BASIC FORMULAE & DEFINITIONS FOR AN INTRODUCTORY COURSE IN BUSINESS MATHEMATICS INTEREST r------__ ~ Amount required to cover operating expenses and make a Exact Number of Days: Interest is Compound Int erest: Interest that C profit calculated over a time period, from when a is paid on both the principal and the 1 • cost of goods is what the retailer pays the supplier or manufacturer loan is given to the end date; day the loan is interest accumulated from past for goods due mayor may not be included in the period periods; interest gained is added to " • selling price of goods is what the customer pays the retailer for • To determine loan period, the actual number the original investment 2 goods (i.e., the buyer's purchasing price) ofdays ill each month should be known • maturity value = p x (I + i)n .. Tip: 30 days hath September, April, • cost + markup = selling price • wherc p = principal; i = interest rate in , June, and November; all the rest • markup (or margin) = selling price - cost percent per period; n = number of have 31, excepting February alone, periods which hath but 28, in fine, till leap • alternative formula: maturity value • percent markup on cost = markup + cost year gives it 29. (future value) = p x (I + rln)nt • percent markup on cost = percent markup on selling price + • Leap years are divisible by 4, which is • where t = time in years; II = number of (I - percent markup on selling price) why 2004 was a leap year and also why an periods per year; r = interest rate in extra day is added to February in every percent per year leap year • percent markup on selling price = markup + selling price compounding interest is • percent markup on selling price = percent markup on cost + Simple Interest: Time is expressed in the period is calculated every n (1 + percent markup on cost) same units as the rate; if r is an annual rate, then t is in years annual • simple interest (i) = principal (p) x rate • markdown = original selling price - marked-down price in percent (r) x time (t) • markdown percent = markdown + original selling price quarter 4 • maturity value = principal + interest • markdown = markdown percent X original selling price IDGIIdl • p is principal of the loan (or borrowed) amount or invested amount (face value); day day 365 • selling price per unit = [total cost of all units - percent markup i is the amount of interest paid for the Present & Future Value .. on cost (total cost of all units)] + [total units - total spoiled loan or earned on the investment; r is the • present value (PV) = value of the , units] percent rate of interest paid for the use loan or investment today C of someone else's money or earned for • future value (FV) or maturity value . lending the money DISCOUNTS = final amount of the loan or l • exact interest: time = t = investment at the end of the last ~ Discounts: Reduction to a basic price exact number of days period II I Trade Discounts: Discounts given to partners in the distribution 365 Simple Interest Future Value 2 channel of goods; also called functional discounts, and are given to • ordinary interest (Banker's Rule): • FV = PV x (I + (i x n)] .. distribution channel members to perform specific tasks . _ _ exact number of days • where i = interest rate per period; n = , • net price = list price - trade discount amount tIme - t - 360 number of periods Cash Discount: Discounts given to customers for paying with cash; Partial Payment Rule: Any partial loan Compound Interest Future Value cash discount is also called sales discount by seller or purchase payment is first used to pay the interest • FV = PV x ( I + i )" discount by buyer that has accrued (total interest to date), and • where i = interest rate in percent per • cash discount = selling price (or invoice amount) x cash the remainder is used to reduce the period; n = number of periods discount rate principal of the loan Interest Earned = FV - PV • amount buyer pays = selling price - cash discount Terms: Cash discount rate is usually stated in credit terms; credit DEPRECIATION term 2/10 on an invoice means that a 2% cash discount is allowed if Depreciation: Loss in value of • residual value (or salvage value, scrap value) = the payment is made within 10 days of the date of the invoice; cash tangible