Operations Research
Final Test Solutions
Course Introduction
Operations Research is a multidisciplinary field that applies advanced analytical methods to help make better decisions. The course covers fundamental concepts such as linear programming, integer programming, network flows, and queuing theory, providing students with the tools to model and solve complex real-world problems. Emphasis is placed on formulating mathematical models, using computational techniques for optimization, and interpreting the results to guide strategic decisions in areas such as logistics, manufacturing, transportation, and service systems. Through theoretical foundations and practical applications, students gain proficiency in both the quantitative and qualitative aspects of decision-making processes.
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) Evaluate: 400(1 + .10 100/365)
Answer: 400 * (1 + .10 .2739726) =400 (1 + .02739726) = 400 (1.02739726) =410.959
Q2) Yanping receives a monthly salary of $1 931.54 paid semi-monthly. The regular workweek is 38 hours.
a) Calculate the hourly rate of pay.
b) If the gross earnings for one pay period is 1 270.75, for how many hours of overtime was Yanping paid at double-time regular pay.
Answer: a) Annual gross earnings = 1931.54 × 12 = 23 178.48
Weekly gross earnings = 11ea8930_327e_fd99_b375_dba43b31ac0c_TB4213_11 = 445.74
Hourly rate of pay = 11ea8930_327e_fd9a_b375_29f1572dda84_TB4213_11 = $11.73 b) Regular semi-monthly gross earnings = 11ea8930_327f_24ab_b375_0398ed6392e5_TB4213_11 = 965.77
Overtime pay = 1270.75 - 965.77 = 304.98 Overtime rate = 11.73 × 2 = 23.46
Overtime hours = 11ea8930_327f_4bbc_b375_55fbf5e8d5ba_TB4213_11 = 13 hours
Q3) Evaluate: 395(2 + .15 290/365)
Answer: 395 * (2 + .15 .7945205) = 395 (2 + .1191781) = 395 (2.1191781) = 837.075
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Page 3
Chapter 2: Review of Basic Algebra
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Sample Questions
Q1) Conor had to pay income taxes of $3 440.00 plus 22% of the amount by which his taxable income exceeded $36 000.00. If his tax bill was $3 684.00, calculate his taxable income.
Answer: Let the taxable income (in dollars) be x.
Then x - 36 000 is the amount that his income is greater than $36 000.
3440 + 0.22(x - 36 000) = 3684
3440 + 0.22x - 7920 = 3684
0.22x = 8164
x = $37 109.09
Q2) Simplify: (5m - 2n)(m - 12n)
Answer: 5m2 - 60mn - 2mn + 24n2 = 5m2 - 62mn + 24n2
Q3) Simplify: (m2)6
A) m12
B) m-4
C) m8
D) m3
E) m2
Answer: A
Q4) Simplify: -9a(-5b)
Answer: 45ab
Page 4
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Chapter 3: Ratio, Proportion, and Percent
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Sample Questions
Q1) $32 is what percent of $26?
Answer: Rate = 11ea8930_32c3_3278_b375_a1bd9aba04b1_TB4213_11 = 1.230769 = 123.0769%
Q2) What is the rate percent if the base is 50 and the percentage is 35?
Answer: Rate = 11ea8930_32c4_1cdf_b375_03b698c792f1_TB4213_11 = .7000 = 70.00%
Q3) Solve: 15:6 = 30:x
Answer: 15:6 = 30:x 11ea8930_32b3_7a7f_b375_213792490fca_TB4213_11 = 11ea8930_32b3_7a80_b375_8bee5f414859_TB4213_11 15x = (30)(6)
x = 11ea8930_32b3_a191_b375_91deddf30da0_TB4213_11 x = 12
Q4) If total deductions on a yearly salary of $28 600 amounted to 16 2/3%, how much was deducted?
Answer: Let the deduction be $x.
x = 16 11ea8930_32c1_f9ee_b375_870a5f679fce_TB4213_11 % of 28600 = 11ea8930_32c2_20ff_b375_81d19658876c_TB4213_11 × 28600 = $4766.67
Q5) The price of an article increased from $15.50 to $17.50. Calculate the percent change in price.
Answer: % change = 11ea8930_32c7_2a3c_b375_a1170198f8d2_TB4213_11 = 0.129032258 = 12.9% increase
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Chapter 4: Linear Systems
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Sample Questions
Q1) Solve the system of equations: 5x = 10 - 2y
5y = 3x - 30
Q2) Two brothers decide to invest in a vacant lot which cost $51000. The first brother contributes $1000 more than one-quarter of what the second brother contributes. How much does each brother contribute?
