Question 1

A psychologist studying the effects of nutrition on the behavior of laboratory rats is feeding one group a combination of three foods: I, II, and III. Each of these foods contains three additives, A, B, and C, that are being used in the study. Each additive is a certain percentage of each of the foods as follows: ​
Foods  ​
If the diet requires 53 g per day of A,  g per day of B, and  g per day of C, find the number of grams per day of each food that must be used.​
A)​ 
B)​ 
C)​ 
D)​ 
E)​

Accepted Answer

The answer of A psychologist studying the effects of nutrition...

Question 2

In the analysis of traffic flow, a certain city estimates the following situation for the "square" of its downtown district. In the following figure, the arrows indicate the flow of traffic. If x₁ represents the number of cars traveling between intersections A and B, x₂ represents the number of cars traveling between B and C, x₃ the number between C and D, and x₄ the number between D and A, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is, for intersection A we obtain Formulate equations for the traffic at B, C, and D. Solve the system of these four equations. A) B), C) D) E)

Accepted Answer

The answer of In the analysis of traffic flow, a...

Question 3

In the analysis of traffic flow, a certain city estimates the following situation for the "square" of its downtown district. In the following figure, the arrows indicate the flow of traffic. If  represents the number of cars traveling between intersections A and B,  represents the number of cars traveling between B and C,  the number between C and D, and  the number between D and A, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is, for intersection A we obtain  Formulate equations for the traffic at B, C, and D. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of In the analysis of traffic flow, a...

Question 4

A man has  invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is  and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property? ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A man has  invested in three...

Question 5

A brokerage house offers three stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. Portfolio III consists of 7 blocks of common stock, 3 municipal bonds, and 3 blocks of preferred stock. A customer wants  blocks of common stock,  municipal bonds, and  blocks of preferred stock. How many units of each portfolio should be offered? ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

Let the number of units of each portfolio be x, y, z respectively. Then we have the following system of equations:

2x + 4y + 7z = 11
x + 2y + 3z = 11
y + 3z = 11

Solving for x, y, and z, we get:
x = 1
y = 2
z = 3

Therefore, the customer should be offered 1 unit of Portfolio I, 2 units of Portfolio II, and 3 units of Portfolio III, which corresponds to choice C.

Question 6

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y. ​ 
A)​ 
​
B)​​ 
C)​ 
​
​​
D)​ 
​
​​
E)​ 
​
​

Accepted Answer

The answer of Solve the following system of equations for...

Question 7

Use technology to solve the system of equations, if a solution exists. 
A) 
, 
, 
, 
, 
B) 
, 
, 
, 
, 
C) 
, 
, 
, 
, 
D) 
, 
, 
, 
, 
E)no solution

Accepted Answer

The answer of Use technology to solve the system of...

Question 8

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?
A)$33,840 at 12%, $28,200 at 10%, $22,560 at 8%
B)$88,800 at 12%, $44,400 at 10%, $148,800 at 8%
C)$76,800 at 12%, $68,400 at 10%, $136,800 at 8%
D)$55,500 at 12%, $111,000 at 10%, $115,500 at 8%
E)$60,000 at 12%, $102,000 at 10%, $120,000 at 8%

Accepted Answer

The total investment is $282,000, and the income is $27,000. Let x be the amount invested at 12%, then 2x is invested at 8%, and the rest is at 10%. Solving the equations: x + y + 2x = 282,000 and 0.12x + 0.10y + 0.08(2x) = 27,000 gives x = 60,000, y = 102,000, and 2x = 120,000.

Question 9

Each ounce of substance A supplies  of the required nutrition a patient needs. Substance B supplies  of the required nutrition per ounce and substance C supplies  of required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts and that the amount of substance B be one-fifth of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% nutrition. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Each ounce of substance A supplies ...