business assets or cash value of asset at end of useful life discount period can a/so begin when the buyer receives the goods property (excluding land) over its (ROG = receipt of goods) Straight line Method: Depreciation expense is useful life due to deterioration, equal over eaeh year of its useful life obsolescence, etc.; also called • depreciation expense per year = PAYROLL depreciation expense; periodically (asset cost - residual value) charged to operating expenses; total estimated useful life of asset in years Pay Based on Hourly Wages: Pay is based on hours ofwork done depreciation limited to cost of • gross pay = (number of regular hours x regular hourly rate) + • partial-year depreciation expense = property ~ (number of overtime hours x overtime hourly rate) depreciation expense per year x • accumulated depreciation = number of months of useful life in the year C Pay Based on Piece Work: Pay is based on acceptable pieces total amount of depreciation 12 of work produced to date 1 Units of Production Method: Depreciation III • gross pay = total number of acceptable pieces produced x • cost of asset = cost paid for expense based on the usage of the asset II i piece work rate asset, including freight • depreciation rate per unit = 2 Pay Based on Commission: Commission is a percentage of sales; • book value of asset = (asset cost - residual value) .. cost of asset - accumulated usually paid to the person generating the sales estimated total number of units , • gross pay = commission = sales x commission rate depreciation produced over useful life of asset 1 • depreciation expense per year = depreciation rate per unit X number of units produced per year Cost of Goods Sold Service Hours Method: Depreciation expense based on hours of useful service • cost of goods sold = cost of goods available for sale - cost of ending inventory • depreciation rate per service hour = Weighted Average Method: Used to calculate cost of ending inventory when the (asset cost - residual value) goods available for sale were purchased at different costs at different points in time estimated total number of hours of . . cost of goods available for sale useful service over useful life of asset • weIghted average cost per umt = number of units available for sale • depreciation expense per year = depreciation rate per service hour • cost of ending inventory = units in ending inventory x weighted average x number of service hours per year cost per unit Sum of Years-Digits Method: Depreciation expense is greater for First In-First Out (FIFO) Method: Used to calculate cost ofending inventory when earlier years than for later years the goods available for sale were purchased at different costs and at different points in • sum of years-digits = sum of the digits representing year ofuseful life time; assumption is that goods purchased earliest into inventory are the ones that are OR sold first; goods in ending inventOlY are those that were purchased most recent~v sum of years-digits = N (~+1), where N is the number of years of • cost of ending inventory = units in ending inventory X their corresponding costs useful life Last In-First Out (LIFO) Method: Used to calculate cost of ending inventory • for an asset with six years of useful life, sum of years-digits = 1 + 2 + when the goods available for sale were purchased at different costs and at different 3 + 4 + 5 + 6 = 21, or 6(;+1) =4{=21 points in time; assumption is that goods purchased most recently into inventory are the ones that are sold first; goods in ending inventory are those that were purchased • depreciation expense per year = (asset cost - residual value) x the earliest remaining useful life in years • cost of ending inventory = units in ending inventory x their corresponding sum of years-digits costs Declining Balance Method: Depreciation expense declines steadily over the useful life of the asset Inventory Turnover: How often a business sells and replaces its inventory; usually over a year • depreciation rate for double declining balance method = (100% 7 estimated number of years of useful life of the asset) x 2 • inventory turnover at retail = .net sales . average IOventory at retaIl • depreciation expense per year = book value of asset at the beginning of the year x depreciation rate • average inventory at retail = • book value of asset at the beginning of a year = book value of asset beginning inventory at retail + ending inventory at retail at the end of the previous year 2 • book va~ue at the end of a year = asset cost x (1 - depreciation . cost of goods sold • mventory turnover at cost = . t t t rate)n; n = estimated number of years of useful life of the asset average IOven orya cos • average inventory at cost = beginning inventory at cost +ending inventory at cost 2

