Rakesh purchased 4 bats and 10 balls for 3,650. He sold all the bats at the profit of 30% and balls at a loss of 20%. If his overall profit is 470, what was the cost price of one bat?​

To find the cost price of one bat, we can follow these steps:

1. Set up the cost price equation for bats and balls.
2. Set up the selling price equation for bats and balls with the given profit and loss percentages.
3. Solve the system of linear equations to find the cost price of one bat.

Let’s go through these steps in detail:

1. Set up the cost price equation for bats and balls:

Let the cost price of one bat be ‘B’ and the cost price of one ball be ‘b’. Rakesh purchased 4 bats and 10 balls, so we can write the equation:

4B + 10b = 3,650

2. Set up the selling price equation for bats and balls with the given profit and loss percentages:

Rakesh sold the bats at a 30% profit and balls at a 20% loss, and his overall profit was Rs 470. The total selling price can be represented as:

4 × 1.3B + 10 × 0.8b = 3,650 + 470

Which simplifies to:

5.2B + 8b = 4,120

3. Solve the system of linear equations to find the cost price of one bat:

Now we have a system of two linear equations in two variables:

(1) 4B + 10b = 3,650 (2) 5.2B + 8b = 4,120

To eliminate the variable ‘b’, multiply equation (1) by 0.8 and equation (2) by 1:

(1a) 3.2B + 8b = 2,920 (2a) 5.2B + 8b = 4,120

Now, subtract equation (1a) from equation (2a):

(3) 2B = 1,200

Now, we can find the cost price of one bat:

B = 1,200 / 2 B = Rs 600

So, the cost price of one bat is Rs 600