A and B started a business in the ratio of 7 : 9 respectively and after six months, C joined the business with two times of the capital of A. After one year, the total profit of the business is Rs. 92000. Find the sum of profit of B and C (in Rs.)?

The correct option is 64000.

To find the sum of the profit shares of B and C, we first need to determine the ratio in which the total profit is distributed among A, B, and C. In a partnership, the profit share is proportional to the product of the capital invested and the time period of the investment.

Let the initial capitals invested by A and B be $7x$ and $9x$ respectively.
Since A and B started the business, they invested their capitals for the entire year (12 months).
C joined after 6 months with twice the capital of A. Therefore:
C's capital = $2\times 7x=14x$
C's investment duration = $12-6=6\text{months}$

Now, we calculate the profit-sharing ratio of A, B, and C by multiplying their respective capital by their investment time:
A's profit share ratio = $7x\times 12=84x$
B's profit share ratio = $9x\times 12=108x$
C's profit share ratio = $14x\times 6=84x$

The ratio of profit distribution A : B : C is:
$84x:108x:84x$
Dividing each term by $12x$ to simplify the ratio:
$A:B:C=7:9:7$

Let the profit shares of A, B, and C be $7k$, $9k$, and $7k$ respectively. The total profit of the business after one year is Rs. 92000. Therefore:
$7k+9k+7k=92000$
$23k=92000$
$k=\frac{92000}{23}=4000$

We need to find the sum of the profits of B and C:
Sum of profit of B and C = $9k+7k=16k$
Substituting the value of $k$:
Sum of profit of B and C = $16\times 4000=64000$

Thus, the sum of the profit of B and C is Rs. 64000.