BASIC FINANCIAt REPORTS Inco me St ateme nt (Profit and Loss [P & L! Statement): A financial report of a business that shows netprofit or loss for a specific period by reporting revenue and expense items during that period of operations Income St atement Items Sales Tax: Tax paid on purchase of most goods and services, though • revenue from sales (or revenues, sales, income, turnover) some are exempt from sales tax; it is applied to the net price (selling • sales = number of items x Oist price - trade discount) price - trade discounts) but not to shipping charges; sales tax varies between states; collected by the business and paid to the state • net sales = sales - sales discount or cash discount • cost of goods sold COGS (or cost of sales) is the amount a product cost to government produce • sales tax = net price x sales tax rate • COGS = net purchase price + cost of acquiring, preparing and placement of • purchase price = net price (1 + sales tax rate) goods for sale • actual sales = total sales • gross profit on sales (or gross profit) = net sales - cost of goods sold 1+sales tax rate . gross profit Excise Tax: Tax paid on specific goods and services, such as luxury • gross margm percent = net sales x 100 automobiles, gasoline and air travel • excise tax = net price x excise tax rate • operating expenses (including general and administrative expenses IG & AI) = expenses to manage the business, and include salaries, legal and professional fees, Property Tax: Levied on the assessed value of property by local utilities, insurance, stationery supplies, property and payroll taxes government to pay for services such as schools, fire and police services; • sales and marketing expenses = expenses needed to sell products, and include assessed value is a fraction of actual market value of the property that is sales, salaries and commissions, advertising, freight and shipping used for tax purposes • R&D expenses = expenses incurred in research and development • property tax rate = estimated revenue from tax • operating expense = G & A expense + sales & marketing expense + R&D total taxable assessed value expense OR • earnings before interest, taxes, depreciation and amortization (EBITDA) budgeted need of local government OR property tax rate = total taxable assessed value operating income = gross profit - operating expense • assessed value = market value x assessment rate • operating margins percent = x 100 • property tax = assessed value x property tax rate ;!IJa?:S • mill rate: a mill is 1/1000 of a dollar or 0.001 dollar; tax rate in mills • earnings before interest and taxes (EBIT) = EBITDA - depreciation and is the tax per $1,000.00 of assessed value amortization expenses 2 • earning before taxes (EBT) or pretax net income = EBIT - interest expenses • taxes include federal, state and local government taxes on income Series of periodic payments usually made in equal amounts; payments computed by compound interest methods; payments made at equal intervals oftime • net income (or earnings) = EBT - taxes • profit margin = net income x 100 Period of time between two successive net sales payment dates Balance Sheet Time between the beginning ofthe first payment period and the • assets - liabilities = owner's equity or shareholder's equity end of the last payment period • assets are items on a company's books that have a positive monetary Future dollar amount value; they typically include items of obvious value, such as cash or of a series of annuity payments and the accrued interest equivalent investments (treasuries, CDs, money market), accounts receivable, prepaid expenses, inventory of finished goods that are ready • annuity certain: term of annuity begins and ends on definite dates; has a for