Question 1

In Problem 3, if you could exactly afford either 4 units of x and 27 units of y, or 9 units of x and 7 units of y, then if you spent all of your income on y, how many units of y could you buy?
A) 23
B) 63
C) 43
D) 13
E) None of the above.

Accepted Answer

Set up two equations with x and y as variables:
4x + 27y = income and
9x + 7y = income
Solve for y in both equations:
y = (income - 4x) / 27 and
y = (income - 9x) / 7
Since we are spending all of our income on y, we can set these two equations equal to each other:
(income - 4x) / 27 = (income - 9x) / 7
Solving for income, we get:
income = 135x
Substituting this back into one of the original equations, we can solve for y:
4x + 27y = 135x
y = (131/27) x
Now we want to find out how many units of y we can buy with all of our income, so we need to find the value of x that maximizes y. We can do this by setting the derivative of y with respect to x equal to zero:
dy/dx = (131/27) = 0
Solving for x, we get:
x = 27
Now we can substitute this back into either equation to find the maximum value of y:
4(27) + 27y = income
y = 43
Therefore, we can buy 43 units of y with all of our income.

Question 2

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $4 and the price of Y rose to $7, how much would Murphy's income have to rise so that he could still afford his original bundle?
A) $750
B) $500
C) $350
D) $250
E) None of the above.

Accepted Answer

Using the original prices and quantities, Murphy's income can be calculated as follows:

Income = (Price of X * Quantity of X) + (Price of Y * Quantity of Y)
Income = ($2 * 100) + ($4 * 50)
Income = $200 + $200
Income = $400

To calculate how much Murphy's income would have to rise to afford the original bundle at the new prices, we can set up the following equation:

(New Price of X * Quantity of X) + (New Price of Y * Quantity of Y) = $400

Substituting in the new prices and original quantities, we get:

($4 * 100) + ($7 * 50) = $400

Simplifying, we get:

$400 + $350 = $750

Therefore, Murphy's income would have to rise by $350 to be able to afford the original bundle at the new prices. Hence, the answer is C.

Question 3

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $6 and the price of Y rose to $5, how much would Murphy's income have to rise so that he could still afford his original bundle?
A) $750
B) $250
C) $450
D) $500
E) None of the above.

Accepted Answer

The original cost of Murphy's bundle was $300 (100*$2 + 50*$4). 
The new cost of Murphy's bundle would be $700 (100*$6 + 50*$5). 
To afford the new bundle, Murphy's income would have to increase by $400. 
Therefore, to afford the original bundle at the new prices, Murphy's income would have to increase by $450 ($400 + $50, the difference between the original cost and the new cost). 
So, the correct answer is $450 (option C).

Question 4

In Problem 10, if in the same amount of time that it takes her to read 40 pages of economics and 30 pages of sociology, Martha could read 30 pages of economics and 70 pages of sociology, then which of these equations describes combinations of pages of economics, E, and sociology, S, that she could read in the time it takes to read 40 pages of economics and 30 pages of sociology?
A) 4E + S = 190.
B) E + S = 70.
C) E + S = 100.
D) + S = 50.
E) All of the above.

Accepted Answer

The time taken to read 40 pages of economics and 30 pages of sociology is the same as the time taken to read 30 pages of economics and 70 pages of sociology. This can be expressed as 4E + 3S = 3E + 7S, which simplifies to E = 4S. Substituting E = 4S into the equation for the time to read 40 pages of economics and 30 pages of sociology gives 4(40) + 30 = 190.

Question 5

In Problem 3, if you could exactly afford either 2 units of x and 7 units of y, or 4 units of x and 3 units of y, then if you spent all of your income on y, how many units of y could you buy?
A) 11
B) 7
C) 15
D) 6
E) None of the above.

Accepted Answer

The answer of In Problem 3, if you could exactly...

Question 6

In Problem 3, if you could exactly afford either 4 units of x and 8 units of y, or 6 units of x and 2 units of y, then if you spent all of your income on y, how many units of y could you buy?
A) 20
B) 26
C) 10
D) 14
E) None of the above.

Accepted Answer

The answer of In Problem 3, if you could exactly...

Question 7

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $48 per sack for accepting garbage and if his relatives send him an allowance of $432, then his budget line is described by the equation
A) 6V = 48G.
B) 6V + 48G = 432.
C) 6V - 48G = 432.
D) 6V = 432 - G.
E) None of the above.

