Question

Amit borrowed Rs$16000$ at $17\dfrac{1}{2}\%$ per annum simple interest. On the same day, he lent it to Ashu at the same rate but compounded annually. What does he gain at the end of $2$ years?

Answer 1

Hint: First we will find the simple interest that has to be paid by Amit at an interest rate of $17\dfrac{1}{2}\%$ per annum for the time period of $2$ years. Amit led the same money to Ashu for the same interest rate at the same date but compounded annually. So here we will find the total amount paid by Ashu to Amit and from that total amount we will find the compounded interest. Now the difference of compounded interest and simple interest gives the result.

Complete step-by-step answer:
Given that, Amit borrowed Rs$16000$ at $17\dfrac{1}{2}\%$ per annum simple interest and the time period is $2$ years. Here
Principle amount $\left( P \right)=16000$
Rate of Interest per annum $\left( R \right)=17\dfrac{1}{2}\%=17.5\%$
Time period $\left( T \right)=2$ years
We know that the simple interest for a time period $T$ for a principal amount $P$ at an interest rate of $R$ is given by $SI=\dfrac{PRT}{100}$
Now the simple interest for the time period $2$ years for a principal amount of $16000$ at $17.5\%$ interest rate is given by
$\begin{align}
  & SI=\dfrac{PRT}{100} \\
 & =\dfrac{16000\times 17.5\times 2}{100} \\
 & =5,600
\end{align}$
Hence the simple interest that has to be paid by Amit for a period of $2$ years is Rs.$5,600$.

Now Amit lends the same amount of money at the same date to Ashu at the same interest rate but it is compounded annually. Then
Principle amount $\left( P \right)=16000$
Rate of Interest per annum $\left( R \right)=\dfrac{17.5}{100}=0.175$
Time period $\left( T \right)=2$ years
In compounded method we directly get the total amount to be paid for a particular time period $T$ years at an interest rate of $R$ for a principal amount $P$ by the below formula
$A=P{{\left( 1+\dfrac{R}{n} \right)}^{nt}}$
Where $n$ is the number of times that the interest is compounded yearly.
Now the total amount to be paid by Ashu to Amit is
$\begin{align}
  & A=P{{\left( 1+\dfrac{R}{n} \right)}^{nt}} \\
 & =16000{{\left( 1+\dfrac{0.175}{1} \right)}^{1\left( 2 \right)}} \\
 & =16000{{\left( 1.175 \right)}^{2}} \\
 & =22,090
\end{align}$
Compounded Interest that has to be paid by Ashu to Amit is
$\begin{align}
  & CI=A-P \\
 & =22,090-16,000 \\
 & =6,090
\end{align}$
The amount that Amit gain from Ashu is $CI=6,090$ and the amount paid by Amit is $SI=5,600$, now the remaining amount is
$\begin{align}
  & CI-SI=6,090-5,600 \\
 & =490
\end{align}$
Hence Amit will gain an amount Rs.$490$.

Note: While using the formula for total amount in a compounded method please remember that the rate of interest should be in decimals so we need to divide the rate of interest with $100$. Also, the value of $n$ is the number of times that interest is compounded annually. In the given problem they mentioned clearly that interest is compounded annually i.e. $n=1$. If they mention the interest is compounded half a yearly then take $n=2$