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

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

Let’s go through these steps in detail:

- 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

- 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

- 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**