Exam III Summary Material – Closing Costs and Points (only needed for Take- home quiz): When you buy a house, you pay a lot • F.1 of extra legal fees up front called closing costs. Points are extra fees you pay up front to lower – Interest is a fee charged for the use of someone your interest rate slightly. (In both cases ex- else's money (bank charges you when you borrow; tra means they do NOT change the price of your bank charges themselves when you save) house!) – The simple interest I = P rt (P =principal, or initial amount, r=interest rate as a decimal, • 4.3 t=time) – A linear equation is an equation that can be writ- – The future value F = P (1 + rt) ten as F.2 • A1x1 + A2x2 + ··· + Anxn = b Compound Interest is interest applied to in- – – The matrix form of a linear system is AX = B r mt terest over time (formula: A = P 1 + m ∗ A is the coe cient matrix ( =number of compoundings per year) m ∗ X is the variable matrix – Continuous Compound Interest formula: F = ∗ B is the constant matrix P ert – Gaussian Elimination: a set of operations you – Eective Yield is the actual rate of interest can perform on a system of equations (rows of the earned per year when compounded matrix) to create an equivalent system of equa- – TVM Solver: tions: ∗ N=mt 1. Switch rows (equations) ∗ I%=r (as a percent) 2. Multiply or divide a row by a nonzero num- ber ∗ PV=P Mult/Div a row by a (nonzero) number and ∗ PMT=0 (*in this case, since nothing is added 3. to or removed from the account throughout); add/subtract to another row. ∗ FV=A – General Strategy (Goal-1's on diagonal, 0's rest of the column): ∗ P/Y=C/Y=m 1. Divide the diagonal row by the given number • F.3 (if zero, switch rst) Multiply the diagonal row by the number in A future value annuity is an investment where 2. – another row and subtract from that row. (do you make given periodic deposits to an account (in- this for each of the other rows in the matrix) terest earned at the same time as each deposit). (EXAMPLES: IRA, 401(k)) – To use the calculator, put the augmented matrix in the calculator, then rref the matrix – A sinking fund is set up (often by a company) to [A|B] make periodic deposits into an account to achieve 4.4 a given nal amount over a given time period. • – (Both cases are handled similarly: In the TVM – If you get a false statement (0 = 1) after rref, the Solver, PV=0 (usually); one of PMT or FV is system of equations has no solution given, and we solve for the other) – If you get more variables than equations, the sys- tem of equations has an in nite number of solutions • F.4 ∗ Choose free variables (all columns which do – Present Value Annuities- Given periodic with- not have the Goal stated in 4.3 above) drawals on an account for a given amount of time, ∗ Solve all other variables in terms of the free at least how much money must initially be in the variables account to accomplish this? (Given PMT with FV=0, nd PV) • 5.1-5.2 Amortization-how much in periodic payments in – An matrix has rows and columns. order to kill a given debt over a given amount of – m × n m n time (Given PV with FV=0, nd PMT) ∗ Row matrix: m = 1 – Equity: The amount you have already paid on ∗ Column matrix: n = 1 your house (Cost of house − Remaining Balance) ∗ Square matrix: m = n
1 – The Aij element of a matrix is the number in the • F.3 ith row and jth column of A. When you were born, your grandparents deposited – Two matrices A and B are equal if A = B for 1. ij ij $140/quarter into an account which earned about all i and j 3.5% per year compounded quarterly. – Add and subtract matrices by adding/subtracting the corresponding elements (matrices must be the (a) How much money was in the account when same size) you turned 18? – Multiply a matrix by a scalar by multiplying every (b) How much interest did the account earn? element by the scalar 2. (Give up this day our daily Starbucks...) You – The transpose of a matrix is formed by interchang- forego your 3 lattes/week and contribute the ing the rows and columns $50/month you save into a high-risk fund which earns 7.5% per year compounded monthly. – The dot product of a row matrix [a1 a2 ··· an] b1 (a) How much money will you have in the account b2 when you retire in 45 years? and a column matrix is given by a1b1 + ··· (b) How much interest will you have earned total bn on the account? a b + ··· + a b (must be the same length) 2 2 n n (c) How much interest did the account earn dur- – Multiply matrices: ABij is the dot product of the ing the 8th month of the 42nd year? ith row of A and the jth column of B (number of columns in A must equal the number of rows in B) • F.4
• 5.3 1. On Sept 6, the Lotto Texas advertised jackpot was $8.25M, to be paid in 30 equal annual payments. – The inverse of a square matrix A is a matrix B Their interest factor is 1.4108%. How much money such that AB = BA = I (where I is the n × n needs to be in their account initially in order to identity matrix). We write B = A−1. pay a winner? – If a square matrix A does not have an inverse, it is 2. You decide to purchase a $29,000 new car. The a singular matrix. car dealership oers you a no down payment op- – Given a matrix equation AX = B, if A has an tion, nancing the entire amount at 4.9% per year inverse, then X = A−1B compounded monthly for 6 years. What are your monthly payments? Exam III Summary of Examples in Class (a) (b) How much do you end up paying in interest? • F.1 (c) Construct the start of an Amortization table for the car you bought in the previous 1. You borrow money at 9% simple interest, and at example: the end of 2 years, you pay back $531. How much Period Payment Interest Principal Balance did you originally borrow? 0 You've bought a house using a $135,000 30-year 2. 1 mortgage at 3.75%. Later, we will learn that your monthly payments are $625.21. What is the new 2 balance on your loan after one month? 3 (d) How much of the 20th payment goes toward • F.2 the principal? 1. When you were born, your grandparents deposited 3. You start working for SuperMegaConglo- $10,000 in an account which earned about 3.5% moMonopolCorp.com at 22 with a monthly per year compounded quarterly. How much money salary of $4500. was in the account when you turned 18? How much Assuming an average of 2% per year in ation, interest did the account earn? (a) what will your monthly salary be when you 2. What is the eective yield of the account above? retire in 48 years? 3. Nineteen years ago, estimated annual tuition to (b) If you want to draw this monthly salary af- attend A&M was $1,948. Today it is about ter retirement for 15 years, how much money $10,400 (Sources: www.collegecalc.org and tu- should you have in your 401(k) when you re- ition.tamu.edu). To the nearest 0.1%, what is the tire if it is in a conservative fund which earns annual increase in tuition? 3% per year compounded monthly?
