e e e all? depo ts, uch in r would r c iv for

4. I f a man has e os t a sta sa s a Jan . 1 125 on d p i in po l ving b nk on , $ , and he decides to withdraw $120 and purchase United States bonds terest all ha bearing in , how much will his money ve earned in inter est for him 1 yr . later?

128 BUSI NESS MATHEMATI CS

The distribution of profits must naturally be compute d in order that ea ch share sha ll h ave its pro rata share of these profits a dde d to the amount of mone y pai d in by the owner

a h ha e to Obta h Th i a of e c S r , in t e actual book value . is s lso

e ssa the a a e h s at h h n ce ry for nnu l r ports of t e as oci ion , w ic

' ma be e e b its memb e s b the state ba y r quir d y r , or y nking

“

e a tme t b b th . d p r n , or y o

Th e S e e s an — Th s a i h 125. ri Pl i pl n s to sell w atever

umbe Of sha es the ass at sha eem e e sa n r r oci ion ll d n c s ry , say

n . 1 he . 1 to se h on Ja ; t n on Apr , ll anot er series of shares

ate tha h which sha ll mature 3 mo . l r n t e first series ; then 1 n h Th he se es a d a t e Oct . 1 . anot r ri July , no r ese may be

but t e e e e a ea the ass issued wic or v n onc y r , if ociation thinks b the e is tt e a m e b est or if r li l c ll for on y for uilding purposes .

n r a m n E arni gs a e usu lly deter ined a d divided semiannually . There are three types Of problem s which are of interest to

an ass a the a vera ge person in oci tion , or conne cted with it as an employee or director .

2 . To n th e W thd a a a 1 6 Fi d i r w l V lue .

man 10 Illustrative E xample . A owns shares in a building andloan s at andh as a his es at the ate 1 er as oci ion p id du r of $ p mo . for e ach is e t a e 5r . e he e o t a . I f h i sh r for y , wh n comp ll d wi hdr w e s allowed ts at 5 er a to hat a t is he e t t ? profi p nnum . w moun n i led EX L N TI O N : 10 s a es at 1 er t ea h sha e P A A On h r $ p mon h for c r . the dues 60 The would amount to $600 (5yr . first dues have h e a e ts 60 . t e se s ha e ea e rn d profi for mo , cond du v rned profits for ha e ea . n so t the ast es T 59 a d e ts 1 . h s mo , on un il l du v rn d profi for mo i gives an a rithmetical series of numbers which has 60 for the first term and 1 the ast te and i the m e te s is 6 T for l rm, in wh ch nu b r of rm 0 . he algebraic sum of such a series of numbers is equal to the sum of the first n ast te s u t e o ne- a the e te a d l rm m l ipli d by h lf numb r of rms, or in this example (1 60) X (60 2) The general method Of comput BUILDI NG AND LOAN ASSOCIATIONS 129 mg the interest is to calculate it on the total dues paid in for the average T time . he average time is obtained by dividing the total number of ths 60 the e a e ee mon , by , number of months in which du s h v b n i a h e a T . at 5 s p id, whic qu ls he interest on $600 for 30%mo % which added to the amount paid in in dues gives the t a wi hdr wal value . The ave ra ge time may be easily calcul ated by taking o ne-half the e t da s an to it . numb r of mon h dding mo .

SOL UTI ON

60 10 600 t ta a t a es X $ $ , o l moun p id in , in du 1 307 600 $ x x . 05 profits 12 $600 withdrawal value

WR ITTE N EXER CISES

1 . man has a e 1 r er s a e 20 sha es 6 A p id du s of $ pe mo . p h r on r for i t he s e ed o t r a an ass . If the ts . e at yr wh n , comp ll wi hd w from oci ion profi e e 4 er a to at a t is he e t t e ? w r % p nnum , wh moun n i l d State eas s a e s mi t b t t 2 . r on why p r on gh e compelled o wi hdraw from n a a building a d loan ssociation .

’

n at th n . h an as 3. A ma e e dof 7 yr finds that e must withdraw from h s H has a e 1 a s a r . r a e . H e a sociation . e c rri d 5Sh re t $1 pe mo pe sh r unpaid fines a gainst him of If his profits are calcul ated at

ndall n a es are e te fin dhis t a a a e . a u p id fin d duc d, wi hdr w l v lu 4 man h s s a es an ass at t at a s H e has . A old 5h r in oci ion h p y t n r r t . the a been in for 8 yr . a dhas paid $1 pe Share pe mon h Find moun to which he is entitled if he withdraws . I f I 12 S a es a i and a ass at and st 5. own h r in build ng lo n oci ion mu l I e t e d 6 . to ho w be t at the end 5. an withdraw of yr mo , much wi l n i l d if the association allows

m tati ts Shar es — Whe the es 127 . Co pu on of Profi on n du and profits combine d amount to the p ar value of the stock (usually all shares are canceled if the borrower has

hase a h se ea h membe s a ash built or purc d ou , or c r i p id in c

' h i i ri To be ab e to if he has not borrowed from t e asso c at o . l know when each share Shall amount to $200 with the profits

9 130 BUSINESS MATHEMATI CS a e to the am he membe it is dd d ount of money paid in by t r , necessary to know ho w to compute the profits for each share

Of h Th is a e : eac series . is done in the following m nn r

I st ati e Exam sh es as llu r v ple . An association has issued 4 series of ar follows :

1st se 1 es 500 s a es ate Jan . 1 19 9 ri of h r , d d , 2d 400 1 1919 July , 3d 300 Jan . 1, 1920 4th 400 u 1 1920 J ly ,

The s ea se es If th e t due in ch ri were $1 per share per month . e n ire

ts Jan . 1 1921 e e hat l be the a e 1 s a e profi on . w r w wou d v lu of h r of ea ch series at that time ? E XP LAN ATI ON : Dues of $1 a month have been paid on each of the h 500 shares for 24 mo . in t e fir st series (the average time is whi ch makes an ave rage investment of $24 X 500 x or for 1 mo . I n the same manner the average investments in the other series are found n i T a dthe total for 1 mo . s he share Of the profits belonging to ea se es is the sa e t t The t. ch ri in m ra io as hese average investments . profi for 1 Share in each series is obtained by dividing the number of shares in a se h i The a e 1 a e ea h ries into t e ent re profit of e ach serie s . v lu of Sh r in c series is the sum of all dues paid in on that share andthe profit for tha t share .

FOR M OF SOLUTI ON

24 500 12 3 x x s 1st series invest . for 1 mo . 18 x 400 x 9% 2d 1

12 x 300 x 617; 3d 1 6 x 400 x 3% 4th 1

total

1 s a e l st se e of $ , h r of ri s

, of 2d of 3d of 4th

132 BUSINESS MATHE MATI CS

h the net ea ni s ea h s are of the net e arnings . Add r ng in c series to the principal of that series investment and divide h se es the sum by the number Of shares outstanding in suc ri , and the result will be the net result of each share in such seri es .

F r t e . se se es 1 . . Il lustrative Example . i s s ries 3yr old ; cond ri yr old

PR I NC I PAL AVE R AGE N I M E S H AR E S MO TH S NVE S T. TI 35 1 l st, x x 4 12 2d. x x 5

Net P ts rofi , x l st series profits

l st serie s shares X

.2d series profits

2d series Shares

WR ITTE N EXER CI SES

and a ass at ss e a new se 1 . A buildi ng lo n oci ion i u d ries at the be ginning

ea h ea . The l st se es has 300 sha es the 2d500 the 3d4 of c y r ri r , , 00, and I f h are 1 er th er sh the 4th 500 . t e dues $ p mon p are andthe profits at the endof the 5th year are find the value of 1 Share in each series at the end of the 5th year . T t ut t the se a e t e a 2 . ry his o wi h cond pl n m n ion d bove and compare your resul ts . BUILDING AND LOAN ASSOCIATIONS 133

3. Tr h h x an a e es ts . y t e first plan on t e se cond plan e ample, dcomp r r ul Which appears to be the better plan for the members of the association?

2 D —I e essa 1 8 . istribution of P rofits Statement. t is n c ry for an association to publish such a statement at certain

te a s e the sem a a a The in rv l , i r i nnu lly or nnually . following

a se b s me ass at s e a membe a co m pl n , u d y o oci ion , will giv r

h s e h pre en iv idea of t e standing of the association .

a e Illustr tiv Exampl e . Suppose that a new series is Opened each 3

es 25er Wk . er s a e . Se e h o . t ! m , wi h du 1 p p h r ri s number 49 as been open

520 . andt e e are sha e s the se es Dec . 31 1920 . The t ta wk h r r in ri , o l s bs ti s ai in se ies m e 49 is e al to 25er 520 u crip on p d on r nu b r qu ¢ p wk . for 1 0 e h k . 3 r s are and s a es 2 w , or $ p , on h r During 19 0 the

s r t s e a 52 . at ea h ha e sub c ip ion qu l ( wk on c s r of shares) . Subtracting from gives the total subscrip

a Dec . 31 1919 andt s a a t l tions p id in , , , hi mount e rns profi s for a l of the 2 Th ts 1920 are a - year 19 0 . e profi for p id on o ne halfof the subscriptions a 1920 andt s a e to es 151 p id in , in , or on hi dd d giv $ the total amount on which profits are allowed for 1920 in series 4 The er e t ts is the t tal t all 9 . p c n ofprofi found by dividing o profi on the se es the t tal s bs ri t s all the se es ha o ri by o u c p ion in ri s ring in pr fits , as “ shown by the total of the column heade d Total Pro fit-Sharing Sub i D 1 the sa ri o ns ec . 3 e ate al l e . c pt , , Allow m r on seri s

49 520 50 507 51 494 52 48 1 53 468 54 455 55442 134 BUSI NESS MATHE MATICS

WRI TTEN EXERCI SES

1 . Us the sa e ate er e ts ete the a e . ing m r p c nt of profi , compl bov form

2 . a es the 2nd G e the 1 t se e 4 . and ha n s iv n s ri s yr old vi g h r ,

se es 3 . and a Sha es net assets : ri yr old h ving r , find

Net profits (b) First series profits (0 ) Value o f l st se rie s Shares Se cond series profits (e ) Va lue of 2udseries shares

136 BUSI NESS MATHE MATI CS

3. The fluctuations or general trend in a series of Similar

ma ni t es s ate b ate a g ud (or izes) , arranged d y d for

given period of time .

1 — 30 . Kinds of Gr aph s or Char ts Graphs used to Show the true proportion of the component parts Of a group total are :

1 . Th e circle .

2 . Th e re ctangle .

3. The st a ht e e ese t the t ta an r ig lin , r pr n ing o l , d divided into segments (or se cts) of proportionate lengths to

s t th m k h repre en e quantities a ing up t e total .

Graphs used to Show the relation of o ne part of a group to other parts of the same group are :

h 1 . a a es ba s Of t e am h P r llel lin or r s e widt , drawn either

h ta e t a m a mm orizon lly or v r ic lly fro co on b ase line .

ms st at s 2 . t a e s e t e h Pic ogr , or illu r ion in p r p c iv , s owing the relative va lues re presented by the quantities h i compared . T is form s very unsatisfactory be cause the relative values are not comprehended h easily by t e reader . n h h h 3. es s a es a es t e Circl , qu r , or y figur in w ic relative

a es are e ese te m e tha o ne me v lu r pr n d in or n di nsion , the attempt being to show relative values by the

h ffe e t es etc . Th Size of t e di r n circl , is form is nu satisfactory be cause the eye cannot grasp th e true

relation from these sizes .

Graphs used to Show the fluctuations or general trend in

m a ma t es a a e ate b a series of Si il r gni ud , rr ng d d y date for a

e t me are : given p riod of i , GRAPHICAL REP RESENTATION 137

1 . The curve (Form connecting points located at

sta es to the ht a e t a ax s as ete di nc rig of v r ic l i , d r

m e b the t me a ab e and at a sta e u in d y i v ri l , di nc p

a m a h ta ax s as ete m e b the w rd fro orizon l i , d r in d y t quan ity variable .

2 m a at e s m h m t me . Co p r iv curve (For wit a com on i variable but with different quantity variable deter mining the location of the points in the respective h gra p s .

t — - ate 131 . Cons ruction of Graphs Squared (or co ordin )

e- a e is e h paper is necessary . Loos le f Siz pr ferred for sc ool

e the a hs e a e sh b e t as a a t the use, Sinc gr p pr p r d ould e k p p r of

a e e t hes andte ths required notebook work . P p r rul d in o inc n

an h s a be m st e e t the es of inc will u u lly found o conv ni n , lin

- at the inches and half inches being heavier than the rest . All rulings Should be of such a color as will bring the graph

If h met s e e e . es e a e t into prop r r li f d ir d , p p r wi ric ruling

h e th may be used instead of that on an inc scale . A rul r wi the same graduations as that of the paper can be used to

Se e t a s a e h h e advantage . l c c l w ic will work w ll on your — a e 10 100 etc . at a e a e e p p r , , , will n ur lly work w ll on p p r rul d

The st t e a hs in tenths or twentieths . con ruc ion of curv gr p will be fa cilitated if the steps are completed in the following order :

a e the a t t es e the stat st a tab e 1 . Arr ng qu n i i giv n in i ic l l

f h the ea est ate st anduse I n the order O t e time units, rli d fir ,

mbe s e . . use etc . round nu r only , g , for

Ma o ff the h ta ax s the t me ts m 2 . rk on orizon l i i poin fro

h x s as the ea l ht s t e e t a a est ate e . left to rig , u ing v r ic l i rli d in

M a the e t al ax s a m the t Of 3. rk on v r ic i upw rd fro poin

h a e s the h x intersection , t e quantity sc l , u ing orizontal a is as 138 BUSINESS MATHEMATICS

the O e The te is ete (or z ro) line . quantity scale selec d d r m e b the a in d y l rgest quantity in the series ; the time scale , b the mbe y nu r of years or months .

4 . Locate the points at the correct distance from the two axes to re present the quantities given in the series with reference to their respective dates andconne ct these points . The gra ph will then be plotted (or drawn) in the first quad

a t that is in the s a e to th h f h x n r n , , p c e rig t O t e vertical a is a d h above the orizontal axis .

e e s to te et th n 5. Add l g nd in rpr e quantitative scale a d time points .

If two m e s ar h 6 . or or curve e plotted on the same c art

e the e e is the same ex e t that a se s a e fi ld , proc dur c p cond c l ,

h i o n ma be at th h Th if t ere s e, y indic ed on e rig t . e curves Should be distinguished by the character of the line or by

n n h b color a d a explanatory legend s ould e given .

h m a ts Sh b t e C ent a t a i e . 132 . F c own y o pon P r s of C rcl This is a for m of gra ph quite commonly used by business

Th e is e t arts h h Sh ho w men . e circl divid d in o p w ic will ow m h the t ta is e ese te b ea h Of its a t uc of group o l r pr n d y c p r s . I n the following example the division Of the circle is deter

s : 15 Of it is sh e t mined as follow % own , r presenting he T amount that might be used for clothing . here are 360

the h e e . 15 360 e es 1 degrees in w ol circl % of d gre . 5x ° the use the t a t h h i 360 By of pro r c or , w ic s ex

ha t XI X it is an eas matt plained in C p er , y er to lay o ff The divisions should always be from the center of the circle .

S e a th t e Ill ustrative Example . om good u ori i s claim that the follow ing per cents are the largest per cents which should be expended o ut of x the family income for the various e penses of the home .

140 BUSINESS MATHEMATI CS

a e a a t to s t M k ch r how hese facts . ’ e s 3. The payments o ut of the milk dollar as reporte d by the Bord n a P C . I ts o nc . 1916 e e as s F rm roduc , in w r follow

To dairyme n To labor

To shareholders ate a s i and x es For m ri l , suppl es, e pens of tt es x etc bo l , bo es,

Cha t t es r h e facts . f 4 . I the total sales are cost of the goods sold returns 150 ss t se n ex e ses 800 and $ , gro profi lli g p n $ , e e a ex 400 the net t and a h g n r l penses $ , find profi gr p th e es facts in o ne circle .

t ar ts of a 133. Graph s by Compone n P — R e ctangle This plan is to divide a given

re ctangle up into its proportionate parts . The length Of the re ctangle should first be

s a ts and determined from the busines f c , then subdivided in proportion to the amounts which shall represent the correct

l a e e to parts of the tota group . P p r rul d

I t is tenths is very useful for this purpose . well illustrated in the following

The ss e e e the Illustrative Example . gro r v nu of

Bell Telephone System for 1 yr . was disposed of in the followin g manner :

Salaries and wages 50% Interest 19% Surplus 65% Taxes . % a s e t t a e 20 Materi l , r n , r v ling, %

This is shown by the component parts of a re ctangle (Form GRAPHI CAL REPRESENTATI ON 141

WR ITTEN EXE RCI SES

1 . Graph E xe rcise 1 under § 132 by thi s method . Th 2 . e disposition of a 51! carfare paid to a certain city railroad in a year was as follows

G eneral expenses includi ng pensions and Cost of power Wages and conducting transportation Other transportation e xpenses a te M in nance of way . Mainte nance of equi pment e D preciation . a a e and x D m g s legal e penses . Taxes

R e ta s s a s and t e s . n l , ubw y , unn l I nterest Re ta s r a e es n l , u f c lin

Dividends . Sur plus

a a h M ke a gr ph by t e above method which will show this . T n ’ 3. he utili zation a daccompanying waste of 1 ye ar s coal supply locomotives on the railroads of the United States w as as follows :

MI LLI O NS MI LLI ONS O F TO NS OF OF COAL DO LLAR S

C s e sta t es ee e e ho t on um d in r ing fir , k ping ngin i e sta a nd e t fire bo x at the end wh l nding , l f in of

run .

Utilizedbytheboiler . Lost In vaporizing moisture in coal Lost through the company Lost I n gases discharged from the h Of s e e Ci de s Lost in t e form uncon um d fu l in n r ,

sparks ” I n h h Lost throughunconsumed fuel t e as . r a at ea a e stea etc Lost th ough r di ion , l k g of m,

HI NT : Make 90 equal spaces on o ne side of the re ctangle to represent l t s a and 170 e a s a es the the s e to re re mil ions of on of co l , qu l p c on o r id p sent millions o f dollars andplot by the above method . he a t E xe se 2 e 132 t e e . 4 . Graph rci und r by bov m hod 142 BUSI NESS MATHE MATI CS

5. Make a chart of the following :

DI STR I B UTI ON OF R A I LR O A D $ 100 I NCOME F O R A C E RT A I N YE A R

Labor Fuel and shop supplies Ma te rial

Division supe rintendent Bette rment R entals I nterest h 6 . G raph t e following data

SO U R CE OF R AI LROAD $1 I NCO ME F OR ONE YE AR

P assengers Product of mines a a t es l M nuf c ur 15. ¢ P roduct of agriculture P roduct of forests 7¢

Merchandise 19 16 E xpress Miscellaneous (balance )

E x se 4 1 2 th a t G a h e e 3 e e e h . 7 . r p rci und r by bov m od Th f e as e te a te te t 8 . e causes o crim r por d by no d de c ive are as follows .

Poverty G ambling and debt Strict parents Having to o much money

E asy going parents .

P lot this by the above method or by that of 132 .

S m e C m a s s b Gra h s . The a h 134 . i pl o p ri on y p f gr p shown in Form - 10 is often better than any other form for some kinds of business facts .

144 BUSI NESS MATHE MATICS

Brights diseas e n Stomach a dbowe l (unde r 2 yr . ) Apoplexy Broncho-pneumonia Cance r o f stomach andlive r Diphtheria andcro tip

a e a h h fl t s 3. M k c art of t e following uctua ion h i t e Un ted States .

P E R CE NT OF A VE R AGE P E R M O N TH 1909 19 1 0 19 11 19 12

n . 4 . I nc. 20% I nc . 42% I c 1% Inc 38%

31 De . 1 Dec Apr 20% De c 11% Dec . % c 8% July 12% 15% 32% 22%

In . Oct 8% 5% I nc . 8 % 3% c

Th f l h s th a t a s that m st be 4 . e ol owing table s ow e moun of v rious food u eaten to secure the same number of calories that are found in 100 grams

a e a a h to sh h i m a s . of ordi nary white bread . M k gr p ew t is s mple co p ri on

r o . o r GR MS No . o G R AMS N A E QUI VALE NT E UI VALENT FOOD I N CALOR I E S To FOOD I N ALORI ES To 100 G R AMS o r 100 G R AMS OF W R I TE B R E AD WR I TE BREAD

B tte 35 100 u r . Wheat flour 70 73 70 Smoked ham 74 70 90 Macaroni 70 Sli ced mutton 90 70 100 75 Sirloin of beef 138 ” War bread 70 Chicken 170 70 Chicken 175 73 Veal loin 180 73 Unsalte d herring 330 70 370 74 380 70 500 74 600 100 954 GRAPHICAL REPRESENTATION 145

5. a e a a t Of the s the a ses ea s M k ch r following, howing c u of l ving po i t ions .

R E AS ON S Not enough money Never started W orking conditions .

Discharged . La id o ff . Dissatisfied B etter job Needed at home Living Failed to report P ersonal reasons

e — h n he 135. Curv Plotting T is plan is considered o e of t best to present business facts in such a manner that the y

Be we ta e th s may b e e asily grasped by the reader . fore k i s e t eta h e e it be e to maste cer ubj c up in d il , ow v r , will w ll r tain well - definedrules which are very important if o ne is to

h These es . be successful in grap work . rul follow

R ules for graphical representation of facts : h a e t e t t e the a t e ete and ea . 1 . M k i l of ch r v ry compl cl r

2 a e the e e a a a e e t a a t ea e t to t . . M k g n r l rr ng m n of ch r r d from l f righ r he tt the a t 3. The ta s a e es a e a e at t horizon l c l figur pl c d bo om of ch r , andfigures may also be used at the to p if needed . Th h a e a r to be a e at the e t the 4 . e figures for t e vertical sc l e pl c d l f of

- t a es ma be a e ee e . chart . R igh h nd figur y dd d if n d d

I e t the a t the ata h it w as a e . 5. nclud wi h ch r d from whic m d P a the e tte and e s so t at t e ma be ea the 6 . l ce l ring figur h h y y r d from - bottom or from the right hand side of the chart . E a est ate s be S at the e t and ate ates to the 7 . rli d hould hown l f l r d right . C ts s s a ea e t to t and the tt 8 . har hould u u lly r d from l f righ from bo om to the to p . ee a b e se to ex ess es a e eat es and red to in 9 . Gr n m y u d pr d ir bl f ur , dicate undesirable features . h ! e e sh sh t e a t e e e ss e . 10 . ro lin ould ow on ch r wh n v r po ibl 146 BUSINESS MATHE MATICS

11 . a e the e e M k z ro lin much hea vier than the squared paper lines .

12 . The tt l bo om ine should be wavy if the zero line cannot be shown .

13. I f the a t e e s to e e ta es h 1 be a ch r r f r p rc n g , t e 00% line Should bro d e th lik e zero line .

“ o “ a. “ O n) a.

First Year Second Year Thi rd Year

Form 11 . Curve G raph

W e the ta s a e e s t h 14 . e t e h n horizon l c l b gin wi h z ro , vertical line at the left which represents zero should be broad .

5If the h ta s a e e tes t e the - 1 . e t and orizon l c l d no im , l f right hand lines are a as the e andend t e a t be sh w vy , b ginning of im c nno own . I f es are to be te be a e no t to 16 . curv prin d, c r ful Show any more lines

- - of the co ordinate paper than is necessary . Lines o ne fourth of an inch apart are bette r .

148 BUSINESS MATHE MATI CS

2 et a . Make a curve to show the rise of the New York City budg for period of 20 yr . from the following data :

l st yea r 2d U

3. a e a e to S the ease the e s ts o f a e ta n na M k curv how incr . in d po i c r i l tiona bank .

T a t a a a i 4 . he following annu l re por of r ilro d s based on a certain year ll f o wm I t . a e a e andis for the 4 yr . o g M k curv for each of the ite ms men tio nedon o ne chart fie ld whose vertical column represents per cent and whose horizontal line represents years .

I N TE RMS O F P E R CE N T

'

SE Y R . l s T Y R . 2ND Y R . S RO Y R BA . 4TH YR .

e at i e 1 . Op r ing ncom Net ope rating revenue after

deducting taxes .

G ss O e at e e e . 2 . ro p r ing r v nu

at ex ses . 3. Oper ing pen

a e to s the e flat ess e e r 5. Make curv how prop r in ion pr ur p square inch

- ut 20 . i on diffe rent sized automobile tires . Abo lb s allowed to the s quare inch of se ction . GRAPHI CAL REP RESENTATION 149

I NF LATI ON P R E SS UR E N LB , P E R SQ. I .

