Question

A invested Rs. 76000 in a business. After a few months B joined him with Rs 57000. At the end of the year the profit was divided between them in the ratio of 2:1. After how many months did B join?
(a) 4
(b) 8
(c) 9
(d) None of these

Answer 1

Hint: Let the number of months after which B joined be x. So, for the first x months, only A earns the profit on his share of Rs. 76000. After x months, B joins with his Rs 57000 and the profit earned in the remaining months will be divided among A and B on the basis of their investment. The condition which governs the question is that the sum of the profit earned by A in the first x months and his share of profit of the remaining (12 – x) months should be twice the share of profit earned by B in the remaining (12 – x) months.

Complete step-by-step answer:
Let us assume that the total profit percentage earned per annum is 12%. This means profit per month will be $\dfrac{12}{12}=1%$.
Therefore, profit earned by A in the first x months will be x%. x% of 76000 will be$76000\times \dfrac{x}{100}=760x$
After x months, the total investment is the sum of investment of A and B.
76000 + 57000 = 133000
Now, in the remaining (12 – x) months, profit will be (12 – x) % of 133000.
$133000\times \dfrac{\left( 12-x \right)}{100}=1330\left( 12-x \right)$
This Rs. 1330(12 – x) will be divided among A and B on the basis of their investment.
The fraction of their profit is given by the relation \[\dfrac{I}{T}\], where I is the individual investment and T is the total investment.
Fraction of profit received by A will be $\dfrac{76000}{133000}=0.57$.
Fraction of profit received by B will be $\dfrac{57000}{133000}=0.43$.
So, total profit earned by A = profit earned in the first x months + 0.57(total profit in (12 – x) months)
Total profit earned by A = 760x + 0.57(1330(12 – x)) = 760x + 758.1(12 – x)
Total profit earned by B = 0.43(total profit earned in (12 – x) months)
Total profit earned by B = 0.43(1330(12 – x)) = 571.9(12 – x)
Now, according to the condition given in the question, the ratio of profits of A and B is 2:1. This means $\begin{align}
  & \Rightarrow \dfrac{\text{Profi}{{\text{t}}_{\text{A}}}}{\text{Profi}{{\text{t}}_{\text{B}}}}=\dfrac{2}{1} \\
 & \Rightarrow \dfrac{760x+758.1\left( 12-x \right)}{571.9\left( 12-x \right)}=\dfrac{2}{1} \\
\end{align}$
Now, we shall verify the options given to us.
We will substitute x = 4 in the LHS of the equation and see whether it verifies with the right hand side or not.
$\begin{align}
  & \dfrac{760\left( 4 \right)+758.1\left( 12-4 \right)}{571.9\left( 12-4 \right)}=\dfrac{3040+758.1\left( 8 \right)}{571.9\left( 8 \right)} \\
 & \dfrac{3040+758.1\left( 8 \right)}{571.9\left( 8 \right)}=\dfrac{9104.8}{4575.2} \\
 & =\dfrac{2}{1}
\end{align}$
 Therefore, option (a) verifies. Hence, option (a) is the correct option.

Note: This question can be solved by simplifying the equation $\dfrac{760x+758.1\left( 12-x \right)}{571.9\left( 12-x \right)}=\dfrac{2}{1}$ and solving for x. But that will be tedious , so to save time, option verification method is preferred. Students are advised to go through the question carefully as this question is a little tricky. Any other value for profit percentage can be assumed, but 12% is a convenient value as there are 12 months in a year.