A certain economic indicator is proportional to the square of the unemployment rate. If the unemployment rate increases by 10...

1. Translate the problem requirements: Understand that "proportional to the square" means the indicator equals some constant times the unemployment rate squared. We need to find the percent increase in the indicator when unemployment rate increases by 10%.
2. Set up the proportional relationship: Express the indicator in terms of the unemployment rate using a simple variable approach that avoids complex fractions.
3. Calculate the new indicator value: Determine what happens to the indicator when the unemployment rate increases by 10% by substituting the new rate into our relationship.
4. Find the percent increase: Compare the original and new indicator values to calculate the percentage change, using the answer choices to guide our level of precision.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is really asking in everyday terms.

When we say the economic indicator is "proportional to the square" of the unemployment rate, we mean: if the unemployment rate goes up, the indicator goes up by the square of that change. Think of it like this - if unemployment rate doubles, the indicator doesn't just double, it goes up by \(2^2 = 4\) times as much.

Our goal is simple: if unemployment rate increases by 10%, how much does the indicator increase?

2. Set up the proportional relationship

Let's use concrete numbers to make this crystal clear. Imagine the unemployment rate starts at some value - let's call it U.

Since the indicator is proportional to the square of the unemployment rate, we can say:

Original Indicator = \(\mathrm{k} \times \mathrm{U}^2\)

where k is just some constant number that doesn't change.

Now, when unemployment increases by 10%, the new unemployment rate becomes:

New Unemployment Rate = \(\mathrm{U} + 10\% \text{ of } \mathrm{U} = \mathrm{U} + 0.1\mathrm{U} = 1.1\mathrm{U}\)

So the new indicator will be:

New Indicator = \(\mathrm{k} \times (1.1\mathrm{U})^2\)

3. Calculate the new indicator value

Let's work out what \((1.1\mathrm{U})^2\) equals in plain English terms.

\((1.1\mathrm{U})^2 = (1.1)^2 \times \mathrm{U}^2 = 1.21 \times \mathrm{U}^2\)

So our New Indicator = \(\mathrm{k} \times 1.21 \times \mathrm{U}^2\)

But remember, our Original Indicator = \(\mathrm{k} \times \mathrm{U}^2\)

This means: New Indicator = \(1.21 \times \text{(Original Indicator)}\)

In other words, the new indicator is 1.21 times the original indicator.

4. Find the percent increase

Now we can easily find the percentage increase:

If the new value is 1.21 times the original, then the increase is:

\(1.21 - 1.00 = 0.21\)

Converting to percentage: \(0.21 \times 100\% = 21\%\)

Let's verify this makes sense: We started with some indicator value, and now we have 21% more than that original value.

Final Answer

The economic indicator will increase by 21%.

This matches answer choice D, which confirms our solution is correct.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "proportional to the square"
Students often confuse "proportional to the square" with simple proportion. They might think if unemployment increases by 10%, the indicator also increases by 10%, missing the crucial "square" relationship entirely.

2. Setting up incorrect relationship
Students may incorrectly set up the relationship as Indicator = \(\mathrm{U} + \mathrm{k}\) instead of Indicator = \(\mathrm{k} \times \mathrm{U}^2\). They fail to recognize that proportional relationships involve multiplication, not addition.

3. Misunderstanding what "10% increase" means
Students might interpret "increases by 10%" as the new value being \(0.1\mathrm{U}\) instead of \(1.1\mathrm{U}\), forgetting that a 10% increase means the new value is 110% of the original.

Errors while executing the approach

1. Calculation error with \((1.1)^2\)
Students frequently miscalculate \((1.1)^2 = 1.21\), often getting 1.11 or 1.20 instead. This is a critical arithmetic error that leads to wrong percentage calculations.

2. Incorrect percentage increase formula
Even when getting 1.21 correctly, students may calculate percentage increase as \((1.21/1.00) \times 100\% = 121\%\) instead of \((1.21 - 1.00) \times 100\% = 21\%\). They confuse "new value as percentage of original" with "percentage increase."

Errors while selecting the answer

1. Selecting percentage of new value instead of increase
Students who calculate 121% (new value as percentage of original) might look for this in answer choices and mistakenly select 21% without realizing they calculated the wrong thing, or worse, select a completely different choice thinking their approach was wrong.

Alternate Solutions

Smart Numbers Approach

Instead of working with variables, let's choose a convenient concrete value for the unemployment rate and work with actual numbers throughout.

Step 1: Choose a smart number for the initial unemployment rate
Let's say the initial unemployment rate is 10%. This is a convenient choice because: • It makes the 10% increase calculation easy • Round numbers simplify our arithmetic • The specific value doesn't matter since we're looking for a percentage relationship

Step 2: Set up the initial indicator value
Since the indicator is proportional to the square of unemployment rate: • Initial unemployment rate = 10% • Initial indicator = \(\mathrm{k} \times (10)^2 = \mathrm{k} \times 100 = 100\mathrm{k}\) (where k is some proportionality constant)

Step 3: Calculate the new unemployment rate
After a 10% increase: • New unemployment rate = 10% + (10% of 10%) = 10% + 1% = 11%

Step 4: Calculate the new indicator value
New indicator = \(\mathrm{k} \times (11)^2 = \mathrm{k} \times 121 = 121\mathrm{k}\)

Step 5: Find the percent increase in the indicator
• Original indicator: \(100\mathrm{k}\) • New indicator: \(121\mathrm{k}\) • Increase: \(121\mathrm{k} - 100\mathrm{k} = 21\mathrm{k}\) • Percent increase = \((21\mathrm{k} \div 100\mathrm{k}) \times 100\% = 21\%\)

Answer: D. 21%

Key insight: This concrete approach demonstrates that regardless of the initial unemployment rate we choose, a 10% increase always results in a 21% increase in the indicator due to the squared relationship.