60 lbs. 70 U 80 90 95 100 105

the ti a to t ati a s 6. Using ver c l line represe n rel ve value of cOals in doll r , and th h ta li e to e ese t n e thr te a in ars e orizon l n r pr n p c of an aci co l doll , and si o ne s a e an se e t 1 the base i e and u ng p c (of y cho n l ng h) for $ on l n , one-ha that s a e 1 the e t a l e a es to sh lf of p c for $ on v r ic l in , dr w four lin ow the comparative values . 5 0 0 N U 5 0

0 H H 9

n s a e se e t the h ta scal to re re 7 . Using o e p c (of cho n l ng h) on orizon l e p n a e the e t a s ale to e ese t 2 sent years ando e of chosen sp c on v r ic l c r pr n $ ,

Show the increasing demand upon the New York City transit lines .

o o o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

c c c c c c c c c c c c c c c c c

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

c c c c c c c c c c c c c c c c c 150 BUSI NESS MATHE MATICS

— rv 136 . C m a s s I I t is te ne cessa o p ri on nvolving Time . qui at times to Show a comparison from o ne year to another on the same e a t e s ch as sa es a es st lin or r icl , u l , w g , co of food r Th m b the use p oduct, etc . is is a ccomplished very si ply y

Of th h h in 1 te that the same am t e grap S own Form 2 . No oun is pla ced on the to p side of the rectangle as on th e bottom side on corresponding lines .

WRITTE N EXE R CI SES

e 0 . 1 . Show the chang s after 1 yr in costs of materials and in freight ates W t a a s a to t at s 12 s 100 r i h gr ph imil r h hown in Form , u ing $ as the basis for each at the be ginning of that time .

Labor . . From $100 to $120 I nterest 100 125 Fuel 100 130 R ailroad rates 100 95 Tracks per mile 100 145 100 1 Rails . 50 Pine 100 183

S th s a e s a e a a a 1 2. how e ri ing w g c l of r ilro d l bor from 899 to 19 11 s ea t to e ese t 2 5and sta t t r to 1920 u ing ch uni r pr n $ r ing wi h $1 pe da .

- in both the left and right hand columns .

WAG E S I N WAG E S I N WAG E S I N 1899 19 1 1 1920

Tra ckmen Station age nt Trainme n Ma chinists

G eneral office cle rks .

Conductors . 70 0 E ngine me n

a at e s e a 3. Obtain simil r inform ion conc rning om r ilroad or large manufacturing plant and make a graph to S how the results of the ln formation which you obtain .

h t — I t is O te a sab e t 137 . Pe riod C ar s f n dvi l o arrange the

h s a mbe em ees ha t working our of nu r of ploy in c r form.

152 BUSINESS MATHEMATI CS

' This is particularly true when the y work in shifts and thus

‘ relieve one a nother a t stated intervals of time or at certain h I t is a s a ab e the a a eme t the ours . l o pplic l in rr ng n of ir va cations as will be shown in the following .

’ t to ea 16 men Il lustrative Example . As sign 2 weeks va ca ion ch of

e t a 2 s a be a a at e ands a hi all . so that no mor h n h ll w y onc , how gr p c y

22 29 512 19 26 3 10 17 24 May June Jul y P Form 13. eriod Cha rt

WRI TTEN EXER CI SE S

ha t s a to the a e 25men . 1 . Make a c r imil r bov for a e a ha t 60 men h e t a 3sha 2 . M k c r for in w ich no mor h n ll be away at

he sa e t e be n in Ma 17 1920 . t m im , gi n g y ,

3. Th ee at l e a e ta t ta e the 8 r p ro m n in c r in ci y k work from to 4, 4 to man 12 and 12 to 8 es e t e . E a ets 1 da . o ff 2 , r p c iv ly ch g in 7 Arrange

a hart to Sh t s . P a o ne Off etc . a te a e tai e c ow hi l n , , f r c r n numb r of days . GRAPHI CAL REPR ESENTATION 153

1 Cur — Th a s f a m a 38 . Comparison of ve s e Offici l O co p ny frequently wish to be able to see a comparison Of certain t m Th s m a b as m she b i e s of the business . i y e e ily a cco pli d y

a These dr wing two or more curves on the same chart field .

es mme ate a e t Sh the m a curv will i di ly , if dr wn corr c ly , ow co p ri s s e m 14 i a h m Of a h on d sired . For s good type of t is for gr p

I st ati E The ea i s andex e ses a e ta a a llu r ve xample . rn ng p n of c r in r ilro d by years are given below :

a s e t e ea s and a tte e the ex e ses Using olid lin for h rning do d lin for p n , these are shown on the same chart (Form

1916 1917 1918 1919 1920 .

s te C a t sh R e et ee I e and t Form 14 . Compo i h r owing lation b w n ncom Ou go

iv c urve s ho wn in r These c urve s a re simil ar t o the c o mpara t e S F o m 12 154 BUSI NESS MATHE MATICS

WR I TTEN EXE R CI SE S

are as T . a a 1 . he monthly earnings and expenses of a r ilro d follows

M R RI L MA Y F E B . A CH AP

E a r nings 3 3 E xpenses

E . AU G . S PT No v .

Make two curves on the same chart to Show the contrast .

a two es to e ese t the a e a e ate a andthe a e 2. Dr w curv r pr n v r g r p id v r e Of h es use a t the l ata age numb r p on in in ci y , from fo lowing d

120 1 13 Av . ra te pa id . $0 $ $ $69 $62

Av . No . u sed 0

19 15

60 56 $52 $46 40 3 a e aid . $ $ $ 9 AV . r t p $ d Av . No . use

ese t the er e ts the e t a and th 3. Re pr n ing p c n on v r ic l column e grades a e th ee es to S on the horizontal column, m k r curv how the pe rcentage Of pupils attaining the various grades as given by three teachers in the same subj ect .

156 BUSINESS MATHEMATICS

NE T DE BT O F TH E UN I TE D STATES millions

900 900

1, 100

7 . Th e following facts Show advances in rentals from 1900 to 1913. The y are based on a study of charges for the same properties in 48 cities ha a t r r t ese . ving a popul ation of or over . M ke h ee cu ves to show h Use a s e the s e the first- s st i ts a as he olid lin for to r s in class busines di r c , d d i e se - ass i st t and a t e th r - ass st ts l n for cond cl d ric s, do ted lin for i d cl di ric , or use colors .

R e nts l st cl a ss

2nd

3rd

o o o o o

m nen arts Sh w b Cur — 139 . Co po t P o n y ves The chart illustrated in Form 15shows the total cost plus the profit in

Th s is es a mamifacturing business . i of pecial value to the

he ma ee at a a e ho w th exe cutive, for y s gl nc e manufa cturing

I t a s Sh endof the business is working . will l o ow if there are GRAPHICAL REP RESENTATION 157

e a ext a hea ex e ses a t irr gul r , r vy p n in some particular p r of the manufacturing costs and allow him to take means to

Dollar s

Ma une ul Au . Se t . Oct. Nov. Dec . J an . F b Mar . A r. e . p y J J y g p

ha sh Form 15. C rt owing Component P arts

The princ ipl es underl ying the construc tio n o f th is chart are simil ar to th o se disc ussed in co nnec tion w ith th e re c t angl e c hart (Fo rm 9)

h N te that e ach te m th e x se is reme dy t em . o i of e pen a dded h so that th t e s the t ta . on , e op curv will ow o l

WR ITTEN E XE R CISES

s the ata st t es s m a to t se 1 . U ing following d , con ruc curv i il r ho shown in Form 15.

L MAR . APRI MAY JUNE

Materials in pro duct io n 800 Skilled la bor

l ed l abor 1 200 Unskil , Ma terial s fo r o ffice 300 300 300 350 300 400 Supervi sio n a nd 500 500 475 500 400 450 Fixe d cha rge s 350 350 400 400 375 380 400 400 450 500 Estima t ed pro fit . . 450 500 158 BUSINESS MATHEMATI CS

2 the ex . Make a graph showing the percenta ge of distribution of T ta penses of operating the railroads of the Unite d States p er year . o l a ll er ts of p cen should be 100 .

1890 189 1 1892 1893 1894 18951896 1897

G e ne ra l E xpe nse s Ma int e na nce o f e q uipme nt

wa y .

Fi xe d ch arge s ” C o nducting tra nspo rta tio n

1898 1899 1900 190 1 1902 1903 1904

G e nera l E x pe nse s M a inte nance o f eq uipment

4 6 4 . wa y . Fixed Conduc tin g t ra nspo rta tio n

a e an m — h u 140 . Corre l tiv d Cu ulative Curve s Suc c rves are constructed so that the y Show a re lation as well as the 1 t . o h total o date For instance , in F rm 6 t e sales up to April are while the total collections to April are

m a the t ta sa es to Se tembe are and the Si il rly , o l l p r total colle ctions to September are The collections

M a h sh a a st the sa es a a c If the for rc ow g in l for J nu ry , et . b s ess t s the firm are ea he t es u in condi ion of id l , t wo curv

a e to e a h h - l a should run nearly p rall l c ot er . A 60 da . g h h that h a es are 60 da . a ea means t e s l d of t e collections .

WRITTEN EXER CI SES

1 . C st t a a h s a to t at s on ruc gr p imil r h hown in Form 16, to represent the following data on production in a manufacturing plant :

160 BUSI NESS MATHEMATICS

ES TI M ATE D COS T P E R ACTUAL COS T P E R WE I GH T o r TR UCK CAR - M I L E CAR -MI LE

41! 5¢

e e sen a — 2 141 . Map R pr t tions M ap graphs are very use ’ h h h ful in the executive s office . T e y s ow at a glance t e loca

all h bs a m a tion of t e su idi ry offices or anufacturing pl nts.

Form 17 shows the location of the ] various cantonments for h h ds the United States Army in 19 18 . Pins wit colored ea

an Th andletters or figures c well b e used with such maps . e

a h letter could st nd for t e n ame of the city or town . A tabulation showing the n ames of the places opposite its

mbe a m a a ha corresponding nu r could cco p ny c rt .

W RI TTEN EXE R CI SE S

the te States and at th 1 . Make a map of Uni d loc e e following selling agencies of a prominent article

St L s . Sa t La e tah Boston , Mass . . oui , Mo l k , U

New e a s La . B tte t . Albany, N . Y . Orl n , u . Mon

i a l as Texas P rt a . New York C ty D l , o l nd, Ore

P . Des nes I a San ra Cal . Harrisburg, a Moi , ow F ncisco ;

T sa a . Sa t . c ame t Cal . Detroi , Mich ul , Okl r n o,

C . 11. en e Chi cago, 1 D v r, olo

Two types of maps should be use d in the tea chi n of map graphs a tl e ma the ass e tat andth the bla ckbo rd ou in p for cl r ci ion, e desk outline h a the teb t e . a the map for no ook work by individu l pupil M ps of world, t States and state be s ffi e t the Uni ed , your own will u ci n for this work . s tli e ma s be he to fit the teb The de k ou n ps hould punc d no ook cover, in . order that the maps prepared by the p 11l may be made a part of his notebook work . ’ The neral use of th e pm ma p I n busmess o fli ces ustifies the author s belief t at they belon in this work . Any good wej1 maps mounted on nd Ta med t ethe t a bo x co rk composition a , og r wi h of map pins of assorted E xe i ses colors and sizes will se rve . rc in which pupils are required to the ma the at o f 1m ortant e a t e show on p loc ion p comm rci l ci i s, industrial _ h e a e I te est to t s s t . s e t . sections, e c , ould giv dd d n r hi ubj c GRAPHI CAL REPRESENTATION 161

2 C . onstruct a map of New York andNew Jersey andon it show the ele at the n ts the v ion of followi g poin on New York, Ontario, andWestern Ra i lroad .

Oneida

Apex

M t Form 17 . ap Char

e — Th 142 . Fr e que ncy Ch ar ts or Curv s ese curves are

te e o h h he e ta th in nd d t S ow o w often or w n c r in ings occur . The curve given in Form 18 shows when the commodities

Th ea a e Sh h rise and fall in price . e p k of curv will ow w en the particul ar thing will rea ch the greatest amount or the

eatest e ea etc . gr numb r of y rs,

WRI TTE N EXER CI SES

1 t a e to S the a e at a a e 439 a i es h . Plo curv how g m rri g of l d w o were t college gradua es .

I I 162 BUSINESS MATHEMATI CS

0 0 0 0 0 0 0 0 0 0 0

o o o o o o

Y E A RS

e s wi ha e Form 18 . Frequency Curv ho ng C ng s in Costs

the e th a e 2 . Construct a curve to show d a rate ch ng s of Americans

a s a es si e 1880 as e te an ins a e m a . v riou g nc , r po r d by ur nc co p ny

DECR EAS E I NCRE ASE

o o o o o o

164 BUSINESS MATHEMATICS

The above-mentioned sales we re actually made by di fierent men selling - - es i im a house to house article on a commission basis . The curv w ll mediately show up the compa rison of their sales . l sh l st th 2 . The foll owing data shows that an automobi e ou d op wi in h i r s ee the b a es t e given distances according to the m les p e hour of p d, if r k

t e f this ata . are in proper order . Construc a curv rom d

AT SPE E D o r A C AR S H OULD STOP I N

9 . 2 ft 15H M N 20 37 25 58 30 35 104 40 148 50 231

in i f at s s the ax m e ce ta es 3. The follow g n orm ion how m i um p r n g of di f erent family annual incomes which ought to be expended in normal x times for the variously named parts of a family e pense a ccount .

PE R CE N TA GE OF ANNUAL I NCOME VARI OUS I TE MS $800 3500- 8800

Miscella neous o pera ting . her ivin bo o s H ig l g . k , s i nsurance re saving , ,

ligi o us, etc

Water rent

Construct a circle or a rectangle for each income anddivide it up into Th a e the les e t its component parts . en pl c circ or r c angles side by side to show comparisons . GRAP HI CAL REP RESENTATION 165

' 4 Th Y k Ti mes Anna lzst . e following fa cts taken from the New or Show the t e P t a e r nd of bond prices ove r a certain period of time . lo curv from the following data 19 17

De c

E a h a as e the New L e I s a e Co m 5. c doll r of c h incom of York if n ur nc a w as e x e e as s a to a e e t ea e t the p ny, p nd d follow ccording r c n y rly r por of

t t a a h to e t t . company . Cons ruc gr p r presen hese facts

P aid for death claims .

P aid to living policyholde rs . n Set aside for reserve a ddividends . P aid to agents a ex e ses a e s e s and For br nch p n , g ncy up rvi ion , e a inspe ction ‘ m dic l For administration andinvestment expenses s a e e t taxes ees and e se s For in ur nc , d b , , f , lic n

Total

t t t ee es the sa e a t e e a t ffe e t 6 . Cons ruc hr curv on m ch r (pr f r bly wi h di r n t the a es the at New colored inks) o Show ch ng in popul ion of York, Chi a and P tts es e t e the ata : c go , i burgh r p c iv ly, from following d

PO PULA TI ON OF POPULATI ON OF PO PULA TI ON OP NEW YO R K C H I C A GO P I TTS B UR GH 166 BUSINESS MATHE MATICS

7 . The U te States i e a 10 e t the ni d l f t bles, Census 19 , r por following Con death rates per among the white males at the various ages . struct the curve from the data andcompare the numbers of different ages .

DE ATH R A TE DE A TH R A TE AG E P E R P E R 12 69 137 245 386 585

60 .

R e e to 1 an a 8 . f r § 38 dconstruct a chart showing the following inform t a e a e es ta e t a e a at s the ion from v r g d figur , k n from many r d org niz ion ,

a a B ea B s ess and a e t at s . H rv rd ur u of u in , individu l inv s ig ion

PE R CE NTAG E O F TO TAL S A LE S COS T OF DOI NG KI ND o r BUS I NE S S CAS H DI S CO UN TS BUSI NE S S Variety goods 3 19 4 23 Clothing Furniture Jewelry 25 25 Drugs . 3

2 Shoes . 3 5 De partment stores O 25 I mplements andvehicles Groceries 2 17

’ to 1 and st t a t th 9 . Re fer 38 con ruc a ch r from e following information t sts and s ts es ne t n t s ts o e e . of profi , co , di coun by lin ( profi for urnov r)

P E RCE NTAG E 0 1: TO TAL SALE S COST O F DOI NG LI NE C ASH BU S I NE SS Books 22 Corsets 24

Furs . 26 Gloves ‘ 24 Hosiery

Handke rchiefs .

Laces .

Linens . 24

Millinery . 25

168 BUSINESS MATHEMATICS

’ “ 12 . m the i ata ta e m P a l N strom s b The Fro follow ng d , k n fro u y ook , ” E s R eta l ma ten a e ten e ta l es conomic of i ing, ke circles of e qu l siz or r c ng e a e sh i th the a t me t of qu l l ngth , ow ng e compone nt parts for ppor ion n of th e t 1 St and o m a e e r n in 0 stores of different numbers of floo rs . udy c p r your results .

PE RCE NTA GE O F R ENT 4 5 6 7 8

35 25 15 10 10 15 65 6550 60 454550 40 35 25 30 25 25 20 20 10 15 10 10 15 10 10 10

C t t two e t e ese t th e it t wo e at rs 13. ons ruc curv s o r pr n e tim akes t op r o t the t me h a to the t ta to do various opera ions . Add i for ea c oper tion o l t the re e in O e at s th min a m lat e time time spen on p c d g p r ion , us for g cu u iv for the whole number of operations TI M E I N SE CONDS FI R S T S E C OND KINDS OF OPE RATI ONS OPE R ATOR OPE RA TOR ? O 5 O Reaches for labe l . 0 Re a ches for brush . 0 W 0 D Wipes brush on glue pot C 0 J Brings brush to label 1 O 1 D Covers label with glue 0 C 57 N Re places brush . 3 P uts label o n 0 hi Adjusts andsmooths label 9 1N

h ee e the sa e ha t e to sh the er 14 . Construct t r curv s on m c r fi ld ow p ’ centage of the pupils of each teacher s class which attained th e various val ues (or percenta ges) in the same grade of work .

P E R CE NT o r? VAL UES NUMBE R OF S TUDE NTS TOTAL NUMB E R IN CLASS First Teacher 2 2 4 20 10 2 GRAP HI CAL REP RESENTATI ON 169

P E R CE NT OF VALUE S NUMBE R OF STUDE NTS TOTAL NUMBE R I N CLASS

Se cond Teacher 2 2 15 10 11

Third Teacher 50 70 0 70 80 20 80 90 10 90—100 10

C st t 15. on ruc a chart to show the comparison of the expenditure of $1 2 s ess e ea s the a a s the U te Sta th in ucc iv y r , of r ilro d of ni d tes, from e following data

I N CE NTS

I S T Y R . 2D Y R . Operating expenses Taxes Excess of fixed charges over non-operat

ing income . . Dividends

C the f n at th ee e a 16 . onstruct from ollowing i form ion r curv s on e ch of th ee ha ts the sa e he t ands a e to sh a e ta s s r c r of m igh c l , ow how c r in bu ine s is running .

MAR . AP R .

l st Chart

R eceip ts Purchases 925 850 950 Total expe nse 600 525 520 550 500 475

2d Ch art

Total e xpense Total salar y expenses

3d Chart

G ross

E xpense Net pro fit 170 BUSINESS MATHE MATI CS

C a h h a e -s e es 17 . h rt t e following facts by the use of t e s m iz d circl or ’ parts of a circle to show the dollar s buyi ng power andits changes during a part Of the World War .

BUY IN G P OW E R 100

C st t two e ta es the sa e e th and e ea t 18 . on ruc r c ngl of m l ng divid ch in o T its compone nt parts to show how rents vary in different lines . hese figures were compiled by the Bureau of Business R esearch of Harvard

the th t n . University . Note how rents of e wo respe ctive li es vary

R ATI O BE TW E E N R E N T A ND NE T SALE S H I G H L o w COMMON OR TYPIC AL 3 Groceries . . %

. 8 5 Shoes . % %

a t t ee u es the sa e e to s the a ts 19 . Ch r hr c rv on m fi ld how following f c T concerning the railroads Of the United States for the year . hese fa cts

r h r Of Ra r a E s Was t . C . a e from t e Bu eau il o d conomic , hing on , D

M A Y MAR .

850 825 860 400 400 440

SE T . No v . AU G . P

850 890 890 900 890 900 470 520 510 570 510 450

n te 4 m es er 20 . ma a s to a a e at the a A w lk pl c r of il p hour, remains 1

t e es a at the ate Of 10 es er . H e w hr . as a , h n rid b ck r mil p hour bsent 8 3 far he a ? hr . How did w lk

3 . C E e e ts e at . E e at s . B st From H obb , l m n of Appli d M h m ic o on ,

nn and C a 1911 . Gi omp ny,

CHAPTER XI I

SH O RT M ETH ODS AND CH ECKS

4 e h - e k n M eth s — A 1 3. Valu of S ort Cut and Ch c i g od computer natur ally should be the master of short-cuts and

h hi The atte is e ha s Of methods for c e cking s work . l r p r p

m ta e th a the me be a se o ne m st be more i por nc n for r , c u u able to show his employ er that he knows that his work is

corre ct . The exe cutive should a lso know these methods in order that he may be able to che ck the work of his employ ees . H e must a lso b e sure of the fa cts presented as a basis for a satisfa ctory andsuccessful business . I t is the Obj ect of this chapter to give to any person who

f i ea h da s h a will devote a small amount O time to t c y, uc knowledge that he will b e well equipped to do computation

h a h b wa andat the same work in the s ortest as well s t e est y, time be able to make an a ccurate check on these computa tions .

thi th is th 144 . Addition . Any ng wor doing wor doing ” he e e we sh no t be sat s e th a e e well , t r for ould i fi d wi pi c of

h i me wa if it is at all s b e work unless we c e ck t in so y, pos i l ,

The he tha are and with most work it is possible . c cks t quite common in addition are

the exam e m to to b tt m the a 1 . Adding pl fro p o o , n dding

b tt m to to e e sa . from o o p , or vic v r

2 b m s . . Adding y colu n SHORT METHODS AND CHECKS 173

I st llu rative Exampl e . (a) 1456 7238 3564 or

12258 12258

EX L N TI ON : a ea se a ate e n at the P A A ( ) Add ch column p r ly, b gin ing T r t and set the es t ea tse . he es t igh , r ul for ch column down by i lf r ul for the tens column should be pla ced o ne pla ce to the left of that for the T T e add the t ta s . s et i a a e e units column . h n o l hi m hod s v lu bl wh n

a s es e a o ne is te te e a . dding long column . p ci lly if in rrup d whil dding a s a a e e i at the e t and (b) Add in imil r m nn r , b g nning l f column moving each result o ne place to the right .

’ ’

E x s O e a ast o ut s . . e 3 cesses of 9 , ft n c ll d c ing 9

a E xa 457 Illustr tive mple . 1 7238 3564

12259 . 1 1 1

E X L N TI ON : the ts es ea e (as P A A Add digi (or figur ) in, ch numb r for example in 1 4 5 7 17 ; divide 17 by 9 andthe a mount

a 8 i s a e o ut at the t . the sa e all the e s . rem ining , , pl c d righ Do m for numb r Then add these e xcesses (or a mounts left over) and the result is 10 in x e the ex esses 9 t s sum i 1 the the above e ampl ; find c of in hi , giv ng in T e the ex esses 9 the t ta h h we a s above . h n find c of in o l , w ic l o

find is 1 .

a a satisfacto r e . T s e h e e is no t This gives f irly v ch ck hi ch ck, ow v r, always to be depende d upon be cause o ne may make a mistake of 9 (or some multiple of 9) or 0 in the additio n which will no t show as an e rror in ’ the excesses of 9 s in the total or a mistake in arranging the figures in a

tea 1457 a e . i . . s number, e , in d , bov 174 BUSINESS MATHEMATICS

WRI TTEN EXER CISES

Try o ut e xamples

3.

H I NT : I n all addition try finding numbers group

10 5 s e th simi a . , or , or om ing l r

Exam le : 7 p [10 3 l 2 4

b a n — The mm h i h 145. Su tr ctio co on c eck s t e process

n i a he ema s te to the s a o e . e . dd t e to Oppo i u u l , , r ind r the subtrahend and it will produce the minuend if the work is corre ct .

I ll ustrative Example . 567

198 CH ECK : 369 198 567

WRITTEN EXER CI SES

Find the difference in the foll owing examples and che ck

2 . 3. 1 .

’ h i the s bt act the ex esses Another c eck s u r ion of c 9 s .

176 BUSI NESS MATHE MATI CS

WRI TTE N EXER CI SE S

“ Try this plan on the following examples and find the bal ances in the a at the e nd t b nk of he day .