sale, depreciated real estate and equipment, and other intangibles, specified number of payments such as goodwill, copyrights, trademarks and patents • contingent annuity: term of annuity begins on a definite date, but ending date • liabilities are monies owed; they typically include accounts payable, bank is dependent on a future or uncertain event; no fixed number of payments and bond short-term debt (to be paid off within a year), and long-term debt • perpetual annuity: term of annuity begins on a definite date, but has no ending Basic Financial St at ement Ratios date; length of term is infinite • liquidity ratios: measures ofability jilr a business to meet short-term D obligations • ordinary annuity: periodic payments are made at the end of each payment • current ratio = current assets -;- current liabilities period • quick ratio = cash + accounts receivable -;- current liabilities • deferred annuity: periodic payments are made at the end of each payment • activity ratios: measures ofefficiency in generating sales with assets period, but the term of the annuity begins after a specified period of time • days inventory = . 365 lOventory turnover • annuity due: periodic payments are made at the beginning of each payment period • collection period = acco~nts receivable credIt sales per day Je of an Ordinary Annuity • using annuity tables . cost of goods sold = . • future value of ordinary annuity annuity payment per period x ordinary • mventory turnover average lOven t ory = annuity table factor • asset turnover = net sales • using formula total assets • future value of ordinary annuity = annuity payment amount per period x • profitabili!y ratios: measures ofreturns J .return on sales = net income [<1+ij"-1 net sales • where i = interest rate per period; n = number of payments during term of .return on assets (ROA) = net income annuity total assets Je of an Annuitv Due • return on equity (ROE) = net in~ome eqUIty • using annuity tables • add 1 to the number of periods, and then read the table earnings available to • future value of annuity due = (annuity payment per period x ordinary common stockholders • earnings per share = -=-n~u=-=-m~b-"'e':":r--=o"-f:;;=sh;="'a-"'re-"s~o"'f';C'---- annuity table factor) - (one annuity payment amount) common stock outstanding • using formula price per share of common stock • future value of annuity due = annuity payment amount per period x • price to earnings (P/E) ratio = earnings per share O+i)O+I-IJ [ i - (one annuity payment amount)

• where i = interest rate per period; n = number of payments during term of LIFE INSURANCE annuity Life Insurance: Insurance that pays a specified sum to the policyholder's Ordinary Annuity beneficiary at the time of the policyholder's death • using annuity tables • present value of ordinary annuity = annuity payment per period x present • insured: person covered by policy • policyholder/policy owner: person who owns policy value of ordinary annuity table factor • premium: periodic payments made for insurance coverage • using formula • face amount: proceeds received on the death o(the insured • present value of ordinary annuity = annuity payment amount per period x • beneficiary(ies): person(s) who receivers) the face amount [t-(IiO-oJ Types of Life Insurance • where i = interest rate per period; n = number of payments during term of • term life is life insurance coverage for a specified period oftime; can be annuity at a guaranteed rate or a guaranteed rate for a period of time and then a projected rate; no cash value except face amount in event of death of Fund into which periodic deposits are made so that th e principal insured within the period of the insurance is repaid on the maturity date (i.e., the amount of the annuity is the value of the • whole life is life insurance that has a guaranteed level premium (i.e., no principal of the debt on the maturity date); deposits need /lot be ofequal amounts increases in premium) and a guaranteed cash value; also called or made at equal intervals oftime; interest for the debt is not paid from the fund straight life or ordinary life • using sinking fund tables • universal life is life insurance that is permanent; premiums are not • sinking fund payment per period = future value x sinking