Question

A, B and C entering into a partnership by investing in the ratio of 3:2:4. After one year, B invests another Rs 2, 70,000 and C at the end of 2 years also invests Rs 2, 70,000. At the end of three years, profit is shared in the ratio of 3:4:5. Find the initial investment of C.
(a) Rs 2, 70,000
(b) Rs 1, 80,000
(c) Rs 3, 60,000
(d) None of these

Answer 1

Hint: To convert the ratio of initial investments of A, B and C into rupees we are multiplying this ratio 3:2:4 by x so the investment of A, B and C are 3x, 2x and 4x respectively. Now, using this initial investment calculate the profit share of A, B and C at the end of three years and take the ratio of A, B and C and equate it to 3:4:5. After solving this equation, we can calculate the value of x and then substitute this value of x in 4x we will get the initial investment of C.

Complete step-by-step answer:
We have given the ratio of initial investment as 3:2:4. Converting this ratio into number we are going to multiply this ratio by x so:
The initial investment of A is equal to 3x.
The initial investment of B is equal to 2x.
The initial investment of C is equal to 4x.
The share of A at the end of three years is equal to the multiplication of 3x by 3 which is equal to 9x.
The share of B at the end of three years is equal to the addition of 2x in the result of multiplication of $\left( 2x+2,70,000 \right)$ with 2.
$\begin{align}
  & 2x+\left( 2x+2,70,000 \right)\left( 2 \right) \\
 & =2x+4x+5,40,000 \\
 & =6x+5,40,000 \\
\end{align}$
The share of C at the end of three years is equal to the addition of the result of multiplication of 4x by 2 with the $\left( 4x+2,70,000 \right)$.
$\begin{align}
  & 4x\left( 2 \right)+\left( 4x+2,70,000 \right) \\
 & =8x+4x+2,70,000 \\
 & =12x+2,70,000 \\
\end{align}$
Now, it is given that at the end of three years profit share of A, B and C is in the ratio of 3:4:5 so equating the ratio of share of A, B and C that we have found at the end of 3 years to 3:4:5 we get,
The ratio of share of A, B and C at the end of three years is equal to:
$9x:\left( 6x+5,40,000 \right):\left( 12x+2,70,00 \right)$
Now, to make this ratio equal to 3:4:5 we are dividing the above ratio by 3x.
$3:\left( \dfrac{6x+5,40,000}{3x} \right):\left( \dfrac{12x+2,70,00}{3x} \right)$
As the above ratio is equal to 3:4:5 so comparing the above ratio by 3:4:5 we get,
We are equating $\dfrac{6x+5,40,000}{3x}$ to 4.
$4=\dfrac{6x+5,40,000}{3x}$
On cross multiplying the above equation we get,
$\begin{align}
  & 12x=6x+5,40,000 \\
 & \Rightarrow 6x=5,40,000 \\
\end{align}$
Dividing 6 on both the sides we get,
$\begin{align}
  & x=\dfrac{5,40,000}{6} \\
 & \Rightarrow x=90,000 \\
\end{align}$
Now to get the initial investment of C we are going to substitute the above value of x in 4x.
$\begin{align}
  & 4\left( 90,000 \right) \\
 & =3,60,000 \\
\end{align}$
Hence, the initial investment of C is Rs 3,60,000.
Hence, the correct option is (c).

Note: In the above solution you can check whether the value of x that you are getting is correct or not by comparing the ratio of C that we have obtained with the given ratio.
The obtained ratio of share of A, B and C at the end of three years is:
$3:\left( \dfrac{6x+5,40,000}{3x} \right):\left( \dfrac{12x+2,70,00}{3x} \right)$
Comparing the third ratio with 5 we get,
$\dfrac{12x+2,70,00}{3x}=5$
On cross multiplication of the above equation we get,
$\begin{align}
  & 12x+2,70,000=15x \\
 & \Rightarrow 2,70,000=3x \\
\end{align}$
Dividing 3 on both the sides we get,
$\begin{align}
  & \dfrac{2,70,000}{3}=x \\
 & \Rightarrow x=90,000 \\
\end{align}$
So, you can see that we are getting the value of x as same as that we have got in the above solution.