2 4 1 . $ 3 5balance in the bank in the morning

C e s e out the da 367 h ck giv n during y 568

BALANC E C H E C KS LAST BALANCE

2. 3 3

— at n M eth s 146 . Mul tiplic io od of checking multiplica tion are a s follows :

D e the t b the m t e to h 1 . ivid produc y ul ipli r obtain t e

h m t a a b t e to Obta th m . multiplic nd , or y ul iplic nd in e ultiplier

2 e eat the m t at and the es t is a e . R p ul iplic ion if r ul s befor h h i we m ay assume t at t e work s correct . ’ t o ut 9 s as s : . as 3 C , follow

43 21 Illu strative Example . x 43

’ EXP L AN ATI ON : Find the e xcesses of 9 s in the multiplier andthe multi ’ u t t ese e x esses t ethe and find the e x ess plicand. M l iply h c og r c of 9 s in this product .

WRITTEN EXERCISES

Multiply the following and check :

56 5 3. 4 2 6 2. 34 5. 45 1 . 5 1 123 SHORT METHODS AND CHECKS 177

’ B the use the ex esses 9 s t t a t a t i o ut y of c of , wi hou c u lly mul iply ng , state which of the foll owing results are corre ct

456 . 7 376 8 . 415 25 75 16

Sh t m h Th h or et ods prove to be very valuable . oroug ly understand each method as you go along and pra ctice it

whenever possible .

— 147. To t b An M t 1 M e the e Mul iply y y ul iple of 0 . ov d ci mal point as many pla ces to the right as there are zeros in

h m It is Ob s e essa o a ex e s t e ultiplier . viou ly n c ry t nn z ro if the multiplicand is a W hole numb er as a decimal point is

understood at the end of each whole number .

I s a Ex e . 4 1 llu tr tive ampl 1 . 56 X 00

SOL TI : e the e a t two a es to the ht U ON Mov d cim l poin pl c rig , giving

Il x 2 lustrative E ample . x

SOL TI N : e the e a t a e s to the t U O Mov d cim l poin four pl c righ , giving

15 Illustrative Example 3. X

SOL UTI ON : Since there is a de cimal point understood at the right of n t e e t s ase we e it t ee a es to the a y whole number, her for in hi c mov hr pl c ht i rig , giv ng

ORAL EXER CI SES

Multiply the following mentally

456 10 6 . 1 . x x

2 x 100 7 . . 00035x

3. x 8 . x

4 x 100 9 . x

5 10 . 310 5 . 0000 x x 178 BUSINESS MATHE MATICS

14 E dn s . 8. To Mul tiply Number s Having ! eros as n i g

M t b the s a t es the a ex as ma ul iply y ignific n figur , n nn ny zeros as there are in both the multiplicand and the multi plier .

Ill strat Ex 430 400 u ive ample 1 . X

SOL TION : ti 43 4 i 172 the annex three zer s U Mul ply by , g ving , n o , giving

ORAL EXERCI SES

Multiply the following

1 . 236 X 20 6 x 900 52 300 7 2 . 3 x x 12

456 8 . 140 3. x x 4 x 9 230 x

5 120 X 400 10 . x

256 20 Illustrative Example 2 . . x

SOLUTI ON

Multiply the following

1 . 53 x 300

2 . x 400 56 120 3. . 0 x

4 . . 0027 x

5. x

99 999 e — ul t b 9 tc . T 149. To M iply y , , , o multiply a

m t the mbe b 10 an h number by 9, ul iply nu r y , d t en sub tract the original number from the result of the multi

plication .

180 BUSINESS MATHE MATICS

Illustrative Example 2 . 576 X 75

SOLUTI ON

WRI TTE N EXER CI SES

i 7 50 . t? 1 . Multiply 6 9 by How would you do ti 125? 2. How would you mul ply by Find the results of the following by the a bove methods

50 11 . 25 7 . 67 3. 32 X X X

25 8 . 88 75 12. 4 . 76 X X X

4 25 9 . 145 50 13. 5. 8 X X X

4 50 10 . 726 250 14 . 496 6 . 3 X X X

Two N mbe Ea h E M l t s in 5. 51 . To 1 u iply u r , c nding

65 65 Illustrative Exampl e 1 . X

X L N TI ON : Whe the sa e e s are to be t e te E P A A n m numb r mul ipli d , wri T e d 25for the last two figures at the right . h n a d 1 to the te ns figure (6 l 7) andmultiply by the othe r tens figure (7 X 6 and h 2 Th write this result at the left of t e 5just written . e product is

W e the sum the te Illustrative Example 2 . h n of ns figures is an even 55 75 number . X

e the 25as the rst a t h E XPLANATI ON : S t down fi p r of t e product. Then find of the sum of the tens figure s of 12 andaddit to the

5 7 35. 35 6 es . P product of the tens figur X 41 . lace the 41 at the left of the 25previously written .

3 W e the sum the te Illustrative Example . h n of ns figures is an odd 35 25 875 number . X

e 75, as the st a t th T EXPLANATI ON : S t down fir p r of e product . hen find5of the sum of the te ns figures (5of 5 Drop the andadd the 2 to the product of the tens figures . SHORT METHODS AND CHECKS 181

ORAL EXER CISES

Multiply the following

e — 152. To Squar Any Numbe r of Two Figures This is based upon the principle that the s quare Of the sum of two 2 a tit es e a b is e a to the s qu n i , lik ( ) , qu l quare of the first

a e the r s qu ntity, plus twic fi t quantity times the second

he s a e the quantity, plus t qu r of second quantity .

Illu strative Example 32 + 24 + 4 ’ 49

WRI TTEN EXE R CISE S

Square the following numbers by the above method

4 . 52 1 . 64

5. 124 Cal it 12 te s . 2. 36 ( l n )

3. 27

it e Time s Th eir Di w a t s fi erence . 153. Sum of T o Qu n i Certain number products come under the principle that the sum Of two quanti ties times the differe nce Of those same two

a s a b a b e a s the s a e the s products, ( ) ( ) qu l qu r of fir t 2 2 h a e the se a quantity minus t e squ r of cond qu ntity, or a b

m e Illustrative Exa pl .

400 — 1 182 BUSI NESS MATHE MATICS

WR I TTEN EXE RCI SE S

Find the product of the following numbers by the above method 56 1 . 22 x 18 4 . 64 x

2 . 58 36 X 44 5. 62 X

3. 53 X 47

M e 11 . 154 . To t b 11 22 M t ul iply y , , or Any ul ipl of

I st at e Exam 264 1 1 llu r iv ple 1 . X

SO LUTI ON 264 1 1

EX PL AN ATI ON : Set down the units figure of the given number in the dt u t 4 . the n ts an e s es 6 4 Se t produc , Add u i n fig r ( down the th t nd zero and carry 1 . Add e ens a hundreds figures (6 2 add

the I a e ak 9 . t the ast re n e s in t x c rri d , m ing Mul iply l figu (hu dr d his e

a e 1 set the es t 2 . Res t 2 904 . mpl ) by , r ul , , down ul

x 2 4 16 22 Illustrative E ample .

SOLUTI ON : 416

TI N T t th E XPL AN A O : wo imes e unit figure 12 . S et down the 2 and the ts andte s u es 6 1 t carry 1 . Add uni n fig r + Mul iply the 7 by h 5. Set t e 5and a 2 14 and 1 a e a es 1 1. . c rri d m k down c rry Add the tens andhundre ds figures (1 4 Multiply the 5by 2 andaddthe

Set the 1 and a 1 . I r a 11 . t car ied , m king down c rry Mul iply the last 2 andaddth I a figure (4 in this example) by e c rried, making 9 . Re sult

WRITTE N EXER CI SES

Multiply the following :

1 . 346 X 11 562 22 2 . X

3. 124 X 22

4 . 214 X 33

184 BUSINESS MATHEMATICS

a 1 . 157. To Mul tiply by Use of Aliquot P rts of 0

x 28 1 Illustrative E ample 1 . x 4

SOLUTION : 280 8 35

L N TI ON : S e t e to t b I m ti EXP A A inc H herefor mul iply y }, ul ply

m e 10 and i e 8 . the nu b r by , div d by 2 Illustrative Example 2 . 7 X 6§

SOLUTI ON : 3 720 240

480 — 10 3% 10 (%o f 10)

ti 10 e 3 ands t a t. Mul ply by , divid by , ub r c

WRI TTEN EXER CI SES

a to t 7 ? 1 . How couldyou pl n mul iply by 3 Find the product of the following by the above method

2. 124 x 24 7 . x 24

3. x 35 8 . x 14

9 . 56 4 . 144 x 6% 3 x 74

56 7 10 . 6 5. 2 x 4 x §

1 1 . 6. x 14 x 34

b se o f l t a t 100 . 158 . To Multiply y U A iquo P r s of

624 25 Illustrative Example 1 . x

SOLUTI ON : 4

EXPLANATION : Since 25 therefore multiply by divide by 4 .

4 6 7 Illustrative Example 2 . 5X 5

SOLUTI ON : 4

5 100 25 100 EXPLANATI ON : Since 7 . (i of 100) SHORT METHODS AND CHECKS 185

WRITTEN EXERCI SES

the to tal value of each of the following

1 . 2 . 2 5 . at 44 yd ¢ 25lb . at 451!

33% 36¢ 37}7 72¢ 12% 48¢ 3l i 32¢ 16% 601! 16% 42¢

4 . 5. 6 .

6 l . a 2 0 b t 1 0 yd . at 16 articles at $12% 90 300 12 $1 48 Gi st 32 a rticles at $25 $125

'

56 37 996 bu . at $623 $1

159 . To D e b 1 1 ivid y 0, 00, e tc .

1 Illustrative Example . 0

E X PL AN ATI ON : Move the de cimal point as many places to the left r as there a e zeros in the divisor .

ORAL EXE R CI SES

e ea t ese e s 10 100 Divid ch of h numb r by , ,

1 . 5. . 124 9.

2. 6 . 000643 10 .

3. 7 . 11 . 649

4 . 8 . 12.

550 75e tc . 160. T0 Divi de by 2 , , ,

25 124 4 I llustrative Example 1 . X

—1 2 N : S e 25 t e e e to e i e E X P LAN ATI O inc 2 , h r for divid div d

1 0 and t 4 . 0 , mul iply by 75 Il lustrative E xample 2. 100 0 3, t 00 3 30 (3of 30) 40 186 BUSINESS MATHEMATICS

WRITTEN EXER CI SE S

Perform each of the foll owing divisions 25 1. 6.

2 . 334 7.

. 50 3 6.

4 . 75 9 .

5. 5 . 14 0 10. 0 5

To D 2 161 . ivide by 4, e tc.

2 Illustrative Exampl e 1 . 50 }

SOLUTI ON 2 1

E xample 2.

WR ITTE N EXERCI SE S

Perform the following divisions

2; 4 . 60 6§

50 3 5. . 624 5 2. 1 ; 2 3. 240 75 6 . ;

a Numbe C m se dof act rs . 162. To Divi de by r o po F o

Il lustrative E xample .

SOLUTI ON

EXPLANATI ON : 42 6 X 7

BUSINESS MATHE MATICS

165. dt N m Ad i ion of u be rs Containing Fracti ons .

Il strati e Exam e lu v pl 1 . g g g SOLUTI ON: 45 i 48 Write the least common denominator 40 g only at the end Of the work . it 50

60

Ill ustrati ve Example 2 . 15% 14; 65% SOLUTI ON : 15% 2 143 8 65% 9

57 94 i 9 T2

a Illustrative Example 3. How m ny yards in 4 pieces of cloth contain 21 2 3 433 ing , 6 ,

2 SOLUTION : 21 1 36 (The small numbers at the right and 3 4 a e e ese t a te 3 bov r pr n qu r r yards. ) 56 I

157 3

p 7 I 4 . Ans Ill ustrative Example 4 4 . «o

NOTE : It will be note d tha t the nume rator is the sum of the denom th e i at is the t the e m inators, and e d nom n or produc of d no inato rs .

Illustrative Example 5. g g%or

NOTE : Observe that the numerator is 2 times what it was in Example

the e minat the sa e . 4, and d no or m

To add a t - Il lustrative Example 6 . fr c ions by cross multiplying the 9 denominators andnumerators . }

SOLUTION 5X 7 35 3 X 6 18 35 8 53 the new e at 1 . , num r or 42 the new e 7 X 6 , d nominator

I 1! I result SHORT METHODS AND CHECKS 189

166. Subtraction of Fracti ons .

he e a r i e . Example . When t num r tors a e al k

5X 3 15 4 X 3 12

15 12 3 the new me at , nu r or 4 20 the ne e ato X 5 . w d nomin r {is res ult

M xe d N mbe . 167 . Multiplicati on of i u rs

Il lustrative Example . 144 X 8§

SOLUTI ON 143

4 = 4 x 5 4 8 X 5 X 14 112 8 x 14

result

a n M xe N mbe s Wh se a ti s 168 . Mul tiplic tio of i d u r o Fr c on Are

I‘ Il lustrative Example 1 . 85X 84

St 8}

l = i >< i

72} result

192 BUSINESS MATHE MATI CS

h i the a e a e ht a e e s Illustrative Example 1 . W at s v r ge w ig of doz n gg weighing 666 grams?

SOLUTI ON :

666 grams 12 grams

xa W at is the a e a e a e the l i men Il lustrative E mpl e 2 . h v r g w g of fol ow ng if 10 men re ceive $4 per day ; 12 men receive per day ; 8 men re ceive pe r day; a nd5men receive per day?

EX P L AN ATI ON : I t is first necessary to find the total wages earned men t e e the t ta t se to ta s the t tal by each group of , h n divid o l of ho l by o number of men .

SOLU'rrON :

NUM B E R OF ME N DAY WAGE 10 12 85

35

Average wage

WRITTEN EXERCI SES

s e t e 118 104 168 156 132 and 2 l f ix s e e 11 b . 1 . I s boy w igh r p c iv ly , , , , , , what is their average weight? ffe th 37 a ha t xes 2 . e 3 . th 2 . I f a me rc n mi lb of co wor ¢ pound, lb wor 42 a hat is a he and1 . t t xt e 39¢ a pound , lb wor h ¢ pound , w pound of mi ur worth? - ll sh s 12 ha s are e e at er da 14 ha s 3. A pay ro ow nd mploy d p y, nd r da 18 ha s at er da and 6 a s at er a t pe y, nd p y, h nd p

the a e a e a a es . day. Find v r g d ily w g e s a st e a e these sa es o ne da 4 . The following cl rk in or m d l in y

S LE S C LE R K No . A

123 124 125 2608 0

h Find the average sales of these clerks for t at day . Which sold above the average? Which below the average? ER GES SI LE AND WEIGHTEI) 193 AV A , MP

I 5. f the expense of running the city of New York is in 1920 and h t is , t e population of the city is approximately wha the per capita expense?

6 . I n a f a e 800 14 e ta s s 350 are 13 . O c r in chool of pupil , yr g ; ,

. a 40 7 0 . a e e 515 . Of a e 600 16 . a e 550 17 yr of g ; , yr g ; , yr of g ; , yr of g , ,

18 . a e n h a e the s . a d10 19 . a e . t a yr of g ; , yr of g Find e ave r ge g of chool

7 . the f e e s to a e Find di f r nce in co t for a trip from New York Or ng ,

- N. J if a l O trip ticket cost and a 50 trip ticket costs

e h — m 172. W ig te d Ave r age s These take their name fro the fa ct that the y must be weighte d or reduced to a common bas s e to Obta a e a e e and ess th s is i in ord r in corr ct v rag , unl i

n h a done a entirely erroneous result will be found . T is pl n is il a h x l well lustr ted in t e following e amp e .

S h s s tes Il lustrative Example . John mith a given to William Jone no H e as follows :$150 due May 14; $200 due June 29 ; $500 due July 20 . s t e due ? wishes to pay the m all at o ne time . When hall hey be consider d

E XPL ANATION : I n order to arrive at a corre ct solution it is necessary - ll the tes e e ai to e e ea t ese to a 1 da . as s a r duc ch of h b i , for if no w r p d

Ma 14 S th l se the use 200 46 da . and 500 67 da. y mi wou d lo of $ for , of $ for ; - : or reduced to a 1 da . bas is we woul d have

$200 X 46 for 1 da . $500 x 67 1

Total 1

h h e se at is e i a e t to 1 da . Smit , t er fore, would lo wh qu v l n for andis entitled to keep the $150 $200 $500 $850 as many days th use 850 to e a the use 42 afte r May 14 as are required for e of $ qu l of $ , 1 7 ” he e ated t e a da . e e t 700 1 da . 530 for , or H nc qu im for p ying Ma 14 3 a a e as s i 50 da . a te all the notes s f r y or July , or rr ng d follow $150 X 0 2' 00 X 46 500 x 67 194 BUSINESS MATHEMATICS

WRI TTEN EXER CI SES

1 . Fi n th e ate th e fo r the t 250 due 3mo . 4 d e qu d paymen of $ in , $ 00 n 700 due in due in 6 mo . , a d$ 8 mo .

ate t me h ue . 50 2. Find the equ d i for t e payment of $300 d in 30 da , $ 0 200 due in . due 60 da , and$ in 90 da . h ate ti e the a e t Of 2 5due e 21 17 3. Fin t e e 7 5 d qu d m for p ym n $ Jun , $

ue 16 2 due . 6 and 150 due Se t . . d July , $ 00 Aug , $ p 3 due in If h 4. es B 200 10 . e a s 120 in 4 mo . h A ow $ mo p y $ , w en sho uld he p ay the balance ? 2 he SOLUTI ON : By paying $1 0 in 4 mo . A loses t use of $120 for 6

h h a the use 720 1 . Th h mo . , w ic is equ l to of $ for mo erefore, e is entitled

e b l e . e the . to keep th a anc , mo , or (r ducing fraction) 9 mo after its maturity .

5 es B a a e 4 . but at the end 1 mo . he . A ow p y bl in mo , of pays

h nd 2 . 00 an h him 500 at t e e 5 d at t e end mo . 5. I n $ , of mo $ , of 3 $ 00 how many months is the bala nce due ?

Feb . 1 1 a s a ntin 6 . A man bought , bill of good mou g to on 4 r 22 4 m . a Ma . 00 . 20 220 andM 1 e t u h a . o cr di ; b t e p id , $ ; Apr , $ ; y 0, $300 When is the balance due? t me at t e h th 7 . Findthe e quated i of m uri y of ac of e following bills, andthe amount due at settle ment includin g interest at

H OE To WILLI M PR I CE JR . JO N D A ,

4 . e t 5To m se . Apr . d on mo cr di T ( 6 U 3 U I t Apr . 15o May 7 To 3 May 21 To 4

INT : Findthe e atedti e a me t ec in m 15 H qu m for p y n , r kon g fro July , ies ate that an te e es due an which is the earl t d y i m b com , dfind the inte r

the e ate ti e to O t . est o u the whole bill from qu d m c 18 . en be s ess t ethe th b e 8 . If two m gin bu in og r wi orrow d capital the es rces are hat was th andat the endof 6 yr . ir r ou w eir aver age annual gain?

s de s ts . 1 a an that a 9. If Mr . William po i on Apr , in b k p ys interest on he sum a es t . 2 average monthly bal nc , of on Apr 1 he reduces e hi s a e a e bal a e this sum to determin v r g nc for April .

196 BUSINESS MATHEMATICS

’ T ta ea n e 680 270 950 h A s i estme t o l r ing pow r, $ $ $ , of w ich nv n g ’ re presents ga andB s

12. X d X ests 9 mo . and an Y form a partnershi p . inv for , then adds Y invests but withdraws at the endof 4 t the end a ea th a as s : mo . A of y r eir ccounts stand follow

DE PT . A

On hand

B DE PT .

Sales . .

G OODS OF DE PT . C

Cost $1 Sales On hand

G E NE R AL EXPE NSE

Apportion the profits a ccording to the average investment . T i fl he i s a e a m st e B . . a 13. follow ng m mor ndu of our or d by G J ckson 4 r t th e hts St a e Co . at e . er 0 a . wi h e H ig or g ¢ p bbl p term of 3 d , average hat was the a t the ? sto ra ge . W moun of bill

SOLUTI ON

I N S TOR AGE DATE R E CE I P TS DE LI vE R I E S B ALAIs c E F OR 1 DA .

200 bbl .

100 bbl .

t to the st The st a e te s are e a e a e 1 . or g i m quiv l n or g of bbl for da .

550 te s 30 da. ea h . At 4 er term th da . e t rm of c ¢ p , s orage

550 X 415$22 . AVE RAGES SI MPLE AND WEIGHTED 197

14. The C e n e e a t Dobson Storage o . r ceived a d deliv r d on ccoun of

N . 10 W . T . h s s e fl : e ei e ov Jo n on undry barr ls of our as follows r c v d ,

. N v . 20 . Dec . 15800 bbl . Jan . 20 . e e e bbl , o , bbl , , , , bbl d liv r d

D . 2 . . 4 eb . 4 800 . D 2 . F ec . e 28 Jan . 500 , bbl , c , bbl , , bbl , , bbl ,

M r . 00 If th ha e e e er . r da. a 30 3 . e s e te m 30 , bbl c rg w r p bbl p r of t e hat w the a f he ? average s orag , w as mount o t bill CHAPTE R XI V

TH E PR O GRESSI O NS

hmet r — h i 173. Arit ic P ogression T is name s given to a

e ta se es mbe s ea h te m h h is me b c r in ri of nu r , c r of w ic for d y a a sta t a t t a e the ffe e e to he dding con n qu n i y , c ll d di r nc , t m e e te exam e 1 4 1 etc . The e e o pr c ding r ; for pl , , , 7, 0, r for , t

the ffe e e s bt a t an e m m he i find di r nc , u r c y t r fro t follow ng

I t is se at t mes e h a term . u ful i in ord r to find t e total of given series of numbers or to find any particular term Of that

m e if a car n a e se es . exa ri For pl , , going down a inclined pl n ,

f . . 4 f c . a e s s es e se 2 f . tr v l in ucc siv conds, t , 6 t , 10 ft , 1 t . , et , ho w far will it go in 30 se conds? The long method would be to set down all the 30 numbers and addthem ; but we shall h i n see t at t ca be done in a much quicker andeasier way.

a t ie and m — h 174 . Qu n it s Sy b ols T ere are certain well established symbols which are use d in the consideration of

n thmet e es m h h r a ari ic s ri of nu bers, w ic a e

a th e first te rm d th e common difi erence I th e last term 11 th e number of tenne s the sum of the terms

r m an r m — 175. G ene al For of A ith etic P rogression This

a m is a a d a 2d a 3d Th gener l for , , , erefore, coeffi cient Of din each term is 1 less than the number of the term.

200 BUSI NESS MATHEMATI CS

9 . Co a e a s e E xe ise 6 and mp r your n w r in Exercise 8 with that in rc , the a tte t r x se ath tha find g in if lo ry ickets a e sold a s by E erci 8, r er n by Ex ercise 6 . 10 n e hi and a . A ma inv sts s savings in the shares of a building lo n

as t h he 2 . s at e s t e l . he e t d oci ion, d po i ing st yr At t b ginning of yr h e is e te th 60 te est th a t s te the l st r . cr di d wi $ in r on e moun de po i d y , and 40 hi At th n pays in only $9 , making s total credit e begin ing

o f the 3d . h e is e te t 120 te est and a s 880 as yr cr di d wi h $ in r , p y in $ c h,

i hi the n . n h h etc . What s s credit at e dof 10 yr a dhow much cash as e paid in?

e me a e — Th 176. G o tric l Progr ssion is is a series Of num be s ea h te m h h is me b m t the r r , c r of w ic for d y ul iplying p e h a ceding term by a constant called t e r tio .

h 2 . i n h h h T e se es 1 3 9 7 etc s o e t e at i . ri , , , , , in w ic r io s 3 The same symbols except d are used as in the arithmetic h h h progression wit t e addition of r for t e ratio . z 3 4 The t e se e is a ar ar ar ar e e th yp ri s , , , , H nc e exponent of r in each term is 1 less than the number Of the term .

I llustrative Example 1 .

ar9

ar I 4

I af '“ 1

n the sum we a s I n deriving a equation for , know, l o

“ 3 4 ”' a or o r ar ar ar (1) No w mul tipl y (1) by-r “ 3 “ " ' " o r o r M ar ar ar (2)

" a ar a Subtracting (1) from (2) " — 1) ar a. — ar n a 8 r 1

l r we et rl ar" Multiplying by , g ” I I Substituting rl for o r in ( ) .

rI —a 8 = r —l THE PROGRESSIONS 201

Illustrat e Exa 2 th h iv mple . Find e 8t term andthe sum terms the e et ess 1 3 9 2 of g om ric progr ion , , , , 7

” I SOLUTI ON : 1 ar

rl a

r 1 3 X

2

xam th l h Illustrative E ple 3. Find e ot te rm andthe sum te rms the eo metical ess 4 —2 1 of g progr ion , , , ,

SOLUTI ON l ar " 1 — 4( 5i z )

4

}7 1

WRITTEN EXER CI SE S

'

1 . Findr the the 6th te the se es 2 , n find rm in ri , 6, 18

2. th e 9th te the se es 2 Find rm in ri , 4 . at a st r 3. A shi p was built co of H e owners at the end Of each year deducted 10 % from her value as estimated a t the beginning

h . a her est ate a at th n r ? of t e year Wh t is im d v lue e e dof 10 y .

