A and B entered into partnership with capitals in the ratio 4: 5. After 3 months, A withdrew ¼ of his capital and B withdrew 1/5 of his capital. The gain at the end of 10 months was Rs. 760.Find A's share in this profit ?

To find A's share in the profit, we need to follow these steps:

Step 1: Determine the initial capital of A and B.

Let's assume that A's initial capital is 4x and B's initial capital is 5x.

Step 2: Calculate the capital left by each partner after withdrawal.

After 3 months, A withdrew 1/4 of his capital. So, A's capital left is (4x - (1/4)(4x)) = (4x - x) = 3x.

After 3 months, B withdrew 1/5 of his capital. So, B's capital left is (5x - (1/5)(5x)) = (5x - x) = 4x.

Step 3: Calculate the ratio of the time period for which A and B invested their capital.

Since A and B initially invested in the ratio of 4:5, their ratio of the time period for which they invested will be inverse, i.e., 5:4.

Step 4: Calculate the profit sharing ratio considering the time period.

The profit sharing ratio will be the product of the capital ratio (3x:4x) and the time ratio (5:4), as follows:

A's profit sharing ratio = (3x * 5) = 15x
B's profit sharing ratio = (4x * 4) = 16x

Step 5: Calculate A's share in the profit.

To find A's share in the profit, we need to calculate the fraction of the total profit that belongs to A. This can be done using the profit sharing ratio.

Let the total profit be Rs. P.

A's share in the profit = (A's profit sharing ratio / Total profit sharing ratio) * Total profit
= (15x / (15x + 16x)) * P
= (15x / 31x) * P
= 15/31 * P

Given that the profit at the end of 10 months was Rs. 760, we can substitute P = 760.

A's share in the profit = 15/31 * 760
= (15/31) * 760
= 360

Therefore, A's share in the profit is Rs. 360.