A merchant purchased a jacket for $60 and then determined a selling price that equaled the purchase price of the...

1. Translate the problem requirements: A jacket costs \(\$60\) to purchase. The selling price equals purchase price plus a markup that is \(25\%\) of the selling price (not \(25\%\) of purchase price). During sale, this selling price gets \(20\%\) discount. We need to find gross profit after the discounted sale.
2. Determine the original selling price: Set up the relationship where selling price equals purchase price plus \(25\%\) of selling price, then solve for the selling price.
3. Calculate the discounted sale price: Apply the \(20\%\) discount to the original selling price to find the actual amount received from the customer.
4. Find the gross profit: Subtract the original purchase cost from the final sale amount to determine the merchant's profit.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening step by step in everyday language:

A merchant buys a jacket for \(\$60\). This is the cost price - what the merchant paid.

Now here's the tricky part: The merchant wants to set a selling price that includes a markup, but this markup is calculated as \(25\%\) of the selling price itself, not \(25\%\) of what he paid.

Think of it this way: If the selling price is some amount, then \(25\%\) of that amount gets added to the \(\$60\) cost to create that same selling price. It's like the selling price creates its own markup!

Then during a sale, the merchant reduces this selling price by \(20\%\) and actually sells the jacket at this discounted amount.

We need to find the gross profit, which is simply: How much he received from the customer minus what he originally paid (\(\$60\)).

Process Skill: TRANSLATE - Converting the complex markup language into a clear mathematical relationship

2. Determine the original selling price

Let's call the selling price 'S' for now and work through this in plain English:

The selling price equals the purchase price plus the markup.
The markup is \(25\%\) of the selling price.

So: Selling price = \(\$60 + (25\% \text{ of the selling price})\)

In everyday terms: \(\mathrm{S} = \$60 + 0.25\mathrm{S}\)

Now we solve this step by step:
\(\mathrm{S} = \$60 + 0.25\mathrm{S}\)
\(\mathrm{S} - 0.25\mathrm{S} = \$60\)
\(0.75\mathrm{S} = \$60\)
\(\mathrm{S} = \$60 ÷ 0.75\)
\(\mathrm{S} = \$80\)

Let's verify this makes sense: If selling price is \(\$80\), then \(25\%\) of \(\$80\) is \(\$20\). So the markup is \(\$20\), and \(\$60 + \$20 = \$80\). ✓

3. Calculate the discounted sale price

During the sale, the merchant discounts the original selling price by \(20\%\).

Original selling price: \(\$80\)
Discount: \(20\%\) of \(\$80 = 0.20 × \$80 = \$16\)

Actual sale price = \(\$80 - \$16 = \$64\)

Or we can think of it as: The customer pays \(80\%\) of the original price
Sale price = \(80\%\) of \(\$80 = 0.80 × \$80 = \$64\)

Both methods give us \(\$64\) as the amount the merchant actually received.

4. Find the gross profit

Gross profit is simply what the merchant received minus what the merchant originally paid:

Gross profit = Sale price - Purchase price
Gross profit = \(\$64 - \$60 = \$4\)

Let's double-check our work:
• Purchase price: \(\$60\)
• Original selling price: \(\$80\) (with \(\$20\) markup that's \(25\%\) of \(\$80\))
• Discounted sale price: \(\$64\) (\(20\%\) off the \(\$80\))
• Profit: \(\$64 - \$60 = \$4\)

Final Answer

The merchant's gross profit on this sale was \(\$4\).

Looking at our answer choices, this corresponds to choice C: \(\$4\).

Common Faltering Points

Errors while devising the approach

Students often misunderstand that the markup is \(25\%\) of the selling price (not \(25\%\) of the purchase price). They might set up the equation as: Selling price = \(\$60 + (25\% \text{ of } \$60) = \$60 + \$15 = \$75\), which leads to an incorrect selling price.

Students may get confused about which price gets discounted during the sale. They might think the discount applies to the purchase price or try to apply both the markup and discount simultaneously, rather than understanding that first the selling price is established, then it gets discounted.

Errors while executing the approach

When solving \(\mathrm{S} = \$60 + 0.25\mathrm{S}\), students commonly make mistakes like forgetting to subtract \(0.25\mathrm{S}\) from both sides, or incorrectly calculating \(0.75\mathrm{S} = \$60\) to get \(\mathrm{S} = \$45\) instead of \(\mathrm{S} = \$80\) by multiplying instead of dividing.

Students may correctly identify that they need to find \(20\%\) of \(\$80\) for the discount, but then calculate it incorrectly (like getting \(\$20\) instead of \(\$16\)), or they might subtract the percentage (20) directly from the price instead of calculating \(20\%\) of the price.

Errors while selecting the answer

Students might select \(\$20\) (the original markup amount) or \(\$16\) (the discount amount) instead of the actual profit of \(\$4\), confusing these intermediate values with the final profit calculation.