A store bought 5 dozen lamps at $30 per dozen and sold them all at $15 per lamp. The profit...

1. Translate the problem requirements: The store bought lamps in dozens at a certain price per dozen, sold them individually at a different price per lamp, and we need to find what percent the profit on each lamp represents of its selling price (not cost price).
2. Calculate the cost price per individual lamp: Convert the bulk purchase price from per-dozen to per-lamp basis.
3. Determine profit per lamp: Find the difference between selling price and cost price for each lamp.
4. Express profit as percentage of selling price: Calculate what fraction the profit represents of the selling price and convert to percentage form.

Execution of Strategic Approach

1. Translate the problem requirements

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

• The store bought lamps in bulk - 5 dozen lamps (that's \(5 \times 12 = 60\) lamps total)
• They paid $30 for each dozen lamps
• They sold each individual lamp for $15
• We need to find what percent the profit represents of the selling price (not the cost)

The key insight here is that we're looking for profit as a percentage of selling price, which is different from the more common "markup" calculation.

2. Calculate the cost price per individual lamp

Since the store bought lamps by the dozen, let's figure out what each individual lamp actually cost them:

Cost per dozen = $30
Number of lamps in a dozen = 12
So, cost per individual lamp = \(\$30 \div 12 = \$2.50\)

This means each lamp cost the store $2.50 to purchase.

3. Determine profit per lamp

Now we can calculate how much profit the store made on each lamp:

Selling price per lamp = $15
Cost price per lamp = $2.50
Profit per lamp = \(\$15 - \$2.50 = \$12.50\)

So the store made $12.50 profit on each lamp they sold.

4. Express profit as percentage of selling price

Here's the crucial part - we need profit as a percentage of the selling price:

Profit per lamp = $12.50
Selling price per lamp = $15
Profit as fraction of selling price = \(\$12.50 \div \$15 = 12.5 \div 15\)

Let's simplify this fraction:
\(12.5 \div 15 = 125 \div 150 = 5 \div 6\)

Converting to percentage: \(\frac{5}{6} = 0.8333... = 83\frac{1}{3}\%\)

Final Answer

The profit on each lamp was \(83\frac{1}{3}\%\) of its selling price.

This matches answer choice C. \(83\frac{1}{3}\%\)

Verification: Let's double-check: \(83\frac{1}{3}\%\) of $15 = \(\left(\frac{5}{6}\right) \times \$15 = \$12.50\), which is indeed our calculated profit per lamp.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Confusing "profit as percentage of selling price" with "profit as percentage of cost price" (markup). Students often default to calculating markup since it's more commonly encountered, but this question specifically asks for profit as a percentage of selling price.

Faltering Point 2: Misunderstanding the bulk purchase structure. Students might get confused by the "5 dozen at $30 per dozen" phrasing and incorrectly think they need to work with the total 5 dozen quantity rather than finding the per-unit cost for individual lamps.

Faltering Point 3: Overlooking the unit conversion requirement. Students might miss that they need to convert from "cost per dozen" to "cost per individual lamp" since the selling price is given per individual lamp.

Errors while executing the approach

Faltering Point 1: Arithmetic error in the division \(\$30 \div 12\). Students might calculate this as $2.50 incorrectly, perhaps getting $3.00 or $2.40, which would throw off all subsequent calculations.

Faltering Point 2: Fraction simplification errors when converting \(\frac{12.5}{15}\) to \(\frac{5}{6}\). Students might struggle with the decimal-to-fraction conversion or make mistakes in reducing the fraction to its simplest form.

Faltering Point 3: Percentage conversion mistakes when converting \(\frac{5}{6}\) to \(83\frac{1}{3}\%\). Students might calculate this as 83% (forgetting the â…“) or make errors in the decimal-to-percentage conversion.

Errors while selecting the answer

Faltering Point 1: Selecting 50% (choice B) if they calculated profit as percentage of cost price instead of selling price. Since profit ($12.50) is exactly 5 times the cost ($2.50), this common mistake would lead them to choice B.

Faltering Point 2: Choosing 500% (choice E) if they confused the ratio direction and calculated selling price as a percentage of profit, or made a decimal place error in their percentage calculation.