Last year 12.6% of the total value of all merchandise sold by a certain department store was from the store's...

1. Translate the problem requirements: We know the store's brand represents \(12.6\%\) of total merchandise value and generated \(\$7.2\) million. We need to find revenue from non-store brand merchandise (the remaining \(87.4\%\) of total sales).
2. Find the total revenue using the part-to-whole relationship: Since store brand (\(\$7.2\) million) represents \(12.6\%\) of total sales, we can determine what \(100\%\) (total sales) equals.
3. Calculate non-store brand revenue: Subtract the store brand revenue from total revenue to find revenue from merchandise that was not the store's brand.
4. Verify using answer choices: Check our result against the given options, using approximation since the question asks for an approximate answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what we need to find.

What we know:
• The store's brand merchandise made up \(12.6\%\) of all merchandise sold
• The store's brand generated \(\$7.2\) million in revenue

What we need to find:
• Revenue from merchandise that was NOT the store's brand

Think of it this way: if the store's brand is \(12.6\%\) of total sales, then everything else (non-store brand) must be the remaining percentage of total sales.

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

2. Find the total revenue using the part-to-whole relationship

Here's the key insight: if \(\$7.2\) million represents \(12.6\%\) of total sales, we can figure out what \(100\%\) (the total) must be.

Let's use everyday reasoning first:
If \(12.6\% = \$7.2\) million, then we need to find what \(100\%\) equals.

We can think of this as: "\(\$7.2\) million is to \(12.6\%\) as Total Revenue is to \(100\%\)"

To find the total, we calculate:
Total Revenue = \(\$7.2 \text{ million} ÷ 12.6\% = \$7.2 \text{ million} ÷ 0.126\)

Let's compute this step by step:
\(\$7.2 ÷ 0.126 = \$57.14 \text{ million}\) (approximately)

So the total revenue from all merchandise was approximately \(\$57.14\) million.

3. Calculate non-store brand revenue

Now that we know the total revenue, finding the non-store brand revenue is straightforward.

Non-store brand revenue = Total revenue - Store brand revenue
Non-store brand revenue = \(\$57.14 \text{ million} - \$7.2 \text{ million} = \$49.94 \text{ million}\)

Since we're looking for an approximate answer, this rounds to about \(\$50\) million.

We can double-check this makes sense:
• Store brand: \(\$7.2\) million (\(12.6\%\) of total)
• Non-store brand: \(\$50\) million (\(87.4\%\) of total)
• Total: \(\$57.2\) million ✓

4. Verify using answer choices

Looking at our calculated answer of approximately \(\$50\) million, let's check against the given choices:

1. 1 - Too small
2. 6 - Too small
3. 36 - Too small
4. 50 - Matches our calculation!
5. 58 - Too large

Final Answer

The store earned approximately \(\$50\) million from the sale of merchandise that was not the store's brand.

The answer is (D) 50.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what needs to be calculated: Students may confuse the question and try to find the total revenue or the percentage of non-store brand merchandise, rather than focusing on the dollar amount of non-store brand revenue.

2. Setting up the wrong relationship: Students might incorrectly think they need to directly calculate \(87.4\%\) of something, without first recognizing that they need to find the total revenue using the given part-to-whole relationship (\(12.6\% = \$7.2\) million).

3. Confusing percentage relationships: Students may mistakenly try to find what percentage \(\$7.2\) million is of the non-store brand revenue, rather than understanding that \(\$7.2\) million represents \(12.6\%\) of the total revenue.

Errors while executing the approach

1. Arithmetic errors in percentage division: When calculating \(\$7.2 ÷ 0.126\), students often make computational mistakes, such as incorrect decimal placement or division errors, leading to wrong total revenue figures.

2. Using the wrong percentage in calculations: Students might accidentally use \(87.4\%\) instead of \(12.6\%\) when setting up their calculations, or convert percentages to decimals incorrectly (e.g., using 12.6 instead of 0.126).

3. Forgetting the final subtraction step: After correctly finding the total revenue, students may forget to subtract the store brand revenue (\(\$7.2\) million) to get the non-store brand revenue, and mistakenly select the total revenue as their answer.

Errors while selecting the answer

1. Selecting the total revenue instead of non-store brand revenue: Students who calculated the total revenue as approximately \(\$57\) million might select choice (E) 58, forgetting that the question asks for non-store brand revenue only.

2. Rounding confusion: Students might calculate the correct value (\(∼\$49.94\) million) but struggle with which answer choice represents the best approximation, potentially selecting choice (C) 36 if they made an error in their mental rounding.

3. Misreading the answer choices: Students may misinterpret the scale of the answer choices, not realizing that all options represent millions of dollars, potentially leading them to select an inappropriately small value like choices (A) or (B).