Question

Rs.44,200 are divided between Ram and john so that Ram’s share at the end of 6 years may be equal to john’s share at the end of 4 years, the rate of interest is 10 percent compounded yearly. Find Ram’s share.

Answer 1

Hint: Assume that the share of Ram is Rs. \[x\] . Since Rs. 44,200 has to be distributed between Ram and John so, the share of John must be Rs. 44,200. Now, use the formula for the amount, Amount = Principal \[\times \]\[{{\left( 1+\dfrac{rate}{100} \right)}^{n}}\] and calculate the amount of Ram after 6 years. Similarly, calculate the amount of John after 4 years. Since the amount of Ram and john are equal so, equality must hold between both of the amounts. At last, solve it further and get the value of x.

Complete step-by-step solution
According to the question, we are given that Rs. 44,200 has to be divided between Ram and John so that Ram’s share at the end of 6 years may be equal to John’s share at the end of 4 years.
We are also given the rate of interest.
The rate of interest compounded yearly = 10% …………………………………….(1)
First of all, let us assume that Ram’s share is Rs. \[x\] ……………………………………….(2)
Since Rs. 44,200 has to be divided between Ram and John so the summation of the share of Ram and John must be equal to Rs. 44,200. That is,
Share of Ram + Share of John = Rs. 44,200 …………………………………………(3)
Now, from equation (2) and equation (3), we get
\[\Rightarrow x\] + Share of John = Rs. 44,200
\[\Rightarrow \] Share of John = Rs. \[\left( 44,200-x \right)\] …………………………………………………(4)
We know the formula for the amount when the rate is compounded yearly, Amount = Principal \[\times \]\[{{\left( 1+\dfrac{rate}{100} \right)}^{n}}\] where n is the number of years ………………………………..(5)
Using the formula shown in equation (5), and from equation (1) and equation (2), let us calculate the amount of Ram after 6 years.
Ram’s amount after 6 years = \[x{{\left( 1+\dfrac{10}{100} \right)}^{6}}\] ……………………………………………(6)
Similarly, let us calculate John’s amount after 4 years.
Now, from equation (1), equation (4), and equation (5), we get
John’s amount after 4 years = \[\left( 44,200-x \right){{\left( 1+\dfrac{10}{100} \right)}^{4}}\] ……………………………………………(7)
We are given that the amount of Ram after 6 years is equal to the amount of John after 4 years …………………………………………………(8)
Now, from equation (6), equation (7), and equation (8), we get
\[\begin{align}
  & \Rightarrow x{{\left( 1+\dfrac{10}{100} \right)}^{6}}=\left( 44200-x \right){{\left( 1+\dfrac{10}{100} \right)}^{4}} \\
 & \Rightarrow x{{\left( 1+\dfrac{10}{100} \right)}^{2}}=\left( 44200-x \right) \\
 & \Rightarrow x{{\left( \dfrac{110}{100} \right)}^{2}}=\left( 44200-x \right) \\
 & \Rightarrow \dfrac{121x}{100}+x=44200 \\
 & \Rightarrow \dfrac{221}{100}x=44200 \\
 & \Rightarrow x=\dfrac{44200\times 100}{221} \\
 & \Rightarrow x=20000 \\
\end{align}\]
In equation (2), we have assumed x as the share of Ram.
Therefore, the share of Ram is Rs. 20,000.

Note: For this type of question where a portion of the money is compounded yearly at some rate and we require the amount after some years. Always approach this type of question by using the formula, Amount = Principal \[\times \] \[{{\left( 1+\dfrac{rate}{100} \right)}^{n}}\] .