Arjun and Bhavesh launch a joint business venture. Arjun contributes 25% more initial capital than Bhavesh, but keeps his funds invested for 4 months less than Bhavesh. At the end of the term, Bhavesh receives Rs.4000 as his profit share, which exceeds Arjun's share by Rs.1000. For how many months did Arjun invest his capital?

The correct answer is 6 months.

Step 1: Define variables for initial capital and investment duration.

Let Bhavesh's initial capital be $C$.

Arjun contributes 25% more initial capital than Bhavesh, so Arjun's capital is:

$C_{\text{Arjun}}=C+0.25C=1.25C=\frac{5}{4}C$

Let the duration of Bhavesh's investment be $t$ months.

Since Arjun keeps his funds invested for 4 months less than Bhavesh, the duration of Arjun's investment is:

$T_{\text{Arjun}}=t-4$ months

Step 2: Calculate the profit shares and their ratio.

Bhavesh's profit share = Rs. 4000

Bhavesh's share exceeds Arjun's share by Rs. 1000. Therefore, Arjun's profit share is:

$P_{\text{Arjun}}=4000-1000=3000$

The ratio of Arjun's profit share to Bhavesh's profit share is:

$\frac{P_{\text{Arjun}}}{P_{\text{Bhavesh}}}=\frac{3000}{4000}=\frac{3}{4}$

Step 3: Formulate the ratio of profit shares based on investment.

In a business partnership, profit is shared in proportion to the product of capital invested and time period of investment:

$\frac{P_{\text{Arjun}}}{P_{\text{Bhavesh}}}=\frac{C_{\text{Arjun}}\times T_{\text{Arjun}}}{C_{\text{Bhavesh}}\times T_{\text{Bhavesh}}}$

Substituting the values into the formula:

$\frac{3}{4}=\frac{1.25C\times (t-4)}{C\times t}$

Canceling out $C$ from the numerator and denominator:

$\frac{3}{4}=\frac{1.25(t-4)}{t}$

Multiplying both sides by 4:

$3=\frac{5(t-4)}{t}$

Cross-multiplying and solving for $t$:

$3t=5(t-4)$

$3t=5t-20$

$2t=20$

$t=10$

Step 4: Calculate Arjun's investment duration.

Bhavesh invested his capital for $t=10$ months.

Therefore, Arjun invested his capital for:

$T_{\text{Arjun}}=10-4=6$ months

Hence, Arjun invested his capital for 6 months.