Understanding the Problem: Cost Price vs Selling Price

The problem states a relationship between the cost price (CP) and the selling price (SP) of shirts. We are told that the total cost price of 15 shirts is exactly equal to the total selling price of 10 shirts. Our goal is to determine if this scenario results in a gain (profit) or a loss, and calculate the percentage of that gain or loss.

Setting up the Equation

Let CP represent the cost price of one shirt and SP represent the selling price of one shirt.

According to the question:

Cost Price of 15 shirts = Selling Price of 10 shirts

We can write this mathematically as:

\(15 \times \text{CP} = 10 \times \text{SP}\)

Finding the Relationship between SP and CP

From the equation \(15 \times \text{CP} = 10 \times \text{SP}\), we can find the ratio of SP to CP.

\(\frac{\text{SP}}{\text{CP}} = \frac{15}{10}\)

Simplifying the fraction:

\(\frac{\text{SP}}{\text{CP}} = \frac{3}{2}\)

This ratio tells us that the selling price (SP) is $\frac{3}{2}$ times the cost price (CP). Since $\frac{3}{2} = 1.5$, this means the selling price is 1.5 times the cost price. Because SP > CP, there is a gain.

Calculating the Gain

The gain is the difference between the selling price and the cost price:

\(\text{Gain} = \text{SP} - \text{CP}\)

Substitute \(\text{SP} = \frac{3}{2} \text{CP}\) into the gain formula:

\(\text{Gain} = \frac{3}{2} \text{CP} - \text{CP}\)

\(\text{Gain} = \left(\frac{3}{2} - 1\right) \text{CP}\)

\(\text{Gain} = \left(\frac{3}{2} - \frac{2}{2}\right) \text{CP}\)

\(\text{Gain} = \frac{1}{2} \text{CP}\)

So, the gain is equal to half of the cost price.

Calculating the Gain Percentage

The gain percentage is calculated using the formula:

\(\text{Gain \%} = \left(\frac{\text{Gain}}{\text{CP}}\right) \times 100\%\)

Substitute the value of Gain we found:

\(\text{Gain \%} = \left(\frac{\frac{1}{2} \text{CP}}{\text{CP}}\right) \times 100\%\)

\(\text{Gain \%} = \left(\frac{1}{2}\right) \times 100\%\)

\(\text{Gain \%} = 50\%\)

Alternatively, we can think about the number of shirts.

Let the cost price of 1 shirt be $C$.

Total CP of 15 shirts = \(15 \times C\)

Let the selling price of 1 shirt be $S$.

Total SP of 10 shirts = \(10 \times S\)

Given: \(15 \times C = 10 \times S\)

From this, we can find the relationship between S and C:

\(\frac{S}{C} = \frac{15}{10} = \frac{3}{2}\)

This means \(S = \frac{3}{2}C\). The selling price per shirt is greater than the cost price per shirt, so there is a gain.

Consider the transaction from the perspective of selling 10 shirts. The selling price of 10 shirts is equal to the cost price of 15 shirts. This means that by selling just 10 shirts, the shopkeeper recovers the cost they paid for 15 shirts. The extra 5 shirts effectively represent the profit.

• Cost of 10 shirts = \(10 \times C\)
• Selling price of 10 shirts = \(10 \times S\)

We know \(10 \times S = 15 \times C\).

The gain made on selling 10 shirts is:

Gain on 10 shirts = Selling Price of 10 shirts - Cost Price of 10 shirts

Gain on 10 shirts = \(15 \times C - 10 \times C\)

Gain on 10 shirts = \(5 \times C\)

The gain is \(5 \times C\), and this gain is made on the cost of 10 shirts, which is \(10 \times C\).

Gain Percentage = \(\left(\frac{\text{Gain}}{\text{Cost of 10 shirts}}\right) \times 100\%\)

Gain Percentage = \(\left(\frac{5 \times C}{10 \times C}\right) \times 100\%\)

Gain Percentage = \(\left(\frac{5}{10}\right) \times 100\%\)

Gain Percentage = \(\frac{1}{2} \times 100\%\)

Gain Percentage = \(50\%\)

Conclusion

Based on the calculation, the gain percentage is 50%.

Revision Table: Key Profit and Loss Formulas

Additional Information: Understanding CP and SP Problems

Problems involving cost price and selling price comparisons are common in quantitative aptitude. The key is often to relate the CP and SP of the same number of articles or to find the CP and SP of a single article. In this specific problem, we used the given relationship \(15 \times \text{CP} = 10 \times \text{SP}\) to establish a ratio between the unit CP and unit SP, and then used that ratio to calculate the gain percentage based on the cost price.

It's important to always calculate profit or loss percentage with respect to the Cost Price (CP) unless otherwise stated in the question.