Solving for the Cost Price of the First Toy Gun

This problem involves calculating the cost price (CP) of two items given their total cost, individual selling conditions (loss/gain percentages), and the overall profit/loss.

Define Variables

• Let $CP_1$ be the cost price of the first toy gun.
• Let $CP_2$ be the cost price of the second toy gun.
• Total Cost Price = $₹1,200$.

Set Up Equations Based on Given Information

The total cost is the sum of the individual cost prices:

$ CP_1 + CP_2 = 1200 \quad (1) $

The first gun was sold at an 8% loss. Its selling price ($SP_1$) is:

$ SP_1 = CP_1 \times (1 - \frac{8}{100}) = CP_1 \times (1 - 0.08) = 0.92 \times CP_1 $

The second gun was sold at a 16% gain. Its selling price ($SP_2$) is:

$ SP_2 = CP_2 \times (1 + \frac{16}{100}) = CP_2 \times (1 + 0.16) = 1.16 \times CP_2 $

There was neither a loss nor a gain overall. This means the total selling price equals the total cost price:

$ SP_1 + SP_2 = CP_1 + CP_2 $

$ SP_1 + SP_2 = 1200 \quad (2) $

Calculate the Cost Price of the First Toy Gun

Substitute the expressions for $SP_1$ and $SP_2$ into equation (2):

$ (0.92 \times CP_1) + (1.16 \times CP_2) = 1200 \quad (3) $

From equation (1), express $CP_2$ in terms of $CP_1$:

$ CP_2 = 1200 - CP_1 $

Substitute this expression for $CP_2$ into equation (3):

$ 0.92 \times CP_1 + 1.16 \times (1200 - CP_1) = 1200 $

Now, solve for $CP_1$:

$ 0.92 \times CP_1 + (1.16 \times 1200) - (1.16 \times CP_1) = 1200 $

$ 0.92 \times CP_1 - 1.16 \times CP_1 = 1200 - (1.16 \times 1200) $

$ -0.24 \times CP_1 = 1200 \times (1 - 1.16) $

$ -0.24 \times CP_1 = 1200 \times (-0.16) $

$ 0.24 \times CP_1 = 1200 \times 0.16 $

$ CP_1 = \frac{1200 \times 0.16}{0.24} $

$ CP_1 = \frac{1200 \times 16}{24} $

$ CP_1 = \frac{19200}{24} $

$ CP_1 = 800 $

Therefore, the cost price of the first toy gun is $₹800$.