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Applied Finite Mathematics: Practice Exam

Module by: Rupinder Sekhon. E-mail the author

Summary: This module contains the practice final exam for Applied Finite Mathematics.

Exercise 1

What will the future value of $5000 be in 5 years at 8% compounded quarterly?

Solution

7,429.74

Exercise 2

A bank pays 8% compounded quarterly, what is the effective interest rate?

Solution

8.24%

Exercise 3

Each year a sum of $2200 is placed in an IRA account paying 12%. After 10 years what will the final amount be?

Solution

38,607.22

Exercise 4

If $5000 is invested at 8% compounded daily for 3 years, find the final amount.

Solution

6,356.08

Exercise 5

Find the present value of $200 per month at 12% for 4 years.

Solution

7,594.79

Exercise 6

A $12,000 car loan is amortized at 6% over 6 years. Find the monthly payment.

Solution

198.87

Exercise 7

In problem 6, what will the balance of the loan be after 4 years?

Solution

4,487.08

Exercise 8

If in problem 6, the loan is amortized over 3 years, what will the monthly payment be?

Solution

365.06

Exercise 9

A $24,000 car has a useful life of 6 years. It can be leased for $400 per month. If the current interest rate is 6%, is it better to lease or to buy?

Solution

buy

Exercise 10

What monthly payment will amount to $10,000 in 5 years at 11%?

Solution

125.75

Exercise 11

A bond has a face value of $1000 and is due in 5 years. It pays $45 interest every 6 months. If the current interest rate is 12%, what is the fair market value of the bond?

Solution

559.39+331.20=889.59559.39+331.20=889.59 size 12{"559" "." "39"+"331" "." "20"="889" "." "59"} {}
(1)

Exercise 12

A company produces solar panels. The total cost of 20 panels is $2100, and the cost of 60 panels is $3900. Express the cost y in terms of the number of panels xx size 12{x} {}.

Solution

y=45x+1200y=45x+1200 size 12{y="45"x+"1200"} {}
(2)

For problems 13, 14 and 15, the final tableaux is given below

Figure 1
the final simplex tableau to be used for the following three problems.

Exercise 13

If the initial problem is a maximization problem, what is the maximum value?

Solution

2500

Exercise 14

If the initial problem is a maximization one, at what point does the maximum value occur?

Solution

200,400,100200,400,100 size 12{ left ("200","400","100" right )} {}

Exercise 15

If the initial problem is a minimization one, at what point does the minimum value occur?

Solution

75,5,9075,5,90 size 12{ left ("75",5,"90" right )} {}

Exercise 16

At a price of $3, 100 units are demanded of a product. At a price of $5, 50 units are demanded. If xx size 12{x} {} is the price and DD size 12{D} {} the number of units demanded, write the demand equation.

Solution

D=−25x+175D=−25x+175 size 12{D= - "25"x+"175"} {}

For problems 17 and 18, use the following system of equations.

2x + 4y = 10 3x + 6y = 15 5x + 10 y = 27 2x + 4y = 10 3x + 6y = 15 5x + 10 y = 27 size 12{ matrix { 2x {} # +{} {} # 4y {} # ={} {} # "10" {} ## 3x {} # +{} {} # 6y {} # ={} {} # "15" {} ## 5x {} # +{} {} # "10"y {} # ={} {} # "27"{} } } {}

Exercise 17

If x=1x=1 size 12{x=1} {}, what is the value of yy size 12{y} {}?

Solution

No solution

Exercise 18

How many solutions does this system have?

Solution

None

Problems 19 - 22 refer to the maximization problem below to be done using the simplex method.

A company makes two types of widgets, Regular and Deluxe. Each type of widget requires the use of three machines for its production. The Regular widget requires three hours on machine I, one hour on machine II, and one hour on machine III, and sells for $10. The Deluxe widget requires one hour on machine I, two hours on machine II, and one hour on machine III, and sells for $20. The maximum number hours available on Machines I, II, and III are 120, 100, and 40, respectively.

Exercise 19

What are the coefficients of the objective function?

Solution

10, 20

Exercise 20

What is the constraint imposed by Machine III?

Solution

x1+x2≤40x1+x2≤40 size 12{x rSub { size 8{1} } +x rSub { size 8{2} } <= "40"} {}

Exercise 21

In solving this problem using the simplex method, how many slack variables are needed?

Solution

3

Exercise 22

After the first full pivot operation, what is the current revenue?

Solution

800

Exercise 23

A firm produces floppies at a variable cost of $1.20 per disk and a fixed cost of $1800. If the disk sells for $3 each, find the break-even point.

Solution

1000,30001000,3000 size 12{ left ("1000","3000" right )} {}

Problems 24 - 25 refer to the following minimization problem:

A diet must contain at least 60 units of protein, and 30 units of fat. Food A provides 2 grams of protein and 4 grams of fat, and costs 30 cents. Food B provides 6 grams of protein and 2 grams of fat, and costs 20 cents.

Exercise 24

Graph the constraints, and shade the feasibility region.

Solution

Graph

Exercise 25

Write the cost function.

Solution

C=.30x+.20yC=.30x+.20y size 12{C= "." "30"x+ "." "20"y} {}

For problems 26 and 27, the supply and demand equations are given as follows:

S=2/3x−100S=2/3x−100 size 12{S=2/3x - "100"} {}, D=−4/3x+500D=−4/3x+500 size 12{D= - 4/3x+"500"} {}, where xx size 12{x} {} is the price.

Exercise 26

Find the equilibrium price.

Solution

300

Exercise 27

How many items will be demanded at that price?

Solution

100

Exercise 28

If there are 5 people in a room, what is the probability that no two have the same birthday?

