A used car dealer sold one car at a profit of 25% of the dealers purchase price for that car...

1. Translate the problem requirements: Convert the profit/loss percentages into equations to find the original purchase prices for each car, then calculate the total profit or loss from both transactions
2. Find the purchase price of the profitable car: Use the fact that selling price equals purchase price plus 25% profit to work backwards from $20,000
3. Find the purchase price of the loss-making car: Use the fact that selling price equals purchase price minus 20% loss to work backwards from $20,000
4. Calculate total profit or loss: Compare the sum of both purchase prices to the sum of both selling prices to determine net result

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what happened here in everyday terms. The dealer sold two cars, both for exactly $20,000 each. But here's the key - one car was sold at a profit and one at a loss.

For the first car: The dealer made a 25% profit on what he originally paid for it, and ended up selling it for $20,000.

For the second car: The dealer took a 20% loss on what he originally paid for it, but still managed to sell it for $20,000.

To find the total profit or loss, we need to figure out how much the dealer originally paid for each car (the purchase prices), then compare that total to what he received ($40,000 total).

Process Skill: TRANSLATE

2. Find the purchase price of the profitable car

Let's think about the first car logically. If the dealer bought this car for some amount and then sold it at a 25% profit for $20,000, what did he originally pay?

In plain English: The selling price equals the original price plus 25% of the original price.

So if we call the original purchase price \(\mathrm{P_1}\):

• Selling price = Purchase price + 25% profit
• \(\$20{,}000 = \mathrm{P_1} + 0.25 \times \mathrm{P_1}\)
• \(\$20{,}000 = \mathrm{P_1} \times (1 + 0.25)\)
• \(\$20{,}000 = \mathrm{P_1} \times 1.25\)
• \(\mathrm{P_1} = \$20{,}000 \div 1.25 = \$16{,}000\)

The dealer originally paid $16,000 for the first car.

3. Find the purchase price of the loss-making car

Now for the second car. The dealer sold this car at a 20% loss for $20,000. What did he originally pay?

In plain English: The selling price equals the original price minus 20% of the original price.

So if we call the original purchase price \(\mathrm{P_2}\):

• Selling price = Purchase price - 20% loss
• \(\$20{,}000 = \mathrm{P_2} - 0.20 \times \mathrm{P_2}\)
• \(\$20{,}000 = \mathrm{P_2} \times (1 - 0.20)\)
• \(\$20{,}000 = \mathrm{P_2} \times 0.80\)
• \(\mathrm{P_2} = \$20{,}000 \div 0.80 = \$25{,}000\)

The dealer originally paid $25,000 for the second car.

4. Calculate total profit or loss

Now we can see the complete picture:

What the dealer paid in total:

• First car: $16,000
• Second car: $25,000
• Total paid: $16,000 + $25,000 = $41,000

What the dealer received in total:

• Both cars sold for $20,000 each
• Total received: $20,000 + $20,000 = $40,000

Overall result:

• Total received - Total paid = $40,000 - $41,000 = \(-\$1{,}000\)

The negative result means the dealer had a loss of $1,000 overall.

Final Answer

The dealer's total result for both transactions combined was a loss of $1,000.

This matches answer choice C: 1000 loss.

Common Faltering Points

Errors while devising the approach

Students often confuse whether the percentage is calculated on the purchase price or selling price. The question states "25 percent of the dealer's purchase price" and "20 percent of the dealer's purchase price," but students might incorrectly think these percentages apply to the selling price of $20,000.

Students may struggle to correctly translate "sold at a profit of 25%" into the equation: Selling Price = Purchase Price × 1.25. They might incorrectly write: Purchase Price = Selling Price × 1.25, which would lead to completely wrong purchase prices.

Errors while executing the approach

When calculating \(\mathrm{P_2} = \$20{,}000 \div 0.80 = \$25{,}000\), students often make division errors with decimals. They might incorrectly calculate this as $16,000 instead of $25,000, or struggle with the division altogether.

Students may correctly find the purchase prices but then make errors in the final subtraction. They might calculate $41,000 - $40,000 instead of $40,000 - $41,000, leading them to think there was a $1,000 profit instead of a $1,000 loss.

Errors while selecting the answer

Even when students correctly calculate that the total is -$1,000, they may misinterpret the negative sign and select "1000 profit" (choice A) instead of "1000 loss" (choice C), not realizing that a negative result means a loss, not a profit.