MATH 106 Finite Mathematics

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MATH 106 FINAL EXAMINATION

This is an open-book exam. You may refer to your text and other course materials as you work

on the exam, and you may use a calculator. You must complete the exam individually.

Collaboration or consultation with others is NOT allowed. Use of instructors’ solutions

manuals and/or online problem solving services is NOT allowed.

Record your answers and work on the separate answer sheet provided.

There are 25 problems.

Problems #1–12 are Multiple Choice.

Problems #13–15 are Short Answer. (Work not required to be shown)

Problems #16–25 are Short Answer with work required to be shown.

MULTIPLE CHOICE

1. Sheila purchases a car for $24,000, makes a down payment of 25%, and finances the rest with

a 60-month car loan at an annual interest rate of 5.4% compounded monthly. What is the amount

of her monthly loan payment?

1. _______

A. $457.32 C. $508.00

B. $381.00 D. $342.99

2. Find result of performing row operation 0.5�1 + �2 → �2 on matrix K:

2. _______

� = [ 8 −2

−6 5 | 10

−9 ]

A. [ 8 −2

−2 4 | 10

−4 ] B. [ 4 −1

−2 4 | 5

−4 ]

C. [ 4 −1

−6 5 | 5

−9 ] D. [ −2 4

−6 5 | −4

−9 ]

3. Customers shopping at a particular supermarket spend a mean time shopping of 47 minutes,

with a standard deviation of 11 minutes. Assuming a normal distribution, what is the probability

that a randomly chosen customer will spend between 36 and 58 minutes shopping in the

supermarket?

3. ______

A. 0.3413 C. 0.6826

B. 0.9544 D. 0.7580

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4. Find the values of x and y that maximize the objective function P = 7x + 5y for the feasible

region shown below. 4. _______

A. (x, y) = (10, 0)

B. (x, y) = (8, 10)

C. (x, y) = (5, 15)

D. (x, y) = (0, 20)

5. Two balls are drawn in succession out of a box containing 2 red and 5 white balls. The balls

are drawn without replacement. What is the probability that both balls drawn are red?

5. _______

A. 19

42 B. 1

21 C. 13

42 D. 2

7

6. Which of the following statements is NOT true? 6. ______

A. If all of the data values in a data set are identical, then the standard deviation is 0.

B. The variance can be a negative number.

C. The standard deviation is the square root of the variance.

D. The variance is a measure of the dispersion or spread of a distribution about its mean.

7. If K = {3, 7, 11, 15} and M = {7, 12, 15, 18}, list {�|� ∈ � �� � ∈ �}

7. ______

A. { 3, 7, 11, 12, 15, 18 } C. { ∅ }

B. { 3, 7, 7, 11, 12, 15, 15, 18 } D. { 7, 15 }

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8. Determine which shaded region corresponds to the solution region of the system of linear

inequalities

x + y t 2

x + 3y ≤ 3

x t 0

y t 0 8. ____

GRAPH A. GRAPH B.

GRAPH C. GRAPH D.

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9 – 10. At Burger Heaven a “double” contains 2 meat patties and 6 pickles, whereas a “triple”

contains (wait for it!) 3 meat patties and 3 pickles. Near closing time one day, only 24 meat

patties and 48 pickles are available. If a “double” sells for $1.50 and a “triple” sells for $2.00,

then how many of each should be made in order to maximize profit? Let x represent number of

“double” burgers and y represent number of “triple” burgers.

9. Identify the production constraint for pickles:

9. _______

A. 6� + 3� ≥ 24 C. 6� + 3� ≤ 24

B. 6� + 3� ≥ 48 D. 6� + 3� ≤ 48

10. State the objective function.

10. _______

A. � = 24� + 48� C. � = 1.5� + 2�

B. � = 48� + 24� D. � = 2� + 1.5�

11. You can win Transylkota’s “Deep-6” lottery jackpot if you correctly choose 6 non-repeating

integer numbers between 1 and 30 (in any order) and those numbers are drawn. You buy one

ticket. What is the probability that it’s the jackpot winner?

11. ______

A. �(�) = 1⁄�30,6 C. �(�) = 1 306 ⁄

B. �(�) = 1/�30,6 D. �(�) = 6 306 ⁄

12. Find the equation of the line passing through (7, 5) and ( – 1 , – 1): 12. ______

A. 2x – 3y = – 1 B. 3x – 4y = 1 C. 3x + 4y = 41 D. 2x + 3y = – 5

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______________________________________________________________________________

SHORT ANSWER (Work not required to be shown)

13. Consider the following graph of a line.

(a) State the x-intercept. ______________

(b) State the y-intercept. ______________

Answer: ______________

(c) Determine the slope. ______________ Answer: ______________

(d) Find the slope-intercept form of the equation

of the line.

