Mathematics for Economics and Business
Final Exam
Course Introduction
This course provides a comprehensive introduction to essential mathematical concepts and techniques used in economics and business. Topics include functions, equations, differentiation, integration, matrix algebra, and optimization, with a focus on real-world applications such as market analysis, cost minimization, and profit maximization. Through problem-solving and practical examples, students will develop quantitative skills necessary for analyzing economic models, making informed business decisions, and understanding the mathematical foundations of various economic theories. This course is designed for students with little prior mathematical background and emphasizes the practical use of mathematics in economic and business contexts.
Recommended Textbook
Contemporary Business Mathematics with Canadian Applications 9th Edition by S. A. Hummelbrunner
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Chapter 1: Review of Arithmetic
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Sample Questions
Q1) A salesperson receives a weekly base salary of $800.00 on a quota of $2 900. On the next $2100, she receives a commission of 14%. On any additional sales, the commission rate is 19%. Find her gross earnings for a week in which her sales total $8455.
Answer: Base salary on first $2900 = $800.00
Commission on next $2100 = 0.14 × 2100 = 294.00
Commission on additional sales = (8455 - 5000) × 0.19 = 3455 × .19 = 656.45
Gross earnings = $1750.45
Q2) A retailer returned 300 defective items to the manufacturer and received a credit for the retail price of $0.75 less a discount of 1/3 of the retail price. What was the amount of the credit received by the retailer?
Answer: Retail value =300($0.75) = $225
Credit = (1-1/3) $225 = (2/3) $225 = $150
Q3) Evaluate: 8600(1 - .27 * 226/360)
Answer: 8600 (1 - .27 .6277778) = 8600 (1 - .1695) = 8600(.8305) = 7142.3
Q4) Simplify: 5(4 + 3)
Answer: 5 7 = 35
Q5) Simplify: 3 + 8 4
Answer: 3 + 32 = 35
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Chapter 2: Review of Basic Algebra
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Sample Questions
Q1) Simplify: (1 - r)2(1 - r)3(1 - r)
Answer: (1 - r)6
Q2) Simplify: (-64ab) ÷ (8ab)
Answer: -8
Q3) Solve the following and check your solutions: 12x + 5(9 - x) = 429
A) 58.86
B) -58.86
C) 54.14
D) -54.14
E) -5.14
Answer: A
Q4) Evaluate: ln[400(1.17<sup>7</sup>)]
Answer: ln[400(1.17<sup>7</sup>)] = ln 400 + ln 1.17<sup>7</sup> = ln 400 + 7(ln 1.17)
= 5.9914645 + 7(.1570038) = 5.9914645 + 1.0990262 = 7.090491
Q5) Simplify: -6m(-3m)
Answer: 18m2
Q6) Evaluate: (-5)<sup>-3</sup>
Answer: - 11ea8930_328f_509d_b375_25ffa7f29bb0_TB4213_11 or -.008
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Chapter 3: Ratio, Proportion, and Percent
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Sample Questions
Q1) What sum of money decreased by 16 2/3% will equal $475?
Answer: x - 16 11ea8930_32c6_18c3_b375_099b1dbd40c7_TB4213_11 % of x = 475
x - 11ea8930_32c6_3fd4_b375_b5adec82634f_TB4213_11 x = 475 11ea8930_32c6_3fd5_b375_db92deb10548_TB4213_11 x = 475
x = 475 11ea8930_32c6_3fd6_b375_77326e5de56e_TB4213_11 = 570
Q2) Your gas bill for March is $274.40. If you pay after the due date, a late payment penalty of $10.72 is added. What is the percent penalty?
Answer: Penalty = 11ea8930_32c7_032b_b375_97f47560dc06_TB4213_11 = 3.9067%
Q3) A special purpose index has increased 155% during the last ten years. If the index is now 310, what was the index ten years ago?
Answer: Let the index ten years ago be x.
x + 155% of x = 310
x + 1.55x = 310
2.55x = 310
x = 121.56863
The index ten years ago was 121.56863.
Q4) $189 is what percent of $4150?
