Question

Sita and Gita invested the same capital in business. At the year-end they share the profit in the ratio $3:2$. If Sita has invested her capital for the whole year, for how many months Gita has invested her capital?
A. $8$ months.
B. $7$ months.
C. $5$ months.
D. $9$ months.

Answer 1

Hint: We will assume the capital as $X$ and in the problem they have mentioned that the capital is the same for both Sita and Gita. Now we will assume the months invested by Gita as $Y$months. Now we will calculate the total amounts invested by Sita and Gita. By adding the total shares invested by Sita and Gita we will get the value of total investment by both of them. Now the profit of the Sita and Gita can be calculated by dividing their shares over their respective time period with the total investment of both of them. In the problem they have mentioned the ratio of the profits of both Sita and Gita, so we will calculate the ratio of the profits of both Sita and Gita and equate it to the given value to find the value of $Y$.

Complete step by step answer:
Given that,
Sita and Gita invested the same capital in business.
Let the capital amount invested by Sita and Gita be Rs. $x$
Given that Sita invested whole the year. So, the total amount invested by Sita in a year ($12$ months) is given by
Investment of Sita $=x\times 12$
Let Gita invested for $y$ months, then the total amount invested by Gita is given by
Investment of Gita $=x\times y$.
Now the total amount invested by both of them is given by
Total Share $=x\times 12+x\times y=x\left( y+12 \right)$.
Profit of Sita can be given by ratio of the total investment of Sita to the total share i.e.
$\begin{align}
  & {{P}_{s}}=\dfrac{12x}{x\left( y+12 \right)} \\
 & \Rightarrow {{P}_{s}}=\dfrac{12}{y+12}...\left( \text{i} \right) \\
\end{align}$
Profit of Gita can be given by ratio of the total investment of Gita to the total share i.e.
$\begin{align}
  & {{P}_{g}}=\dfrac{xy}{x\left( y+12 \right)} \\
 & \Rightarrow {{P}_{g}}=\dfrac{y}{y+12}...\left( \text{ii} \right) \\
\end{align}$
In the problem they have mentioned that the ratio of profits of Sita and Gita are \[3:2\].
$\begin{align}
  & \therefore {{P}_{s}}:{{P}_{g}}=3:2 \\
 & \Rightarrow \dfrac{{{P}_{s}}}{{{P}_{g}}}=\dfrac{3}{2} \\
\end{align}$
From equations $\left( \text{i} \right)$ and $\left( \text{ii} \right)$ substituting the values of ${{P}_{g}}$ and ${{P}_{s}}$ in the above equation, then we will get
$\begin{align}
  & \dfrac{\dfrac{12}{y+12}}{\dfrac{y}{y+12}}=\dfrac{3}{2} \\
 & \Rightarrow \dfrac{12}{y+12}\times \dfrac{y+12}{y}=\dfrac{3}{2} \\
 & \Rightarrow y=\dfrac{12\times 2}{3} \\
 & \Rightarrow y=8 \\
\end{align}$

So, the correct answer is “Option A”.

Note: We can solve the equation ${{P}_{s}}:{{P}_{g}}=3:2$ in another manner i.e.
Let ${{P}_{s}}=3p$, ${{P}_{g}}=2p$
$\begin{align}
  & \Rightarrow \dfrac{12}{y+12}=3p \\
 & \Rightarrow y+12=\dfrac{12}{3p} \\
 & \Rightarrow y=\dfrac{4}{p}-12 \\
\end{align}$
Now substituting this value in ${{P}_{g}}$, then we will get
$\begin{align}
  & \dfrac{y}{y+12}=2p \\
 & \Rightarrow \dfrac{\dfrac{4}{p}-12}{\dfrac{4}{p}-12+12}=2p \\
 & \Rightarrow \dfrac{\dfrac{4-12p}{p}}{\dfrac{4}{p}}=2p \\
 & \Rightarrow \dfrac{4-12p}{4}=2p \\
 & \Rightarrow 8p=4-12p \\
 & \Rightarrow p=\dfrac{4}{20}=\dfrac{1}{5} \\
\end{align}$
Now substitute the value of $p$ in $y=\dfrac{4}{p}-12$, then we will get
$\begin{align}
  & y=\dfrac{4}{\dfrac{1}{5}}-12 \\
 & \Rightarrow y=4\times 5-12 \\
 & \Rightarrow y=8 \\
\end{align}$
From both the methods we got the same result but the second method is a little bit complicated and time consuming so we will use the first method.