Solution

.97286.97286 size 12{ "." "97286"} {}

Exercise 29

AA size 12{A} {} and BB size 12{B} {} are mutually exclusive, PA=.4PA=.4 size 12{P left (A right )= "." 4} {}, PB=.5PB=.5 size 12{P left (B right )= "." 5} {} find PA and BPA and B size 12{P left (A" and "B right )} {}.

Solution

0

Exercise 30

AA size 12{A} {} and BB size 12{B} {} are independent. PA=.4PA=.4 size 12{P left (A right )= "." 4} {}, PA and B=.24PA and B=.24 size 12{P left (A" and "B right )= "." "24"} {}, find PBPB size 12{P left (B right )} {}.

Solution

.6.6 size 12{ "." 6} {}

Exercise 31

What is the probability of getting 3 heads if a coin is tossed 5 times?

Solution

.3125.3125 size 12{ "." "3125"} {}

Exercise 32

If PA=.5PA=.5 size 12{P left (A right )= "." 5} {}, PB=.4PB=.4 size 12{P left (B right )= "." 4} {} and PA and B=.2PA and B=.2 size 12{P left (A" and "B right )= "." 2} {}, find PA or BPA or B size 12{P left (A" or "B right )} {}.

Solution

.7.7 size 12{ "." 7} {}

Exercise 33

How many different ways can two boys and three girls be chosen from a total of 6 boys and 8 girls?

Solution

840

Problems 34 - 36 refer to the following information.

Companies AA size 12{A} {}, BB size 12{B} {}, and CC size 12{C} {} produce 15%, 40%, and 45% respectively of the major appliances in an area. One percent of company AA size 12{A} {} appliances, 2% of company BB size 12{B} {} appliances, and 3% of company CC size 12{C} {} appliances require service within the first year.

Exercise 34

What is the probability that an appliance chosen at random is defective?

Solution

.023.023 size 12{ "." "023"} {}

Exercise 35

If an appliance is chosen at random and found to be defective, what is the probability that it came from company BB size 12{B} {}?

Solution

.3478.3478 size 12{ "." "3478"} {}

Exercise 36

Suppose it was manufactured by company BB size 12{B} {}, what is the probability it is a defective appliance?

Solution

.02.02 size 12{ "." "02"} {}

Exercise 37

On 30% of his quizzes a student receives a score of 8, and on 70% his score is 9, what is his average?

Solution

8.78.7 size 12{8 "." 7} {}

Problems 38 - 40 refer to the following.

An urn contains 3 red, 4 white and 5 blue marbles, and two marbles are drawn at random.

Exercise 38

What is the chance of getting a blue marble on the second draw given that a red has been drawn on the first?

Solution

5/115/11 size 12{5/"11"} {}

Exercise 39

What is the probability of obtaining one white and one other marble?

Solution

.4848.4848 size 12{ "." "4848"} {}

Exercise 40

What is the probability of obtaining at least one white marble?

Solution

.5758.5758 size 12{ "." "5758"} {}

For problems 41 - 43, consider the following transition matrix giving the probabilities for the next purchase of Tide and Brand XX size 12{X} {}.

Table 1
  Next purchase
    Tide Brand XX size 12{X} {}
Present Tide .8 .2
Purchase Brand XX size 12{X} {} .4 .6

Exercise 41

What percentage of the Tide people will buy Brand XX size 12{X} {} next month?

Solution

.2.2 size 12{ "." 2} {}

Exercise 42

If the original share of the market is .25.75.25.75 size 12{ left ( "." "25", "." "75" right )} {}, what will the share be two months from now?

Solution

.6.4.6.4 size 12{ left [ matrix { "." 6 {} # "." 4{} } right ]} {}

Exercise 43

What will the long term share of the market be?

Solution

2/31/32/31/3 size 12{ left [ matrix { 2/3 {} # 1/3{} } right ]} {}

For problems 44 - 46, consider the following transition matrix for an absorbing Markov Chain.

Figure 2
This matrix shows the probability of switching from one state to another.

Exercise 44

Identify the absorbing states.

Solution

1 and 4

Exercise 45

Write the solution matrix.

Solution

Figure 3
This matrix shows the probability of switching from states 2 or 3 to states 1 and 4.

Exercise 46

Find the probability of ending in state 4, given one started in state 2.

Solution

.4.4 size 12{ "." 4} {}

Exercise 47

Given the 3×33×3 size 12{3 times 3} {} game 23310−1 0 0 4 23310−1 0 0 4 size 12{ left [ matrix { 2 {} # 3 {} # 3 {} ## 1 {} # 0 {} # - 1{} } right ]} {} , find the optimal strategy for the column player.

Solution

10 0 10 0 size 12{ left [ matrix { 1 {} ## 0 } right ]} {}

For problems 48 - 50, consider the following 2×22×2 size 12{2 times 2} {} payoff matrix.

− 1 0 1 / 4 − 1 / 4 − 1 0 1 / 4 − 1 / 4 size 12{ left [ matrix { - 1 {} # 0 {} ## 1/4 {} # - 1/4{} } right ]} {}

Exercise 48

Find the row player's optimal strategy.

Solution

1/32/31/32/3 size 12{ left [ matrix { 1/3 {} # 2/3{} } right ]} {}

Exercise 49

Find the column player's optimal strategy.

Solution

1/65/61/65/6 size 12{ left [ matrix { 1/6 {} ## 5/6 } right ]} {}

Exercise 50

Find the value of the game if the row and column players' strategies are .7.3.7.3 size 12{ left [ matrix { "." 7 {} # "." 3{} } right ]} {}, and .5.5.5.5 size 12{ left [ matrix { "." 5 {} ## "." 5 } right ]} {} , respectively.

Solution

– .35 –.35

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