Between 1990 and 2000, the number of Americans aged 45 to 64 was projected to increase from 47 million to...

1. Translate the problem requirements: We need to find the ratio of percent increase (45-64 age group) to percent decrease (18-24 age group). Percent increase = \(\frac{\mathrm{New} - \mathrm{Old}}{\mathrm{Old}} \times 100\), and we want the ratio of these two percentages.
2. Calculate the percent increase for the 45-64 age group: Find how much this group increased as a percentage of the original number.
3. Calculate the percent decrease for the 18-24 age group: Find how much this group decreased as a percentage of the original number.
4. Form the ratio and compare to answer choices: Divide the percent increase by the percent decrease and match to the closest ratio among the given options.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're being asked to find. We have two age groups with different population changes:

• The 45-64 age group: growing from 47 million to 61 million

• The 18-24 age group: shrinking from 26 million to 25 million

We need to find how these changes compare to each other. Specifically, we want to know: "How many times bigger is the percent increase of the first group compared to the percent decrease of the second group?"

Think of it this way: if one group grows by 30% and another shrinks by 10%, then the ratio would be 30:10, which simplifies to 3:1.

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical comparison

2. Calculate the percent increase for the 45-64 age group

Let's figure out by what percentage this age group increased.

Starting point: 47 million people

Ending point: 61 million people

Increase in actual numbers: \(61 - 47 = 14\) million people

Now, to find the percent increase, we need to ask: "14 million is what percent of the original 47 million?"

Percent increase = \(\frac{\mathrm{Increase}}{\mathrm{Original}} \times 100\)

Percent increase = \(\frac{14}{47} \times 100\)

Let's calculate: \(14 \div 47 = 0.298\ldots\)

So the percent increase ≈ 29.8%, which we can round to about 30% for easier calculation.

3. Calculate the percent decrease for the 18-24 age group

Now let's find the percentage decrease for the younger age group.

Starting point: 26 million people

Ending point: 25 million people

Decrease in actual numbers: \(26 - 25 = 1\) million people

To find the percent decrease: "1 million is what percent of the original 26 million?"

Percent decrease = \(\frac{\mathrm{Decrease}}{\mathrm{Original}} \times 100\)

Percent decrease = \(\frac{1}{26} \times 100\)

Let's calculate: \(1 \div 26 = 0.0385\ldots\)

So the percent decrease ≈ 3.85%, which we can round to about 4% for easier calculation.

4. Form the ratio and compare to answer choices

Now we need to find the ratio of the percent increase to the percent decrease.

Ratio = Percent increase ÷ Percent decrease

Using our more precise calculations:

Ratio = \(\frac{14}{47} \div \frac{1}{26}\)

When dividing fractions, we multiply by the reciprocal:

Ratio = \(\frac{14}{47} \times \frac{26}{1} = \frac{14 \times 26}{47} = \frac{364}{47}\)

Let's calculate: \(364 \div 47 \approx 7.74\)

This means the ratio is approximately 7.74 to 1.

Looking at our answer choices:

1. 5 to 1 - too low
2. 6 to 1 - too low
3. 8 to 1 - closest to our 7.74
4. 10 to 1 - too high
5. 14 to 1 - much too high

Final Answer

The ratio of the percent increase in the 45-to-64 age group to the percent decrease in the 18-to-24 age group is closest to 8 to 1.

Answer: C) 8 to 1

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "ratio" means in context

Students may confuse this with finding the simple ratio of the actual population changes (14 million to 1 million = 14:1) rather than understanding they need to compare the percent changes. The question specifically asks for the ratio of percent increase to percent decrease, not the ratio of absolute changes.

2. Confusion about which group corresponds to increase vs. decrease

Students might mix up which age group is increasing and which is decreasing, especially since both involve changes over time. This could lead them to calculate the ratio in reverse (percent decrease to percent increase instead of percent increase to percent decrease).

Errors while executing the approach

1. Incorrect percent change formula application

Students frequently make errors when calculating percentage changes, such as using the wrong denominator. For example, they might calculate the percent increase as \(\frac{61-47}{61}\) instead of \(\frac{61-47}{47}\), or use the final value instead of the original value as the base.

2. Arithmetic errors in division and ratio calculation

When calculating \(\frac{14}{47} \div \frac{1}{26}\), students may incorrectly handle the division of fractions, either by not flipping the second fraction correctly or making computational errors when multiplying \(14 \times 26\) or dividing by 47.

3. Premature rounding leading to significant errors

Students might round intermediate calculations too aggressively (like rounding \(\frac{14}{47}\) to 0.3 instead of keeping more precision), which can lead to a final answer that's noticeably different from the correct ratio of approximately 7.74.

Errors while selecting the answer

1. Selecting based on incomplete calculation

Students who calculate the ratio as approximately 7.74 might second-guess themselves and choose a "rounder" number like 10:1 (choice D) instead of recognizing that 8:1 (choice C) is actually the closest option to their calculated value.