Th at a t ases 4 . 4 . e popul ion of ci y incre in yr from to What is the rate of increase if

a the te States the ea 1 5. The popul tion of Uni d in y r 900 was

If thi s s d ease 50 e e 25r . at l the ula 000 . houl incr % v ry y , wh wou d pop tion be in the year 2000? at a is est ate a 4 6. If the annual depreci ion of building im d t % of its cost andthe building cost what is its estimated value at the end ? of 20 yr . 202 BUSINESS MATHEMATICS

7 . A machine costing depreciate s 7 of its cost each year . Fi its esti ate a e at the end f r nd m d v lu O 8 y .

8 . I n a n an e t a h ea a. a fa t re m ki g inv n ory t t e close of each y r, m nu c u r e e t d ducted 10% from the value of his ma chine ry at the previous inv n ory, e Th at was the b ca use of deterioration . e machine cost Wh a t he en th th r ? v lu e a t dof e 5y .

INT : a e at end 1st r 90 H V lu of y . . X U ‘ l N 2d U 902 x

Wh at was the common ratio?

9 . ts 100 a sa s hi h a s 3 nte A boy pu $ in ving bank, w c p y % compound i r

h endOf 6 . ? est u e a a . W at es i a nt to at t e r , compo nd d nnu lly h do t mou y 1 H I NT : Value endof 1st yr . X $ 00 10 Th l at a t n s 1 ea h ea . e popu ion of ci y i s a dincrease 0% c y r 1 th t e a 10 . for 0 yr . Find popul ion in yr 11 s has two a e ts ea Of his a e ts has two a e ts . A per on p r n , ch p r n p r n ,

and . a est h as a ten e e a. so on How m ny anc ors a person , going b ck g n r t s t his a a a th st e e at and ass m n ion , coun ing gr nd p rents s e fir g n r ion u i g that each ancestor is an ancestor in only one line of descent?

204 BUSINESS MATHE MATICS

EXER CI SES

64 the use the x by of table .

16 64 16 X" 64 2“‘ X 26 1 ° b 2 Same as su 2: (FTOIn the ta ble )

2 . Ca ul ate 8 128 16 b the use the ta e lc X X y :of bl .

3. Ca ate 16 84 256 the use o f the lcul , 3 by table .

SOLUT I ON : 2 1 4 256 28 256 2 i 4 2 8

1 Sa me as x 4 (From the tab le)

the ta 4 . Apply ble to

(a ) 16 (b) 512 64 (0)

2 the use ta e . 5. Square 3 by of bl

SOLUTI ON : 32 2 1 ° 2 Same as 3: (From the table) LOGARI THMS 205

6 . Apply the table to the following

(a) 323 (b) 64 2 (c) 324

7 . the s a e t 2 6 th u Find qu r roo of 5by e se of the table .

SOLUTI ON V256 24 Same as 16

x P L ATI N : e th t 2 t th E AN O Divid e roo ( ) in o e power (8) andit gives 4.

the tab e to the s a e t 8 . Apply l find qu r roo of e 9 . cub t 10. four h th 11 . fif 64 X 256 X 16 12 Solve by the tab1e 32 x 512

a 1 13. C lculate X 6 a 12 64 14 . C lculate 5X

x t — 179 . Logarithms Ar e E ponen s . The logarithm of a given number is the exponent Of the power to which a base h number must be raised to produce t is number . I n logarithms three different numbers are always involve d :

1 mbe . . A nu r

2 a thm . . I ts log ri

3 The base se . . u d

2 tr xam l e. I n 3 9 we can sa 2 is the o arith 9 Illus ative E p , y l g m of h as m a si e 34 81 we can sa 4 is the a t m to t e b e 3. Si il rly, nc , y log ri h of L Th e B N we can sa L is the a thm 8 1 to the base 3. erefor , , y log ri of

N to the bas e B .

em of arithms —Th s is a set mbe s 180 . A Syst Log i of nu r h with their logarithms all ta ke n to t e same base. Notice o

1 an s i s e a 1 . that the logarithm of in y ystem s 0, inc 206 BUSINESS MATHEMATICS

SYSTE M o r LO GAR I I' ‘ H MS W I TH BAS E 2

LOGAR I I‘ 'B M RE ASO N NUMB E R Lo c AR ITaM R E ASON

2

The student need not know e xa ctly how decimal loga

a h rithms like are found . Origin lly t ey were found by

x a ts a long process of e tr cting roo . Since logarithms are

x e ts the ma be te ete as h . h e pon n , y y in rpr d suc T us in the

a 2 88 3 we see that the 15850th e f equ tion 8 8 , , pow r O the 10 000th t 2 e a s 3 and these at s , roo of qu l , if oper ion were

e 2 the es t b a ctually perform d on , r ul would e 3.

e ms — T 181 . Notation andT r o avoid writing long expo n n s h n e at as i h e ts, uc a qu ion 3 s c anged into “ 3 and is ea a thm 3to the b log 2 r d log ri of ase 2 equals The subscri pt indicating the base is usually omitted when 10 is the base . The integral part of the logarithm is called its character

he manti istic andthe decimal part t ssa.

WRITTEN EXER CISES

following in the l anguage of logarithms

Lo 16 4 . Read a thm 16 to b 2 g 3 , log ri of ase is 4.

a. s. 3 t 3= s. 10 7. 100 z = x 4 . 4 16 — , , 5= 4 = 5. 2 32 9 . 10 .0001

208 BUSINESS MATHEMATICS

2 a e ma andb et e 1 1 1 a ma and d ci l ; w en 0 and 00, deci l ;

mb e b e n a t ee 1 a d 1 h ma . e e nu r w n 0 as 0 a deci l H nc , in

e e a the ha a te st h hm Of an mbe g n r l , c r c ri ic of t e logarit y nu r

ea ha 1 gr ter t n , in the Briggsian or common sy stem of loga rithms is 1 ess tha the mb e f a es at the e t the , l n nu r O pl c l f of

e m Th s h ha te s i 2 d ci al point . u t e c rac ri tic of s ; of i i s 3; of s 0 .

184 . No Change in Mantissa Wh en De cimal Point i s

M e d— I n he mm stem wh h the base i 1 ov t co on sy , in ic s 0, the mantissas do not change when the de cima l po int is

‘ m a h th s is t e k m e . TO e ta w we a e 10 ov d und rs nd y i ru , t and multiply or divide both members of this e qua 2 I t b 10 100 b 10 10 . R e a that he ion y , or y c lling w n 2 s ? “ 7 x is m ltiplie by a we obtain a or $ and he u d — , w n 8 8 3 x is divided by we obtain 33 or then by the same process of reasoning we have

127 or log 127 log 27 — . 127 . 1 l o38 10 log i , or 2 0127 —10 01 7 log . or

The mi nus Sign ove r the characteristic at the right he

h e s Th b e a h longs to the c aract ri tic only . us, y r g rding t e

h ma ss h chara cteristic only as c anging in signs, nti as sta y t e same no matter where the de cima l point in the number is

and ma t ssas ar e alwa s siti e . changed to , n i y po v

a e Ch ar a te sti s — mbe b 185. Neg tiv c ri c Any nu r etween

n 0 1 ha two he s e s be e the fi st 001 a d . , ving cip r (or z ro ) for r

r e the tha 0 h signi ficant first figu o r n ) figure, as 3for i ts

e its ar thm es bet ee 3 an 2 an characteristic , sinc log i li w n d d LOGARI THMS 209

1 the ma t ssa a e i mbe bet ee . 0 n i dd d s positive . Any nu r w n

n ha 2 i e a d . 1 s ts ha a i a thm s for c r cteristic , since ts log ri li b et ee 2 an 1 and the ma t s a a i s t e a s w n d , n i s dded s po i iv ; l o ,

n mb h h e its a e bet ee . 1 a nd1 t e e b e b e y nu r w n , r eing no cip r for

st s fi a e has 1 i h e its fir igni c nt figur , for ts c aracteristic , sinc logarithm lies between 1 and 0 and the mantissa a dde d is h h n n m s t e . e e e e a t e a a c a u po i iv H nc , in g n r l, c r cteristi of y ber less than 1 is o ne mor e than the number of ciphers be t ee the e ma t andthe e and w n d ci l poin first significant figur ,

is e at e . Th s the ha a h hm n g iv u , c r cteristic of t e logarit of

4 is i l 396 s . i 00 68 3 . 7 ; of ; of 000076 s 5.

1 — 86 . Explanation of a Logarithm Table I n the logarithm

h - table t e left hand column is a column of ordinary numbers . The first two figures of the given number whose mantissa i s h re h m I n the to are the s oug t a found in t is colu n . p row h figures from 0 to 9 . The t ird number is found there .

e e to Obta the ma t ssa 364 we ta e 36 the H nc , in n i of , k in first column andlook along the row beginning with 36 until

h h Th ma t ssa th s o h we come to t e column eaded 4 . e n i u T h h tained is 5611 . o find t e mantissa of we find t e ma t ssa 2 1 and h ma i he ame as that n i of 7 , t e ntissa of 7 s t s of

70 or 700 .

To i th e M a a nta nin M e 187 . F nd ntiss of a Numbe r Co i g or

— he ma t ssa than Thre e Figure s (Interpolation) . Find t n i for the first three figures andadda correction for the remaining

Th e i h a figures . is corr ction s computed on t e ssumption that the differences in logarithms are proportional to the h Th h differences in t e numbers to which they belong . oug

hi t is no t st t it is s f t s propor ion ric ly a ccurate, u ficiently

a e a t a s a ccur t for pr c ic l purpo es .

1 4 210 BUSINESS MATHEMATICS

Il lustrative Example : Find the mantissa for

ma tissa 1 2 1 158 n for 59 . 0 4 mantissa for

5 9 2 . 147 1 8 .1 87 .00 7 X

ffe e e 1 2 di r nc for .00 7 mantissa for

The difference between the mantissas of two successive mb o e s is . e e t nu r called the tabular diff e rence H nc , find from the table the mantissa for a number containing more than three figures : Obtain from the table the mantissa for the st th ee es an h he ext hi he fir r figur , d also t at for t n g r

mbe and s b a t M ffe e e b tween nu r, u tr c . ultiply th e di r nc e th e two mantissas by th e remaining figure s with a de cimal

r point at th eir left, and add th e r e sult to th e mantis sa fo th e fir st th r ee figur e s .

— 188 . To ndthe ar thm a N Fi Log i of Given umber . Deter h h mine the characteristic . Neglect t e decimal point (in t e . given number) and obtain from the table the mantissa for th e given figure .

Il tra th lus tive Example 1 . Fi nd e logarithm of

' SOL TI ON : The hara te st is 0 s e 1 e f U c c ri ic of , inc (the numb r o a places at the left of the decim l point) 1 0 .

5599 l o of mantissa of 363 . g

558 . 0012 . 362 . 7 X 57 . 0007

0012 l o difference for 1 . g of

he a t a Exam e 2. t of . 07 Illustr tive pl Find log ri hm 8546. _ 546 i I N : The ha a te st . 078 s si SOLUT O c r c ri ic of 2 , nce 1 (the number h h e a t 1 of ciphers at the rig t of t e d cim l poin ) 2 . 4 f 0 mantissa of 786 895 log o . 785 4 - 785 . 89 9 $3949 10

0 005 -46 005 X .0002 difference for 1 .0 — log of .078546 10

EXPLANATI ON : Instead of the we change the - 2 to

hi h it e als . 8 10, w c qu

212 BUSI NESS MATHE MATI CS

— 190 . to n S e the ha a te How Fi d Antilogarithms . inc c r c r istic depends only on the position of the decimal point and not the es m the e mbe the cha a te on figur for ing giv n nu r , r c r isti c is negle cted at the outset of the process Of finding the antilogarithm .

If h ma n Ta e 1 . t e given ntissa ca b e found in the table : k from the table the figures corresponding to the mantissa of the given logarithm ; use the characteristic of the given logarithm to fix the de cimal point in the numb er obtaine d from the table .

in th i I llustr ative Exampl e . F d e number whose logarithm s

442 r 2 . SOL UTI ON : The figures corresponding to the mantissa . 5a e 77 S e the a a t t is 1 the e are two es at th t h inc ch r c eris ic , r figur e lef of t e deci mal point . The e if refor ,

If the e ma t ssa es no t th a 2 . giv n n i do occur in e t ble Obtain from the table the next lower mantissa with the cor

h es the a t a hm S he responding t ree figur of n ilog rit . ubtra ct t tabular mantissa from the given mantissa ; divide the latter difference by the difference between the next lower and the next higher mantissa in the table ; anne x this quotient to the three figures of the antilogarithm already obtained from

se the ha a te st to a e the the table . U c r c ri ic pl c decimal point in the result .

Fi the Illustrative E xam pl e 1 . nd number (or antilogarithm) whose logarithm is is no t the ta e the ext S L TI ON . 4237 e i O s . 42 2 U in bl ; n low r 3 . The

i 0005. I f a ffe e e f difference between the m s . di r nc O 17 in the last two figures of the mantissa mak es a difference of 1 in the third figure of the e a t a ffe e e 5 the ast e Of the ma ' numb r ( n ilog) , di r nc of in l figur ntissa will 5 e a f e e e 1 . 294 t es e t to the thi mak di f r nc of 1 7 of , or , wi h r p c rd figure of the number . LOGARITHMS 213

Hence if

I st at Exam e 2 . I f a: 1 e 0 as. llu r iv pl log , find

SOL TI ON : The a t 660 h the e e es ess a t ssa is . 2 U n r l m n i , of w ich numb r

is 1845.

The tabul ar difference 55 x .00184

191 . To indth e N mb e b e P F u r y Us of roportionate Parts . If the logarithm is giv en the number may be found by

a Th h i use of proportionate p rts . e met od s outlined in the following

h Il lustrative E xample . Find t e number of which the logarithm is

SOL UTI ON : Fi the a t ssa st e w th s and ut nd m n i ju b lo i , p down the es st t ee es the e 737 corr ponding fir hr figur of numb r, or Find the difference betweenthe mantissa just below the given mantissa

and st a e the e a t ssa t s ase is 6 . ju bov giv n m n i , which in hi c Find the dif erence between the given mantissa and the next lower

a t ssa i t s ase is 3. m n i , wh ch in hi c the t ate a ts ea e 6 i Find propor ion p r column h d d , follow down t until e to the e 3 the e ea est to it a e t you com figur (or figur n r in v lu ) , hen follow a ss to the e t- a ea e ext a d t t s cro l f h nd column h d d r igi , which in hi case

is 5.

n x th s e to the st th ee es 737 t s ase a A ne i figur fir r figur ( in hi c ) , m king

th a a te st to t th s Off the e Use e ch r c ri ic poin i , giving numb r whose logarithm is as

— Tab e a thms The is a 192. l of Log ri following table of ,

mb e s s h as ma be a logarithms of nu r , uc y pplied to the

The tab e s problems in thi s book . l of proportionate parts i at the right . 214 BUSINESS MATHEMATICS

r arts P o p . P

30 10 3032 3054 30753096 3118 3139 3160 318 1 320 1 3222 3243 3263 3284 3304 3324 3345336533853404 3424 3444 3464 3483 3502 3522 354 1 3560 3579 3598 Differenc e 36 17 3636 36553674 3692 37 1 1 3729 3747 3766 3784 3802 3820 3838 3856 3874 3892 3909 3927 39453962

3979 3997 40 14 4031 4048 40654082 4099 41 16 4 133 4 150 4 166 4 183 4200 42 16 4232 4249 4265428 1 4298 4314 4330 4346 4362 4378 4393 4409 44254440 4456 4472 4487 4502 4518 4533 4548 4564 4579 4 594 4609 4624 4639 4654 4669 4683 4 698 4713 4728 4742 4757

4771 4786 48 00 48 14 4829 4843 48 57 4871 4886 4900 49 14 4928 4942 49554 969 4983 499 7 50 11 5024 5038 5051 50655079 5092 51055119 5132 51455159 5172 51855198 52 11 5224 5237 5250 5263 5276 5289 5302 53155328 5340 5353 5366 5378 5391 5403 54 16 5428

5441 5453 54655478 5490 5502 5514 5527 5539 5551 5563 55755587 5599 56 11 5623 56355647 5658 5670 5682 5694 57055717 5729 5740 5752 5763 577 55786 5798 5809 5821 5832 5843 585558 66 5877 5888 5900 59 1 1 5922 5933 5944 59555966 5977 5988 5999 6010

6021 6031 6042 6053 6064 60756 0856096 6 107 6 117 6 128 6 138 6 149 6 160 6 1 70 6180 6 19 1 6201 6212 6222 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 63356 34563556365637563856395640564 156425 64356444 6454 6464 6474 6484 6 493 6503 6513 6522

6532 6542 6551 656 1 6571 658 0 6590 6599 6609 6618 6628 6637 6646 6656 666566756684 6693 6702 67 12 672 1 6730 6739 6749 6758 6767 6776 67856794 6803 68 12 6821 6830 6839 6848 6857 6866 68756884 6893 69 02 69 11 6920 6928 6937 6946 69556964 6972 6981

6990 6998 7007 70 16 7024 7033 7 042 7050 7059 7067 7076 7084 7093 7 10 1 7 110 7 1 18 7 126 71357 143 7 152 7 160 7 168 7 177 7 1857 193 7202 7 210 7218 7 226 7235 7243 7251 7259 7267 7275728 4 7292 7300 7308 7316 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396

7404 7 412 7419 7427 74357443 7451 7459 7466 7474 7482 7490 7497 7 5057513 7520 7528 7536 7543 7551 7559 7566 7574 758 2 7589 7597 7604 76 12 76 19 7627 7634 7642 7649 7657 7664 7672 7679 7686 7694 770 1 7709 77 16 7723 7731 7738 77457752 7760 7767 7774

7782 7789 7796 7803 7810 7818 78 257832 7839 7846 7853 7860 7868 78757882 7889 7896 7903 79 10 7917 79 24 7931 7938 79457952 7959 7966 7973 79 80 7987 7993 800 0 8007 8 0 14 8021 8 028 8 035804 1 8 048 8055 8062 8 069 80758082 8089 8 096 8 102 8 109 8 1 16 8 122

216 BUSINESS MATHE MATICS

Pk o p . l %u1 s

0414 04 18 0422 0426 0430 0434 0438 0441 04450449 Difi et ence 0453 0457 0461 04650469 0473 0477 4 4 4 0488 1 0 8 1 0 8 0492 0496 0500 0504 0508 0512 05150519 0523 0527 0531 05350538 0542 0546 0550 0554 0558 056 1 0565 0569 0573 0577 0 580 0584 0588 0592 0596 0599 0603

0607 0611 06 1506 18 0622 0626 0 630 0 633 0637 0 641 06450648 0652 0656 0660 0663 06 67 067 1 0674 06 78 0682 068 6 0689 0693 0697 0 700 0 704 0708 07 1 1 0 71 5 0719 0 72 2 0726 0730 0734 0737 074 1 07450748 0752 07550759 0763 0766 0770 0774 0777 078 1 07850788

0792 07950799 0803 0806 08 10 08 13 08 17 0821 08 24 0828 08 31 08 350839 0842 08 46 0849 08 53 0856 08 60 0864 0867 08 71 0874 08 78 08 8 1 088 508 88 0892 08 9 6 0899 0903 0906 09 10 09 13 09 17 0920 0924 0927 093] 0934 0938 094 1 09450948 0952 09550959 0962 0966

0969 0973 0976 0980 0983 098 6 0990 09 93 0997 1000 1004 1 007 1 0 11 1014 1017 102 1 1 024 1028 1031 1035 1038 1 04 1 1 0451048 1052 1 0551059 1062 1065106 9 1072 1 0751079 1082 1086 1 08 9 1093 1096 1099 1 1 03 1106 1 1 09 1 1 13 1 116 1 1 19 1 123 1126 1129 1133 1 136

1139 1 143 1 146 1 149 1153 1 156 1 159 1 163 1 166 1169 1 173 1 176 1 179 1 183 1 186 1 18 9 1 193 1 196 1 199 120 2 1206 1 209 12 12 12 16 1219 1222 12251 229 1232 1 235 1239 1 242 12 451248 1252 12551258 1261 12651268 127 1 1274 1 278 128 1 128 4 1287 1290 1294 1297 1 300 Ex tra Difi er digit ence 1303 1307 1310 1313 1316 1319 1 323 1326 1329 1332 13351339 1342 13451348 1351 13551358 136 1 1364 1367 1370 1374 1377 1380 1383 1386 1389 139 2 1 396 1399 1402 14051408 14 11 14 14 14 18 142 1 1424 1427 1430 1433 1436 1440 1443 14 46 1449 1452 14551 458

' 146 1 1464 1467 1471 1474 1477 148 0 1483 1486 1489 1492 14951498 150 1 1504 1508 151 1 1514 1517 152 0 1523 1526 1529 1532 15351 538 154 1 1544 154 7 1550 1553 1556 1559 1562 15651569 1572 15751578 1 58 1 1584 1587 1590 1593 1596 1599 1 602 16051608 16 1 1

1614 16 17 1620 1623 1626 1629 1 632 16351638 1 64 1 1644 1647 1649 1652 16551658 166 1 1664 1667 16 70 1673 1676 1679 1682 168 5168 8 169 1 1694 1697 1 700 1703 1706 1708 171 1 17 14 17 17 1 720 1723 1726 17 29 1732 17351738 1741 1744 1746 1749 1752 17551758

1761 1764 1767 1770 1772 17751778 1781 1784 1787 1790 1793 179 6 1798 180 1 1804 1807 18 10 1813 18 16 18 18 18 21 1824 1827 1830 1833 1836 1838 1841 1844 18 47 1850 1853 18551858 186 1 1864 1867 1870 1872 18751878 188 1 1884 1886 1889 18 92 18951898 190 1 LOGA RI THMS 217

t P re p . P ar s

1903 1906 1909 1912 19 1519 17 1920 1923 1926 1928 1931 1934 1937 1940 1942 19451948 1951 1953 1956 1 959 1962 19651967 1970 1973 1976 1978 1981 1984 1987 1989 1 992 19951998 2000 2003 2006 2009 2011 014 2017 2019 2022 20252028 2030 2033 2036 2038 2041 2044 2047 2049 2052 20552057 2060 2063 2066 2068 207 1 2074 2076 2079 2082 2084 208 7 2090 2092 20952098 2 10 1 2103 2 106 2 109 2 1 1 1 2 1 14 2117 21 19 2 122 2 1252127 2130 2133 2 1352 138 2 140 2143 2146 2 148 2151 2154 2156 2159 2 162 2 164 2167 2170 2172 2 1752 177 2180 2183 2 1852188 219 1 2193 2196 2198 2201 2204 2206 2209 22 12 2214 22 17 2219 2222 2225 2227 2230 2232 22352238 2240 224 3 22452248 2251 2253 2256 2258 226 1 2263 2266 2269 22 7 1 2274 2276 2279 228 1 2284 2287 228 9 22 92 2294 2297 2299 2302 2304 2307 2310 2312 23152317 2320 2322 23252327 2330 2333 23352338 2340 2343 23452348 2350 2353 23552358 2360 2363 23652368 2370 2373 23752378 2380 238 3 23852388 2390 2393 23952398 2400 2403 24052408 2410 2413 24 1524 18 2420 2423 24252428 2430 2433 24352438 2440 2443 24452448 2450 2453 24552458 2460 2463 24652467 2470 2472 24752477 2480 2482 24852487 2490 249 2 2494 2497 2499 2502 2504 2507 2509 2512 2514 2516 2519 2521 2524 2526 2529 2531 2533 2536 2538 254 1 2543 25452548 2550

2553 25552558 2560 2562 2 5652567 2570 2572 2574 2577 2579 2 58 2 2584 2586 2589 259 1 2594 2596 2598 260 1 2603 26052608 26 10 26 13 26 1526 17 2620 2622 26252627 2629 2632 2634 2636 2639 2641 2643 2646 2648 2651 2653 26552658 2660 2662 26652667 2669

267 2 2674 2676 2679 268 1 268 3 2686 2688 2690 2693 269 5269 7 2700 2702 2704 2707 2709 27 1 1 27 14 27 16 27 18 2721 2723 27252728 2730 2732 27352737 2739 2742 2744 2746 2749 2751 2753 27552758 2760 2762 27652767 2769 2772 2774 2776 2778 2781 2783 2785

2788 2790 2792 2794 2797 2799 280 1 28 04 2806 2808 28 10 28 13 28 1528 17 28 19 2822 28 24 2826 2828 2831 2833 28352838 2840 2842 2844 2847 28 49 2851 2853 2856 2858 2860 2862 28652867 2869 2871 2874 2876 2878 2880 2883 28852887 28 89 28 91 2894 2896 2898

2900 2903 29052907 2909 29 11 29 14 29 16 29 18 2920 2923 29252927 2929 2931 29 34 2936 2938 2940 2942 29452947 2949 2951 2953 2956 2958 2960 2962 2964 2967 2969 2971 2973 29752978 29 80 2982 2984 2986 298 9 2991 2993 29952997 2999 3002 3004 3006 3008 3010 3012 30153017 3019 302 1 3023 30253028 3030 218 BUSINESS MATHEMATI CS

WRI TTEN EXERCI SES

Find the numbers (or antilogarithm ) whi ch correspo nd lowing logarithms

10

10 . 10 10 5° 10

11 .

