Question

A began a business with Rs.45000 and B joined afterwards with Rs.30000. At the end of a year the profit is divided in the ratio 2:1. After how many months B joined?

Answer 1

Hint: From the question, we were given cost, profit and time, the ratio of product of cost and time of A and B is equal to the ratio of profit of A and B. So, by using this concept this problem can be solved.

Complete step-by-step solution:
From the question, we were given that A began a business with Rs.45000 and B joined afterwards with Rs.30000. At the end of a year the profit is divided in the ratio 2:1.
Let us assume the amount invested by A is x, amount invested by B is y.
Then
\[\begin{align}
  & x=45000.....(1) \\
 & y=30000.....(2) \\
\end{align}\]
Let us assume the profit ratio is equal to \[a:b\].
Then
\[\begin{align}
  & \dfrac{a}{b}=\dfrac{2}{1} \\
 & \Rightarrow a=2b.....(3) \\
\end{align}\]
Then time taken by A is equal to one year to earn the profit.
Let us assume the time taken by A is equal to p, time taken by B is equal to q.
As we were given cost, profit and time, the ratio of product of cost and time of A and B is equal to the ratio of profit of A and B.
\[\Rightarrow \dfrac{xp}{yq}=\dfrac{a}{b}.....(4)\]
Now place equation (1), equation (2), equation (3) in equation (4) we get
\[\begin{align}
  & \Rightarrow \dfrac{(45000)(1)}{(30000)(q)}=2 \\
 & \Rightarrow q=\dfrac{3}{4} \\
\end{align}\]
The value of q is equal to \[\dfrac{3}{4}\].
So, it took \[\dfrac{3}{4}\]years for B to earn the required profit.
We know that one year is equal to 12 months.
So,
Months took by B \[=\dfrac{3}{4}\times 12=9\]
So, 9 months took by B to earn the required profit.
Therefore, B joined after 3 years.

Note: Students may undergo calculation mistakes while solving this problem. If a small mistake is done, then the final answer may get interrupted. So, these mistakes should be avoided while solving the problem. Students should follow a correct calculation procedure. Also, we have been asked to find the time after which B has joined not the time period B took to earn the profit.