'A' invested Rs. 6600 for x months while 'B' get (X+3) invested Rs. 1100 less amount than 'A' for (x + 3) months. If the ratio of the profits received by 'A' and 'B' is 4:5, respectively then find the value of 'X'

The correct option/answer is 6.

Let's break down the problem step-by-step to find the value of $x$:

Step 1: Determine the investment of 'A' and the duration.
Investment of 'A' = Rs. 6600
Time period for 'A' = $x$ months

Step 2: Determine the investment of 'B' and the duration.
B's investment is Rs. 1100 less than A's investment.
Investment of 'B' = Rs. 6600 - Rs. 1100 = Rs. 5500
Time period for 'B' = $(x+3)$ months

Step 3: Relate investment-month product to profit ratio.
The ratio of profits of two partners in a business is equal to the ratio of the products of their investment amounts and their respective time periods.
Profit of 'A' : Profit of 'B' = (Investment of 'A' × Time of 'A') : (Investment of 'B' × Time of 'B')

We are given that the ratio of profits received by 'A' and 'B' is 4:5. Therefore:
$\frac{6600\times x}{5500\times (x+3)}=\frac{4}{5}$

Step 4: Solve the equation for $x$.
Simplify the fraction on the left-hand side by dividing the numbers 6600 and 5500 by 1100:
$\frac{6x}{5(x+3)}=\frac{4}{5}$

We can cancel out 5 from both denominators:
$6x=4(x+3)$

Expand the right-hand side:
$6x=4x+12$

Subtract $4x$ from both sides:
$2x=12$

Divide by 2:
$x=6$

Thus, the value of 'X' is 6.