12. 10

13. 10

I t Co mputati on by the Use Logar ithms .

n al ebra ha show in g t t,

“ andthat,

n h I t is show t at,

andthat

(ax) a px

It can a lso be shown that

220 BUSINESS MATH EMATICS

am e . 2 Illustrative Ex pl 1 x . 03 86

SOLUTI ON

Il lustrative Example 2 .

SOLUTI ON 1 . 6695

11 . 669510 1 log . 0998 0 log Quotient Quotient

6 am e . n the 6th e 7 2 2 Il lustrative Ex pl 3 Fi d pow r of 9 9, or 79 9

2 SOLUTI ON log . 79 9

/ the e of Ill ustrative Example 4. Find cub root or 9

SOLUTI ON log é log a:

WRITTE N EXER CISE S

763b 298 b the use a hms and h 1 . Multiply y y of log rit , c eck the result

actual multiplication .

2. Multiply by and he a a 3. Divide by c ck by ctu l division . R a e to h e a e 4 . is fourt pow r or find v lu of 5

a l/ 2 . the 5th e 9 34. 5. Calculate root of or find v lu of LOGARI THMS 221

Find value of the following

6 . 026 7 X . 8 X 862 X 7 . 336 1984 X

8 . 2 57 . 0 3 X 8 4 he r h we . 073 raised to t fou t po r 9. 6374 62 . 0 14 X X 10 . X 19 005862 X 8271 6 15. . 07

16 . E x a the f h 0329 . tr ct fi t root of .

17 . x a th e h f . 2 E tr ct fourt root O 007 .

Find the value of the following

18 . X X . 0079

19 . X HI NT : Work the same as if the sign were

and ut p in correct sign at end.

_ 28 . 79 47 2 523 X 249 X 0 X . 98

29 . 767 >< 396

30 . . 643 x 7095 21 (1 032) I 5

67 9 x . 462 22: x

31 .

23. g m 32.

. . 2 642 33 03 5x . 5x x [y 0765 34 . The h 0089 sevent root of . 8 26 _ WW 7 35' VI 5 27 529 36. \l67 X 518 x . 058

l

the m e e e a e h e ame e is he Find circu f r nc of circl w os di t r inc s,

1rd. 7C from the e quation C o ( h e e m - x i h ea an e e a a s . a n 38 . Find t e ar of llips w os s i is in d

' m the e a A n: a b. h ax b is . s orter is in fro qu tion , el l i ps e h a e a a a e h e de are a b 39 . Find t e r of tri ngl w os si s h a . e e e t e e : andc in r sp ctiv ly, using qu tion a b 0 — - h e AA x/ s (s b) (s c) w er s 2

ame e a he a a h h is to a e e h 40 . The di t r of sp ric l b lloon w ic lift giv n w ig t is calculated by the e quation :

where 222 BUSINESS MATHEMATI CS

D diameter of balloon in fee t A in weight pounds of a cu . ft . of air ( l u l ‘ u l ‘ M G gas in the balloon W weight to b e raised (including the weight of the balloon) i 2 D if A .0807 G 005 . F nd . ; . 6 ; W lb he 0 2 G 41 . W u in t e ua x 4 A . 08 7 Find , s g q tion of E ercise 0, if ; 0056 ; D

42 . a m a u n b ho t a e es the re u e e h In w r ing b ildi g y w t r pip , q ir d l ngt m pipes 4 in . in di a eter is determined by the equation :

L (P — t) (T — t) 004 ’ h X . 5C w ere P T

L length of pipes in feet P temperature (Fahrenheit) of the pipes T required in the building t of external air ’ m . ac e 0 nu ber Of cu . ft of sp e to b warmed ° Find hen P 1zo ; t T C x 100

" 24 a: 43. If 8 3 , find SOLUTI ON

x 5 as. 44 . I f 3 2 , find “ I x . 46. f 64 4, find

4 as. 6. If find x = 47. If 2 64, find at .

224 BUSINESS MATHE MATICS

ra h h Illust tive Example . Find the cb st or present worth w ic will am 923at 4 m ount to $ % co pound interest in 12 yr .

SOL UTI ON

12 l o g

To fin h n h n n 3. d t e a mou t w en i terest is compou ded q

m e a ea use the ti s y r, formula :

A = r fl +

WR I TTEN EXE R CI SE S

F the am 933 at m e 1 . ind ount of $ 5% co pound d annually for

7 yr . h h h 5r . a t 2 . Find t e principal w ic amounts to in 1 y 5%

co mpounded annually.

e e h f 9 1 e a 1 r . 3. Find the cost or pr s nt wort O $ 8 to b p id in 0 y allowing

5% interest compounded annually .

4 . i he am 700 h h ran 12 r . e e e F nd t ount of $ w ic for y , int r st b ing co m pounded semiannually at

A = p 0 + §

h a m 1 at 6 m e e 2 5. t e 0 r . c Find ount of $ % co pound int r st for y , o m

A the m e e . pounded annually . lso find co pound int r st

the am o i 12 r . at 6 m 6 . Find ount for y % co pounded annu

ally . 2 m the a m of 5 500 r . e a 7 . Find ount $ for y co pound d nnually at

n the am 300 50 r . at 6 m 8 . Fi d ount of $ for y % co pounded a nnu

the am 300 50 r . at 6 m 9 . Find ount of $ for y % co pounded semi

annually .

n the am u 300 50 r . at 6 m e 10 . Fi d o nt of $ for y % co pound d quarterly. ho w ma ea $1 e e at 3 e 11 . In ny y rs will doubl its lf % int rest com poun ded annually? COMME RCIAL APPLI CATI ONS OF LOGARITHM S 225

SOLUTI ON

2 h m 1 . In ow any years will $1 double itse lf at 5% compounded annually? At At h m 13. In o w any years will amount to at 5% ih tarast compounded annually? ho w ma e 12 e t e 14 . In ny y ars will $ double its lf a inter st pounded semiannually?

SOL UTI ON

n yr .

ho ma ea l 100 e e at 6 e e 15. In w ny y rs wi l $ doubl its lf % int r st pounded semI annually? ma a l 100 e el at 6 e e co m 16 . In ho w ny ye rs wi l $ doubl its f % int r st pounded quarterly?

m e am 400 10 r . a e at 17. What sum of on y will ount to $ in y if pl c d interest at if compounded annually?

— in n und Ca u at ns See a T e 195. S ki g F lc l io ( lso h following explains the a pplication of logarithms to sinking fund calculations . If the sum set apart at the endof each year to be put at

n e S an compound interest is represe t d by , d

I S 226 BUSINE SS MATHEMATICS

P principal r e e int r st on $1 fo r 1 yr . R 1 am n ( r) ou t of $ 1 for 1 yr . A am P 11 ount of for yr . then the sum at the end of ‘ the

‘ ‘ I hat is S R + s x a + ff (1) (2) R x (1) (3) (2) (I ) (4) Factor l eft member —and right member (5) 2 b y R 1 (6) R 1 2

(Note that (1) above is a geo metric progression.)

Illu ra be set a a a st tive Example 1 . If p rt nnually, andput ha be the am 6 m 1 r . % co pound interest for 0 y , w t will ount?

SOLUTI ON S H?" 1) r 1) 06 . — (with a 6 place table )

xam A Illustrative E ple 2 . county owes What sum must be set a ar a ua l as a s a ce the e p t nn l y, inking fund, to c n l d bt in 10 yr., e m h i a os ea h e provid d oney is wort F nd tot l c t c y ar .

A! SOLUTI ON an 1

I O (1 .06) 1

1

228 BUSINE SS MATHE MATI CS

WRI TTEN EXER CI SES

1 . An a a e the am un nnu l p nsion of $600 was unpaid for 5yr . Find o t due if inte rest is computed a t

2. A widow receives a pe nsion of $400 from th e United State s govern me a a - r Wha h she e e e a - a if nt ndb ck pay for 8 y . t s ould r c iv in b ck p y inte re st is at 3% per year?

— an Annu . find 198 . Finding th e Pr e sent Value of ity To th n a u an annu hen the me co n e prese t v l e of ity, w ti it is to

n nd the a e er en are en w e use the n ti ne a r t p c t giv , followi g formula :

present value am f P 71 ea the am t the a ui ount O for y rs . or oun of nn ty for n years

amount of P for n years ” P (1 r) P R ”

S (R " 4 1 amount of the unpaid annuity for n R 1

1 ) ” ” H ence P R Si nce A P R

is e e a the ac a If the annuity p rp tu l, fr tion pproaches 1 as its R ” limit .

(whe n ann uity is perp etual)

the e e a Illustrative Example 1 . Find pr s nt v lue of an annual pension

5r . a 4 e e . of for y , t % int r st COMMERCIAL APPLICATI ONS OF LOGARITHMS 229

SOLUTI ON S R " — l x R " R 1 1) X 1 1 x 04

0486644

ll ra i e xam e 2 i I ust t v E pl . F nd the present value a perpetual scholar h ha a 30 a 0 a t . s ip t t p ys 8 nnu lly, a 6 % intere st

SOLUTI ON S

r 300

. 06

WRI TTEN EXE R CI SE S

1 . Find the present value of a n annua l pension of to co ntinue

r at 4 n e . 12 y . , % i t rest 0

2 . A man re e b a a r a a ea e is tir d y r il o d on y rly p nsion of $900 . H e r an ha the a e h if lives 9 y . d6 mo . W t is v lu of suc pension money is worth n the rese a e a e e a h a h 450 r 3. Fi d p nt v lu of p rp tu l sc ol rs ip of $ pe year at

n the e e a e a e t hase a asis 5 4. fi d pr s nt v lu of prop r y purc d on b of $ 00 e i s h paid annually for 15yr . if mon y wort

e sent Val ue of Annui e s — h 199 . Finding Pr ti T e present

a nn hi h h i value of a perpetu l a uity w c s all beg n in a . given en th e me n n number of years, wh ti it is to co ti ue and the

r en are en ma be un the l rate pe c t giv , y fo d by fol owing formula 230 BUSINESS MATHEMATICS

S 0 0 _ (Where p number of years before annui ty begi ns) Rp (R 1)

lus ra i e xam e nui Il t t v E pl 1 . Find the prese nt value of a perpetual an ty f o e n 3 r . at 4 to b gi in y , % inte rest .

SOLUTION S R ? (R 1)

3 (1 . 04) x . 04

' Ill ra l 2 Iun h ust tive Examp e . d t e present value of a term annuity of

to n in 6 r . andto at begi y , co ntinue 12 yr .

SOLUTI ON

S R0 1 9 X R 4 R 1 (where q number of years that 1 annuity is to continue) X 06

WRITTEN EXER CISE S

1 n the ese a e a e e a a nu 500 e . Fi d pr nt v lu of p rp tu l n ity of $ , to b gin in

a 4 e res . 8 yr . t % int t 2 n the rese a e an a n in 1 . Fi d p nt v lu of nnuity of to begi 0 yr . t andcontinue for 15yr. a

55r. a e . H e be e e at 70 an 8 . A man is y of g is to r tir d on annual

S se ha he es he 85. th pension of $900 . uppo t t liv until is Find e present val ue of such a pension at 4% intere st .

e ui — find 200. Finding th Ann ty To this when the present

a ue the e andthe a e er en are en the v l , tim , r t p c t giv , follow ing formula may be appli ed

232 BUSINE SS MATHE MATI CS

To findthe nu h u b a mber of years the premium s o ld e p id , in e a the n h n no the ord r th t compa y s all sustai loss , follow ing formula may be used :

Ar log (1 S o r n lo g R

I n the ca lcula tion of life insurance it is ne cessary to em ploy t ables which shall show for any age the probable dura tion of life .

[The following table gives the number of survivors at the

f e en a e o ut e n a e at the a e Of 1 di f r t g s of p rso s liv g 0 .

SUR VI VOR S

l a e xam le . ak the u e the a e a e I lustr tiv E p T ing fig r s of bov t bl , calculate what the chance is that a person 15yr . of age wil l live t o the age of 35?

SOLUTI ON COMME RCIAL AP PLI CATIONS OF LOGARITHMS 233

ORAL AND WRI TTEN EXER CI SES

1 . Wha th h ? e a e ha a e be 80 r . t is c nc t t person 40 yr . old will liv to y old ha a e 70 r . e b e r ? T t p rson y old will liv to 90 y . old 2 . a e 20 r . a e ha are the ha e ha he If p rson now is y of g , w t c nc s t t will live to be 45? To be 50? To be 65? To be 80?

3. What annual premium should be charged for a policy worth at the n 2 r e h e dof 0 y . if mon y is wort

4 . the a ua em m 50 the am the If nn l pr iu is $ , ount of policy is andmoney is worth for how many years must the premium be paid that no loss shall be sustained by the company? e m h b h h 5. Wh at annual pr miu s ould e c arged for a policy wort

at the en 1 r . e h dof 0 y , if mon y is wort

M I S CELLANE OUS WRI TTEN EXE R CI SE S

Use logarithms in solving the following

ha am 20 r . at 5 m e 1 . To w t will ount in y % co pound d annually?

ha e am 10 r . e a t 6 m e : 2 . To w t do s ount in y if l ft % co pound d ? (a) Annually? (b) Semiannually? (c) Quarterly t m e a l sum m e e a a 30 r . 3. A of on y l ft co pound d nnu ly for y ? amounts to What is the sum At ha er e e e m be e e am 4 . w t p c nt int r st ust l ft in ord r to ount to m e a a ? in 32 yr . co pound d nnu lly h 2 A ha er e m be e a 4 r . am 5. t w t p c nt ust l ft so t t in y it will ount to compounde d annually? h w ma ea a sum e e e at 6 e 6 . In o ny y rs will doubl its lf if l ft % int rest compounded annually?

1 r . at 4 m e h m 8 a . 7 . Find t e a ount of in y % co pound d nnually

Find also the compound interest . m h be a an a n a e a a 8 . What su s ould p id for n u l p nsion of p y ble e h 3 er a m m annually for 20 yr . money b ing wort % p nnu co pound in terest if

Present value

sum am at m e e at 9 . What will ount to if put co pound int r st ? 4% for 15yr . m a a 1 a e at i e e e 3 r . 10 . If is pl c d nt r st s i nnu lly for y , to how much will it amount in that time ? 234 BUSINE SS MATHE MATI CS

11 . A ha person borrows $600 . H o w much must he pay annually t t the h ma be a 4 co m e e 35 . a i e e at w ol d bt y p id in yr , llowing nt r st % pounde d annually?

12 . th u 1 m m u e e am 00 in 25r . at 5 er a Find o nt of $ y % p nnu , co po nd d annually.

13. Wha the e e h th end 100 r . t is pr s nt wort of payable at e of y , interest being at the rate of 5per annum andcompounde d ann ually? the e e a 14 . Find pr s nt v lue of an annui ty of $100 to be paid for 30

r . e i e e at 4 m . y , r ckon ng int r st % co pounde d annually 15 th m e a n 1 1 r . co m . Find ou t of $ in 00 y at 5% compound intere st

pounde d a nnually .

F the n 50 1 . 4 16 . ind amou t of $ 0 in 0 yr at % compounded semi

annually. 17 Wha the e e a e h h is h n . t is pr s nt v lu of w ic to be paid at t e e d i e t m 5r . e e a 3 e a ? of 1 y , r ckoning nt r st % co pound d nnually 18 a the e e a e o an a 5 ha e t the . Wh t is pr s nt v lu f nnuity of $ 00 t t ceas s a i e e e at end of 25yr . nt r st reckon d 19 the ul a a a e n ea e m . If pop tion of st t i cr s s in 10 yr . fro to find the average yearly rate of increase if

A e a e a e 1 and v r g r t R , population at e nd population at beginni ng

20 If the o u a a a e andth e . p p l tion of st t now is e y arly rate l 1 of increase is findthe popu ation after 0 yr . H ence if

Population at end P 1, (yearly rate P 5 popul ation at beginning

21 A man a sum m e at e e a al n . borrows of on y int r st nnu ly, a d

a e t 5 a e . If hi s a a a 441 th lends the s m a % qu rt rly nnu l g in is $ , find e sum

borrowed . 2 he a a em m is 150 the am the 2 . If t nnu l pr iu $ , ount of policy is andmoney is worth for ho w many years must the premi um be paid that no loss shall be sustained by the company? a he a e th a 23. If city wis s to t k up wor of bonds t the e nd h s set a e ea h ea if th 4 r . ho w m m e a e of y , uc u t it sid c y r, r t of interest is 5% and

C number of dolla rs in debt 1: S (1 r) 1] number of yea where . rs S sum se r t asi de a nnually r rate of interest

236 BUSINE SS MATHE MATICS

— n h hi n e men a 202. Bonds To fi d w at interest on s i v st t

u a e e e the n u a ma be u e : p rch s r will r c ive, followi g form l s y s d

P price of a bond that has 11 years to run r per cent it bears S face of bond (usuall y $100 or q current rate of interest Le t a: rate of interest on the investment

en P 1 x ” al ue of urc ase m o ne a t the end of 11 ea rs Th , ( ) v p h y y ” " 3 S r (1 q ) Sr (1 q) S r S amo unt o f m oney receive d on bo nd if interes t o n b ond is p ut imm e di atel y a t co m poun d i ntere st a t q%

But S r l n S r S 5W . ( + q) + 0) + q S q - l l 1 l + x = ( P ) n q

h Illustrative Example 1 . Wha t is t e rate rate of interest on a 4%

at 114 ha has 26 r . to run m e h bond , t t y , if on y is wort 4 4 1 l x 26

1 14 x . 035 )

1 x

zc . 033 Purchaser receives

t a e xam e 2 . At ha e m 7 be Illus r tiv E pl w t pric ust bonds bought,

n in 12 r . h the e e a a e em a a ru n g y , wit int r st p y bl s i nnu lly, in order that the purchase r may receive on his investment 5% interest semiannually?

— S r S = 100 2 q .0 5(inte res t semiannual S S r 1 " q ( q) ly) ” q (1 x) r . 035 n 24 251 0252 4 x -025 .0 ( 0 COMME RCIAL APPLICATIONS OF LOGARITHMS 237

WRITTEN EXER CI SES

2 2 o r due 1 r . n e em 1 . If $1 6 is paid f bonds in y a d yi lding s i a ha er ce ea e the e m e m e annu lly, w t p nt is r liz d on inv st ent, provid d on y is worth 2 % semiann ually? ' 2 e m e h 2 em a ua ha l 2 r to . Wh n on y is wort % s i nn lly, if bonds ving y . run and bearing semiannual coupons of each are bought at wha t per ce nt is realized on the inve stment?

ma due r . n 3. What y be paid for bonds in 10 y , a dbearing semi annual 4 a h e ea e m coupons of % e c , in ord r to r liz 3% semiannually, if oney is ema a he worth 3% s i nnu lly, w n

P —r S

4 L e o ma 30 r . are o t . If ib rty b nds turing in y b ught a and money is worth what is the yield? hi dma r 5 he U e S a es 10 . . W n nit d t t T r turing in y are bought at and money is worth wha t is the yield?

ma be a U e S a e due 25r . andh 6 . What y p id for nit d t t s bonds in y , ear 4 ea h e l ing semiannual coupons of % c , in ord r to rea ize 45semiannually if money is worth 3% semi annually? ' N Y C 4 due in 4 a the n ew s 5r . 7 . Wh t is yield o ork ity k y at 102} if money is worth CHAPTER XVI I

TH E SLIDE RULE

— f h n and Use . So ar as as ee e e ned 203. H i story b d t rmi , the slide rule as an instrument having o ne piece arranged to

n he was n en e a n a o ri slide along a ot r, i v t d , ccordi g to C j , by

Th esent William Oughtred between 1620 and 1630 . e pr arrangement of scales (see Form 19) was devised by Lieuten

h f he en h m u 1 ant M ann eim O t Fr c ar y abo t 850 .

“ It was originated undoubtedly because of the fa ct that it

u be ne in i n ha in is a time andlabor saver . It sho ld bor m d t t nearly all pra ctical calculations only an approximately

n e ne e a and he i l he O e a correct a sw r is c ss ry, t sk l of t p r tor is often best shown by his ability to approximate to the right

the u as a u a the a a degree of a ccura cy . If res lt is cc rate s d t

n a u a e as o r n e re employed to obtai it , or as cc r t u a sw r is

h n w ha m he an n m me quired to be, t e e ve a cco plis d eco o y of ti

and labor .

n re e a e ha n a a ne en It is e ti ly possibl , ft r vi g tt i d profici cy

n n he e u e a n e u h h ha not in ha dli g t slid r l , to obt i r s lts w ic s ll

h 1 e e a have more t an i of % of error . This is p rf ctly s tis

factory for many of the problems of the business world .

h u e in the e e e he u e T e slide r le is us d offic , ith r to c ck fig r s,

n a u a n . u e men u a n a or for origi al c lc l tio It comp t s s r tio , p y

l n e e e en a e a e un and ro l, i t r st , p rc t g r t s, disco t , profit loss ,

n ex han e e h a n m un n e e and foreig c g , fr ig t , pror ti g, co po d i t r st , has many other applications of the kind in the business

field .

240 BUSI NE SS MATHEMATI CS

e nn n at 1 a a n and a n 1 1 2 b gi i g g i c lli g it 0, read it 0, 0

100 hen e nn n at 1 a a n ea 1 2 ; t b gi i g g i r d it 00, 00 Thi s is allowable because the mantissa for 10 is the same as 1 a 00 etc . b n e ha e e a de th t for , It will e ot d t t th r is

n h h h he e crease i t e lengt s of t e spa ces from left to right . T s decreases in lengths correspond exa ctly to the differences

n h hm 1 1 e can a u in betwee t e logarit s from to 0 . W lso p t marks to show the mantissa s for the logarithms of

n e lo h no a h f n etc . Now si c g is t is is t h lf t e di fere ce

een l o 1 and lo 2 he e e th a n t ex betw g g , t r for e m rk does o actly bise ct the li ne from 1 to 2

— 205. H ow to e ad th e l de ul e n BB e R S i R . O it will b noted that the distance from 1 to 2 is divided into what we shall call 10 large divisions a nd they will be read from 1 at

h e a the h as 11 12 19 2 t e l ft (tow rd rig t) follows : , ,

h n num e o ne- n n - ea as e e e o e o e etc . un e (r d t l p o b rs , two, , or d r stood as It will also b e noted tha t ea ch of these

e n a a n e n 5a e a larg divisio s is g i divid d i to p rts , ch of which

2 ha h n h en e . 0 t e e n a e t e d ot s , so t t s co d divisio ft r first mark of the large division would be read andthe fourth small division after the mark denoting the fifth large divi n h sion would b e read etc . It will be oted t at the num

e n m 2 3 a 10 but a h ber of larg divisio s fro to is lso , that e c

n u e a a n n n 2 ma n large divisio is s bdivid d g i i to o ly s ll divisio s, so that the small mark after the first large division between 2

The am h and3would b e read etc . s e sc eme works from 3

m 56 b e e e a e e are but 2 a to 5. Fro to it will obs rv d th t th r l rge n n 5ma ne in ea h the a e divisio s a d s ll o s c of l rg divisions . The first large mark there would b e read and the first small mark following thi s large mark would be read 1 h The divisions are the same from 6 to 0 . If t e little runner THE SLI DE RULE 241

n h h o findo n h u h u be having a hair line o it , w ic y u t e r ler , s o ld used andthe hair line should fall half way between the first

ma n 5andthe e n ma ne a e 5 u s ll li e after s co d s ll li ft r , it wo ld

The ame an u n be un on be read etc . s pl of r li g will fo d the right - half of the ruler commencing with the second 1 h marked o n t e ruler .