fund table factor guaranteed (i.e., may go up or down) • using formula Calculating Premiums: Using insurance tables; read tables according to • sinking fund payment per period = futUre value x [( i)1I J 1+1 -I age and gender of insured; insurance rates are generally per $1,000.00 of coverage • where i = interest rate per period; n = number of payments during term of • premium = (coverage amountll,OOO) x insurance rate annuity A Used mainly for consumcr loans; where m = number of payments in one year, n = total number of scheduled payments in life of loan, C • monthly payment = [rate + ( r~~~n1h' 1X principal • 1+ rate -1 = finance charges per payment period, P = principal or original loan amount Ratio of the jinance charge • constant ratio method: APR = P~:~O to the average amount 0.( credit in use during the life ofthe loan; ted by expressed as a percentage rate per year; is a true cost of a loan; • direct ratio method: APR = 3P(n + (n + I ) ~H-CC a meant to prevent lenders from advertising a low rate by hiding . mC(95n+9) fees; rules to compute APR are not clearly defined • n ratio method: APR = 12n(n+l)(4P+C)

'fS are Stocks: Shares of ownership in a company • yearly interest = face value of bond x yearly interest rate 6 • common stock gives the holder voting rights . Id - yearly interest • preferred stock does not allow the holder to have voting rights, but • current Yle - bond price instead, otTers preference in dividend payments 1. 6.7, b d . Id total yearly interest • dividends are payments to shareholders from projits • on Yle = bond price . h earnings available to shareholders • earnmgs per s are = total number of shares outstanding Mutual Funds: Monies invested in multiple entities (shares = ownership, similar to stocks) PIE' . I . . closing common stock price • net asset value (NAV) is dollar cost of one share of the mutual fund or • ratIO = pnce earnmg ratIO = earnings per share price per share ofthe mutualfund ur the • then k . Id d"d d . Id yearly dividends per share • NAV = total value of portfolio • stoc Yle = IVI en Yle = common s t oc k prIce . number of shares outstanding in the fund I - ending price+total dividend income received _ 1 • If d . Id _ income distribution per share • tota return - b egmnmg.. prIce. mutua un Yle - NA V Bonds: Promises of payment for monies loaned • tIt _ ending NAV +total distribution per NAV _ I to a re urn - initial NAV • bondholders are creditors

• EX: 5/ 2 is 5 divided by 2, which is 2 with a remainder of 1, Whole Numbers: Set of all positive integers (1, 2, 3, ... ), zero (0), and resulting in a mixed number 21/2 negative integers (-I, -2, -3, ... ); integers are whole numbers • EX: 313 is 3 divided by 3, which is 1 with a remainder of 0, • numeric representation: $8,614,757,210,943.36 was the U.S. National resulting in a whole number 1 Debt on 12/27/06; National Debt is the amount of money that the U.S. Mixed Number to Improper Fraction Treasury Department has borrowed to date in order to meet Congress's • improper fraction = expenditures beyond its income eq ual (denominator of fraction part X whole number part) • in words: eight trillion. six hundred and fourteen billion, seven hundred + numerator of fraction part and fifty-seven million, two hundred and ten thousand, nine hundred and denominator of fraction part forty-three dollars and thirty-six cents (6X3)+5 23 8 614 757 210 943. 3 6 • EX: 3~ 6 6

~~"f,{Jj'!N 1 ~ ":\l'4~t!!t!~ H'w,' I\',f,j, I(.~ :: ~~(J!,(i!tt -J" t "fi,?l r" ,m, !" ~'~ 1\'1( ~~ 'IJ' )i~"'IC ;"':-1-' ';1; .t~'J -," ',' :;" N;l Fraction Operations It the (, (. I ~ '!H....J"i. i1(J#IJ."',"A.m'~" i~~~k.W.~'-icf"'1!· ';"-'i."r"GiJ",,'i'1.YlJ!lII"~if.B~JitJuS. {.l.'Jti'-G1l;',::iJ!tii:mt. • reduction: converting the fraction to higher or lower terms by multiplying d has or dividing the numerator and denominator by the same number (any numerator (number written above the line) number other than zero); value ofthe fraction does not change by S2 • denominator (number written below the line) • EX' 3x4_!1. . 1-4-4X4 -16 Types of Fractions • proper fraction: numerator is less than the denominator • EX' 21- 2177_1 . 