A) B1 = 11000, B2 =40000
B) B1 =40000, B2 =11000
C) B1 = 12500, B2 =38500
D) B1 = 50000, B2 =10000
E) B1 = 1000, B2 =50000
Q3) A couple is about the get married. They decide to start a joint bank account to save for a house. They want to have an initial investment of $9000. The first partner contributes $350 more than three-quarters what the second partner contributes. How much does each partner contribute?
A) P1 = 4557.14, P2 = 4442.86
B) P1 = 3557.14, P2 = 5452.86
C) P1 = 4057.14, P2 = 4942.86
D) P1 = 5057.14, P2 = 3942.86
E) P1 = 5857.14, P2 = 3992.86
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Page 6
Chapter 5: Trade Discount, Cash Discount, Markup, and Markdown
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Sample Questions
Q1) A skateboard costing $16.23 was marked up to realize gross profit of 50% of the selling price.
a) What was the selling price?
b) What was the gross profit as a percent of cost?
Q2) For Your Eyes Only Company bought plasma screen TV sets for $11 585 less 4.5%, 3.2%. Overhead is 19% of regular selling price and required profit is 31% of regular selling price. The gas plasma TV sets were marked at a new regular selling price so that the store was able to advertise a discount of 10% while still maintaining its margin. To clear the inventory, the remaining TV sets were marked down 15%. What operating profit or loss is realized at the clearance price?
Q3) Murray's Pro Shop purchased sets of golf clubs for $710.00 less 22%, 17%, and 19%. Expenses are 19% of the regular selling price and the required profit is 19% of the regular selling price. The store decided to change the regular selling price so that it could offer a 47% discount without affecting its margin. At the end of the season, the unsold sets were advertised at a discount of 60%. What operating profit or loss was realized on the sets sold at the end of the season?
Q4) A store advertises a discount of $10.00 on a shirt. If the discount is 30%, for how much was the shirt sold?
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Chapter 6: Break-Even and Cost-Volume-Profit Analysis
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Sample Questions
Q1) A company has variable costs that are 1/8 the value of their sales revenues. Total net income for the most recent period was a profit of $50 400 and sales were $500 000. The company has started a new marketing campaign that they hope will increase sales, but it will require additional advertising of $15 000. How many sales dollars does the company have to generate in order to remain at the same level of profitability as before the new ad campaign?
A) $517 142.86
B) $387 100.00
C) 392 100.00
D) $387 100.86
E) 392 100.86
Q2) Last year a printing company had total sales of $37 500. The total of its variable costs was $15 000, and fixed costs were $18 000. Capacity is at sales maximum of $50 000.
a) Calculate the break-even point in (i) dollars of sales
(ii) as a percent of capacity
b) Draw a detailed break-even chart
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Page 8
Chapter 7: Simple Interest
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Sample Questions
Q1) How many months will it take $639 to grow to $795 at 8.45% p.a.?
Q2) Two obligations of $835 each, due 90 days ago and 35 days ago respectively, are to be settled by two equal payments to be made today and 65 days from now respectively. If interest allowed is 8.75% and the agreed focal date is today, what is the size of the equal payments?
Q3) In how many months will $9100.00 grow to $9818.75 at 6.41% p.a.?
Q4) A debt of $1410.00 due 6 months ago and a second debt of $1815.00 due in 4 months with interest at 6.16% p.a. are to be settled by two equal payments due now and 9 months from now respectively. Find the size of the equal payments at 7.37% p.a. with the agreed focal date now.
Q5) Determine the deposit that must be made to earn $85 in 7 months at 10.5%.
Q6) A loan of $1000 taken out on January 1 requires equal payments on February 1, March 1, and April 1. If the focal date is April 1, what is the size of the equal payments at 6.0%?
Q7) What principal will earn $219.89 interest at 11.25% p.a. from November 16, 2013 to February 7, 2014?
Q8) Determine the exact time between January 20, 2012 and May 10, 2012 by counting days.
Page 9
Q9) What is the present value of $9700.00 due at 6.25% p.a. in 89 days?
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Chapter 8: Simple Interest Applications
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Sample Questions
Q1) Find the present value of a seven-month note for $3 940 dated April 1 with interest at 7.45% if money is worth 6%, on August 15?