— 206. Operations with the Sl ide Rule I t is not difficult to learn to use the slide rule if the student will use small

in how an e a n num e a . b rs t first If doubt to do op r tio , try it

h n h h u an h n first wit small umbers w ic yo c easily c e ck me tally . 4 Mu 2 b . M e th 1 M t . . ultipli ca i on ltiply y ov e slide (the part of the rule in the middle which slides) so as to set the 1 of the B scale directly under 2 of the A scale andread the answer 8 o n the A scale directly above the 4 of the B scale ; or set the 1 of the C scale dire ctly above the 2 of the D scale and read the answer 8 o n the D scale dire ctly below the 4

h n fin th u t e C a e . H e e d e num e of sc l c , to prod ct of two b rs,

h 1 h a e o n o ne the num e o n th D set t e of t e C sc l of b rs e scale , andunder the other number o n the C scale read the product

n h o t e D scale . Sometimes in multiplying we will have to use the 1 at the

- h an en e C a e . exam mu right h d dof t sc l For ple , ltiply 86 by

2 1 h h -han en h n . Set at t e d t e C a e o 86 D n rig t d of sc l of , a d

e m under 2 of C read the product 172 o n D . W si ply use the

1 at the e end the 1 at the h end C a n l ft or rig t of , ccordi g as it brings the other number over scale D . It will be observed in the above example tha t if we had used the 1 o n the left end C u a e u h the 2 C o ff the a e D of , it wo ld h v bro g t. of sc l . n n n hu P lace your decimal poi t by i spectio . T s to multi

n 1 D an un 1 5 ply by set 1 C o 8 , d der 0 C read the

n D h n ma e an a xima number 189 o . T e k ppro te multiplica

1 6 242 BUSINESS MATHE MATICS

n men a 10 2 e e are tio t lly, X 20 ; hence we know that th r

n e a u e in the u n as the e u . two i t gr l fig r s prod ct , givi g r s lt The decimal point will have to be placedby making an ap

x a e a u a n m n pro im t c lc l tio e tally .

'

2 . Di vi si . D n on i e 6 b 2 . 2 e D a d ea iv d y Set C ov r 6 , r d the e u e un e e e v e one r s lt dir ctly der 1 C o n D . Th r for to di id num e b an e t h n h b r y oth r, se t e divisor o scale C over t e divi

' en o n a e D n un 1 n n d d sc l , a d der C read the quotie t o scale D m n n . H e e a a n the n r g i deci al poi t is pla ced by i spectio .

u i e 15set 15C e 2 5D and un e Th s to div d by , ov r 8 , d r 1 C read the quotient 19 o n D ; but we can Observe that 3 15a 3 2 n 19 . is bout 1 5, or or . ; he ce our quotient is .

in the 3. Combined M ultipli cati on and Di vi si on . F d 26 X 4 a ue Set 8 C e 26 D andun e 4 C ea the v l of ov r , d r r d 8

e u 13on D . i the v n 26 8 a e set r s lt F rst di isio of , by is m d by

n 8 C e 26 D andun e 1 C we i ea the u en ti g ov r , d r m ght r d q oti t ;

h n As 1 C al but we want to multiply t is quotie t by 4 . is ready on this quotient we have only to read the product 13

h u e me we can find o n a D un e 4 C . B t e s e sc le d r . y of this sch

the u a n . exa e in th fo rth term of proportio For mpl , e pro

- 26 ' X 4 n 4 x portio : , a: Therefore to find the 8

u e a n et the e m h fo rth t rm of proportio , s first t r over t e sec

n andun er the h ea the u e m o d, d t ird r d fo rth t r .

4 C ti nued M ulti li ca ti on and D v n i . . on p i si on I this work use the li ttle glass (or celluloid) runner which has the hair n line o it .

Il lustrative Example 1 . Find the value of 4 x 6 X 3.

SOL TI ON : Set 1 C at the h e 4 D set the e 6 C U rig t ov r , runn r on , set 1 at the h the u e in as n e 3C ea 72 D C rig t on r nn r (slid g gl s) , u d r r d on .

4 X 6 X 3 72 .

244 BUSINE SS MATHE MATICS

h Illustrative Example 3. Find t e value

SOL TI ON : Set 7 B 6 A an d e 14 B ea the e D . U on , und r r d r sult on

the ea a e h e a 3 Illustrative Example 4 . Find ar of circl w os r dius is h inc es .

L I N Set 1 C 3 D and a e 1! B e a the a ea SO UT O : on , bov on r d r ,

A. sq . in . on NOTE : If the student can do his work on the slide rule so that it is h mal h be a a o m mp corre ct to t e first de ci , t is will s tisf ct ry for ost co uta

EXER CI SE S

Find the value of the following 5 4 1 . 6 X

2. X

3. X

4 . X . 5

5. .08 X 4 8

6. . 28 X .004

. 54 7 . X

8 . X 3 x 4 >< 12

9 . X

10 . . 54 . 92 X 5 7 m 5 x x 11 . X . 00 6 X 14 2 128 64 1 . . X 5 13. 2 ' 7 x 16 x 4 14. 144 24 4e x s4 >< 9 15. . 75 '

16. 128 18 x 27 x 7

. 10 x 35>< 65

17. 3 x s. 4 x s. 6

4 x4 . 6 x 2 .6

16 . 4 x 12 x 4 .2 .65 x 3V6 625 25

. 25 TH E SLI DE RULE 245

38 . and h e Find the area of a rectangle whose length is in . w os h widt is in .

39 . in . Find the area of a circle whose radius is (a) 4 in . ; (b) ; (c)

ft . ; (d) yd.

- 40 . the Find the circumference of a circle whose radius is ft . if “ m e e e e h t circu f r nc quals twice t e radius times .

41 . e he h Find the area of a cylinder whose radius is 3in . andwhos ig t is in .

42 . the a e m n r ha ar the a e If w g s of 6 e for 1 da . a e w t e w g s of 12 men a t the same rate ? 4 e a m the 3. A d p rt ent store offered a sale of 7 articles for find 1 cost of 3articles at the same rate . 44 i ? . 8 s what per cent of 24

HI NT : 8 24 what deci mal what 5 45. Whfi t 1 6 h 46 . w at 4 ha 7 . i t w t h 48 . w at decimal? 9 4 ha de Imal? 9 . 7 2 w t C

h at 2 da . 50 . Find t e interest on $800 5% for 7

HI NT

Principal X R ate X Time in days Interest

th e e 650 at 4 65da . 51 . Compute e int r st on $ % for

h the e e 500 75da . at 4 52 . W at is int r st on $ for %

h h h e 120 5r . at 53. Find t e principal w ic will produc $ in y

HI NT

Intere st p rI np a1 R ate X Time (in yearS )

r Find the principal which will yield in 8 y . Find the square of each of the following :

(a) (f) (b) (g) 15 (c) 12 (h) . (d) 9 (i) (e ) (j) 15 246 BUSINE SS MATHE MATICS

h 56. Find t e square root of each of the following (a) 64 (e) (b) 49 (f) (c) 9 (g) 2 (d) 144 (h) 3

e ro s No r m: Check (g) and (h) by extra cting their squar ot , memorize the result co rre ct to 3de cimals .

the a e h th l 57. Find v lu of ea c of e fol owing (a) 1: x 12 (d) at x “ (b) it x 6 (e ) i t x (c) 1r x 4 2 (f) i t x

h a e h se area 144 . . 58 . Find t e r dius of a circl w o is sq in he a e h e c um ere e 59. Find t radius of circl w os irc f nc is

e h e a . 60. Find the circumference of a circl w os rea is sq in .

and C e ts — n h e 207. To Find Cubes ub Roo To fi dt e cub

s : Set 1 e 4 D and ea the of 4 , work a follows of B ov r of r d 4 B answer 64 on A directly over of .

n h e e e the e exa e To fi dt e cube root , r v rs proc ss . For mpl ,

4 m e the a n to find the cube root of 6 , ov slide b ck a d forth until the number o n B dire ctly under 64 (on A) is the same

D he e - han 1 e n un e 1 C o n . t ot as that d r , If l ft d do s work,

- n 1 n use the right ha d o C .

EXE R CI SES

he e the n m e 1 . Find t cub of followi g nu b rs (a) 2 (f) (b) 3 (g) (c) 5 (h)

(d) 15 (i) .04 (e ) (j) 00 15

' the a roximate cube s the l n 2. Find pp root of fol owi g

(a) 8 (e) 68 (b) 27 (f) 425 (c) 125 (g) (d) 216 (h)

248 BUSINESS MATHE MATICS

Re and ductions conversions .

R e e 24 . m e . 8 . duc ft to et rs (24

. m 9 Cha e 9 . the e 9 16 ng oz to d ci al of a lb . (

- - 10 . . l Change 36 ft b to ft . tons . (36

11 . a e . r . . Convert 8 cubic ft . of w t r to lb pe sq in (8 X 4333

12. 2 m r hr s . 2 . 6 4 Reduce 5iles pe . to knot ( 5X 8 8 - 13. Cha 12 H . . h 2 nge P ours to kilowatt hours . (1 X

14 . the a e 215 Find v lu of X V243. I NT : Set the e the h -ha end A 1 C H rid r to on rig t nd of , bring of the e andm e the e 2 15 C he w th to rid r ov rid r to on , w n under it e find e D answer on .

To find the wages due .

h findthe a ue hr . 4 r 15. If we wis to w ges d for N at $ 8 pe week for 44 N 4 hr. we ha e the i s , v proportion 7

2 hr - the a e due 6 . a man e e e 42 4 hr . 16 . Find w g s for if r c iv s $ for a 8 week . the a e e e at e e 17 . Find r t of int r st on consols n gl cting brok erage .

HI NT 112 100

n 20 at 50 e h r 18 . A article costing $ is sold $ l ss Find t e pe cent h the er e of gain on the cost . W at is p c nt of gain on the selling price?

Given Sales Cost of goo ds sold Selling expenses General expenses Profit

Wha t per cent of the sales is each item? h a m due an em ee he has e 44 hr 20 . Find t e ount ploy if work d % . at

$20 per 48 hr . wk . he e e an a e h 4 h h a 24 21 . Find t s lling pric of rticl boug t for $ on w ic % profit (on cost) is to be made . and be as ma e 2 22. If an article costs is to sold so to k 0% on the selling pri ce find the selling price .

e e set . C HI NT : Cost will be 80 % of s lling pric , so 8 of scale to of h D scale andunder 1 of C scale read t e answer . THE SLI DE RULE 249

Work the following with the slide rule

MAKE ON SE LLI NG

’ P R ICE

22 % 34% 18 % 40% 45%

4 Wha am due an m 2 . e t ount is ployee for 4%hr . of overtime work at m nda hal he th e a e e u a 44 hr . ee a e a e 25? ti f, w n r g l r w kly w g r t is $

25. Wha the a e f 75he the ex h t is v lu in of $ , w n c ange rate is

HI NT : 75X

x ha e th a 26 . If e c ng is find e v lue in dollars of £40 I T Set C 40 D and N : e 1 C ea th . H of to of , und r of r d e answer

27 . t he a e a 75he ex ha Find v lu in fr ncs of $ , w n c nge is the a e a x h 28 . Find v lu of fr ncs if e c ange is

29 . Find the value in lira of $85when exchange is h 30 . Find t e value of lira in dollars if excha nge is

1 . The e 1 10 n 3 5e a d the net . list pric is $ , l ss % Find cost

the e a 4 L B m n . 32 . Find inter st on % iberty ond of $50 fro Ja r to Ma .

HI NT 04 X 50 x 74

R ul e s for Characteristic . a e e e o e ara Multiplic tion . If th slid proj cts t th e l ft, th e ch cteristi c a e sum th e ha a e i i th e a if th e equ ls th of c r ct r st cs of f ctors to right, it equals th e sum 1 . th e i e ec th e e e a th e harac e i Division . If sl d proj ts to l ft, it qu ls c t rist c th e hara e c th e d 1 — of th e dividend, minus c ct risti of ivisor ii to th e right it equals th e diff erence . CHAPTER XVII I

DENOMINATE NUMBERS

en mina e u e — A n n n 208. D o t N mb rs de omi ate umber is a

l . nu e a e name u as 54 d. 6 h 8 e e mb r with sp cific , s ch $ , y , , m t rs , n n n h etc . Soo er or later a y perso is apt to ave occasion to n n A n h h use de omi ate numbers . ccordi gly it is t oug t best to introduce a short chapter of these numbers in this book both as text for the student andas referen ce matter for the busi ne man has a u e h u e a ss , who prob bly st di d t is s bj ct t some

’ e in ch l career and en en m n ll tim his s oo th forgott ost , if o t a , of it .

h r a e e num e n u n h T ere a e sever l tables of th s b rs , i cl di g bot the E nglish and the M etric systems . A few simple exer

in nne n em ha e een n u e cises co ctio with th v b i trod c d , to

m u n in e ha h gether with ethods of sol tio , ord r t t t e reader

a l e may rec l th m .

Tabl es .

LONG ME ASURE

12 in .

3 ft .

16 . 5%yd. or 4ft 2 3 0 rd .

1780 yd.

5280 ft .

252 BUSI NE SS MATHEMATICS

OY WE I H T w e i etc . TR G (used in igh ng gold, )

20 pennyweight s 2 1 oz .

Avo I RnU P oI s WE I GH T

1 hundredweight

“ - - h 2240 . 1 toh se a etc . a sa e a e lb long (u d in co l , tr n ctions w ol s l )

- O H E C R I E S WE I GH T AP T A _

1 scruple

’ heca e 1 lb . troy or ap ot ri s l ‘

4374

a 1 beef 200 1 i h lb . ut 7 al . ft . a er e 62 1 cu . of w t w g s % (abo } g )

Wh ea 60 . 1 bu . t lb

1 bu . oats

LI QUI D. ME AS UR E

4 gills

231 cu . in . DENOMINATE NUM BERS 253

DR Y ME AS URE

1 qt .

1 pk .

1 bu . cu . in .

ME AS UR E S

60 sec . 1 yr .

hr . 1 a r . 6 0 min . 1 commerci l y

1 da . 1 mm r . 24 hr. co on y

1 . 1 ea r . 7 da. wk l p y

mme i a mo . 1 30 da . 1 co rc l century ea s s e b 400 ea e b 4 are Centenni al y r divi ibl y , y rs divisibl y leap years .

ME A S U R E S OF VAL UE

UNI TE D STATE S M ONE Y E NG LI S H MONE v 10 mill s 1 cent 4 farthings 1 penny (d) 10 cents 1 dime 12 pence 1 shil ling (s) 10 dimes 1 dollar 12 shill ings 1 pound sterling 10 dollars 1 eagl e 2 - ” e I 2 e I 24 e . 1 far . 83c nts ; d 59 c nts ; s 4c nts

FR E NCH MON E Y G E R M AN MONE Y 100 centimes 1 franc 100 pfennigs 1 mark

MI S CE LL ANE O US ME AS UR E S

12 things

12 doz . 12 gross 24 shee ts 20 quires

n to L e Den mi nat ns — 210. Reduci g ow r o io I t is sometimes necessary to reduce a given quantity to a lower d enomina

e u e uan e f e en en tion, or to r d c q titi s of di f r t d ominations to the same denomi nation . 254 BUSINE SS MATHE MATICS

? H al . 3 . 1 t . Ill ustrative Example 1 . o w many gills in 5g qt p

L I 20 . SO UT ON 5gal . qt

3 . 23 . 20 qt . qt qt

2 46 . 3qt . pt

1 . 4 . 46 pt . pt 7 pt 1 47 pt . 88 gills

R e e . 2 mi . e e m na . Illustrative Example 2. duc 6 6 to low r d no i tions

320 . SOLUTI ON 1 mi . rd

i . 2 2 . 626 m . 6 6 X 3 0 rd

1 rd . yd.

2 d. 32 rd . . 3 X y

1 yd. 3ft .

76 3 . 76 yd. . X ft

12 . 1 ft . in

2 12 . 28 ft . . 8 X in

2 0 . 1 d. 2 . . Therefore 626 mi . 0 rd y ft in

WRITTEN EXERCI SES

Reduce

7 . 4 . . 56 . . 1 . 5mi . to rd bu to qt

5. . 64 e min . and . l . . 5 e . 5a e sec 2. 7 g to pt d gr s to

4 . 6 . . 374 ha e e a . 3. . 87 mi c ins to low r d nomin tions

H he Den i nat ns — I 211 . Reducing to ig r om io t is some times necessary o r conveni ent to change a given quantity n to a higher denomina tio .

xam e 1 . Cha e hr . h he mi Illustrative E pl ng to ig r deno nations .

SOLUTI ON

24 hr . 1 da .

52 da . and20 hr . hr . , remaining

1 . 7 da . wk

52 7 7 . and3da . ema 52 da . wk r ining — 7 54 1 mo ! and3 . emain n 7 wk . wk r i g

hr 1 mo . . 3da . 20 hr . Therefore . 3wk

256 BUSINE SS MATHE MATI CS

ra e xam e . S t a 5. 10 . m 12 . Illust tiv E pl ub r ct ft in fro ft . 9 in

S L TI 2 O ON : 1 . 9 in . U ft 11 ft 21 in . 5 10

6 ft . 1 1 in .

WRI TTEN EXER CI SE S

Subtract the following

m 14 . . 1 . Fro lb 8 oz

7 lb . 7 oz .

4 . 2 . d F m 36 . . 2 . ro rd y ft 8 in

2 d. 1 . . 26 rd . y ft 9 in A man h ee ea h a i m a e 3. sold t r lots c cont in ng fro fi ld H w m h had he e ? taining 2%acres . o uc l ft

a n n n D 4 M u t l t U O e en inat u e . 21 . l ip ic io si g om e N mb r Denomina te numbers may be multiplied as in the follow

l Mu 6 a . . . Ill ustrative Example . ltiply g 3qt 1 pt by 6

SOLUTI ON

1 . 3 qt . pt 6

l 1 8 6 . 36 ga . qt pt 41 1

W RI TTE N EXER CI SES

Mul tiply the following

2 . . b 7 . d. 8 1 . 4 y ft in y

4 . b 15. 12 . 2 . lb oz y

ha the e h 6 . . a if a 3. W t is w ig t of %cu ft of c st iron c st iron is 7ktimes as

h . th . ? heavy as water andwater weig s lb to e cu ft .

Den ina e — 5. D i s n U n One t u e 21 iv io si g om N mb r . De nominate numbers may b e divided as in the following

D e 356 ll b . 4 . Ill ustrative Example 1 . ivid gi s y DENOMI NATE NUMBERS 257

SOLUTI ON 356 gills 4 89 gills

89 22 t . 1 gills ,p gill

22 pt 11 qt . 4 11 1 l 356 gills qt . gi l

l a x e am e 2 . D e 46 d. 2 I lustr tiv E pl ivid y ft . 8 in. SOLUTI ON :Re duce to inches andthen proceed as

WRITTEN EXER CI SES

Divide the following :

37 d. 2 . S in . b 8 . 1 . y ft y

1 . 7 . b 6 . 2 . 6 lb oz y

hr . a man can a 16 mi . 6 ha his a e Of a e 3. If w lk in w t is r t tr v l?

ma e 23 ml . I n hr . ha h r hr . 4 . an a m e 8 8 t e a e e ? If uto obil k s , w t is r t p

Th e M et em — h the em h 2 16 . ric Syst T is is syst of weig ts and measures in use in France . It is also used quite ex tensively in the United States and o ther countries at the

ea a an a e the a ha ll h present time . Its gr t dv t g is f ct t t a t e tables use a scale of 10 .

er s — The me e the un Of n h 217 . T m t r is it le gt and is in approximately . The lite r is the unit of ca pacity andis equal in volume to

e me e . 1 cu . d ci t r

The r am the un e h an d the e h 1 g is . it of w ig t , is w ig t of cu.

n me e e a e in a a uum a ce ti t r of distill d w t r v c , t its greatest

en ah enhe . e h a n E d sity F r it It w ig s gr i s , ng lish measure .

P refixe s — The h ee La n e x 218 . t r ti pr fi es denote parts of the unit : 258 BUSINE SS MATHEMATICS

milli means o ne o ne-thousandth

i centi o ne-hundredth deci o ne-tenth

Greek prefixes denote multiples of the uni t

deka means ten hecto one hundred o ne thousand myria ten thousand

219 . a T bl es .

LI NE AR ME AS URE (The Unit is the meter)

m e cm. 10 millimeters (mm . ) 1 centi et r ( )

10 centimeters 1 decimete r (dm. )

10 decimeters 1 meter (m. )

10 mete rs 1 dekameter (Dm . )

10 dekameters 1 hectometer (H m . ) 10 hectomete rs 1 kilomete r (Km)

10 kilometer s 1 myriameter (Mm . )

WRI TTE N EXERCI SES

dm . . mm . H m . Km Change 356 m. to D ; to ; to ; to D m . m . Km. R educe 2642 cm. to ; to ; to H w ma me e A rectangle is cm . long . o ny t rs long is it?

m . 2 Dm . 4 cm . In . tCh 5Km. 3H ange to t

Mm. O 25m. Re duce . to

SQUAR E ME ASURE

mm . 1 . m. 100 sq . sq c

m . . dm. 100 sq c 1 sq .

dm . 1 . m . 100 sq . sq

m . 1 . Dm. 100 sq . sq

Dm . . H m. 100 sq . 1 sq

H m . 1 . Km. 100 sq . 86

260 BUSINE SS MATHE MATI CS

092 5 9 sq .

8361 sq . 5 5 sq .

sq . 5

cu . 5

cu . 5 5 7646 cu .

9463 l .

1 101 l .

1.

3524 H .

r . 1 g 0648 g .

1 oz . (troy) g .

a s . 1 oz . ( voirdupoi ) g

2 . 1 lb . (troy) 373 Kg

s K . 1 lb . (avoirdupoi ) 4536 g

1 . dm . o f a e a e and e h 1 cu w t r of w t r, w ig s

1 cm . 3937 in .

in .

6214 mi .

d. sq . y

1 . . 308 cu . yd

7 . 1 . 056 liquid qt

0 dr . . 9 8 y qt gr

21 . . 03 5oz tro y

0 27 . a . 35oz voirdupois

s lb . avoirdupoi

WR ITTE N EXE R CI SE S

1b . a r . R educe Kg . to voi

2 . 5. m R educe ft . in to

m . . . Cha nge 60 sq . to sq ft

in 0 mm . ? H o w many in . 3

2 1. 1 . Re duce gal . 3qt . pt to

ha e K : s and e e mi C ng g to ton low r d no nations . - . m . h w t hs 7 . 1 13 . er c o ma 1 a e h . e a If c s iron w ig , g p cu , ny do s weigh?

h . h 25d. o f at er m 8 . Find t e cost of y clot p

mi . th H o w ma Km. in 25 e e h a h ? 9 . ny (to near st t ous ndt ) DE NOMI NATE NUMBERS 261

m 16 sec . ? 10 . h at the a e 100 . W at is the time of traveling mi . r t of in fl at the a e 11 . e and9 . ee If a stream of water 5ft . wid in d p is owing r t me e 1 d. er sec . find the e h a e of y p , w ig t of w t r in tric tons, suppli d in

2 h r . 1 a . . e h . , if cu ft of water w ig s oz l h e a h 12. h . K . t e Find t e weight in lb andin g of ga . of b st lco ol,

e a . 792 sp cific gr vity .

th ir 1 . r . m . w ma . 13. If e pressure of the a is about Kg pe sq c ho ny lb h ft . ? is that to t e sq . ? Km . 4 . Wha th e ee 5mi . and8 1 t is e difference in yd. b tw n

A bar e a 6 . b 3 . b 4 . 15. of iron (sp cific gr vity is ft y in y in Find its weight in Kg . CHAPTE R XIX

PRACTICAL MEAS UR EMENTS

M — 220. Practical easur ements Such measurements are

h ni m e men a t e e e t. e . a u are wh t t rm sig fi s ; , s re ts which of

a a use an e n an u ne a n pr ctic l to y p rso or y b si ss t a y time . These include the measurements of or a ppertaining to differ n n an e u a e n n n h e t ki ds of gl s ; s rf c s ; polygo s, i cludi g t e paral lelo ram the e an e the uare andthe an e g , r ct gl , sq , tri gl ; circles,

n n h ame e the a u the i cludi g t e di t r, r di s, circumference , and the area ; problems involving square root ; area of irregular

u h as the h shaped figures ; solids, s c cube, t e cylinder, the

ne the a a nd the e e . co , prism toid , sph r

— 221. Th e n e Ah an e is the a un en n A gl . gl mo t of op i g be h ne h h mee tween two straig t li s w ic t at a point . The side s of the angle are the lines whose n m h n h intersectio for s t e a gle . T e vertex oi an angle is the point in whi ch the sides inter

Reading an Angl e .

h e wa ea an an e 1 . T e b st y to r d gl is to place a small letter or figure like a or 1 as in the following

and a an e a an figures, c ll it gl , or gle 1 .

An e wa to use h ee l e e as an 2 . oth r y is t r tt rs , gle ABC n u e u n h in the followi g fig r , p tti g t e vertex letter e in the middl .

264 BUSINE SS MATHE MATI CS

— h n an e e 225. The Straight Angle T is is a gle whos sid s lie in the same strai ght line and ex tend in Opposite direc

x h n in the a an tions from the vert e ; as t e a gle ABC, ccomp y ing figure .

