Question

A began business with Rs. 4,500 and was joined afterwards by B with Rs. 5400. If the profit at the end of the year was divided in the ratio of 2:1, then the time of joining B was after
(a) 5 months
(b) 7 months
(c) 8 months
(d) 9 months

Answer 1

Hint: To solve the given question, we will first assume that the total number of months left in a year when B began business is n. Then we will make use of the fact that the total profit earned by A and B will be proportional to the number of months they did business and the amount of money they started with. After finding their profits, we will take the ratio of the profits and we will equate it to 2:1. From here, we will get the value of n. The answer of the question will be = (12 – n) months.

Complete step-by-step answer:
To start with, we will assume that the number of months B worked in the business started by A is n months. Thus, we can say that in months were left in a year when B joined the business. Now, it is given that A worked for 1 year, i.e. 12 months in that business. Now, we will make use of the fact that the profit earned by A and B will be proportional to the number of months they worked for and the amount of money they started with. Thus, the profit earned by A will be given by
\[\text{Profit earned by A}\propto \left( 12\text{ months} \right)\left( Rs.4500 \right)\]
Let the proportionality constant be K. Thus, we will get,
\[\text{Profit earned by A}=K\left( 12\text{ months} \right)\left( Rs.4500 \right)......\left( i \right)\]
Similarly, the profit earned by B will be given, by
\[\text{Profit earned by B}=K\left( \text{n months} \right)\left( Rs.5400 \right)......\left( ii \right)\]
Now, it is given in the question that the ratio of their profits will be 2:1. Thus, we will get the following equation.
\[\dfrac{\text{Profit earned by A}}{\text{Profit earned by B}}=\dfrac{K\left( 12\text{ months} \right)\left( Rs.4500 \right)}{K\left( \text{n months} \right)\left( Rs.5400 \right)}=\dfrac{2}{1}\]
\[\Rightarrow \dfrac{K\left( 12\text{ months} \right)\left( Rs.4500 \right)}{K\left( \text{n months} \right)\left( Rs.5400 \right)}=\dfrac{2}{1}\]
\[\Rightarrow \dfrac{12\times 45}{n\times 54}=2\]
\[\Rightarrow n=\dfrac{12\times 45}{2\times 54}\]
\[\Rightarrow n=5\text{ months}\]
Thus, B worked for 5 months in that business. Thus, we can say that he joined (12 – 5) months after A started business. Thus, he joined 7 months after A started business.
Hence, option (b) is the right answer.

Note: The above question can also be solved in an alternate way as shown. Let x be the total months after which B joined the business. Thus, we can say that,
\[\dfrac{12\times 4500}{\left( 12-x \right)\times 5400}=\dfrac{2}{1}\]
\[\Rightarrow \dfrac{\left( 12-x \right)\times 54}{12\times 45}=\dfrac{1}{2}\]
\[\Rightarrow \dfrac{12\times 54}{12\times 45}-\dfrac{x\times 54}{12\times 45}=\dfrac{1}{2}\]
\[\Rightarrow \dfrac{6}{5}-\dfrac{x}{10}=\dfrac{1}{2}\]
\[\Rightarrow \dfrac{x}{10}=\dfrac{6}{5}-\dfrac{1}{2}\]
\[\Rightarrow \dfrac{x}{10}=\dfrac{7}{10}\]
\[\Rightarrow x=7\text{ months}\]