Question

Excluding purchases by businesses, the average amount spent on a factory-new car has risen 30 percent in the last five years. In the average household budget, the proportion spent on car purchases has remained unchanged in that period. Therefore the average household budget must have increased by 30 percent over the last five years.
Which of the following is an assumption on which the argument relies?

• A. The average number of factory-new cars purchased per household has remained unchanged over the last five years.
• B. The average amount spent per car by businesses buying factory-new cars has risen 30 percent in the last five years.
• C. The proportion of the average household budget spent on all car-related expenses has remained unchanged over the last five years.
• D. The proportion of the average household budget spent on food and housing has remained unchanged over the last five years.
• E. The total amount spent nationwide on factory-new cars has increased by 30 percent over the last five years.

Hint:

When an argument involves percentages and proportions, translating it into a simple algebraic equation can make the underlying assumptions very clear. The missing piece needed to make the equation balance is often the unstated assumption.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is an assumption question about a mathematical argument. We need to identify the hidden premise that is necessary to make the conclusion logically follow from the other premises.
Step 2: Detailed Explanation:
Let's define some variables to represent the argument mathematically:
- \(C\): Average amount spent on one new car. - \(B\): Average household budget. - \(P\): Proportion (percentage) of the budget spent on car purchases. - \(N\): Average number of new cars purchased per household.
The total amount a household spends on cars is \(C \times N\).
The proportion of the budget spent on cars is \(P = \frac{C \times N}{B}\).
Premises of the argument:
1. The price of one car (\(C\)) has increased by 30%. So, \(C_{new} = 1.30 \times C_{old}\). 2. The proportion (\(P\)) has remained unchanged. So, \(P_{new} = P_{old}\).
Conclusion of the argument:
The budget (\(B\)) must have increased by 30%. So, \(B_{new} = 1.30 \times B_{old}\).
Let's plug these into our proportion equation: \[ P_{old} = \frac{C_{old} \times N_{old}}{B_{old}} \quad \text{and} \quad P_{new} = \frac{C_{new} \times N_{new}}{B_{new}} \] Since \(P_{new} = P_{old}\): \[ \frac{C_{old} \times N_{old}}{B_{old}} = \frac{C_{new} \times N_{new}}{B_{new}} \] Substitute the known changes: \[ \frac{C_{old} \times N_{old}}{B_{old}} = \frac{(1.30 \times C_{old}) \times N_{new}}{B_{new}} \] The argument concludes that \(B_{new} = 1.30 \times B_{old}\). For this to be true, all the other terms must cancel out. This can only happen if \(N_{new} = N_{old}\).
If \(N_{new} = N_{old}\), then the equation becomes: \[ \frac{C_{old} \times N_{old}}{B_{old}} = \frac{1.30 \times C_{old} \times N_{old}}{B_{new}} \] \[ \frac{1}{B_{old}} = \frac{1.30}{B_{new}} \implies B_{new} = 1.30 \times B_{old} \] This shows that the conclusion only holds if we assume that \(N\), the average number of cars purchased per household, has remained unchanged.
Step 3: Final Answer:
The argument implicitly assumes that households are buying the same number of cars as they were five years ago.