Answer: R = 11ea8930_32c0_c163_b375_3b0c2bcc0b31_TB4213_11 = 4.5542%
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Page 5
Chapter 4: Linear Systems
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Sample Questions
Q1) Find the slope and y-intercept: 4x + 5y = 31
Q2) Solve the system of equations: 28 - 7x = 4y 36 - 4y = 3x
Q3) Solve the system of equations: 6.6x + 3.5y = 110
2.9x + 4.7y = 106
Q4) Solve graphically: x = y, 2y = x - 200
Q5) A machine requires 3 hours to make a unit of Product A, and 6 hours to make a unit of Product B. The machine operated for 195 hours producing a total of 40 units. How many hours were used to manufacture units of Product A?
Q6) Find the slope and y-intercept: 4.5x + 9y = 81
Q7) Twice the larger number is greater than three times the smaller number by 19. The sum of the larger number plus one-third of the smaller number is 26. Find the two numbers.
Q8) Solve the system of equations: 2.2x + 0.85y = 2.25 1.5x + 1.75y = 2.75
Q9) Hamidah's sales last week were $251 less than three times Colleen's sales. Together they sold $1123. Determine how much each person sold last week.
Q10) Solve graphically: 5x - 2y = 10 and y = 5
Q11) Find the slope and y-intercept: 3x + 2y = 10 Page 6
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Page 7
Chapter 5: Trade Discount, Cash Discount, Markup, and Markdown
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Sample Questions
Q1) Big Boy Clothing Store lists an article for $99.00 less 20% and 15%. A competitor carries the same article for $115.00 less 18%. What further discount (correct to the nearest 1/10 of 1%) must the competitor allow so that its net price is the same as Deep Discount's?
Q2) A television set was sold during a clearance sale for $299.00. If the television was sold at a discount of 70%, what was the list price?
Q3) Determine the equivalent single rate of discount for the following series of discounts. Discount of 17.25%, 14.7% and 16.37%.
A) 40.969%
B) 42.838%
C) 41.223%
D) 41.162% E) 43.084%
Q4) A fridge was sold during a clearance sale for $657.00. If the fridge was sold at a discount of 24%, what was the list price?
Q5) The regular selling price of merchandise sold in a store includes a margin of 62% based on selling price. During a sale, an item which cost the store $231.25 was marked down 47%. For how much was the item sold?
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Chapter 6: Break-Even and Cost-Volume-Profit Analysis
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Sample Questions
Q1) The gas division of Power-U-Up plans to introduce a new gas delivery system based on the following accounting information.
Fixed costs per period are $4 236; variable cost per unit is $168; selling price per unit is $211; and capacity per period is 450 units.
a) Draw a detailed break-even chart
b) Compute the break-even point (i) in units;
(ii) as a percent of capacity; (iii) in dollars.
c) Determine the break-even point as a percent of capacity
(i) if fixed costs are reduced to $3 788;
(ii) if fixed costs are increases to $5 577 and variable costs are reduced to 75% of the selling price;
(iii) if the selling price is reduced to $191.
Q2) Trevor, the new owner of the vehicle accessory shop is considering buying sets of winter tires for $299 per set and selling them at $520 each. Fixed costs related to this operation amount to $3 250 per month. It is expected that 18 sets per month could be sold. How much profit will Trevor make each month?
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9
Chapter 7: Simple Interest
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Sample Questions
Q1) Determine the exact time between January 30, 2011 and May 20, 2011 by counting days.
Q2) A loan of $10 000.00 is to be repaid in four equal payments due in three months, six months, nine months, and one year respectively after the date of the loan. Calculate the size of the equal payments if interest is 8.5%.
Q3) You have $5000 invested in a 30-day savings certificate at an interest rate of 1.00%. How much money will you have when the certificate matures?
A) $5004.11
B) $5040.11
C) $5400.11
D) $4.11
E) $4500.11
Q4) How many months will it take $639 to grow to $795 at 8.45% p.a.?
Q5) Determine the exact time between January 20, 2012 and May 10, 2012 by counting days.
Q6) Debts of $1480.00 due four months ago and $1385.00 due in one month are to be settled by two equal payments due now and nine months from now respectively. Find the size of the equal payments at 12% p.a. with the agreed focal date now.
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Chapter 8: Simple Interest Applications
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Sample Questions
Q1) You bought a $100,000 364-day T-bill. The T-bill was discounted at a rate of 5%. If you paid $99,900.00 for the T-bill, how many days before maturity did you buy it?
A) 7.3 days
B) 3.7 days
C) 37 days
D) 73 days
E) 1 day
Q2) An investor purchased $250 000 in 91-day T-bills on the issue date for $248 157.56. After holding the T-bills for 37 days, she sold them for a yield of 3.25%.
a) What was the original yield of the T-bills?
b) For how much did the investor sell the T-bills?
c) What rate of return (per annum) did the investor realize while holding this T-bill?