28 - 2877 - 4 • EX: tori • lowest terms: when the numerator and denominator of a fraction do • improper fraction: numerator is greater than or equal to the denominator not have a common divisor; also called simplest form • EX: ~ori • EX: i or 1~ • complex fraction: either the numerator, the denominator, or both are a • GCD or HCF: divide the numerator and denominator of a fraction by fraction their greatest common divisor (GCD) to reduce it to its lowest terms; GCD is also called the highest common factor (HCF) • EX' Ys • EX: GCD or HCF of 63 and 294 is 21 '% • to calculate: Mixed Number: Consists of a whole number and a fraction • step 1: divide the larger number in the fraction by the smaller number • the sum of the two numbers (whole number + fraction) (divide 294 by 63, quotient 4, remainder 42) • EX: A sum of 5 and 3/4 is written as the mixed number 53/4 • step 2: if there is a remainder in step I, then divide the smaller number in the fraction by the remainder in step I (divide 63 by 42, Converting Fractions: Improper fractions may be turned into whole quotient I, remainder 21) -9 numbers or mixed numbers • step 3: ifthere is a remainder in step 2, then divide the remainder in step • divide the numerator by the denominator; if there is a remainder, then the I by the remainder in step 2 (divide 42 by 21, quotient 2, remainder 0) result is a mixed fraction; if the remainder is zero, then the result is a • step 4: continue dividing each remainder by its succeeding whole number remainder until the remainder is zero lilill Review Skills Basics • step 5: the last divisor (the last non-zero remainder) is the GCD or • division by multiples of 10: move the decimal to the left by the same HC F (which is 2 1) number of spaces as the number of zeros • 63 = 63721 ~21-l • EX: 0.27 -i- 1,000 = 0.00027 294 294' -14 • multiplication by multiples of 10: move the decimal to the right by the • LCD or LCM: lowest common denominator (LCD) of a group of same number of spaces as the number of zeros; add zeros if there are no fractions is the least common multiple (LCM) of the denominators of digits to the right those fractions • EX: 0.27 X 1,000 = 270 • EX: Calculate the LCD or LCM of 8, 24 and 45; when there is no Percent: To convert any whole number or decimal number to a common factor in a group of numbers, then the LCM is the product percentage, move the decimal point two places to the right (adding zeros ofthe numbers; LCM is 8 X 24 X 45 = 8,640 if necessary) and add a percentage symbol (%) at the end of the number • EX: Calculate the LCD or LCM of 6, 15, 42; when there are • rounding percents follows the same rules as rounding decimals [see common factors in a group of numbers, then the numbers are Decimals] repeatedly divided by their common prime factor; at least two • EX: 2 is 200%, 0.15 is 15%,0.2846 is 28.46% numbers should be divided in each step; the LCM is the product of Conversions the prime numbers and the final quotients: 2) 6, 15,42 From To Rule Example 3) 3,5,21 Fractions Decimals Divide and round as needed 1,5,7 Move decimal poiat two P to LCM = 2 X 3 X 1 X 5 X 7 = 210 DecimaIa Percents the riabt aDd add za'OI if ~ Adding Fractions: The denominator of the sum is the least common tMa Idd perceat symbol ( mUltiple (LCM) of the individual denominators, and the numerator of Move decimal point two pl aces to the sum is the sum of the individual numerators Percents Decimals the left and add zeros when needed, 64.48% is 0.6448 • add the integers and the fractions separately when adding mixed then delete percent symbol (%) numbers • EX: 3/4 + 2/3 + 6/7 3/4 = 3 X 21/4 X 21 = 63/84 , 2/3 = 2 X 28/3 X 28 = 56/84 , 617 = 6 X 12/7 X 12 = 72/84 Basic Algebra 3/ + 2/ + 6/ = 63/ + 56/ + 72/ = 63+56+72 = 191/ = 223/ 4 3 7 84 84 84 84 84 84 Basic Terms: While arithmetic operations use numbers and fractions based (LCM of 4,3, 7 = 84) on the 10 Arabic numerals 0 through 9, algebra uses letters, symbols, numerals and equations Subtracting Fractions: The denominator of the difference is the least common mUltiple (LCM) of the individual denominators, and the Signs: Plus (+) sign is used to represent positive numbers (greater than zero); numerator of the difference is the difference of the individual numerators the minus (- ) sign is used to represent negative numbers (less than zero) • subtract the integers and the fractions separately when