Q2) A promissory note has a face value of $6 000 and it has a date of issue of June1 this year. The term is for 6 months. The rate of interest is 8.00%. What is the maturity value of the note?
A) $6242.60
B) $6246.60
C) $6200.60
D) $6248.60
E) $6244.60
Q3) The owner of Easy Clips borrowed $8800.00 from Red Deer Community Credit Union on June 17. The loan was secured by a demand note with interest calculated on the daily balance and charged to the store's account on the 5th day of each month. The loan was repaid by payments of $2500.00 on July 25, $2300.00 on October 17, and $3500.00 on December 30. The rate of interest charged by the credit union was 6.5% on June 17. The rate was changed to 6.65% effective July 1 and to 6.95% effective November 1. Determine the total interest cost on the loan, up to and including Dec. 30.
Q4) Find the maturity value of a $473 note issued on October 4 at 8.5% for 190 days.
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Chapter 9: Compound Interest - Future Value and Present
Value
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Sample Questions
Q1) Find the present value and the compound discount of $6 600.00 due in seven years, three months, if interest is 7.2% compounded quarterly.
Q2) Determine the compound discount on $18875 due in 7.75 years if interest is 9.72% compounded monthly.
Q3) What sum of money will grow to $23000.00 in seven years at 9.612% compounded monthly?
Q4) A debt payment in the amount of $2000.00 due today, is to be settled by a payment of $1500.00 nine months from now and a final payment in 18 months. Determine the size of the final payment if the money is worth 12% p.a. compounded quarterly. Use the Banker's Method.
Q5) Find the maturity value of a promissory note for $1400.00 dated March 31, 2001, and due on August 31, 2006, if interest is 7.64% compounded quarterly.
Q6) Calculate the accumulated value of $3000.00 at 8% compounded quarterly for fifteen years. How much of the amount is interest?
Q7) Calculate the present value of $12 500.00 due in two years and nine months if interest is 7.8% p.a. compounded semi-annually.
Page 11
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Chapter 10: Compound Interest - Further Topics
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Sample Questions
Q1) A six-year, $1650.00 note bearing interest at 9.51% compounded annually was discounted at 11.2% compounded semi-annually yielding proceeds of $1916.75. How many months before the due date was the discount date?
Q2) At what nominal rate of interest compounded quarterly will $2000.00 earn $400.00 interest in three years?
Q3) If the effective rate of interest on an investment is 6.52%, what is the nominal rate of interest compounded monthly?
Q4) A financial obligation requires the payment of $250.00 in eighteen months, $350.00 in thirty months, and $300.00 in fifty-four months. When can the obligation be discharged by a single payment of $800.00 if interest is 5% compounded semi-annually?
Q5) In how many days will $770.00 grow to $880.00 at 11.5% p.a. compounded monthly?
Q6) Calculate the effective annual rate for 5% p.a. compounded semi-annually.
A) 0.050625
B) 5.0625%
C) 0.0255%
D) 2.55%
E) 15.0625%
Q7) In how many years will money triple at 12% compounded semi-annually?
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Chapter 11: Ordinary Simple Annuities
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Sample Questions
Q1) Suppose $726.56 is deposited at the end of every six months into an account earning 6.45% compounded semi-annually. If the balance in the account four years after the last deposit is to be $31 300.00, how many deposits are needed?
Q2) Note: The calculations for this question were done using Excel's RATE function. What is the effective annual rate of interest on a loan of $9200.00 repaid in semi-annual installments of $810.00 in nine years?
Q3) How much interest is included in the accumulated value of $250 paid at the end of every month for 5 years if the interest rate is 4.0% compounded monthly?
Q4) Calculate the accumulated value of quarterly payments of $100.00 made at the end of each quarter for ten years just after the last payment has been made if interest is 8% p.a. compounded quarterly.
A) $4003.91
B) $6040.20
C) $2040.00
D) $4415.88
E) $2015.88
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Chapter 12: Ordinary General Annuities
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Sample Questions
Q1) Calculate the accumulated value after ten years of payments of $1000.00 made at the end of each month if interest is 4% compounded semi-annually.
Q2) The loan will have a term of 54 months and monthly payments of $385.13. The interest rate on the loan is 8.84% compounded quarterly. What is the amount of money that was borrowed?
A) $17310.71
B) $17430.71
C) $17030.71
D) $17130.71
E) 20368.27
Q3) What is the accumulated value of deposits of $1120.00 made at the end of every six months for three years if interest is 8.48% compounded quarterly?