Q3) A six-month promissory note dated November 8, 2013 is made at 6% for $2 900.00. What is the present value of the note thirty-eight days later if money is worth 7.2%?
Q4) Determine the maturity value of a 120-day note for $2 260.00 dated May 9 and bearing interest at 5.66%.
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Page 11
Chapter 9: Compound Interest - Future Value and Present Value
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Sample Questions
Q1) Calculate the accumulated value of $1000.00 at 8% compounded monthly for 12 years.
A) $2603.39
B) $1603.39
C) $2600.39
D) $2003.39
E) $603.39
Q2) A ten-year promissory note discounted after seven years at 10% compounded quarterly has a maturity value of $40000. Find the proceeds.
Q3) You borrow $4750 at a rate of 6.72% compounded monthly. You make a partial payment of $1400 in 3 months. You also agree to make two equal payments 3 and 5 months after the partial payment. What is the size of the equal payments? Use a focal date of 8 months.
A) $1573.84
B) $1754.02
C) $1735.84
D) $1784.35
E) $1788.02
Q4) Determine the maturity value of $5400 due in 91 months compounding annually at 8.75%.
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Chapter 10: Compound Interest - Further Topics
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Sample Questions
Q1) A five-year, $4500.00 promissory note with interest at 7.5% compounded semi-annually was discounted at 8% compounded quarterly yielding proceeds of $6150.00. How many months before the due date was the discount date?
Q2) How long, in compounding periods, will it take for $3327 to increase by $3799 if you are able to earn 6.2% compounded semi-annually?
A) 42.95
B) 24.95
C) 4.56
D) 6.45
E) 4.95
Q3) Calculate the nominal rate of interest p.a. compounded semi-annually if $5000.00 accumulates to $7002.64 in 78 months.
A) 0.8656%
B) 0.02625%
C) 2.625%
D) 0.0525%
E) 5.25%
Q4) Calculate the number of years for money to triple at 3.6% p.a. compounded monthly.
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Page 13
Chapter 11: Ordinary Simple Annuities
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Sample Questions
Q1) Note: The calculations for this question were done using Excel's RATE function. A property worth $191 000.00 can be purchased for 15% down and quarterly mortgage payments of $1972.00 for 30 years. What effective rate of interest is charged?
Q2) Planning for their child's college education, David and Carol Roberts opened an account paying 6.36% compounded monthly. If ordinary annuity payments of $700.00 per month are to be paid out of the account for three years starting seven years from now, how much did the Roberts' deposit?
Q3) Calculate the accumulated value of monthly payments of $100.00 made at the end of each month for ten years just after the last payment has been made if interest is 12% p.a. compounded monthly.
A) $23 003.87
B) $33 003.87
C) $12 000
D) $1 046.22
E) $330 003.87
Q4) Jason deposited $100.00 at the end of each month for five years into an account paying 6% compounded monthly. What will be the balance in the account at the end of the five-year term?
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Chapter 12: Ordinary General Annuities
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Sample Questions
Q1) Note: The calculations for this question were done using Excel's RATE function. A property worth $15 000.00 is purchased for 10% down and semi-annual payments of $1750.00 for twelve years. What is the effective rate of interest if interest is compounded semi-annually?
Q2) You won $150 000.00 in a lottery and you want to set some of that sum of money aside for 6 years. After 6 years you would like to receive $2710.00 at the end of every 6 months for 8 years. If interest is 6% compounded quarterly, how much of your winnings must you set aside?
Q3) A $ 120 000.00 mortgage is amortized over 25 years. If interest on the mortgage is 8.5% compounded semi-annually, calculate the size of monthly payments made at the end of each month.
Q4) Sean Paul borrowed $20 000.00 from his father to finance his new business. The loan agreement calls for equal payments at the end of each month for 10 years. The payments are deferred for 4 years and interest is 4.00% compounded semi-annually. What is the size of the monthly payments?
Q5) What is the accumulated value of deposits of $2000.00 made at the end of every year for three years if interest is 6.00% compounded quarterly?
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Chapter 13: Annuities Due, Deferred Annuities, and Perpetuities
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Sample Questions
Q1) What sum of money must be deposited in a trust fund to provide a scholarship of $630.00 payable at the end of each month if interest is 7.8% compounded monthly?
