A furniture dealer purchased a sofa and an armchair for the same amount and sold the sofa for 20 percent...

1. Translate the problem requirements: Clarify that "gross profit" means selling price minus cost, and we need to find by what percent the sofa's profit exceeds the armchair's profit
2. Set up with concrete values: Use simple dollar amounts for both purchase prices to avoid working with variables and fractions
3. Calculate selling prices using given relationships: Find armchair selling price using \(10\%\) profit margin, then find sofa selling price using the "\(20\%\) more" condition
4. Compare profits to find percentage difference: Calculate both profits and determine by what percent the sofa's profit exceeds the armchair's profit

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in plain English. When the problem mentions "gross profit," this simply means how much money the store made on each item - which is the selling price minus what they paid for it.

The question asks: "the store's gross profit on the sofa was what percent greater than its gross profit on the armchair?" This means we need to find by how much the sofa's profit exceeds the armchair's profit, expressed as a percentage of the armchair's profit.

Key relationships given:
• Both items were purchased for the same amount
• The sofa was sold for \(20\%\) more than the armchair's selling price
• The armchair's gross profit was \(10\%\) of what the store paid for it

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

2. Set up with concrete values

To make this easier to work with, let's use actual dollar amounts instead of variables. Since both items were purchased for the same amount, let's say:

Purchase price of sofa = \(\$100\)
Purchase price of armchair = \(\$100\)

Using concrete numbers helps us avoid getting confused with variables and makes the arithmetic much cleaner. We can always verify our percentage answer works for any dollar amount.

3. Calculate selling prices using given relationships

Now let's find the selling prices step by step.

For the armchair:
We know the gross profit on the armchair was \(10\%\) of what the store paid for it.
Armchair's gross profit = \(10\% \times \$100 = \$10\)
Since gross profit = selling price - purchase price:
Armchair's selling price = \(\$100 + \$10 = \$110\)

For the sofa:
The sofa was sold for \(20\%\) more than the armchair's selling price.
Sofa's selling price = \(\$110 + (20\% \times \$110)\)
Sofa's selling price = \(\$110 + \$22 = \$132\)

Therefore:
Sofa's gross profit = \(\$132 - \$100 = \$32\)

4. Compare profits to find percentage difference

Now we can compare the profits:
Armchair's gross profit = \(\$10\)
Sofa's gross profit = \(\$32\)

The question asks by what percent the sofa's gross profit is greater than the armchair's gross profit.

Difference in profits = \(\$32 - \$10 = \$22\)

Percentage increase = \(\frac{\text{Difference}}{\text{Armchair's profit}} \times 100\%\)
Percentage increase = \(\frac{\$22}{\$10} \times 100\% = 220\%\)

This means the sofa's gross profit is \(220\%\) greater than the armchair's gross profit.

Final Answer

The store's gross profit on the sofa was \(220\%\) greater than its gross profit on the armchair.

Verification: Armchair profit \(\$10\), Sofa profit \(\$32\). The sofa's profit is \(\$22\) more, which is \(\frac{\$22}{\$10} = 2.2 = 220\%\) greater.

The answer is E. \(220\%\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "20% more than the armchair"

Students often confuse what the sofa's selling price is compared to. The problem states the sofa was sold for "\(20\%\) more than the armchair," which means \(20\%\) more than the armchair's selling price, not its purchase price. Students might incorrectly calculate the sofa's selling price as \(20\%\) more than its \(\$100\) purchase price, leading to a sofa selling price of \(\$120\) instead of the correct \(\$132\).

2. Confusing "percent greater than" with "percent of"

The question asks for "what percent greater than" the armchair's profit, not what percent the sofa's profit is "of" the armchair's profit. If the sofa's profit is \(\$32\) and armchair's is \(\$10\), students might incorrectly calculate \(\frac{32}{10} = 320\%\) and select that as their answer, when the correct calculation for "percent greater" requires finding the difference first: \(\frac{32-10}{10} = 220\%\).

Errors while executing the approach

1. Calculation errors with percentages

When calculating \(20\%\) of the armchair's selling price (\(\$110\)), students might make arithmetic errors: \(20\% \times \$110 = 0.20 \times \$110 = \$22\). Common mistakes include calculating this as \(\$11\) (using \(10\%\) instead of \(20\%\)) or \(\$20\) (rough approximation), leading to incorrect sofa selling prices.

2. Setting up the profit calculations incorrectly

Students might forget that gross profit = selling price - purchase price. Some may accidentally calculate profit as selling price only, or confuse which values to subtract from which, especially when working with multiple items and prices.

Errors while selecting the answer

1. Selecting 320% instead of 220%

After correctly calculating that the sofa's profit (\(\$32\)) is \(3.2\) times the armchair's profit (\(\$10\)), students might select \(320\%\) as their answer. However, when something is \(3.2\) times another value, it's actually \(220\%\) greater than the original (since \(3.2 = 1 + 2.2\), and the \(2.2\) represents the "greater than" portion).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient concrete values
Since both items were purchased for the same amount, let's use \(\$100\) as the purchase price for each item. This makes percentage calculations straightforward.

Purchase price of sofa = \(\$100\)
Purchase price of armchair = \(\$100\)

Step 2: Calculate armchair selling price
The armchair's gross profit was \(10\%\) of what the store paid:
Armchair profit = \(10\% \times \$100 = \$10\)
Armchair selling price = \(\$100 + \$10 = \$110\)

Step 3: Calculate sofa selling price
The sofa was sold for \(20\%\) more than the armchair:
Sofa selling price = \(\$110 + (20\% \times \$110) = \$110 + \$22 = \$132\)

Step 4: Calculate sofa profit
Sofa profit = Selling price - Purchase price = \(\$132 - \$100 = \$32\)

Step 5: Compare the profits
Armchair profit = \(\$10\)
Sofa profit = \(\$32\)
Difference = \(\$32 - \$10 = \$22\)

Step 6: Find percentage increase
Percent increase = \(\frac{\text{Difference}}{\text{Armchair profit}} \times 100\%\)
= \(\frac{\$22}{\$10} \times 100\% = 220\%\)

The sofa's gross profit was \(220\%\) greater than the armchair's gross profit.