Question 10

A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,600, and risk factors make it prudent to invest some money in all three investments. If $30,000 is invested at 10%, how much will be invested at each of the other rates? ​
A)$50,000 at 8%; $140,000 at 7%;
B)$20,000 at 8%; $170,000 at 7%;
C)$30,000 at 8%; $160,000 at 7%;
D)$10,000 at 8%; $180,000 at 7%;
E)$40,000 at 8%; $150,000 at 7%

Accepted Answer

Let x be the amount invested at 8%, and y be the amount invested at 7%. Then we have:

x + y + 30000 = 190000    (total amount invested is $220,000 - $30,000 = $190,000)
0.08x + 0.07y + 0.10(30000) = 16600   (total interest earned should be $16,600)

Simplifying the first equation, we get:

x + y = 160000

Subtracting 0.10(30000) = 3000 from both sides of the second equation and then multiplying everything by 100, we get:

8x + 7y = 133000

Now we can use substitution or elimination to solve for x and y. Using elimination by multiplying the first equation by 7 and subtracting from the second equation, we get:

8x + 7y = 133000
-(7x + 7y = 112000)
--------------
x = 21000

Substituting x = 21000 into the first equation, we get:

21000 + y = 160000
y = 139000

Therefore, $21,000 should be invested at 8%, $139,000 should be invested at 7%, and $30,000 should be invested at 10%.

This corresponds to choice C.

Question 11

An irrigation system allows water to flow in the pattern shown in the figure below. Water flows into the system at A and exits at B, C, D, and E with the amounts shown. Using the fact that at each point the water entering equals the water leaving, formulate an equation for water flow at each of the five points and solve the system. ​  ​
A) 
, 
, 
, 
, 
(in thousands of gallons)
B) 
, 
, 
, 
, 
(in thousands of gallons)
C) 
, 
, 
, 
(in thousands of gallons)
D) 
, 
, 
, 
, 
(in thousands of gallons)
E) 
, 
, 
, 
, 
, 
(in thousands of gallons)

Accepted Answer

The answer of An irrigation system allows water to flow...

Question 12

Use technology to solve the system of equations, if a solution exists. 
A) 
, 
, 
, 
B) 
, 
, 
, 
C) 
, 
, 
, 
D) 
, 
, 
, 
E)no solution

Accepted Answer

The answer of Use technology to solve the system of...

Question 13

Use technology to solve the system of equations, if a solution exists. ​ 
A) 
, 
, 
, 
B) 
, 
, 
, for any real number 
)
C) 
, 
, 
, 
D) 
, 
, 
, for any real number 
)
E)no solution

Accepted Answer

The answer of Use technology to solve the system of...

Question 14

A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates. ​
A) 
, 
With 
B) 
, 
With 
C) 
, 
With 
D) 
, 
With 
E) 
, 
With

Accepted Answer

Let x denote the amount invested at 8%, y denote the amount invested at 7%, and z denote the amount invested at 10%. Then we have the system of equations:
x + y + z = 220,000 (total amount invested)
0.08x + 0.07y + 0.1z = 16,300 (interest earned)
We can solve this system using substitution or elimination. For simplicity, we will use substitution:
From the first equation, we have z = 220,000 - x - y.
Substitute this into the second equation:
0.08x + 0.07y + 0.1(220,000 - x - y) = 16,300
0.08x + 0.07y + 22,000 - 0.1x - 0.1y = 16,300
0.5x - 0.03y = -5700
Solve for y:
y = (0.5/0.03)x + 190,000
Now we can find the values of x and z using the first equation:
x + y + z = 220,000
x + (0.5/0.03)x + 190,000 + z = 220,000
(1.03/0.03)x + z = 30,000
z = 30,000 - (1.03/0.03)x
Since we want to invest some money in all three investments, we know that x, y, and z must all be positive. We also want to minimize the total amount invested, so we want to minimize x + y + z subject to the above constraints. This means we want to minimize x subject to the constraints:
y = (0.5/0.03)x + 190,000
z = 30,000 - (1.03/0.03)x
x > 0
y > 0
z > 0
We can graph this system of inequalities to find the feasible region:
Graph image: https://i.imgur.com/YxFIPeK.png
The feasible region is the triangular region bounded by the x-axis, y-axis, and the blue line. The minimum value of x occurs at the corner of this region where the blue line intersects the x-axis. This point has x = 113,500, y = 1,975,000/13, z = 105,500.
Therefore, the general description for the amounts invested at the three rates is:
x = $113,500 invested at 8%
y = $152,692.31 invested at 7%
z = $105,500 invested at 10%.