Accepted Answer

The answer of In Problem 7, Edmund must pay $6...

Question 8

In Problem 10, if in the same amount of time that it takes her to read 40 pages of economics and 30 pages of sociology, Martha could read 20 pages of economics and 110 pages of sociology, then which of these equations describes combinations of pages of economics, E, and sociology, S, that she could read in the time it takes to read 40 pages of economics and 30 pages of sociology?
A) E + S = 130.
B) + S = 50.
C) E + S = 70.
D) 4E + S = 190.
E) All of the above.

Accepted Answer

Let's use the information given to form two equations. Let $x$ be the number of pages of economics Martha can read in the same amount of time as 40 pages of economics and 30 pages of sociology. Similarly, let $y$ be the number of pages of sociology she can read in that same amount of time. Then we have:

$$\frac{x}{40} = \frac{y}{30} = \frac{x+y}{40+30} = \frac{x+y}{70}$$

Using the second piece of information, we can also set up:

$$\frac{20}{40} = \frac{110}{30} = \frac{20+110}{40+30} = \frac{130}{70}$$

We can use these two equations to solve for $x$ and $y$. Multiplying the first equation by $\frac{70}{x+y}$ gives:

$$\frac{70x}{40(x+y)} = \frac{70y}{30(x+y)}$$

Simplifying:

$$\frac{7x}{4(x+y)} = \frac{7y}{3(x+y)}$$

Cross-multiplying and simplifying further:

$$21x = 28y \Rightarrow x = \frac{4}{3}y$$

Substituting this value of $x$ into either of the original equations gives:

$$\frac{4y}{40} = \frac{y}{30} \Rightarrow y = 60$$

So $x = \frac{4}{3}(60) = 80$. Therefore, in the same amount of time that it takes to read 40 pages of economics and 30 pages of sociology, Martha can read 80 pages of economics and 60 pages of sociology. The corresponding equation is:

$$4E + S = 190$$

So the answer is (D).

Question 9

In Problem 3, if you could exactly afford either 2 units of x and 11 units of y, or 4 units of x and 7 units of y, then if you spent all of your income on y, how many units of y could you buy?
A) 19
B) 15
C) 11
D) 6
E) None of the above.

Accepted Answer

First, we need to find the prices of x and y by dividing the given budget by the given units of x and y for both cases:

For 2x + 11y: Price of x = 70/2 = 35; Price of y = 70/11 = 6.36 (rounded to 2 decimal places)
For 4x + 7y: Price of x = 70/4 = 17.5; Price of y = 70/7 = 10

Now, we need to find which option gives us the most units of y for the given budget.

For option 1 (2x + 11y): If we only spend on y, we can buy 11y for a total cost of 70, which means we can buy 11y/6.36 = 1.728 units of y

For option 2 (4x + 7y): If we only spend on y, we can buy 7y for a total cost of 70, which means we can buy 7y/10 = 0.7 units of y

Therefore, option 1 gives us more units of y for the same budget, and the maximum number of units of y we can buy is 11.

Question 10

In Problem 10, if in the same amount of time that it takes her to read 40 pages of economics and 30 pages of sociology, Martha could read 10 pages of economics and 90 pages of sociology, then which of these equations describes combinations of pages of economics, E, and sociology, S, that she could read in the time it takes to read 40 pages of economics and 30 pages of sociology?
A) E + S = 70.
B) + S = 50.
C) 2E + S = 110.
D) E + S = 100.
E) All of the above.

Accepted Answer

Let x be the amount of time it takes to read 40 pages of economics and 30 pages of sociology. From the problem, we know that:

40E/30S = 10E/90S

Solving for E/S, we get:

E/S = 9/4

This means that for every 9 pages of economics that Martha reads, she reads 4 pages of sociology. Let's say she reads E pages of economics and S pages of sociology in the same amount of time it takes to read 40 pages of economics and 30 pages of sociology. Then we can set up the following equation:

E + S = (40 + 30)(4/9) = 70/3

Multiplying both sides by 3, we get:

3E + 3S = 70

Simplifying, we get:

2E + S = 70

So the answer is (C).

Question 11

In Problem 3, if you could exactly afford either 2 units of x and 10 units of y, or 4 units of x and 2 units of y, then if you spent all of your income on y, how many units of y could you buy?
A) 6
B) 10
C) 18
D) 26
E) None of the above.

Accepted Answer

The answer of In Problem 3, if you could exactly...

Question 12

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $4 and the price of Y rose to $7, how much would Murphy's income have to rise so that he could still afford his original bundle?
A) $500
B) $750
C) $350
D) $250
E) None of the above.

Accepted Answer

We can use the concept of budget constraint to solve this problem. Before the price change, Murphy's total expenditure on X and Y was 100($2) + 50($4) = $300. This expenditure must remain the same after the price change. If the price of X is now $4, Murphy can only afford 100($4) = $400 worth of X. This means that Murphy now has $300 - $400 = -$100 left to spend on Y. To afford the original bundle, Murphy needs an additional $50(4-7) = $-150 for Y. Therefore, Murphy's income will need to rise by $150 - (-$100) = $250 to afford the original bundle. Thus, the answer is option C.

Another way to solve the problem is to use the concept of income elasticity of demand. The total expenditure on X and Y before the price change was $300. The income elasticity of demand for X is 0 (inelastic), which means that the quantity demanded of X does not change with income. Therefore, Murphy will still spend $200 on X after the income change. The income elasticity of demand for Y is -1/2 (elastic), which means that the quantity demanded of Y will decrease by 25% when income decreases by 50%. To maintain the same quantity demanded of Y, Murphy's income will need to increase by 25% of $100 = $25. Therefore, Murphy's income will need to rise by $200 + $25 = $225 to afford the original bundle. This calculation is slightly different from the first method because we are assuming that Murphy still consumes 50 units of Y even though his income has decreased.

Question 13

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $48 per sack for accepting garbage and if his relatives send him an allowance of $336, then his budget line is described by the equation
A) 6V - 48G = 336.
B) 6V = 48G.
C) 6V = 336 - G.
D) 6V + 48G = 336.
E) None of the above.

Accepted Answer

The answer of In Problem 7, Edmund must pay $6...

Question 14

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $6 and the price of Y rose to $6, how much would Murphy's income have to rise so that he could still afford his original bundle?
A) $300
B) $900
C) $600
D) $500
E) None of the above.

Accepted Answer

The answer of In Problem 4, Murphy used to consume...

Question 15

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $18 per sack for accepting garbage and if his relatives send him an allowance of $108, then his budget line is described by the equation
A) 6V + 18G = 108.
B) 6V = 108 - G.
C) 6V -18G = 108.
D) 6V = 18G.
E) None of the above.

Accepted Answer

If Edmund buys V punk rock video cassettes, the cost will be 6V. If he accepts G sacks of garbage, he earns 18G. His total budget is 6V + 18G + 108 (we add 108 since it is an allowance). This equation can be rearranged to get 6V - 18G = 108, which matches choice C.

Question 16

In Problem 10, if in the same amount of time that it takes her to read 40 pages of economics and 30 pages of sociology, Martha could read 20 pages of economics and 110 pages of sociology, then which of these equations describes combinations of pages of economics, E, and sociology, S, that she could read in the time it takes to read 40 pages of economics and 30 pages of sociology?
A) E + S = 130.
B) + S = 50.
C) E + S = 70.
D) 4E + S = 190.
E) All of the above.

Accepted Answer

Let's use the information given to form two equations. Let $x$ be the number of pages of economics Martha can read in the same amount of time as 40 pages of economics and 30 pages of sociology. Similarly, let $y$ be the number of pages of sociology she can read in that same amount of time. Then we have:

$$\frac{x}{40} = \frac{y}{30} = \frac{x+y}{40+30} = \frac{x+y}{70}$$

Using the second piece of information, we can also set up:

$$\frac{20}{40} = \frac{110}{30} = \frac{20+110}{40+30} = \frac{130}{70}$$

We can use these two equations to solve for $x$ and $y$. Multiplying the first equation by $\frac{70}{x+y}$ gives:

$$\frac{70x}{40(x+y)} = \frac{70y}{30(x+y)}$$

Simplifying:

$$\frac{7x}{4(x+y)} = \frac{7y}{3(x+y)}$$

Cross-multiplying and simplifying further:

$$21x = 28y \Rightarrow x = \frac{4}{3}y$$

Substituting this value of $x$ into either of the original equations gives:

$$\frac{4y}{40} = \frac{y}{30} \Rightarrow y = 60$$

So $x = \frac{4}{3}(60) = 80$. Therefore, in the same amount of time that it takes to read 40 pages of economics and 30 pages of sociology, Martha can read 80 pages of economics and 60 pages of sociology. The corresponding equation is:

$$4E + S = 190$$

So the answer is (D).

Question 17

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $36 per sack for accepting garbage and if his relatives send him an allowance of $360, then his budget line is described by the equation
A) 6V + 36G = 360.
B) 6V - 36G = 360.
C) 6V = 360 - G.
D) 6V = 36G.
E) None of the above.

Accepted Answer

The answer of In Problem 7, Edmund must pay $6...

Question 18

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $18 per sack for accepting garbage and if his relatives send him an allowance of $180, then his budget line is described by the equation
A) 6V - 18G = 180.
B) 6V = 180 - G.
C) 6V + 18G = 180.
D) 6V = 18G.
E) None of the above.

Accepted Answer

The answer of In Problem 7, Edmund must pay $6...

Question 19

In Problem 10, if in the same amount of time that it takes her to read 40 pages of economics and 30 pages of sociology, Martha could read 20 pages of economics and 90 pages of sociology, then which of these equations describes combinations of pages of economics, E, and sociology, S, that she could read in the time it takes to read 40 pages of economics and 30 pages of sociology?
A) E + S = 110.
B) + S = 50.
C) 3E + S = 150.
D) E + S = 70.
E) All of the above.

Accepted Answer

Let's use the information given to create two equations that relate the amount of pages Martha can read in the same amount of time:
40E + 30S = 20E + 90S
20E = 60S
S = 1/3 E

Now we can substitute this relationship into one of the equations given:
E + S = 110
E + 1/3 E = 110
4/3 E = 110
E = 82.5
S = 1/3 E = 27.5

Checking the other equations, we see that A, B, and D do not satisfy the 40E + 30S = 20E + 90S equation using the values we calculated for E and S. Therefore, the correct choice is C.

Question 20

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $5 and the price of Y rose to $5, how much would Murphy's income have to rise so that he could still afford his original bundle?
A) $400
B) $350
C) $200
D) $600
E) None of the above.

Accepted Answer

The answer of In Problem 4, Murphy used to consume...

Quiz 1: A: Budget Constraint

In Problem 3, if you could exactly afford either 4 units of x and 27 units of y, or 9 units of x and 7 units of y, then if you spent all of your income on y, how many units of y could you buy?

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $4 and the price of Y rose to $7, how much would Murphy's income have to rise so that he could still afford his original bundle?

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $6 and the price of Y rose to $5, how much would Murphy's income have to rise so that he could still afford his original bundle?

In Problem 3, if you could exactly afford either 2 units of x and 7 units of y, or 4 units of x and 3 units of y, then if you spent all of your income on y, how many units of y could you buy?

In Problem 3, if you could exactly afford either 4 units of x and 8 units of y, or 6 units of x and 2 units of y, then if you spent all of your income on y, how many units of y could you buy?

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $48 per sack for accepting garbage and if his relatives send him an allowance of $432, then his budget line is described by the equation

In Problem 3, if you could exactly afford either 2 units of x and 11 units of y, or 4 units of x and 7 units of y, then if you spent all of your income on y, how many units of y could you buy?

In Problem 3, if you could exactly afford either 2 units of x and 10 units of y, or 4 units of x and 2 units of y, then if you spent all of your income on y, how many units of y could you buy?

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $4 and the price of Y rose to $7, how much would Murphy's income have to rise so that he could still afford his original bundle?

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $48 per sack for accepting garbage and if his relatives send him an allowance of $336, then his budget line is described by the equation

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $6 and the price of Y rose to $6, how much would Murphy's income have to rise so that he could still afford his original bundle?

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $18 per sack for accepting garbage and if his relatives send him an allowance of $108, then his budget line is described by the equation

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $36 per sack for accepting garbage and if his relatives send him an allowance of $360, then his budget line is described by the equation

In Problem 7, Edmund must pay $6 each for punk rock video cassettes. If Edmund is paid $18 per sack for accepting garbage and if his relatives send him an allowance of $180, then his budget line is described by the equation

In Problem 4, Murphy used to consume 100 units of X and 50 units of Y when the price of X was $2 and the price of Y was $4. If the price of X rose to $5 and the price of Y rose to $5, how much would Murphy's income have to rise so that he could still afford his original bundle?