______________________

14. 980 randomly-selected respondents to a marketing survey were asked their age in years

(18 to 39 or 40+) and the choice of vacation destination they liked best. Following data

obtained:

Preferred Vacation

Destination

Age 18-39 Age 40 + Total

Theme Park 245 17 262

Seashore 135 183 318

Mountains 151 85 236

Spa Resort 51 113 164

Total 582 398 980

(Report your answers as fractions or as decimal values rounded to the nearest hundredth.)

Find the probability that a randomly-selected respondent:

(a) prefers the mountains or is age 18 – 39: Answer: ______________

(b) prefers the mountains given that the respondent is age 18 – 39: Answer: ______________

(c) prefers the mountains and is age 18 – 39: Answer: ______________

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15. Let U = {1,2,3,4,5,6,7,8,9,10}, A = {1,2,3,4,5}, B = {1,3,4,6}, and C = {2,4,6}.

List the members of the following sets:

a. � ∩ � Answer: ______________

b. � ∪ � Answer: ______________

c. �′ ∩ �′ Answer: ______________

SHORT ANSWER, with work required to be shown, as directed.

16. There is a collection of 13 books. 9 of the books are fiction and 4 of the books are nonfiction.

As an assignment, a student must read 6 of the books over the summer.

(a) In how many ways can 6 of the 13 books be chosen? Show work.

(b) In how many ways can the 6 books be chosen, if 4 of the books must be fiction and 2 of the

books must be non-fiction? Show work.

(c) If 6 books are selected at random from the collection of 13 books, what is the probability that

4 are fiction and 2 are non-fiction? Give answer as a fraction or as a decimal rounded to nearest

ten-thousandth (4 places after decimal) Show work.

______________________________________________________________________________

17. Solve the system of equations using elimination by addition, substitution, or augmented

matrix methods (your choice). Show work.

7� − 4� = 6

2� + 3� = 10

______________________________________________________________________________

18. Cara needs $9,000 in 11 years. What amount can she deposit at the end of each quarter at

8% annual interest compounded quarterly so she will have her $9,000? Show work.

A. $125.19 C. $134.01

B. $129.49 D. $540.69

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19. According to the US Department of Justice’s Bureau of Justice Statistics, 2.4% of US

firearms purchase/transfer applications submitted in 1999 were rejected due to background check

information. In 2009, the rejection rate of applications submitted due to background check

information was 1.4%.

(a) Which of the following linear equations could be used to predict application rejection

rate y in a given year x, where x = 0 represents the year 1999? Explain/show work.

A. y = – 10x + 1999 C. y = – 10x + 2.4

B. y = – 0.1x + 1999 D. y = – 0.1x + 2.4

(b) Use the equation from part (a) to predict the firearms application rejection rate (%) in

the year 2022. Round answer to nearest hundredth of a percent. Show work.

(c) Fill in the blanks to interpret the slope of the equation: The rate of change of

application rejection rate with respect to time is

______________________ per ________________. (Include units of measurement.)

______________________________________________________________________________

20. The feasible region shown below is bounded by lines x + y = 1, 2x + y = 3, and y = 0.

Find the coordinates of corner point A. Show work.

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21. If the probability distribution for the random variable X is given in the table, what is the

expected value of X? Show work.

xi – 30 10 20 60

pi 0.30 0.30 0.25 0.15

______________________________________________________________________________

22. There is a 0.91 probability that MATH 106 students will correctly follow all instructions

on the Final Exam. What is the probability that exactly 91 of the 100 students taking MATH 106

in a particular term correctly follow all Final Exam instructions? Round answer to the nearest

ten thousandth (four places after decimal). Show work.

23. A psychologist studied the number of words spoken by a sample of 8 three-year-olds. The

numbers of words recorded per each 3-year-old were 20, 19, 30, 22, 50, 36, 44, and 19.

(a) State the mode (if one exists. If not, indicate “none”).

(b) Find the median. Show work/explanation.

(c) State the mean. Show work/explanation.

(d) The sample standard deviation is 12.2. What percentage of the data fall within one

standard deviation of the mean? Show work/explanation.

(d) _______

A. 34%

B. 88%

C. 68%

D. 75%

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24. Christina is selling an antique dining room furniture set through a broker. She wants to get

$1500 for herself, but the broker gets 15% of the selling price as commission. What should the

selling price be? Show work.

A. $1650.00 C. $1725.00

B. $1764.71 D. $1783.52

25. A mall developer surveyed 500 customers yesterday to learn what they go shopping for at

the mall. 345 customers said they shopped for clothes. 290 customers said they went shopping

for electronics. 200 customers said they shopped for both clothes and electronics.

(a) What is the probability that a single randomly-selected shopped for either clothing or

electronics yesterday, but not both? Show work.

(b) Let C = {customers shopping for clothes} and E = {customers shopping for electronics}.

Determine the number of customers belonging to each of the regions I, II, III, IV.

Region I: ________ Region II: __________ Region III: _________ Region IV: __________

______________________________________________________________________________

U

C E

II

IV

I III