In a business A invests ₹ 600 more than B . The capital of B remained invested for 7 1/2 months, while the capital of A remained invested for 2 more months. If the total profits be ₹ 620 and B gets ₹ 140 less than what A gets, then A's capital is :

Correct option is C

Given:

1. A 's capital is ₹600 more than B 's capital.
2. B 's capital remained invested for 7.5 months.
3. A 's capital remained invested for 9.5 months.
4. The total profit is ₹620.Formula Used:

1. Profit sharing is proportional to the product of capital and time:

$$\text{Profit ratio of A to B} = \frac{\text{Capital of A} \times \text{Time of A}}{\text{Capital of B} \times \text{Time of B}}$$​​

2. The total profit relation:

$$\text{Profit of A} + \text{Profit of B} = \text{Total Profit}$$​​

Solution:

1. Let B 's capital be x . Then A 's capital is:

Capital of A = x + 600.

2. Profit ratio of A and B using the formula:

$$\frac{\text{Profit of A}}{\text{Profit of B}} = \frac{(x + 600) \times 9.5}{x \times 7.5}$$​

Simplify:

$$\frac{\text{Profit of A}}{\text{Profit of B}} = \frac{9.5(x + 600)}{7.5x}$$​​

3. It is given that:

Profit of A = Profit of B + 140

Let Profit of B = p . Then:

Profit of A = p + 140.

4. Total profit is ₹620, so:

Profit of A + Profit of B = 620.

Substituting:

(p + 140) + p = 620.

Solve for p :

2p + 140 = 620.


$$2p = 480 \implies p = 240$$​​

Therefore:

Profit of B = 240 and Profit of A = 240 + 140 = 380.

5. Using the profit ratio:

$$\frac{\text{Profit of A}}{\text{Profit of B}} = \frac{380}{240} = \frac{19}{12}$$​​

From the capital and time ratio:

$$\frac{(x + 600) \times 9.5}{x \times 7.5} = \frac{19}{12}$$​​

Simplify and solve:

$$12 \times 9.5(x + 600) = 19 \times 7.5x$$​​


114(x + 600) = 142.5x.


68400 = 142.5x - 114x.


$$68400 = 28.5x \implies x = \frac{68400}{28.5} = 2400$$​​

6. A 's capital is:

Capital of A = x + 600 = 2400 + 600 = 3000.

Final Answer:

A 's capital is ₹3000.
**Option C: ₹3000**