2 (c) In order to have this amount of money in your be :) Find the general solution, then nd solutions 401(k) when you retire, how much should you corresponding to the largest and smallest possible deposit each month of your employment if you values of the free variable. put it in an aggressive fund which earns 9% 1 0 0 4 per year compounded monthly? 0 1 −2 6 • 4.3 • 5.1-5.2 2 1 A hospital nutritionist prepares a menu of chicken 1 2 3 1. 1. If A = −1 3 and B = , nd and rice for patients on a low-fat, high-protein diet. 4 −2 0 0 4 It must contain exactly 9 g of fat, 51 g of protein, AB and BA . and 460 cal. Each 3-ounce serving of chicken con- 11 11 2. Find the values of x, y, and z such tains 1.5 g of fat, 24 g of protein, and 120 cal. Each 1/2 cup serving of rice contains 6 g of fat, 3 g of x 10 1 −1 that protein, and 220 cal. How many servings of chicken −4 7 − 2 z 0 = and rice should be included in the menu? −6 4 0 3 4 12 A transit company oers a discount on the rst 2. y − 3 7 . class fare, called an "executive" fare, to those pas- −6 x + z sengers on a train who have no luggage. The com- 3. Let A be a 4 × 6 matrix, B be a 6 × 6 matrix, C be pany must sell 4 times as many economy class tick- a matrix, and be a matrix. Find the ets as rst class and executive class tickets com- 4 × 4 D 6 × 4 dimensions of each of the following, or write Does bined. The company wants to keep their costs to $1180 per train, with the cost being $15 for each Not Exist if applicable. rst class passenger, $10 for each executive passen- (a) A + 2DT ger, and $5 for each economy class passenger. If (b) DA + B the train holds 180 passengers, how many tickets (c) C − 3A of each type should be sold? (d) ADT 3. (Details of example TBA. See Example 7 on p124 (e) (AD)T for a similar one) (f) BAT 4. A company produces slalom water skis and trick • 4.4 water skis. The labor hours for each task are sum- marized in the following table: Solve the following system of equations: 1. Assembly Finishing 2x − 4y + z = −4 Slalom 3 hrs 1 hr 4x − 8y + 7z = 2 Trick 5 hrs 1.5 hrs −2x + 4y − 3z = 5 In addition, the company has two manufacturing plants, and hourly wages are summarized in the 2. Solve the following system of equations: following table: CA MD x1 + 2x2 + 4x3 + x4 − x5 = 1 Assembly $15 $13 2x + 4x + 8x + 3x − 4x = 2 1 2 3 4 5 Finishing $10 $8 x1 + 3x2 + 7x3 + 3x5 = −2 De ne matrices L and H respectively using the above data. Evaluate and interpret the products 3. A company that rents small moving trucks wants to purchase 25 trucks with a combined capacity of HL and LH. 28,000 cubic feet. Three dierent types of trucks • 5.3 are available: a 10-foot truck with a capacity of 350 cubic feet, a 14-foot truck with a capacity of 15 10 5 −1 700 cubic feet, and a 24-foot truck with a capacity 1. Given A = 1 1 1 , nd A if it exists, of 1,400 cubic feet. The number of 10-foot trucks 4 4 −1 must be 30 less than twice the number of 24-foot 1180 trucks. How many of each type of truck should the then multiply it by b = 180 . company purchase? 0 Write the following system of equations in matrix 4. In the row-reduced matrix below, x represents the 2. form. Then nd −1 and use it to solve the system. number of ounces of sardines in a special diet, y A represents the number of ounces of spinach, and 6x + y + 2z = −14 z represents the number of ounces of whole grains 3x − y + 3z = −1 (columns are in alphabetical order-as they should 3x + 5y + 2z = 17
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