— n An e . i is o ne e ua a e 226 . The Right gl Th s of two q l gl s made by one straight line meeting an other straight line . Thus if the line CD meets the line AB so as to make the angle DCA equal to

the an e D B ea h e e an e a gl C , c of th s gl s is

right angle . What part of a straight angle is a A right angle? H o w many degrees in a right angle?

e en i — i n 227 . P rp d cular Line a A l e is said to be perpendi cul ar to another line when it m ee ts it so as to form two

equal angles . What kind of angles do the lines form?

— 228. Th e Kinds of Angl es The a cute angle is n h An s less tha a rig t angle . obtu e angle is an angle grea ter than a right

n e but e han a a h an a gl , l ss t str ig t gle .

ABC an a u e an e . GED an u e an is c t gl is obt s gle .

22 ur ace s — A u fa e ha 9. S f s r c is t t which has length and h breadth but no t ickness .

A plane surface is a level surfa ce such as the surfa ce of

a e . A a h e e o n in an still w t r str ig t dg will fit it y position .

A plane figur e is a figure all of whose points lie in the same

plane . PRACTI CAL ME ASURE MENTS 265

— n n a ane 230 . Polygons A polygo is a portio of pl h un e b a h n a h n u e . T e bo d d y str ig t li es , s t e followi g fig r

e e e h A na p rim t r of a polygon is t e sum of all its sides . diago l

a a h ne n n no n- a a en e e as in is str ig t li j oi i g two dj c t v rtic s if, the u n h u b n ne fig re below , a li e s o ld e drawn from a y o n n cor er to an opposite cor er .

— ua a e a . A ua 231 . Q dril t r ls q dril ateral is a plane surfa ce

a h bounded by four str ig t line s .

— 2 a all e a . A a e a 23 . P r logr ms par ll logr m is a quadrilatera l having its opposite sides parallel .

— an e . A e 233. Re ct gl s re ctangl is a parallelogram all of whose angles are h n rig t a gles .

uar e — A ua e a e an 234 . Th e Sq sq r is r ct gle having four equal Sides .

— an e A an e a ane 235. Th e Tri gl tri gl is pl figure bounded n h n by three sides a ndhavi g t ree a gles .

—A h a Th e ht ian e . n e a 236. Rig Tr gl rig t tri gl is triangle an e ha m that has o ne right angle . No tri gl s ore than o ne right angle . The sum of the three angles of any triangle e quals two right angles or 266 BUSINESS MATHE MATI CS

— 237 . Th e H Th tenu e a h ypotenuse . e hypo s of rig t tri an e the e h gl is sid opposite t e right angle .

— 238 . Th e Equil ateral Triangle An e quilateral triangle is a triangle having all its side s equa l and all its angles equal

2 Th e e e an — An 39 . Isosc l s Tri gle isosce le s triangle is a

an e ha n e e ua and an tri gl vi g two sid s q l two gles equal .

— 240 . A 30 0 t an e a an e o n 6 ri gl is tri gl , e of whose angles is a nother of whose angles ° ° is 60 and the thi rd angle obviously 90 C The hypotenuse is twice the length of the shorter arm BC.

n 241 . The base of a y plane figure is the side on which it is

n as AC in 24 . supposed to sta d , § 0

242. The al titude of any plane figure is th e p e r p e n di c u l a r distance from the 0 p posite point highest from the base to the base extended

268 BUSINE SS MATHE MATI CS

4 . Do the am h a nd the he h h s e if t e length (l) in . ig t or widt , a e a or ltitud ( ) is i in .

5. h nd h e t e a ea 2 a n . a Find r of a re ctangle whose base (b) is . 5of in w os

a 12 in . ltitude (a ) is . 5

6 . A e ni H o w ma a e t n s court is 78 ft.long and 36 ft . wide . ny squ r feet doe s it contain? Wh at part of an a cre is it?

. 12 d. 7 Find the perimeter and the area of a re ctangle 15yd. by y

8 . H o w ma a and5. e be e e ny p ving blocks 1 ft . long in wid will r quir d 2 mi a e a e and . ? to p v str et . long 35ft wide

9 . A h e a a and20 . e . t e r ct ngul r field is 40 rd . long rd wid Find cost of en f cing it at a rod .

10 . the a n n h e a a om 12 . Find cost of p i ti g t e four sid w lls of ro ft long ,

10 . 6 . W e a nd9 . hi h at 12 e er . d. a a e e ft in id , ft g c nts p sq y , no llow nc b ing m ade for openings .

11 . Th andthe h e length of a rectangul ar pie ce of iron is in . widt is in . Find its area and perimeter .

12 . 1 the a e me e e h . ha the If sq . ft . of bov ntion d iron w ig s lb w t is weight of the e ntire piece if of same thi ckness thr oughout?

13. The a n h a b . a dt e e e pl n of slide valve is in . y in pr ssur

a 5. er . e h F the a e t e a e . b ck of it is 8 lb p sq in . ind tot l forc pr ssing v lv 4 h 1 . Find t e area of a channe l iron from the dimensions in the aecom a p nying figure .

th a a h h e a the a 15. Find e re of t e s ad d p rt in ccompanying hollow square . PRACTICAL MEASURE MENTS 269

16 . e and H o w many pie ces of sod will it take to sod a lawn 24 ft . wid

28 . th 4 in . ? ft long if e pie ces are 12 in . by 1 1 7 . The a e d a e 4 d. a a in . n e a e 1 0 . a r of r ct ngl is 8 sq , its b s is y Find the altitude .

18 . Find th e area of a floor from the dimensions in the accompanying

— 24 h e A ea a a a e a . 9 . To Find t r of P r ll logr m The parallelo gram ABCD may b e Shown equal in area to the re ctangle

F AEFD by cutting off thetriangle ABE

and a n o n the n pl ci g it tria gle CDF . This shows that the equa tio n fo r the area of a parallelogram is then the same

ha ha n? as that for the rectangle . W t is t t equatio P(

WRI TTEN EXE R CISES

m h e the a ea a a a e a a e b . 1 . Find r of p r ll logr w os b s ( ) is 8 in andwhose

a e a AB a e e 6 . ltitud ( , or in bov figur ) is in e e the m a a l e am h e 2 . Compl t following for for p r l logr s w os dimensions are

BA SE A LTI TUDE AR E A o r PAR AL LE LOG R AM

A e e me a the m a a a e am has an a ea 3. pi c of t l in for of p r ll logr r of t a . and the a e . e e . sq . in b s is in Find h ltitud 270 BUSINE SS MATHE MATI CS

— the 250 . To Find th e Ar e a of a Triangle If we draw

a na AC in the e an B D hen cut h u h the di go l r ct gle A C , t t ro g

n h fin ha the i an e diago al, we s all d t t tr gl

ABC will exactly fit o n the triangle ACD . A triangle may therefore b e shown to be equal in area to o ne-half of the area of a rectangle with the same base andthe same altitude . Sta te then the equa tion for the area of a tri angle whose

h e a u e a the e mem h base is b andw os ltit d is a . C ll l ft ber of t e equation A A .

WR I TTEN EXER CI SES

h he a ea a a e e as e b 12 . and h 1 . Find t r of tri ngl w os b ( ) is in w ose altitude (a ) is 8 in .

ha the a ea a a e h e a e . and h e a 2 . W t is r of tri ngl w os b s is ft w os lti ? tude is ft .

h e a ea . A 144 . the a e a a e . and 3. Find ltitud of tri ngl w os r ( ) is sq in whose bas e (b) is 48 in .

nd th e C um e en e a C r e — n 251 . To Fi irc f r c of i cl Fi d the length of the circumferen ce of a circle by taking a cardboard

e a n circl , m rki g some point o n

as P he e it , , w r circle touches level as A and roll it along o n a level surfa ce until P again touches the — The an e AB level surface say B . dist c will then represent

the circumference of thi s circle . It will b e found also that this length divided by the di ameter of this circle will give

' ’ a e ll. i . he e a pproxima tely c ll d (p ) T r fore , in the a c

D i t C 7: u e C D . companying fig r , or

272 BUSINE SS MATHE MATI CS

dthe eam 5. The am a e e a e a e 3in . an di eter of l v r s f ty v lv is , st blows

D h a e the a e . ff a 1 . m t e er . e e e e o t 95b . p sq in t r in upw rd pr ssur on v lv An y a e h a ea a e b a m the 6 . Eg ptian obt in d t e r of circl y subtr cting fro Tr h diameter o ne ninth of its length andsquaring the remainder . y t is h it the a a a e h e ame e 9 in . and e e b e pl n on circl w os di t r is , t n solv y bov method andobtain the amount of difference of areas .

7 . The e an e e has 300 e ea h 3 . i ame e boil r of ngin tub s , c in in d t r , for Fi the a e a conducting the heat through the water . nd tot l cross s ction l area .

me e . A r l a a l and 20 . a 8 . A pie ce of land is circu ar ft in di t r ci cu r w lk

e a a n . Wh a the h a at 5r 5ft . wid is l id rou d it t is cost of t is w lk pe sq . ft .

— 253. Th e Trapez oid andits Ar ea The tra pez oid is a quadrilateral having only Th two sides parallel . e area of a trapez oid is the product of o ne-half of the altitude '

sum bases b andb .

A trape z o id

WRITTEN EXER CI SES

i the a ea a a e 1 . F nd r of tr p zoid

and 10 . whose base s are 12 in . in and whose altitude is 4 in . The a e a a e are 2 . b s s of tr p zoid dthe in . an d 2 . 8 6 . an 4 ft . in ft , ' h u the altitude is 1 ft . 8 in . I d area in sq . in . the a ea the a e m 3. Find r of co panying figure . he a ea a a e 66 4 . T r of tr p zoid is , '

he a e . 14 b 8 . t b , Find ltitud

— n uar e t The ua e 254 . Extracti g Sq Roo sq r of a number b mu n m e num the e u a ne . e b is r s lt obt i d y ltiplyi g so b r y itself, 2 5 5 5 25andw e sa ha the ua e 52 5. as x , y t t sq r of is PRACTI CAL ME ASURE MENTS 273

The square root of a number is o ne of the two equal a t h m the a emen a e f c ors of t a t number . Fro st t t bov it is

u ha he 2 hi ma be re obvio s t t t square root of 5is 5. T s y p % e n b h n 25 5. r se ted y t e followi g ways . or

WRITTEN EXER CI SE

C m e e the m 1 . o pl t following for

NUM B E R I TS SQUAR E R o o r

It will b e observed from the above form that the square root of any number between 1 and 100 is between I and 10 ; n n of a number between 100 a d10000 is between 10 a d 100 .

5ma be un as as h n i n The square of 2 y fo d follows, or s ow

n the a ccompanyi g figure .

(20 x 5) 52 20 2 + (20 x 5) 202 + 2 (20 X 5) 52

This may be stated as follows The square of any number of two figures is equa l to the square of the tens plus twice the product of the tens by the units plus the square of the units . By a pplyi ng this principle the square root of any num ber may be obtained .

1 8 274 BUSINESS MATHE MATICS

The square of any number will contain twice as many

u e o n e n fig r s or e l ss tha twice as many figures as the number . Therefore separate the number into groups of two figures each beginning at the d ecimal point and working each w ay h from it . T ere will be as many figures in the square root as

e e are u in h th r gro ps t e number .

ll a i e xam e . Ifind the are 625 2 5 I ustr t v E pl squ root of , or ’

6 25. SOL UTI ON : Begin at the de cimal poin t andseparate the 4 m a The a 25 nu ber into groups o f two figure s e ch . l rgest h 4 5225 a e 6 is 4 andt e a e 2 . ai the squ r in , squ r root of is Obt n 45 emai 2 an x h x r nder d anne t e ne t group giving 225. H a a e the uare the e m the m e he e e th ving t k n sq of t ns fro nu b r, t r for e te maindar (225) must contain twice the product of the tens by the units

Two me 2 2 . plus the square of the units . ti s tens or 0 40 40 is con

tai n d me 22 5he the u ure o ur . Two me e 5ti s in 5. t n is nits fig of root ti s the e me the u the a e the un is the ame as th t ns, ti s units , pl s squ r of its, s e dth n h sum me the e an e me t e n s . H e e of two ti s t ns , u its , ti s u it nc , we h m the s 22 add5 t t 40 and um b 5 a 5. h uni s to e ultiply y , obt ining T ere fore the square root of 625is 25.

PR I NCI PLE : To obtain the square root of a number

Be inn n at the e ma n e a a e the num e n 1 . g i g d ci l poi t , s p r t b r i to

groups of two figures ea ch .

e th ua e t the ea e e e u 2 . Ta k e sq r roo of gr t st p rf ct sq are con tained in the left-hand group for the first root figure ;

u a ua e m the e - an u an s btr ct its sq r fro l ft h d gro p , d to n th the remainder bring dow e next group .

D e the nu e u a ne ex u e u 3. ivid mb r th s obt i d, cl siv of its ni ts, by twice the root figure already found for a second root figure ; place this figure at the right of the root figure alr eady found andal so addthi s figure to the trial divisor

Mul his sum the us u e . ast t u j t s d tiply t by l roo fig re . Subtra ct and proceed in a similar manner until the

root is obtained .

276 BUSI NE SS MATHE MATICS the e the h an e a e as an e ua n two l gs of rig t tri gl , or st t d q tio ,

I z 2 . t may also be proved tha t a x/B b or tha t b

u am n Ill strative Ex ple . Fi d H if a

V12 2 16 2

WRI TTEN EE RCI SES

. andthe a e a a. h a e if a. 20 in. b 25. 1 Find H , r of rig t tri ngl , in m h me a e a ase a 2 The s a e am 0 ft . . di t nc fro o to first b s on b b ll di ond is 9 ,

m e a e 90 ft . the e andthe distance fro first to s cond b s is , find distanc from second base to home in a straight line .

h d 890 d. e a a S a e an b 150 d. 3. If a park is r ct ngul r in p y long y y h a e b a i m o ne e t e wide, how muc is s v d y w lk ng fro corn r to h opposite corner along a diagonal walk instead of walking along the sides?

4 . a at B a e If body is , It ft . bov the surfa ce o f the earth the num ber mi e m at h h of l s , w ic it can be see n, is limi ted by the curvature the e ar h h of t . T is distance is ob tainedby the equation In

Use this in the following

a er a n h h se is 200 . a e the The li ght of c t i lig t ou ft bov sea level . H ow many miles distant can it be seen?

a en e 65. The back stay of susp sion bridg is ft long, andthe distance of m the o ne the e s 54 the anchoring point fro foot of of pi r is ft . Fi ndthe height (a) of the pier . P RACTICAL MEASURE MENTS 277

5. A h a a 1 . the e e r ilw y incline is 1 ft . in 50 ft What is proj ct d or ori z o ntal length (b) ?

F h a h 6 . ind t e total rea of t e accompanying figure by dividing it up

a as n a e h th a ea e a h an h h a . into p rts, i dic t d , t en finding e r s of c dt en t e tot l

- Th h e se a h a e 30 . The 1 . 7 . e ypot nu of rig t tri ngl is in altitude is 8 in

(a) Find the base . (b) What is the side of a square whose area is equal to are a o f this triangle ?

(0 ) Fi nd perimeter of triangle .

° o n he 30 —60 256. Preportio s of t

— The enu e e Triangle . hypot s is twic

The an e o the shorter arm, a . gl p po site the hypotenuse is therefore the principles of thepreceding section

WRITTEN EXER CI SES

° 1 The e e the 30 a e 8 ha the h e e? . sid opposit ngl is , w t is ypot nus Find the base by applying the principle in 255. ° ° e 2 The a e a 30 60 a e 9 in . the h e . . b s of tri ngl is , ypot nus is in

andthe a ea the a e . the Find the other l eg of the triangl e, r of tri ngl Find perimeter of the triangle . 278 BUSINESS MATHE MATICS

° 3. th e h a 60 h ea 1 in . the e h d . If pitc of t r d is , find d pt ( )

4 . n the a f if BD Fi d ltitude AH , o the rhombus depicte d here AC

AB 8 in . and< ABD Fi , nd the area .

the h f 5. Find number of square inc e s in the surface o h h in r . s eet iron , s own accompanying figu e

ian en re e de 57. nd Ar ea a r e G s . 2 To Fi of T gl , iv Th Si

h a ea oi a n n the e u an T e r tria gle, give three sid s witho t alti

me e u e . tude, is so times r q ir d It is ob tained by taking the square root of the product of one-half the sum of the sides by o ne-half the sum of the sid es minus one of the sides by one- half the sum of the sides mi nus the second side by one-half the sum of the sides minus the third

ex e e as a n e ua n side , or, pr ss d worki g q tio ,

280 BUSI NE SS MATHE MATICS

’ SI N- LE A h na the MPSO s R U : dd together t e first ordi te, (

e en u a he a n he um the he p rp dic l r) t l st ordi ate, twice t s of ot r

na e and u me the Oddordi t s, fo r ti s n Mu sumof the e ven ordi a tes . lti ply this sum by the extreme length of the diagram and divide the result by three time s the number of parts into which the diagram is divided .

WRI TTEN EXERCI SES

’ S a e S m e as an e a the 1 . t t i pson s rul qu tion, using following notation :

A area .

L e h o i a am. l ngt . di gr

i a . y I first ord n te

yl last ordinate .

h h . etc . t e e a e y; . y3, ot r ordin t s 72 mb e a hi h the a am nu r of p rts into w c di gr is divided .

the a m a n e 2 . In cco p nyi g figur

Find the area . a e the a m a i u e meas e 3. Tr c cco p ny ng fig r , ur and find area PR ACTI CAL ME ASURE MENTS 281

T 4 . he diagram shows the cross e a un me a s ction of g t l oil ring . Find a ea b the a e m h its r y bov et od , then check by finding the area in some h m ot er anner .

— 259 . Solids . A rectangular solid is bounded by six

e an u a u a . h a r ct g l r s rf ces It is called a prism . T e bases of

m are a a e and pris p r ll l equal . PR I NCI PLE : The volume of a rectangular solid is e qual (in cubic units) to the product of the number of like units in h n its t ree dime sions .

WRI TTEN EXER CI SE S

1 . the me a e e . e m a 6 b 4 . Find volu of pi c of t l in y in by in . 2 h h W a t e ume a a 2 . 1 . b b t is vol of t nk 5ft y 5ft . y 8 ft

3. A e ha an e e De s in . e m e cub dg of t r in its volume I n cu . in .

d . ft . an and a e . . n cu , its surf c in sq in a dsq . ft . A 4 . a a a e . An e e h H cubic l t nk is full of w t r dg of t is cube is 9 in . o w

ma . . a e are the h a ny cu in of w t r in tank? If 1 cu . ft . of t is w ter weighs h h h lb . w at is t e weig t of the wa ter in the tank?

5. A e e ee . a e e h e ma e a h pi c of st l of in squ r s ction is c os n to k a l t e tool . De e m t h e he e if e th 7 1 . . 2 t r in w ig t . its l ng is cu in of stee l weighs . 8

lb . 6 r . A ba ee 2 . e n 2 m e a a of st l %in squar a d ft . long is old d into squ re bar 12 the me the bar a e m e . ft . long . Find di nsions of ft r it is old d

7. A e a a a mea e h e b 13 . b r ct ngul r t nk sur s on t e insid 11%in . y %in y h h 9 in . Compute t e number of gallons w ich the tank contains when

fi e n h n h 2 . l hi h h 1 a . ll d wit n a inc a da alf of t e top if 31 cu in . olds g

A 0 . . air er m e e a h e the a 8 . llowing 3 cu ft of p inut for c p rson in cl ss m ho w m h air m be e the m andho w ma me roo , uc ust driv n into roo ny ti s must the a ir be changed during the recitation period to insure good ventilation? a e h a how ma can be a 9. If 38 cu . ft . of co l w ig ton, ny tons put in bin

nd 6 . ee ? 12 . . e a ft long, 8 ft wid , ft d p he e h ea e h 700 lb . ha be t 10 . If 1 cu . ft . of l d w ig s , w t will w ig t of

2 e and 3 . a 3 . 3 . . 4 . a re ctangul ar mass of le d ft in long, ft in wid . ft high? 282 BUSINESS MATHEMATICS

2 h e — 60. To Find t Volume of a Prism The volume of any prism is equal to the product of its base and its a ltitude.

WRITTEN EXERCI SES

i h me a a l m h e 1 . F nd t e volu of tri ngu a r pris w os base is an e quilateral

d h h 2 . a e h se 8 in . an e e h is 1 tri ngl w o side is , w os ig t in I NT : n the a e the a e a 5257 H Fi d ltitud of tri ngl , or pply

2 the me o f e . Find volu a. st el rod whose base

a hexa h e in. if th is gon wit sid 2< , e length of the

rod is 10 ft . HI NT : A hexagon is composed of 6 equilateral

triangles .

h me o a ee h e as e he 3. Find t e volu f st l rod w os b is t form of a rhombus

h in. nd a e a e the 10 . 5 a . wit sides of , b s ngl if rod is ft long

h e a re e 4. Find t e cubic cont nts of conc t m 20 . the e retaining wall 1 ft long, di nsions of the cross se ction be ing as shown in the

5. all eel r In st const uction work, such as the ct m dern h constru ion of o buildings, t e e h the ee o m w ig t of st l is c puted . In a certain buildi ng the specifications - e u e 100 16 ft. eam a r q ir b s, right cross sec hi h h tion of w c is s own . Find the weight the eam and the at of b s cost 9¢ per lb . when

i n a e . S ee we h 4 put pl c ( t l ig s 90 lb . per

cu . ft . )

284 BUSINE SS MATHE MATI CS

6 . A h An e a ollow cylindrical can is partly fil led with water . irr gul r e e r the pi c of iron o e is placed in the water andcauses it to rise in . in

e . th h the me cylind r If e di ameter of the cylinder is 8 in . w a t is volu of the o re?

7 . ha the A e 2 n . . W cylind r is 0 ft . long a dits volume is cu in t is diameter of the cylinder?

8 . An he e e hi ne . t iron p ip is a cylindrical shell 2 in . in t ck ss If pip is

10 . and h e ame e 12 in . an 1 . . e ft long its inn r di t r is , d cu ft of iron w ig s l findthe 4 0 b . e 8 , w ight of the pipe .

262. ami Pyr ds . (a) The lateral area of any regular pyramid is e qual to the product of one half of the slant height (OH by the peri

o i the meter base . (b) The lateral area of a frustum (lower part) of a regular pyr amid is equal to o ne half of the sum of the perimeters of the

mu e b the an e h MI . bases ltipli d y sl t h ig t , ( ) (0 ) The volume of any regular pyrami d is equal to o ne third of the product of the area of the base by its a ltitude

WRITTEN EXE R CI SES

he h a e a am 24 ft . and the ase is a 1 . The slant ig t of r gul r pyr id is , b h th a e a e a ea . triangl e each side of whi c is 8 ft . Find l t r l r he m e e wa h a ami s 48 1 . h 2 . The Gre t Pyr d of Egypt, w n co pl t d, ft ig ,

e a e was 764 . . H ow ma the andeach side of its squar b s ft long ny sq . ft . in surfa ce of the sides?

re h the a a a e 3. The figu s ows pl n of squ r a u m a ami roof in the form of fr stu of pyr d , fl CD the upper base being a at de ck . is

AB 6 ft . and the he h the 18 ft . , is , ig t of a e AO the m 8 roof, or ltitud of frustu , is the e h ha the a e s AC ft . Find l ngt s t t r ft r andAE must be cut . P RACTICAL MEASURE ME NTS 285

4 . A e e h - w dg , w ose cross se ction is an isosceles triangle , base in . d n 4 ih . a a e . me . ltitud , is in long . Find its volu

The a ea th 0 . . One the 5. r of e base of a regul ar pyramid is 8 sq in of

e e xe e 1 15. andma e an a e lateral dg s (0 A in fig . of E rcis ) is in k s ngl of °

Th h . h . e me t e am 60 with the base . Find t e altitude volu of pyr id h h e a e e a 6 . Find t e volume of a triangular pyramid w os ltitud qu ls 8 ,

e h a e are 14 15and13 e e e . See andthe e , ( 5 sid s of w os b s , r sp ctiv ly h h m 40 . h a exa a 7. A stone spire is a hollow regular pyra id ft ig , on gon l

e ea h e h h 10 . The h ar the e a a bas , c sid of w ic is ft ollow p t of spir is lso h h a a e e h e h h m 6 . h a exa a hexagonal pyra id 3 ft ig , on gon l b s c sid of w ic he S e ? H o ma . . e t is 9 ft . w ny cu ft of ston in pir

263. The Cone . (a) The lateral area of a right circular cone is equal to o ne half the product of the perimeter of the base h n h h andt e sla t eig t . (h) The lateral area of a frustum of a right circular cone i s equal to o ne half the product of the sum of the perimeters of the

h n he h bases andt e sla t ig t . (0 ) The volume of a circular cone is equal to the area of its base multiplied by o ne third of the altitude .

