Question

Mahesh borrowed a certain sum for two years at simple interest from Bhim. Mahesh lent his sum to Vishnu at the same rate for two years compound interest. At the end of two years, Mahesh received Rs. 410 as compound interest but paid Rs. 400 as simple interest. Find the sum and rate of interest.

Answer 1

Hint: We first explain the formulas for simple and compound interest. We assume the values for the interest rate and the principal. We put the values and find the equations for two unknowns. We solve them to find the solutions.

Complete step-by-step solution:
First, we will explain the formulas for compound interest and simple interest.
Let the principal be P, interest rate be r and time period be n, then for the compound interest the formula will be $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}-P$ and simple interest will be $A=\dfrac{Pnr}{100}$.
Mahesh borrowed a certain sum for two years at simple interest from Bhim. Mahesh lent his sum to Vishnu at the same rate for two years compound interest.
Let us assume the amount is P and the rate is r.
So, according to the formula Mahesh will pay for the simple interest ${{A}_{1}}=\dfrac{P\times 2\times r}{100}=400$ and he will get for the compound interest ${{A}_{2}}=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}-P=410$.
Simplifying ${{A}_{2}}=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}-P=410$, we get
$\begin{align}
  & P\left( 1+\dfrac{r}{100}+1 \right)\left( 1+\dfrac{r}{100}-1 \right)=410 \\
 & \Rightarrow \dfrac{Pr}{100}\left( 2+\dfrac{r}{100} \right)=410 \\
\end{align}$
From ${{A}_{1}}=\dfrac{P\times 2\times r}{100}=400$, we get $Pr=20000$.
Putting the value in $\dfrac{Pr}{100}\left( 2+\dfrac{r}{100} \right)=410$, we get
$\begin{align}
  & \dfrac{20000}{100}\left( 2+\dfrac{r}{100} \right)=410 \\
 & \Rightarrow 2+\dfrac{r}{100}=\dfrac{410}{200} \\
 & \Rightarrow \dfrac{r}{100}=\dfrac{410}{200}-2=\dfrac{10}{200} \\
 & \Rightarrow r=5 \\
\end{align}$
Now putting the value of $r=5$ in the equation of $Pr=20000$, we get
$\begin{align}
  & P\times 5=20000 \\
 & \Rightarrow P=\dfrac{20000}{5}=4000 \\
\end{align}$
The values for the rate of interest and the principal are 5 and 4000 respectively.

Note: We also can use the substitution process where we replace the values for one variable in the second equation. The addition of the interest with the principal value is the total payback.