A bookseller sold 4 rare books for $100 each. He made a profit of 25% of his cost on the...

1. Translate the problem requirements: Clarify that "25% profit on cost" means \(\text{selling price} = \text{cost} + 0.25(\text{cost})\), and "20% loss on cost" means \(\text{selling price} = \text{cost} - 0.20(\text{cost})\). We need to find the overall profit or loss in dollars.
2. Calculate individual cost prices from known selling prices: Work backwards from the $100 selling price to find what each book originally cost using the profit/loss percentages.
3. Determine total cost versus total revenue: Compare the sum of all cost prices against the total selling price of $400 to find the net result.
4. Calculate final profit or loss amount: Subtract total cost from total revenue to determine if there was an overall profit or loss.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what the profit and loss percentages actually mean in everyday terms.

When we say "25% profit on cost," this means the bookseller sold the book for 25% more than what he paid for it. In other words, if he bought a book for some amount, he sold it for that amount plus 25% of that amount.

Similarly, "20% loss on cost" means he sold the book for 20% less than what he paid for it.

Since we know all books sold for $100 each, we can work backwards to find what he originally paid for each book.

For books with 25% profit: \(\text{Selling Price} = \text{Cost Price} + 25\% \text{ of Cost Price} = \text{Cost Price} \times (1 + 0.25) = \text{Cost Price} \times 1.25\)

For books with 20% loss: \(\text{Selling Price} = \text{Cost Price} - 20\% \text{ of Cost Price} = \text{Cost Price} \times (1 - 0.20) = \text{Cost Price} \times 0.80\)

Process Skill: TRANSLATE - Converting percentage profit/loss language into mathematical relationships

2. Calculate individual cost prices from known selling prices

Now let's find what the bookseller originally paid for each type of book.

For the 2 books sold at 25% profit:
We know: \(\$100 = \text{Cost Price} \times 1.25\)
Therefore: \(\text{Cost Price} = \$100 \div 1.25 = \$100 \div \frac{5}{4} = \$100 \times \frac{4}{5} = \$80\)

So each of these 2 books cost him $80 originally.
Total cost for these 2 books = \(2 \times \$80 = \$160\)

For the 2 books sold at 20% loss:
We know: \(\$100 = \text{Cost Price} \times 0.80\)
Therefore: \(\text{Cost Price} = \$100 \div 0.80 = \$100 \div \frac{4}{5} = \$100 \times \frac{5}{4} = \$125\)

So each of these 2 books cost him $125 originally.
Total cost for these 2 books = \(2 \times \$125 = \$250\)

3. Determine total cost versus total revenue

Now let's compare what he paid in total versus what he received in total.

Total amount he paid (total cost):
\(\$160\) (for the 2 profitable books) + \(\$250\) (for the 2 loss-making books) = \(\$410\)

Total amount he received (total revenue):
\(4 \text{ books} \times \$100 \text{ each} = \$400\)

4. Calculate final profit or loss amount

To find his overall profit or loss, we subtract what he paid from what he received:

\(\text{Profit or Loss} = \text{Total Revenue} - \text{Total Cost}\)
\(\text{Profit or Loss} = \$400 - \$410 = -\$10\)

Since the result is negative, this represents a loss of $10.

Final Answer

The bookseller had an overall loss of $10 on the sale of all 4 books.

This matches answer choice E: "Loss of $10"

Common Faltering Points

Errors while devising the approach

Students often confuse "25% profit on cost" with "25% profit on selling price" or think it means the selling price is 25% of the cost price. The key insight is that profit/loss percentages are always calculated based on the original cost price, not the selling price.

When working backwards from selling price to cost price, students may set up the relationship incorrectly. For example, they might think that if there's a 25% profit, then \(\text{Cost Price} = \text{Selling Price} \times 1.25\), when it should be \(\text{Selling Price} = \text{Cost Price} \times 1.25\).

Errors while executing the approach

When calculating \(\text{Cost Price} = \$100 \div 1.25\), students often make arithmetic mistakes, especially when converting 1.25 to the fraction \(\frac{5}{4}\) and then flipping it to multiply by \(\frac{4}{5}\). Similarly, with \(\$100 \div 0.80\), converting 0.80 to \(\frac{4}{5}\) and flipping to \(\frac{5}{4}\) can lead to errors.

Students may correctly find individual cost prices ($80 and $125) but then make arithmetic errors when calculating totals. For example, incorrectly computing \(2 \times \$80 + 2 \times \$125\) or making errors in the final subtraction \(\$400 - \$410\).

Errors while selecting the answer

When students get -$10 as their final calculation, they might incorrectly interpret this as a profit of $10 instead of recognizing that the negative sign indicates a loss of $10. This is especially common when students rush through the final step without carefully considering what the negative result means.