Solving a Profit and Loss Problem

This question involves the concepts of profit and loss in percentages. We are given information about two scenarios involving the selling price of an article at different percentages of loss and gain, and the difference between these selling prices. We need to find the selling price required to achieve a specific gain percentage.

Understanding Key Terms in Profit and Loss

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Profit: When SP > CP. Profit = SP - CP.
• Loss: When SP < CP. Loss = CP - SP.
• Profit Percentage: $\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100$
• Loss Percentage: $\text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$

Setting Up the Problem

Let the Cost Price (CP) of the article be denoted by $C$.

According to the question, the article was initially sold at a loss of 8%. The Selling Price (SP) in this case can be calculated as:

$\text{SP}_1 = \text{CP} - \text{Loss} = C - 8\% \text{ of } C = C - 0.08C = (1 - 0.08)C = 0.92C$

In the second scenario, the article was sold at a gain of 10.5%. The Selling Price (SP) in this case is:

$\text{SP}_2 = \text{CP} + \text{Gain} = C + 10.5\% \text{ of } C = C + 0.105C = (1 + 0.105)C = 1.105C$

Using the Difference in Selling Prices to Find Cost Price

We are told that if the article was sold at a gain of 10.5% instead of a loss of 8%, the seller would have received Rs. 92.50 more. This means the difference between $\text{SP}_2$ and $\text{SP}_1$ is Rs. 92.50.

$\text{SP}_2 - \text{SP}_1 = 92.50$

Substitute the expressions for $\text{SP}_1$ and $\text{SP}_2$:

$1.105C - 0.92C = 92.50$

Combine the terms with $C$:

$(1.105 - 0.92)C = 92.50$

$0.185C = 92.50$

Now, solve for $C$ to find the Cost Price:

$C = \frac{92.50}{0.185}$

To simplify the division, we can multiply both the numerator and denominator by 1000 to remove the decimal places:

$C = \frac{92.50 \times 1000}{0.185 \times 1000} = \frac{92500}{185}$

Now, perform the division:

So, the Cost Price of the article is Rs. 500.

Calculating Selling Price for 12% Gain

The question asks for the selling price required to gain 12%. We now know the Cost Price is Rs. 500.

To gain 12%, the Selling Price (SP) should be:

$\text{SP}_{\text{new}} = \text{CP} + 12\% \text{ of } \text{CP} = C + 0.12C = (1 + 0.12)C = 1.12C$

Substitute the value of $C = 500$:

$\text{SP}_{\text{new}} = 1.12 \times 500$

$\text{SP}_{\text{new}} = \frac{112}{100} \times 500$

$\text{SP}_{\text{new}} = 112 \times \frac{500}{100}$

$\text{SP}_{\text{new}} = 112 \times 5$

$\text{SP}_{\text{new}} = 560$

Therefore, to gain 12%, the person should have sold the article for Rs. 560.

Summary of Steps

1. Identify the given information: initial loss percentage, desired gain percentage, and the monetary difference between the two selling prices.
2. Represent the Cost Price as a variable (e.g., C).
3. Write expressions for the Selling Price under each given condition (loss and gain) in terms of the Cost Price.
4. Set up an equation using the given difference in selling prices.
5. Solve the equation to find the value of the Cost Price.
6. Calculate the Selling Price required to achieve the target gain percentage using the calculated Cost Price.

Revision Table: Profit and Loss Formulas

Additional Information on Profit and Loss Calculations

Problems involving profit and loss percentages often require converting percentages to decimal or fractional forms for calculations. Remember that profit and loss percentages are typically calculated on the Cost Price unless stated otherwise. The difference in selling prices, as seen in this problem, is a common way to set up an equation to find the unknown Cost Price.

In this specific problem, the difference between a sale at an 8% loss and a sale at a 10.5% gain is equivalent to a total percentage difference of $8\% + 10.5\% = 18.5\%$ of the Cost Price. This is because going from an 8% loss to the Cost Price is an increase of 8%, and going from the Cost Price to a 10.5% gain is a further increase of 10.5%. The total change relative to the Cost Price is the sum of the magnitudes of the loss and the gain percentages when one is a loss and the other is a gain.

So, $18.5\%$ of CP corresponds to Rs. 92.50.

$\frac{18.5}{100} \times C = 92.50$

$0.185C = 92.50$, which is the same equation we derived earlier, confirming the logic.