Understanding the Shirt Cost Price Problem

This problem involves calculating the initial cost price of a shirt based on its selling price after a loss, the details of another shirt sold at a profit, and the overall financial outcome (no profit, no loss).

Analyzing the Given Information

• Total cost price of two shirts = ₹1,200.
• The first shirt was sold at a loss of 8%.
• The second shirt was sold at a gain of 24%.
• The overall transaction resulted in neither a loss nor a gain.

Step-by-Step Calculation for First Shirt Cost Price

Let's denote the cost price (CP) and selling price (SP) for the two shirts.

Let the cost price of the first shirt be $C_1$ and the cost price of the second shirt be $C_2$.

We know that the total cost price is $C_1 + C_2 = ₹1,200$.

Calculating Selling Prices

First Shirt:

• Loss = 8%
• Selling Price ($SP_1$) = $C_1 \times (1 - \frac{\text{Loss percentage}}{100})$
• $SP_1 = C_1 \times (1 - \frac{8}{100}) = C_1 \times (1 - 0.08) = 0.92 C_1$

Second Shirt:

• Gain = 24%
• Selling Price ($SP_2$) = $C_2 \times (1 + \frac{\text{Gain percentage}}{100})$
• $SP_2 = C_2 \times (1 + \frac{24}{100}) = C_2 \times (1 + 0.24) = 1.24 C_2$

Applying the Overall Profit/Loss Condition

The problem states that there was neither a loss nor a gain overall. This means the total selling price equals the total cost price.

Total Selling Price = Total Cost Price

$SP_1 + SP_2 = C_1 + C_2$

$0.92 C_1 + 1.24 C_2 = ₹1,200$

Solving the Equations

We have a system of two linear equations:

1. $C_1 + C_2 = 1,200$
2. $0.92 C_1 + 1.24 C_2 = 1,200$

From the first equation, we can express $C_2$ in terms of $C_1$: $C_2 = 1,200 - C_1$

Now, substitute this expression for $C_2$ into the second equation:

$0.92 C_1 + 1.24 (1,200 - C_1) = 1,200$

Distribute $1.24$:

$0.92 C_1 + (1.24 \times 1,200) - 1.24 C_1 = 1,200$

$0.92 C_1 + 1,488 - 1.24 C_1 = 1,200$

Combine the $C_1$ terms:

$(0.92 - 1.24) C_1 + 1,488 = 1,200$

$-0.32 C_1 = 1,200 - 1,488$

$-0.32 C_1 = -288$

Now, solve for $C_1$:

$C_1 = \frac{-288}{-0.32}$

$C_1 = \frac{288}{0.32}$

To simplify the division, we can multiply the numerator and denominator by 100:

$C_1 = \frac{288 \times 100}{32}$

$C_1 = \frac{28,800}{32}$

Performing the division:

$C_1 = 900$

Verification

Let's check if this value of $C_1$ works:

• If $C_1 = ₹900$, then $C_2 = 1,200 - 900 = ₹300$.
• $SP_1 = 0.92 \times 900 = ₹828$.
• $SP_2 = 1.24 \times 300 = ₹372$.
• Total SP = $SP_1 + SP_2 = 828 + 372 = ₹1,200$.

Since the Total SP (₹1,200) equals the Total CP (₹1,200), our calculation is correct.

Conclusion

The cost price of the first shirt is ₹900.