Q2) Determine the accumulated value after 14.5 years of deposits of $172.00 made at the beginning of every 6 months and earning interest at 4.45% compounded monthly?
Q3) Vicki paid for a kitchen set over time, signing an installment contract that requires semi-annual payments of $500.00 for 5 years. The first payment is made on the signing date and interest is 18% compounded semi-annually. What was the cash price?
Q4) Carla Craig invested a retirement gratuity of $17 570.00 in an RRSP paying 6.5% compounded semi-annually for 11.5 years. At the end of 11.5 years, she rolled the RRSP balance over into an RRIF paying $1500.00 at the beginning of each month starting with the date of rollover. If interest on the RRIF is 9.5% compounded monthly, for how long will Carla receive monthly payments?
Q5) A property was purchased for quarterly payments of $1070.00 for 9 years. If the first payment was made on the date of purchase and interest is 5.57% compounded annually, what was the purchase price of the property?
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Chapter 14: Amortization of Loans, Including Residential
Mortgages
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Q1) John has agreed to purchase his partner's share in real estate by making payments of $5000.00 every three months. The agreed transfer value is $40000.00 and interest is 10% compounded annually. If the first payment is due at the date of the agreement, what is the size of the final payment?
Q2) Mr. Lamb borrowed $8321.00 at 11.12% compounded monthly. He agreed to repay the loan in equal monthly payments over five years.
a) What is the size of the monthly payment?
b) How much of the 24th payment is interest?
c) What is the principal repaid in the 37th payment?
d) Prepare a partial amortization schedule showing details of the first three payments, the last three payments, and totals.
Q3) The owner of the Guelph Inc. borrowed $20000.00 at 1.2% compounded semi-annually and agreed to repay the loan by making payments of $1000.00 at the end of every three months.
a) How many payments will be needed to repay the loan?
b) How much will be owed at the end of five years?
Q4) A loan of $10000.00 is repaid by quarterly payments of $1000.00 each at 12% compounded quarterly. What is the principal repaid by the 11th payment?
Page 17
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Chapter 15: Bond Valuation and Sinking Funds
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Q1) A $100 000, 6.0% bond with semi-annual coupons redeemable at par is bought 17.5 years before maturity at 74.25. What was the approximate yield rate?
Q2) Annual sinking fund payments made at the beginning of every year for four years earning 4.5% compounded annually amount to $25 000.00 at the end of four years. Construct a sinking fund schedule showing totals.
Q3) Three $25 000, 6% bonds with semi-annual coupons redeemable at par were bought eight years before maturity to yield 7% compounded semi-annually. Determine the gain or loss if the bonds are sold at 89.625 four years later.
Q4) A 50 000 bond bears interest at 4.5% payable semi-annually and is redeemable at par on November 4, 2022. The bond is sold on March 20, 2013, to yield 5.5% compounded semi-annually. What is the cash price?
Q5) A $25 000, 6% bond redeemable at par is purchased 11 years before maturity to yield 6.9% compounded semi-annually. If the bond interest is payable semi-annually, what is the purchase price of the bond?
Q6) A $150 000 bond bearing interest at 6% payable semi-annually is bought eight years before maturity to yield 4.5% compounded annually. If the bond is redeemable at par, what is the purchase price?
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Page 18
Chapter 16: Investment Decision Applications
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Q1) An investment of $180 000 yields annual net returns of $27 700 for seven years. What is the rate of return on the investment (correct to the nearest tenth of a percent) if part of the initial investment, in the amount of, $80 000 is recovered at the end of 7 years?
a) Use linear interpolation to find the approximate value of the rate of return.
b) Find the answer using Cash Flow and IRR.
Q2) The SHREK Company has to make a decision about expanding its production facilities. Research indicates that the desired expansion would require an immediate outlay of $160 000 and an outlay of a further $60 000 in five years. Net returns are estimated to be $25 000 per year for the first five years and $20 000 per year for the following 9 years. Find the net present value of the project. Should the expansion project be undertaken if the required rate of return is 10% compounded annually?
Q3) A company is considering a project that will require a cost outlay of $30 000 per year for three years. At the end of the project, the company expects to salvage the physical assets for $50 000. The project is estimated to yield net returns of $60 000 in Year 4, $40 000 in Year 5, and $15 000 for each of the following five years. Alternative investments are available yielding a rate of return of 14.5%. Compute the net present value of the project.
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