A furniture dealer purchased a desk for $150 and then set the selling price equal to the purchase price plus...

1. Translate the problem requirements: The dealer buys a desk for \(\$150\). The selling price equals purchase price plus a markup, where the markup is 40% of the selling price (not the purchase price). We need to find the gross profit, which is the difference between selling price and purchase price.
2. Set up the relationship between selling price and markup: Express the selling price in terms of itself using the given markup relationship to create a solvable equation.
3. Solve for the actual selling price: Use algebraic manipulation to find the numerical value of the selling price.
4. Calculate the gross profit: Subtract the purchase price from the selling price to determine the dealer's profit.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English first. A furniture dealer buys a desk - that's straightforward, he pays \(\$150\) for it. Now here's where it gets interesting: when he decides what price to sell it for, he doesn't just add a fixed dollar amount. Instead, he sets up his selling price in a specific way.

The selling price equals the purchase price PLUS a markup. But this markup isn't a percentage of what he paid - it's 40% of what he's going to sell it for. This creates a relationship where the selling price depends on itself.

Our goal is to find the gross profit, which is simply: How much he sold it for minus how much he paid for it.

Process Skill: TRANSLATE - Converting the relationship "markup that was 40% of the selling price" into mathematical understanding

2. Set up the relationship between selling price and markup

Let's call the selling price "S" to keep things simple.

From the problem, we know:

• Purchase price = \(\$150\)
• Selling price = Purchase price + Markup
• Markup = 40% of the selling price

So the markup = 40% of S = \(0.40 \times \mathrm{S}\)

Now we can write: \(\mathrm{S} = \$150 + 0.40\mathrm{S}\)

This equation captures the key relationship: the selling price equals the original cost plus a markup that's 40% of that very selling price.

3. Solve for the actual selling price

Starting with our equation: \(\mathrm{S} = \$150 + 0.40\mathrm{S}\)

To solve this, we need to get all the S terms on one side:
\(\mathrm{S} - 0.40\mathrm{S} = \$150\)
\(0.60\mathrm{S} = \$150\)

Dividing both sides by 0.60:
\(\mathrm{S} = \$150 \div 0.60\)
\(\mathrm{S} = \$150 \div \frac{60}{100}\)
\(\mathrm{S} = \$150 \times \frac{100}{60}\)
\(\mathrm{S} = \$150 \times \frac{5}{3}\)
\(\mathrm{S} = \$250\)

So the selling price is \(\$250\).

Let's verify this makes sense: If selling price is \(\$250\), then 40% of \(\$250 = \$100\). And indeed, \(\$150 + \$100 = \$250\). ✓

4. Calculate the gross profit

Gross profit = Selling price - Purchase price
Gross profit = \(\$250 - \$150 = \$100\)

Let's double-check our work by thinking about it another way: if the selling price is \(\$250\) and the purchase price is \(\$150\), then the markup is \(\$100\). Is \(\$100\) equal to 40% of the selling price? 40% of \(\$250 = 0.40 \times \$250 = \$100\). Yes! ✓

Final Answer

The dealer's gross profit from the purchase and sale of the desk was \(\$100\).

This matches answer choice E. \(\$100\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the markup percentage is based on
Students often misread "markup that was 40% of the selling price" as "markup that was 40% of the purchase price." This is a natural mistake because we're more accustomed to markups being calculated as a percentage of cost. If a student makes this error, they would calculate markup as 40% of \(\$150 = \$60\), leading to a selling price of \(\$150 + \$60 = \$210\) and incorrect profit of \(\$60\).

2. Setting up the relationship incorrectly
Some students may struggle to translate "selling price equals purchase price plus markup that was 40% of selling price" into the correct equation \(\mathrm{S} = 150 + 0.40\mathrm{S}\). They might incorrectly write something like \(\mathrm{S} = 150 + 40\mathrm{S}\) (forgetting to convert percentage) or get confused about which variable represents what in the relationship.

3. Confusing profit with markup
Students may not clearly distinguish between "markup" and "profit" in this context. While both happen to be \(\$100\) in this problem, understanding that markup refers to the amount added to cost price, while profit is the difference between selling and purchase price, is crucial for setting up the problem correctly.

Errors while executing the approach

1. Algebraic manipulation errors when solving \(\mathrm{S} = 150 + 0.40\mathrm{S}\)
When moving terms to isolate S, students commonly make errors like: incorrectly getting \(\mathrm{S} + 0.40\mathrm{S} = 150\) instead of \(\mathrm{S} - 0.40\mathrm{S} = 150\), or miscalculating \(1 - 0.40 = 0.60\). These algebraic slips lead to wrong values for the selling price.

2. Division errors when calculating \(\mathrm{S} = 150 \div 0.60\)
Students often struggle with dividing by decimals. They may incorrectly calculate \(150 \div 0.60\), perhaps getting 90 instead of 250, or make errors when converting 0.60 to fractions. Some might calculate \(150 \times 0.60 = 90\) instead of \(150 \div 0.60 = 250\).

3. Percentage calculation mistakes in verification
When checking their work by calculating 40% of their selling price, students may make basic percentage errors, such as calculating 40% of 250 as \(40 \times 250 = 10,000\) instead of \(0.40 \times 250 = 100\), which could shake their confidence in an otherwise correct solution.

Errors while selecting the answer

1. Selecting the markup amount instead of the profit
Since the markup (\(\$100\)) equals the profit (\(\$100\)) in this problem, students who calculated everything correctly but confused the final question might still get the right numerical answer by coincidence. However, in other similar problems, this conceptual confusion would lead to wrong answers.

2. Choosing an intermediate calculation result
Students might select \(\$60\) (choice B) if they incorrectly calculated markup as 40% of purchase price, or \(\$40\) (choice A) if they made errors in their percentage calculations and happened to arrive at one of the given choices through incorrect methods.