subtracting Absolute Value: Value of any number, disregarding its sign mixed numbers • absolute value is denoted by the sign II • convert the mixed number into an improper fraction before subtracting • EX: 1+51 I-51 5 when the fractional part of the number you are subtracting is larger = = than the fractional part of the number you are subtracting from Expressions & Terms: Any symbol or combination of symbols that • EX: 4/5 - 1/7 = 28/ 35 - 5135 = 23/35 represents a number is called an algebraic expression; when an expression (LeM of 5 and 7 is 35) has many parts, the parts are connected by + and - signs, and each such part, together with its sign, is called a term; a monomial is an expression with Multiplying Fractions: The numerator of the product is the product of one term, a binomial has two terms, and a polynomial is an expression with the individual numerators, and the denominator of the product is the more than one term product of the individual denominators • convert mixed numbers into improper fractions and then Factors & Coefficients: When two or more numbers are mUltiplied, each multiply: of the numbers or their product is called a factor of the resulting term • any individual factor in a term is the coefficient of the remaining factors ~x~ = aXe b d bxd of that term • EX: 5x is a term, 5 is the (numerical) coefficient of x • EX. 3/ X 2/ = 2x3 = 6/ = 2/ . 7 9 7x9 63 21 • EX: Rxy is a term, 8 is the (numerical) coefficient ofxy, y is the (literal) coefficient of Rx Dividing Fractions Power: Product of equal factors is called a power of that factor • dividend -i- divisor = dividend X reciprocal of the divisor • EX: 2 X 2 X 2 = third power of 2 = 23 • ~ -i- ~ = ~ X ~ = a Xd b d b e bXe • EX: a X a X a X a x a = fifth power of a = as

• EX: 3/7 -i- 2/9 = 3/7 X 9/2 = 27/14 = 113/ 14 Basic Algebraic Rules: Consider the numbers a, b, c, d - (- a) = + a Decimals (-a) (-b) = + ab Format: 0.2368 (-a) (+b) = - ab a+b=b+a a + (b + c) = (a + b) + c axb=bxa a x (b x c) = (a x b) x c • rounding: to round 0.2368 to two places, first identify the digit at the jf a = band c = b, then a = c place you want to round (here, it is 3), then identify the next digit to the right (here, it is 6); if this digit is greater than or equal to 5, the digit at the I f a = band c = d, then a + c = b + d place of rounding is increased by I- if not, it remains the same (because If a = band c = d, then a - c = b - d 6 is greater than 5, the digit 3 is increased by I); the decimal 0.2368 is If a = band c = d, then a x c = b x d rounded to 0.24; similarly, rounding 0.1239 to two places is 0.12 Ifa = band c = d, then ale = bId' when c is not equal to zero Review Skills Basics Adding Numbers with Same Sign: Add the absolute Basic Statistics values of the numbers to get the sum and then prefix the Measures of Central Tendency are mean, median and mode common sign • mean (arithmetic mean or average) • EX: (+5) + (+6) = +11; (-3) + (-5) =-8 • mean of a set of numbers = sum of the numbers/number of items Adding Numbers with Opposite Signs: Add the • mean is very sensitive to extreme values among the set of numbers; it is usually represented by absolute values of the numbers with like signs, then the lowercase Greek letter mu (/1) for a set of numbers (population) and x-bar (x) for a subtract smaller absolute value from the larger absolute value, and prefix the sign of the larger value sample (subset of those numbers) • EX: (+5) + (-4) + (+3) + (-2) = (+8) + (-6) = +2 • EX: Mean of 6,4,2,7,9 = (6 + 4 + 2 + 7 + 9) + 5 = 28/5 = 5.6 Adding & Subtracting Algebraic Expressions: Terms • median in an expression with the same factors are called like terms; • median of a set of numbers is the central or middle number in the set when the numbers are adding or subtracting polynomials is done by adding or arranged according to their magnitude or size subtracting the numerical coefficients of like terms • EX: Median value of 6,4,2,7,9 is the middle value among 2, 4, 6, 7, 9; so, median = 6 • EX: (2a + 5b - 6) + (3a - 2b + 8) - (a + 2b - 4) • for an even number of items, the median is the average of the two middle numbers = (2a + 3a - a) + (5b - 2b - 2b) + (-6 + 8 + 4) • EX: Median value of 6,4,3, 2, 7, 9 is the average of the two middle numbers of 