By selling an article for $265, a dealer lost 8%. At what price should it be sold in order to make a profit of 20%?
Understand the Problem
The question is asking for the selling price of an article that will yield a 20% profit, given that it was previously sold at a loss of 8%. To solve this, we first need to determine the cost price of the article using the selling price and the loss percentage, and then calculate the new selling price based on the desired profit percentage.
Answer
The new selling price needs to be approximately $130.43\%$ of the original selling price.
Answer for screen readers
The new selling price, represented as a percentage of the original selling price, is approximately $130.43%$ of $S$.
Steps to Solve
- Identify the cost price from the previous selling price
Let the original selling price (when sold at a loss of 8%) be $S$. The cost price (CP) can be calculated using the formula for loss, which is:
$$ CP = \frac{S}{1 - \frac{\text{Loss%}}{100}} = \frac{S}{1 - 0.08} = \frac{S}{0.92} $$
- Calculate the new selling price for the desired profit
To find the new selling price (SP) that yields a 20% profit, we use the formula for profit, which is:
$$ SP = CP \times (1 + \frac{\text{Profit%}}{100}) = CP \times (1 + 0.20) = CP \times 1.20 $$
- Substitute the cost price into the selling price formula
Now, we substitute the cost price we found in step 1 into the selling price equation:
$$ SP = \left(\frac{S}{0.92}\right) \times 1.20 = \frac{1.20S}{0.92} $$
- Simplify the expression for the new selling price
To make it easier to interpret, we can simplify this expression:
$$ SP = \frac{1.20}{0.92} S $$
- Calculate the selling price as a percentage of the original selling price
Now, we can find out what percentage this represents of the original selling price:
$$ \text{Percentage} = \frac{SP}{S} \times 100 = \frac{1.20}{0.92} \times 100 $$
Now we simplified everything, so we can compute the final selling price based on the original selling price $S$.
The new selling price, represented as a percentage of the original selling price, is approximately $130.43%$ of $S$.
More Information
This answer means that to achieve a 20% profit, the selling price would need to be about 30.43% higher than the initial selling price that incurred an 8% loss.
Tips
- Forgetting to convert the percentage into decimal form when using in calculations (e.g., using 20% directly instead of 0.20).
- Not properly calculating the cost price from the loss selling price, leading to incorrect final selling price estimates.
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