WRITTEN EXE RCISES

he a he h a h a e 8 . andt 1 . The slant ig t of rig t circul r con is in r dius of

the a e a a ea . The a a ea . the base is 6 in . Find l t r l r tot l r b e ma e a a d. a a e u e 2 . H ow many sq . y of c nv s will r q ir d to k conic l

m h 12 . te nt whose altitude is 12 ft . anddi a eter of t e base ft

HI NT : Find slant height by 254 . ' l . " I n, h t : 01 1 6 M C “: U ! (11 1 6 U U I VUU S u a-0 3

The he column is a frustum of a co ne . ight is

th e a es 6 it . and4 . 6 in. e e e . s of b s ft , r sp ctiv ly i ndits weight .

: 2 ndthe one of revolution is 40 sq . in . a radius the slant height (b) the altitude (c) the

— r A he n id. prismatoid is a poly dro

in a a ane he a ons p r llel pl s , called t b ses, nich are e ither

a i ms, or tr p is a rectangle

1. line parallel rectangle the wedge . perpendicular bases of the prisma toid . prismatoid is equal to the sum of its h ea - e n M mu e ar of its mid s ctio ( ) , lti he altitud e or V a (B b 4M)

LITTEN EXERCISES

. b in. e e h e ase me in . 5 tee l w dg w os b as ures 8 y ,

in . one u. . h . in g 6 , if c in of steel weig s oz

BUSI NE SS MATHE MATICS

(b) The volume of a sphere is equal to the product of the a ea u a e o ne h he a u r of its. s rf c by t ird of t r di s, or 3 VS phere i t R . (0 ) The volume of a spheri cal shell (a hollow sphere) is equal to the volume of the outside sphere minus the volume of the inside Sphere whose radi us is r or V = 3 a s pherical s hell 31 7? $7! (R t ) °

WRI TTEN EXER CI SES

1 The u a e of a e me the m a he urface . s rf c til d do , in for of mispherical s h e i am 24 . m e . H man es w os d eter is ft is ade of colore d til s 1 in sq . ow y til are require d to make it? l an 2. If a e h 114 h . ha h cubic foot of ivory w ig s , w t is t e weight of a 2 ame e ? ivory billi rd ball in . in di t r n me e 3. A hollow Spherical steel shell is 1 in . thick, a dits inner dia t r

. h ar 4 the u . ft . ? is 8 in H ow much does it weigh if t ere e 90 lb . to c 2 h . e di ame e 4. If a boiler is in the form of 4 ft cylind r ft . in t r, wit hem he a e ho ma a W i h ? isp ric l nds, w ny g llons ll it old

5. he e ar h ame e r d in . res ec T r e two spheres w ose di t rs a e 4 in . an 8 , p ‘ h l . The me ea tive y Ihnd (1) The area of e a ch sphere . (2) volu of c T h a h he e sphere . (3) he relation between t e reas or t e volumes of t s two

spheres . Th he of e ame e 16 . 6. di t r of an arclamp is in H ow many square inc s u face has the a m as mi be a he s r l p, su ng it to sp re? APPEND I X

TABLES AND FORMULAS

1 . o t n a B nd ab A P r io of o T le andH ow It is Used.

20 YE AR I NTE RE ST P AYAB LE SE MI ANNUALLY

BONDS BE AR ING I NTE RE ST AT TH E RATE o r

3t% 3%

h a 5 e me ha e can I Illustrative Example . 1 . If I wis % inv st nt, w t pric ma n 20 ea ? afford to pay for a 45% bond turi g in y rs 5and the SOLUTI ON : Look in the left hand column for % follow to hea e b The m er he e e right until the column d d y nu b is t r for I can pay as high as a 6 ma i I n 20 ea t Il lustrative Exampl e 2. If % bond tur ng y rs cos s ? how much will this net the buyer 289 1 9 290 APPE NDIX

SOLUTI ON : Look in the 6% column andfollow down to then l a th and ou l fol ow cross to e left until the left hand column is found, y wil find therefore it will net The tables as used by th e bond houses are much more extens ive as to the um e a n b r of ye rs andp er ce nts .

WRI TTEN EXER CI SE S

h 1 . Fro m the above ta ble find the ra te on t e investment on bonds maturing in 20 years if bought as follows

(a ) 4% bonds bought at (b) 6% (0 )

h h n be ur 2 . Find t e price at whic bonds maturing in 20 years ca p chased to produce the following

(a) bond to yield (b) 7% 4i % (c) 5%

h h the e me o n h h a ure in 30 3. W at is t e rate on inv st nt bonds w ic m t

. h e the n : da , if boug t und r following conditio s

(a) 7% bonds bought at (b) 3% (c) 5% (d)

e at h h ma u n 2 . an 4 . Find the pric w ic bonds t ri g in 0 yr c be bought, to produce the following :

(a) 5% bond to yield (b ) 5i % (0 ) 7% (d) 6 % (e ) 5% (f) 4%

292 AP PENDIX

a a ti n 3. T ble of Decimal Equival ents of Some of th e Fr c o s of 1 Inch .

F R AC TI ON o r F R ACTI ON o r DE C IM AL ONE I NC R O N E I Nc u E QUI VA LE N T

e D — H a 4 . Table of Wag s by th e ay 8 our s to a D y.

H o u ns $2

’ h a e er 5h a be E XPLANATI ON : At t e r t of p day, ours w ges will

Similar tables may be constructe d for any commercial house at their prevailing wages .

a u as for Use in C er al 5. T ble of Form l omm ci Work .

rinci le I interest I P rt. p p ; ; 7 rate ” time

la st term se ri of ari th. es — l st term = a + (n 1) d number of terms

commo n difi erence TABLE S AND FORMULAS 293

1 d s ries ) ] sum of arith . e

rati o in a g eo m . series

sum of a gem . series

a rea 1r ab se mi-maj o r a xis Of a n ell ip se se mi-minor a xi s of a n ellipse a b nd c a re the si es of . , a d

a b c

tria ngle , s 2

c ircum ference

' dia met er Of the circl e ra di us o f the circl e dia m eter o f a ballo on

o f 1 c u . ft . o f a ir weig ht in l b .

o 1 cu . t . a s weig ht in l b . f f o f g in ball oo n

we i t t o b e ra ise inclu in gh d , d g b all o on

number o f seconds req uired fo r a b omb to fa ll from a n

13(a) . T aerop l a ne height in feet P (1 a m o unt ; P principal ; x

ra te ; 71 n o . Of yeat s a mo unt o f the princi pa l for 1: years

amo unt o f $1 for 1 yr . at the gi ve n ra te

i nte rest o n $1 fo r 1 yr . sum to b e se t aside annua ll y present val ue o f a n annua l pensio n R a nd n a s i n F o rmula 15

‘ 1; numb er of yr . before pe nsio n b egins

numb er of yr . it is t o b e paid V 5R a nd P . . a s in Fo rmula 16 . .

o r . m no . f y pre ium sho ul d b e ai in o r er t a t L p d d h ife I ns . s a ll susta in n C o . h o l o ss 1 r amount t o be paid immediately after l ast premium amo unt of premium paid a nnua fl y present val ue amount of a nnua l pensio n rat e 294 APPENDIX

po pula tio n a t e nd po pul a ti o n a t b eg inning 1 r ra te o f i ncrea se o f po pul a ti o n numb er o f do llars in de bt numb er o f years sum set a sid e a nnually rat e o f interest

t o t al v a lue pre mium pa id e a c h yea r 1 7

n t a t n pri ce o f a b o d h ha s yr. t o run

1 te it b ea rs ”— ra (S q +S r (1 +q) S r ) ?a fa ce o f b o nd (usua ll y $ 100 or P q

c ur re nt ra te of interest rate o f interest yiel d base altitude

ba

2 b b A (trapez o id) 2 2

'

A rre ula r fi ure sum o s of trian l es tra e z oi s etc . (i g g ) f A g . p d , ‘ {H hypo te nuse of a right tria ngle a al ti tude o f a rig ht tria ngle when b i s b a se

no . o f ft . o bj ec t i s a b o ve surface o f earth

no . o f mi . Obj e ct ca n b e seen

o tenuse Of a 30-60 ri t hyp . gh tri angl e ° arm o ppo site the 30 a ngl e l st e e b 2d e e dg , dg . V (rectangula r so lid) 3d e e V-vo um dg , l e area of ba se a nd a a ltit ude V (pyramid) o f pyra mid - Lateral a rea (pyramid) Qsl ant be ight x P 1, P 1, perimeter o f ba se V (c o ne) 1 area o f ba se X altitude A co ne sl a nt ei t circu ere n e o L . . ( ) a h g h X mf c f base a a l t . ; B a rea of l o wer ba se V (prismato id) a a (B b 4 M ) area o f upper ba se ar ea o f mid-sectio n A (sphere) ra dius o f the sphe re V (sphere ) R radi us o f o utsi de of — 3 V (spherical shell) 7T(R 3 r ) radius o f insi de of shell

296 APPENDIX

that is inch ; inches a . insur nce instant ; the present month quart interest rod i 'e . nvoic received

i e e am . nv ntory r R e h ma Ma . ic s rk ; rk keg kegs shilling; Shillings link ; links South d . sec . e lb pound ; pounds o o o o o o . s con lis t price September March settlement

m ha h me . erc ndise s ip nt S h e . Gentlemen ; irs . s ipp d m m a e . ile ; iles sign ture ; sign d a e ha minute ; minute s . squ r c in month ; months square foot Mister s quare mile a e Mistress . squ r rod North -square yard number torn November tub October township

e ma a e o o o o o o o o o o on d nd tr nsf r corre ct treasurer Oll Il cc; ounces las t month b a page via . y w y of

a me ame . p y nt n ly ; to wit piece ; pieces volume paid wee k h e h . b t e b . y ; y w ig t per centum ; by the yard ; yards hundred year : years TABLE S AND FORMULAS 297

TAB LE SYMB OLS

thousand inch; inches ; o nds greater than less than andSo on multiplication at ; to number if wri tten care of before a figure ; cent ; cents pounds if written che ck mark after a figure degree One and one four th r division . pe cent

. te dollar ; dollars . pounds s rling eq ual ; e quals since fo ot ; feet ; minutes subtraction

300 INDEX

C mm e a e e h and D a o oditi s , t bl of w ig ts r fts , mea e 252 Dr mea e a e 253 sur s , y sur s, t bl , C m e e a a b o pound int r st, c lcul tion y a hm 223 log rit s , E C e 285 on , C e a e en 9 1 ffi e a e em 2 onv rsion t bl s, curr cy, E ci ncy w g syst , 9 — 253 me age em 29 — E rson w syst , C e 13 21 h e h and mea e ost of s lling , Englis w ig ts sur s , C e 250—253 ub roots , c m e b e l e 246 m a e me 259 261 o put d y slid ru , co p r d to tric , , a e h e and x ha e t bl s owing pow rs roots , E c ng , 29 1 me 77 do stic , C m ea es a c e a e 90 ubic sur , c pt nc s , h 251 l s ex ha e 93 Englis , bi l of c ng , m e r 259 a 100 t ic , bills of l ding , — — C e a e a e 9 1 92 a 83 90 urr ncy, t bl of v lu s , , dr fts , 253 ex e m e e pr ss on y ord rs , 79 C n e me 283 me h o f a me 77 yli d r, volu of, t ods p y nt, a m e e 78 8 post l on y ord rs , , 9 e e a h m e e 8 1 t l g r p on y ord rs , e m 8586 t r s , , Dai ba a e e e 124 e n 8586 ly l nc s , int r st, for ig , , Da - a e a e em 23 l ex ha e 93 y r t w g syst , bi ls of c ng , e ma a e e a e e a es 9 1 D ci ls, t bl of quiv l nts, curr ncy v lu , 292 mme a 100 — co rci l bills , Den m a e m e 250 26 1 e a e 9 1—92 250 o in t nu b rs , conv rsion t bl s , , a 255 253 ddition , ' 256 a e 93 division , b nk rs bills , mu a 256 o f a 100 ltiplic tion , bills l ding , e h h e 254 e e e 96 r ducing to ig r, l tt rs of cr dit, ed e 253 ar ex ha e 92 r ucing to low r, p of c ng , a 255 s a m e e 98 subtr ction , po t l on y ord rs, a e h i e s and a 94 t bl s ow ng pow r roots , quot tions , 29 1 a e 94 r t s of , ’ De a 46 a el e h e 98 posits , b nks , tr v rs c cks , De e a x e e i e em 11 pr ci tion , E p ns s , bus n ss probl , m a 50 x e e e 13—17 co put tion , E p ns s , s lling , me h s x e a hm 203—205 t od , E pon nts , log rit s , ec ea a e a x es m e e 7 d r sing r t on origin l E pr s on y ord rs , 9 a e 52 v lu , xe a e e ea a e fi d r t on d cr sing v lu , F 53 xe a e i a a e e s a e 57- 64 fi d r t on or gin l v lu , Fir in ur nc , 52 rm a Fo ul s , a h e 51 a e 292—2 4 str ig t lin , t bl of, 9 ifi erential a e a e em 25 e n ex ha e 9 1 10 D r t w g syst , For ig c ng , 0 (See " a x ha e e Division , lso E c ng , for ign ) m e b e e 242 a co put d y slid rul , Fr ctions , en m a e m e 256 h rrietho ds 189 d o in t nu b rs , s ort , e a e 67 a l e e ma e lif insur nc , t b of d ci l quivalents me h 187 292 short t ods, e mea e 253 a e al a e 7 - Doz n sur s, Fr t rn insur nc , 0 72 INDEX 301 — Insura—nce Conti nued life Conti nued e me r e s — G o tric p ogr ssion , 200 202 a c e 68 — lo ns on poli i s, e me 262 288 a 5 G o try, ed 72 7 “ occup tion of insur , , Se e a Sa e a —u es 68 Goods ( lso l s p id p polici , ma ng 13 a me ene ia es 69 rki up , p nts to b fic ri , ra hi e e a 135- 171 i G p c pr s nt tion , po ic es , 64 m 136 c es a e 7 For s , poli i l ps d , 0 c e 138 l c e e 73 cir l s , po i i s, scop of, c m a em m 66 o p risons, pr iu s , e 153 is 74 curv s , r ks, me 150 me 74 involving ti , wo n , m e 142 c a a ha a 73 si pl , o cup tion l z rds, 137 e a e 75 con ruc ion of , st t — ov rinsur nc , e 145156 160 e e 34—4 curv s, , Int r st, 9 e e ha s 161 a 118—126 fr qu ncy c rt , ccounts , ma 160 a 118—126 ps , b nk , e 135 a e 289 obj cts of, bond t bl s , ec a e 140 236 r t ngl s , bonds, m 43—49 co pound , H a a e t 46 nnu l d posi s , rn utat n b a hm co p io y log rit s , H a e -R a a e em 28 ls y ow n w g syst , z23 H a a c a na 73 48 z rds, oc up tio l , sinking funds, H h e e e 254 a e 44 ig r numb r, r ducing to , t bl , H e e e e 266 a a a e 124 ypot nus d fin d , d ily b nk bal nc s , e e 34 d fin d, I m h 35 for ont s , ea 35 for y rs , c me tax 111—117 al a a a i In o , post s vings b nk ccount ng , he a e a xes 108 124 In rit nc t , — e e a 110 a an a 1 18 126 f d r l, s vings b k ccounts , a e 108- 110 h me h m 36 st t , s ort t ods of co puting , su a e 56—76 m e In r nc , si pl , a e 72 e ee a e 42 ccid nt, b tw n d t s, 57—6 4 b me 35142 fire, y ti , , em m 59 e a 209 computation of pr iu , Int rpol tion , e 59 e e 9 polici s , Inv ntori s , m a e 59 premiu r t , h a e a e 6 1 s ort r t —t bl , a e a 70 72 fr t rn l , al 75 Lan mea e industri , d sur s , 56 h 250 251 Englis , , kinds o—f e 64 76 e 259 lif , mtric , re Le e ed 96 age of insu d , tt rs of cr it, — — e a a 227 235 L e a e 64 76 annuiti s , c lcul tion , if insur nc , c a e 73 a e a l a 227- 235 benefi i ri s , nnuiti s c lcu tion , o licl eS 68 L e a e 233 ca sh surrender p , if t bl s , em m 66 L e er e u a a e 264 computation of pr iu s, in , p p ndic l r of ngl , es 74 L nea mea e crippl , i r sur s , m al c m a sh 250 dividends of utu o p Eng li , e 67 me 258 -261 ni s , tric, 302 I NDEX

L mea e Mea e — Conti nued iquid sur s, sur s h 252 me 257 258 Englis , tric , , me 259 me 253 tric , of ti , L a a a See B a o n ssoci tions ( uilding qu ntity, ” and a a s a a 251 lo n s oci tions ) s ilors , L a i a e 10 1e 68 a e o ns , on nsur nc pol s , squ r , L a i hm h 251 og r t s, Englis , a e a a e b 227—235 me 258 nnuiti s, c lcul t d y, tric , a a hm 2 12 e an ntilog rit s , surv yors long or l d , a a 223—237 h 2 51 pplic tions, Englis , e e s ma ri 236 me 259 bond int r t to tu ty, tric , B a mm em e a e 251 riggsi n or co on syst , surv yors squ r , 207 a e t bl s , ha a e e e 206 h 250—253 c r ct ristic d fin d , Englis , m e e a a me 258—261 co pound int r st c lcul tions, tric , 223 Me em 257 258 tric syst , , “ " ex e 203—205 M e ee e pon nts , on y (S Curr ncy ) er a 209 M e e 78—82 int pol tion , on y ord rs , ma a e e 206 M h e e a a s ntiss d fin d , ont s, int r st c lcul tion a 206 5 not tion , 3 a a 225 M a e tax 1 11 sinking fund c lcul tions, ortg g , — em 205209 M u a syst s , ltiplic tion , em h ame a e 205 m e b e e 241 syst s wi—t s b s , co put d y slid rul , a e 2 14 217 de m a e m e 256 t bl s, no in t nu b rs , ex a a 209 h me h 177—186 pl n tion , s ort t ods , a e a 2 11 proportion t p rts , e m 206 t r s , “ Loss or gain (See Profit and loss " a eme st t nts ) N a a hm 206 ot tion log rit s , L e n m e e g 253 ow r u b r, r ducin to , Numbers (See Denomina te num " M bers )

Man a e e 206 tiss , d fin d , M a ri e mea s e 251 n ur s , a Ma -u 13 Occup tions , rking p goods, ha a 73 M ea e z rds of sur s, e 72 75 a a 251 of insur d , , ngul r , e h ea m a es co a a Ov r d , co puting s l st, c p city, h 252 13 Englis , me 259 tric ,

cubic , i h 251 Engl s , me 259 a al e am tric , P r l logr , e 253 a ea 269 curr ncy, r of , 253 e e 265 dry, d fin d , ea P a - l 32 lin r, y ro l slips , n h 250 e en a e E glis , P rc t g , me 258 a e a e a e 5 tric , v r g s l s , u ea e and e ea e a e 4 liq id , incr s d cr s of s l s , h 252 and 1 Englis , profit loss , met 259 a e 20 ric , profit on s l s,

INDEX — Sa e Conti nued S a e 272—275 l s qu r root, e hea ex e e 13 m e b e e 243 ov r d , p ns , co put d y slid rul , er e 20 e h w e s and p c nt of profit on , tabl s o ing pow r roots , percentag e of increa se and de 29 1 ea e 4 S a cr s , ubtr ction , and a eme 1—12 h me h s 174 —177 profit loss st t nts , s ort t od , e e e 5 e mi a e m e 255 r turn d goods r cord , d no n t nu b rs , e r e 13 S a es s lling p ic , urf c , a a e e 2 a ea 279 t bul t d r cords , r of , Sa in a 1 18 e e 264 v g s b nks , d fin d , n e e a c 1 18—126 S e n mea e i t r st on c ounts , urv yors lo g sur , ta 124 h 251‘ pos l , Englis , Se r e a a 13—2 1 me 258 lling p ic , c lcul tion of , tric , Sha e S e a e m ea s e 251 r s , urv yors squ r ur , l and a S m " a e 297 profit on , bui ding lo n y bols, t bl of, a a 128 ssoci tions , e es a a d a s ri pl n , building n lo n a a 128 ssoci tions , w h a a a e and it dr w l v lu , building a a a 128 a e lo n ssoci tions , T bl s , Sh me h 172—1 a e a 295—296 ort t ods , 90 bbr vi tions , a 172 a l a mea es 251 ddition , ngu r sur , 187 A he a e e h 252 division , pot c ri s w ig ts , ‘ a 189 a e h 252 f r ctions, voirdupois w ig ts , e e 36 289 int r st , bond , m a 177—186 m a a e e h 252 ultiplic tion , co p r tiv w ig ts , s a 174—177 m net 18 ubtr ction , co puting profit , S m 20 inking funds , co puting profit, a a b a hms 225 mea s e c lcul tions y log rit , cubic ur s , cu a b m er h 251 cal l tion y co pound int Englis , est 48 me 259 , tric , S e e 238- 249 e a es 253 lid rul , curr ncy v lu , e and e b 246 ec ma e a e s me cub s cub roots y, d i l quiv l nts of o e 239 the a 1 h d scription , of fr ctions of inc , b 242 292 division y, h and use 238 dr m a e 253 istory , y e sur s , m a b 24 1 e e 19 ultiplic tion y, finding s lling pric , ea 240 fi re a e h a e 62 r ding of, insur nc s ort r t , — a e b 243 rm a 292 294 squ r root y, fo ul s, S 28 1—288 e e 4547 olids , int r st , , e 285—286 e 233 con , lif , e 283 ea mea es cylind r, lin r sur , r ma 286—287 h 250 p is toid , Englis , m 28 1—282 me 258 pris s , tric , am 284—285 mea e pyr ids, liquid sur s , he e 287—288 h 252 sp r , Englis , 8 —288 me 259 S he e , 2 7 ric , p r t — a e e e 265 a hm 2 14 217 Squ r , d fin d , log rit s , S a e mea e mea e h 250—253 qu r sur s , sur s , Englis , l h 251 me r 258—261 Eng is , t ic , me 258 me and h a e co m tric , tric Englis v lu s — e 251 a e 259 261 surv yors , p r d , INDE X 305 — — able s Conti nued Triangle Conti nued — e and 29 1 r h 275277 pow rs roots , propo tions of rig t , a mea e 251 i h 265 s ilors sur s , r g t, a e mea e ei h 252 squ r sur s , Troy w g t, l h 251 Eng is , me 258 V tric, mea e ur eyor long or land ur , s v s s alua ion (See De recia ion l h 250 V t p t Eng is , e ha e i and a Valu , s r s , build ng lo n me 259 tric , a a 128 ssoci tions, e a e mea e 251 surv yors sq u r sur , e c e 285 o um of on , m 297 V l sy bols, me e 283 Volu of cylind r, tax a e 106 — t bl s , me sma o 286 287 Volu of pri t id , me mea es 253 ti sur , me am 284—285 Volu of pyr id, r e h 252 T oy w ig t, me he e 288 Volu of sp r , e S h w da 292 wag s for o y, ei h 250- 253 w g ts, W h 250—253 Englis , — me 258 261 e a me t tric, Wag p y n s, mm e 252 29 of co oditi s , bonus, a e em 27 Task and bonus w g syst , mm 31 — “ co issions , axe 102 117 S ee a me e mem a m 33 T s , ( lso Inco curr ncy or ndu , " “ " tax h er a e tax , da - ra e em 23 , In it nc y t syst , " M a e tax eren a a e em ortg g ) diff ti l r t syst ,

as es sme a e me h , e e fi e em 29 s nts, st t t ods Em rson f ci ncy syst , 103 l - em m a e H a ey Rowan pr iu r t , e e 102 g d fin d , é r e m a 103 - i 32 p op rty, co put tion of, pay roll sl ps, 108 e e em 24 pi c work syst , e 102 e 8-h da 292 purpos of, tabl for our y, e e a h m e e 81 and em 27 T l gr p ic on y ord rs , ta sk bonus syst , me We h Ti , ig ts , e e a u a 3542 heca es 252 int r st c lc l tion for, , apot ri , a e meas e 253 252 t bl of ur s, avoirdupois, e a ea 272 m e 252 Trap zoid , r of, com oditi s , ' a e e he 98 m a a e 252 Tr v l rs c ck , co p r tiv , a e ish 250—253 Tri ng l , Eng l , ea 270 278 e i 258—261 ar of, , m tr c , e e 265 a e 250—253 d fin d, t bl s , a e a 266 252 equil t r l , Troy, e 266 hypo tenus , 6 Y isocel es, 26 30 ° 266 e es a l a 35 o f 277 Y ea rs , int r t c lcu tion for, proportions ,