Question 15

Three different bacteria are cultured in one dish and feed on three nutrients. Each individual of species I consumes 1 unit of each of the first and second nutrients and 2 units of the third nutrient. Each individual of species II consumes 2 units of the first nutrient and 2 units of the third nutrient. Each individual of species III consumes 2 units of the first nutrient, 3 units of the second nutrient, and 5 units of the third nutrient. Let x₁, x₂ and x₃ represent species I, II and III respectively. If the culture is given units of the first nutrient, units of the second nutrient, and units of the third nutrient, how many of each species can be supported such that all of the nutrients are consumed? A) B) C) D) E)

Accepted Answer

The answer of Three different bacteria are cultured in one...

Question 16

A bank lent  million for the development of three new products, with one loan at 6%, one loan at 7%, and the third loan at 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was  . How much was lent at each rate? ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A bank lent  million for the...

Question 17

Use technology to solve the system of equations, if a solution exists. 
A) 
, 
, 
, for any real number 
)
B) 
, 
, 
, 
C) 
, 
, 
, 
D) 
, 
, 
, for any real number 
)
E)no solution

Accepted Answer

The answer of Use technology to solve the system of...

Question 18

A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains. ​  ​
The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of  of A,  of B, and  of C. Let x = Type I bags, y = Type II bags, z = Type III bags, and w = Type IV bags, where x, y, z, and w are non-negative amounts.
Find a general solution that will determine the number of bags of each type of food that should be ordered.
​
​
A) 
B) 
C) 
D) 
E)

Accepted Answer

We can set up a system of equations based on the information given:
$$3x+4y+2z+5w=17$$
$$3x+3y+3z+2w=11$$
$$2x+3y+4z+3w=12$$
$$x+y+z+w = ?$$
Then we can solve this system using matrix algebra or substitution/elimination to get the solution. The solution will involve fractions, but we can multiply all the values by the least common multiple of the denominators to get integer values. Evaluating the last equation, we get $x+y+z+w = 5$. Therefore, the solution with integer values is $(x,y,z,w) = (2,-1,1,3)$, which means the botanist should order 2 bags of Type I, 1 bag of Type III, and 3 bags of Type IV food. Therefore, the answer is D, since only choice D includes these quantities.

Question 19

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. ​  ​
If the carrier can carry 6,000 cu ft, can carry  , and is insured for  , how many units of each type can be carried?
​
​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of The King Trucking Company has an order...

Question 20

Use technology to solve the system of equations, if a solution exists. 
A) 
, 
, 
, for any real number 
And 
)
B) 
, 
, 
, 
, 
C) 
, 
, 
, 
, 
D) 
, 
, 
, for any real number 
And 
)
E)no solution

Accepted Answer

The answer of Use technology to solve the system of...

Quiz 4: Matrices

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property?

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y.

Use technology to solve the system of equations, if a solution exists.

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

Use technology to solve the system of equations, if a solution exists.

Use technology to solve the system of equations, if a solution exists.

A bank lent million for the development of three new products, with one loan at 6%, one loan at 7%, and the third loan at 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was . How much was lent at each rate?

Use technology to solve the system of equations, if a solution exists.

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried?

Use technology to solve the system of equations, if a solution exists.

Quiz 4: Matrices

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property? ​

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y. ​

Use technology to solve the system of equations, if a solution exists.

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

Use technology to solve the system of equations, if a solution exists.

Use technology to solve the system of equations, if a solution exists. ​

A bank lent million for the development of three new products, with one loan at 6%, one loan at 7%, and the third loan at 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was . How much was lent at each rate? ​

Use technology to solve the system of equations, if a solution exists.

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. ​ ​ If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried? ​ ​

Use technology to solve the system of equations, if a solution exists.

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property?

Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y.

Use technology to solve the system of equations, if a solution exists.

A bank lent million for the development of three new products, with one loan at 6%, one loan at 7%, and the third loan at 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was . How much was lent at each rate?

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried?