Question

The compound interest on a certain sum of 2 years is Rs. 786 and S.I. is Rs. 750. If the sum is invested such that the S.I. is Rs. 1296 and the number of years is equal to the rate per cent per annum, find the rate of interest.

Answer 1

Hint: Here, we will first find the principal amount from the first two given conditions. Then, we will calculate the rate percent per annum using the value of the principal amount obtained. We will use the formulas for simple interest $S.I.=P\times R\times T$ and the formula for compound interest $C.I.=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}-P$.

Complete step-by-step answer:
Let us consider that initially the principal amount is P and the annual rate is R.
Since, the total time is given to be 2 years. So, we can write using the formulas of simple interest and compound interest as:
$SI=P\times R\times 2=750...........\left( 1 \right)$
And,
$CI=P{{\left( 1+\dfrac{R}{n} \right)}^{2n}}-P$
Since, the interest is compounded annually, so n = 1.
$\Rightarrow CI=P{{\left( 1+R \right)}^{2}}-P=786...........\left( 2 \right)$
On dividing equation (2) by equation (1), we get:
$\begin{align}
  & \dfrac{P{{\left( 1+R \right)}^{2}}-P}{P\times R\times 2}=\dfrac{786}{750} \\
 & \Rightarrow \dfrac{1+{{R}^{2}}+2R-1}{R\times 2}=\dfrac{786}{750} \\
 & \Rightarrow 750\left( {{R}^{2}}+2R \right)=786\times 2R \\
 & \Rightarrow 750{{R}^{2}}+1500R=1572R \\
 & \Rightarrow 750{{R}^{2}}-72R=0 \\
 & \Rightarrow R\left( 750R-72 \right)=0 \\
\end{align}$
So, R=0
Or,
$\begin{align}
  & 750R-72=0 \\
 & \Rightarrow R=\dfrac{72}{750}=0.096 \\
\end{align}$
Since, R cannot be 0. So R = 0.096
On putting R = 0.096 in equation (1), we get:
$\begin{align}
  & P\times 0.096\times 2=750 \\
 & \Rightarrow P=\dfrac{750}{0.192}=3906.25 \\
\end{align}$
So, P = Rs. 3906.25
Now, it is given that the same amount gives a simple interest of Rs. 1296 when the rate percent is equal to the time. Since, here we are taking rate percent, so the formula for simple interest will be:
$SI=\dfrac{P\times R\times T}{100}$
Let the rate percent be = r, then the time is also equal to r. So, we can write:
$\begin{align}
  & \dfrac{P\times r\times r}{100}=1296 \\
 & \Rightarrow {{r}^{2}}=\dfrac{1296\times 100}{3906.25}=33.1776 \\
 & \Rightarrow r=\sqrt{33.1776}=5.76\% \\
\end{align}$
Hence, the rate percent is 5.76%.

Note: Students should note here that the formula for simple interest is $SI=P\times R\times T$ when the annual rate of interest is given. But when we talk of rate percent, the formula for simple interest becomes $SI=\dfrac{P\times R\times T}{100}$.