Q4) 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.
Q5) Mrs. Engleder has made deposits of $750.00 at the end of every six months for twenty-three years. If interest is 3.72% compounded monthly, how much will Mrs. Engleder have accumulated six years after the last deposit?
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Chapter 13: Annuities Due, Deferred Annuities, and Perpetuities
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Sample Questions
Q1) Calculate the present value of payments of $100.00 made at the beginning of each quarter for ten years if the interest is 6% compounded quarterly.
Q2) Quarterly deposits of $650.00 were made at the beginning of every 3 months for 11.25 years. If the interest is 4.92% compounded monthly, what amount can be withdrawn 6 years after the last deposit?
Q3) A trust fund will start to pay you $15200 per year at the beginning of each year once you turn 18. The prevailing interest rate is 8.79% compounded annually. For how long can you receive payments if you are celebrating your 14th birthday today and the current value of your trust fund is $111345?
A) 20.97 years
B) 29.07 years
C) 20.07 years
D) 29.77 years
E) 24.97 years
Q4) Sheila intends to retire in 14 years and would like to receive $2472.00 every six months for 9 years starting on the date of her retirement. How much must Sheila deposit in an account today if interest is 6.65% compounded semi-annually?
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Chapter 14: Amortization of Loans, Including Residential
Mortgages
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Sample Questions
Q1) What is the size of the final unequal payment of a loan that has 69.47 payments and the equal payment size is $215.64? The interest rate is 3.1% per compounding period.
A) $120.17
B) $210.17
C) $102.17
D) $99.10
E) $99.17
Q2) A debt of $43 100.00 is repaid by payments of $3350.00 made at the end of every six months. Interest is 8.4% compounded quarterly.
a) What is the number of payments needed to retire the debt?
b) What is the cost of the debt for the first five years?
c) What is the interest paid in the 11th payment?
d) Construct a partial amortization schedule showing details of the first three payments, the last three payments, and totals.
Q3) A loan of $8380.00 is repaid by equal payments made at the end of every three months for 3 years. If interest is 7% compounded quarterly, find the size of the quarterly payments and construct an amortization schedule showing the total paid and the total cost of the loan.
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Chapter 15: Bond Valuation and Sinking Funds
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Sample Questions
Q1) A $250 000, 6.5% bond with semi-annual coupons redeemable at par is bought 17.5 years before maturity at 78.25. What was the approximate yield rate?
Q2) A $100 000 bond is redeemable at 105 in 20 years. If interest on the bond is 5% payable semi-annually, what is the purchase price to yield 6% compounded semi-annually?
Q3) A $100 000.00, 7% bond with semi-annual coupons redeemable at 105 on November 1, 2001, is purchased on July 23, 1999, to yield 8% compounded semi-annually. Determine the quoted price.
Q4) A $100 000.00, 7.2% bond with quarterly coupons redeemable at par is purchased 11 years before maturity to yield 6% compounded semi-annually. Determine the premium or discount.
Q5) A $250 000, 7% bond redeemable at par with interest payable semi-annually is bought 4 years before maturity. Determine the premium or discount and the purchase price if the bond is purchased to yield: a) 6.5% compounded semi-annually b) 8% compounded semi-annually
Q6) A $100 000, 4% bond with semi-annual coupons is redeemable at 108. What is the purchase price to yield 5.5% compounded semi-annually seven years before maturity?
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Chapter 16: Investment Decision Applications
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Sample Questions
Q1) Billy Bob wants to convert his farm into a ski resort. He asked you to determine his rate of return based on the following estimates.
- Development cost for each of the first 2 years, $87 000.
- Construction of a chalet in Year 3, $1240 000.
- Upon his retirement in fifteen years, improvements in the property will yield him $270 000.
- Net returns from the operation of the golf course will be nil for the first three years and $200 000 per year afterwards until his retirement.
Q2) Project A requires an immediate investment of $18 000 and another $16 000 in three years. Net returns are $4500 after two years, $13 000 after four years, and $8900 after six years. Project B requires an immediate investment of $4000, another $6000 after two years, and $4000 after four years. Net returns are $3375 per year for 8 years. Determine the net present value at 11%. Which project is preferable according to the net present value criterion?
Q3) You win a lottery and have a choice of taking $200 000.00 immediately or taking payments of $8000.00 at the end of every three months for ten years. Which offer is preferable if interest is 8% compounded quarterly?
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