The sum of the cost price of two bikes is Rs 50,000. The first bike was sold at a profit of 20%, and the second bike at a loss of 20%. If the selling prices are the same, what is the cost price (in Rs) of the first bike that was sold?

Options

A. 20000
B. 25000
C. 30000
D. 27500

Answer

To solve the problem, let's denote the cost price of the first bike as \( CP_1 \) and the cost price of the second bike as \( CP_2 \). According to the problem, we have the following information:

1. The sum of the cost prices of the two bikes is Rs 50,000:
\[ CP_1 + CP_2 = 50,000 \]

2. The first bike is sold at a profit of 20%. Therefore, the selling price \( SP_1 \) of the first bike can be expressed as:
\[ SP_1 = CP_1 + 0.2 \times CP_1 = 1.2 \times CP_1 \]

3. The second bike is sold at a loss of 20%. Therefore, the selling price \( SP_2 \) of the second bike can be expressed as:
\[ SP_2 = CP_2 - 0.2 \times CP_2 = 0.8 \times CP_2 \]

4. According to the problem, the selling prices of both bikes are the same:
\[ SP_1 = SP_2 \]

Now, substituting the expressions for \( SP_1 \) and \( SP_2 \):
\[ 1.2 \times CP_1 = 0.8 \times CP_2 \]

Now, we can express \( CP_2 \) in terms of \( CP_1 \):
\[ CP_2 = \frac{1.2}{0.8} \times CP_1 = 1.5 \times CP_1 \]

Now, we can substitute \( CP_2 \) back into the first equation:
\[ CP_1 + 1.5 \times CP_1 = 50,000 \]
\[ 2.5 \times CP_1 = 50,000 \]

Now, solving for \( CP_1 \):
\[ CP_1 = \frac{50,000}{2.5} = 20,000 \]

Thus, the cost price of the first bike is Rs 20,000.

### Summary of Steps:
1. Set up equations based on the information given about cost prices and selling prices.
2. Express the selling prices in terms of the cost prices.
3. Set the selling prices equal to each other since they are the same.
4. Substitute one cost price into the sum equation to solve for the other.
5. Calculate the cost price of the first bike.