Question

$A,B$ and $C$ started a business where their initial capital was in the ratio of $4:5:6$. At the end of 8 months, $A$ invested an amount such that his total capital became half to $C's$ initial capital investment. If the annual profit of $B$ is Rs.7500 then what is the total profit.

Answer 1

Hint: This problem deals with business like investments, profits and total amount of capitals, etc. But if we understand it and observe the problem carefully it just deals with the calculations of the investment by each A, B and C annually, and given the annual profit of B, we can calculate or find out the total profit.

Complete step-by-step solution:
Given that there are three persons A, B and C who started business.
A, B and C started this business with their initial capital in the ratio of 4:5:6.
Here investment after a certain month is given by the product of their initial capital investment and the number of months mentioned.
Suppose here the finding the investment of C after an year, which means 12 months, is given by:
$ \Rightarrow 6 \times 12 = 72$
The investment of B after 12 months, is given by:
$ \Rightarrow 5 \times 12 = 60$
Now given that the investment of A at the end of 8 months is half the initial capital of the invested amount of C.
Which means that in that year for 8 months, the investment of A is given by:
\[ \Rightarrow 4 \times 8\]
And from the $8^{th}$ month to the $12^{th}$ month, which is a span of 4 months, the investment of A is the half of initial capital of A, which is half of 6, as given below:
$ \Rightarrow \dfrac{6}{2} \times 4$
$ \Rightarrow 3 \times 4$
Hence the investment of A for 12 months is given by:
$ \Rightarrow 4 \times 8 + 3 \times 4$
$ \Rightarrow 32 + 12 = 44$
The investment of A, B and C are in the ratio of, as given below:
$ \Rightarrow 44:60:72$
$ \Rightarrow 11:15:18$
Now given that the annual profit of B is Rs.7500.
Here annual profit is the profit in 12 months.
We found the investment of each A, B and C for 12 months, well actually their ratios of investments.
Now let the total profit be $x$.
The profit of B is given by:
$ \Rightarrow \dfrac{{15}}{{11 + 15 + 18}}x = 7500$
Now solving for $x$, as given below:
$ \Rightarrow \dfrac{{15x}}{{44}} = 7500$
$ \Rightarrow 15x = 44(7500)$
$ \Rightarrow 15x = 330000$
$ \Rightarrow x = 22000$
$\therefore $The total profit is given by Rs.22,000

The total profit is Rs.22,000

Note: While solving this problem we have to be careful and vigilant while solving the investment of A, here it is given that till the end of 8 months the investment of A is half of the initial capital of C, which means that till the first 8 months, the investment of A is just the product of its own initial capital and eight months, but for the next 4 months, the investment of A is the product of half of the initial capital of C and 4 months.