2,3,4,6,7, = 4a + b + 6 9, which is the average of 4, 6; so, median = 5 Multiplying Algebraic Expressions • mode • monomial x monomial = product of the numerical • mode of a set of numbers is the number that occurs most frequently in the set coefficients x product of literal factors • EX: Mode of 6,4,2,7,7,6,7,4,7,9 is 7 as it occurs the most times • if two numbers occur the most number of times, the set is bimodal; if many numbers occur the • EX: 6ab x 8c = 48 abc most number of times, the set of numbers is multimodal; ifall numbers appear only once, then • polynomial x monomial = each term of the there is no mode polynomial x the monomial, then add the resulting partial products Measures of Dispersion are range, percentile, quartile, variance and standard deviation • range of a set of numbers is the difference between the highest and lowest values • EX: (5a + 8b) x 2c = lOac + 16bc • EX: Range of 2, 6, 4, 8,3,9, 7 is (9 - 2) = 7 • polynomial x polynomial = each term of one • percentile for a value n is found by dividing the number of items less than n by the total number polynomial x each term of the other polynomial; then of items, and then multiplying this by 100 add each of the partial products • quartile • EX: (6a + 4b) x (2c + 5d) = 12ac + 8bc + 30ad + 20bd • QI is the first quartile (25th percentile) when 1/4 of the items are below the value QI Dividing Algebraic Expressions • Q2 is the second quartile (50th percentile) when 1/2 of the items are below the value Q2; Q2 • monomial + monomial = quotient of numerical corresponds to the median coefficients x quotient of literal coefficients • Q3 is the third quartile (or 75 th percentile) when 3/4 of the items are below the value Q3 • EX: 36ac + 6c = e6/6) x (ac/c) = 6a • EX: If 45 out of 50 students in a class have scores less than 85 on an exam, then a student • polynomial + monomial = each term of the with a score of 85 is in the 90th percentile 1(45/50) x 100 = 90%I polynomial + the monomial, then add the partial • EX: What are the quartiles for the set of numbers 2,4,7,3,5,8,9, 10? quotients Arranging the numbers in ascending order: 2, 3, 4, 5, 7, 8, 9, 10; arranging them into four equal • EX: (24ab + 6ac + 42bc) + (6abc) parts: 2, 3; 4, 5; 7, 8; 9,10 = (24ab/6abc) + (6ac/6abc) + (42bc/6abc) QI = (3+4)/2 = 3.5, value under which there are 1/4 of the items = 4/c + I/b + 7/a Q2 = (5+7)/2 = 6, value under which there are 1/2 of the items Q3 = (8+9)/2 = 8.5, value under which there are 3/4 of the items • variance & standard deviation of a set of numbers measures the ,\pread of" the data ahout the mean ofthose numbers; the standard deviation is equal to the square root of the variance, and has the same units as the original numbers • standard deviation is usually represented by the lowercase Greek letter sigma (0'), and variance by S2 • for a set of n numbers (population):

i=1 variance = S2=-k'i(Xi-jJ.)2; standard deviation = (I= n ;=1 • EX: Calculate the standard deviation of 3, 6, 15, 19, 27 n=5,jJ.=7%=14,(I=j3~O =8.72 • Frequency Distribution: A set of numbers or data arranged in ascending order is called an array; the number oftimes a specific number is repeated in a data set or array is called its frequency; when a set of numbers are grouped into several groups, the groups are called classes, the size of the class is called the class interval; the number of items in each class is called its frequency, and the grouped data is called a frequency distribution

US $5.95 CAN $8,95 NOTE: This QuickStudy" guide is intended for infonnational purposes only, Due to its condensed fonnat, this guide cannot cover every aspect of the subject; free dfwn~adS & Author: Ravi Behara, PhD, nun re o..!. titles at mther, it is intended for use in conjunction with course work and assigned texts. qUlc 5 uuy.com Neither BarCharts, Inc., its writers, editors nor design staff, are in any way 11111=111 responsible or liable for the use or misuse ofthe infonnation contained in this guide. All rights reserved. No part of this publication may be reproduced or transmitted in any form, or by any means, electronic or mechanical, including photocopy. recording, or any information storage and retrieval system, without written permission from the publisher. ©2007 BarCharts, Inc. 0308