A and B invest in a partnership business in the ratio of 3:4 and A got Rs.2700 as a profit share out of total profit of Rs. 5100 and the difference between the time period between A and B is 3 months, then find for how many months A kept his investment?

The correct option is 9.

To find the number of months A kept his investment, we can use the concept of partnership, which states that the ratio of profit shares is equal to the ratio of the products of their investments and their time periods.

Let the investment of A be $I_A$ and the investment of B be $I_B$.
The ratio of their investments is given as:
$I_A:I_B=3:4$

We are given:
Total profit = Rs. 5100
A's profit share ($P_A$) = Rs. 2700

Thus, B's profit share ($P_B$) is:
$P_B=5100-2700=2400$

Now, let's find the ratio of their profits:
$P_A:P_B=2700:2400=9:8$

Let A's investment duration be $T_A$ and B's investment duration be $T_B$.
Using the partnership formula:
$\frac{P_A}{P_B}=\frac{I_A\times T_A}{I_B\times T_B}$

Substitute the known ratios into the formula:
$\frac{9}{8}=\frac{3}{4}\times \frac{T_A}{T_B}$

Now, solve for the ratio of their time periods:
$\frac{T_A}{T_B}=\frac{9}{8}\times \frac{4}{3}=\frac{3}{2}$

This means:
$T_A:T_B=3:2$

Let $T_A=3x$ and $T_B=2x$ months.
We are given that the difference between their investment durations is 3 months:
$T_A-T_B=3$
$3x-2x=3$
$x=3$

Therefore, A kept his investment for:
$T_A=3\times 3=9$ months.