Question

A sum of money, at compound interest, yields Rs. 200 and Rs. 220 at the end of the first and second years respectively. The rate % is:
(a) 20
(b) 15
(c) 10
(d) 5

Answer 1

Hint: In this question, we do not have the principal amount or rate of interest of any one of the years, use the number of years and the formula for finding the total amount of money after the number of years at certain compound interest, $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$and create two equations for the first year and second year and divide them to find the rate of interest.

Complete step-by-step solution:
We do not have the principal amount, but we have a sum of money with compound interest at the end of the first year and the second year which is Rs. 200 and Rs. 220 respectively.
We know compound interest formula, $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$
where, A = Total amount of money after n years, P = Principal amount, r = rate of interest, n = no. of years
For the first year, we have a sum of money at a compound interest of Rs. 200, A = 200
$200=P{{\left( 1+\dfrac{r}{100} \right)}^{1}}$………………. (i)
Similarly, for the second year, we have A = 220
$220=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}$………………. (ii)
To find the rate of interest, we have to divide the equation (ii) by equation (i)
We get,
$\dfrac{220}{200}=\dfrac{P{{\left( 1+\dfrac{r}{100} \right)}^{2}}}{P\left( 1+\dfrac{r}{100} \right)}$
After dividing the right-hand side by P in the numerator and denominator, we get
$\dfrac{220}{200}=\dfrac{{{\left( 1+\dfrac{r}{100} \right)}^{2}}}{\left( 1+\dfrac{r}{100} \right)}$
Now, let us divide the right-hand side by $\left( 1+\dfrac{r}{100} \right)$ in the numerator and denominator, we get
$\dfrac{220}{200}=\left( 1+\dfrac{r}{100} \right)$
Let us divide the left-hand side by 2 in the numerator and denominator, we get
$\dfrac{110}{100}=\left( 1+\dfrac{r}{100} \right)$
Now, let us divide by 100 in the numerator and denominator in the left-hand side, we get
$1.1=\left( 1+\dfrac{r}{100} \right)$
Calculate the value of r, which is the value of rate of interest.
$\begin{align}
  & 1.1=\dfrac{100+r}{100} \\
 & 110=100+r \\
 & r=110-100 \\
 & =10
\end{align}$
Therefore, the rate of interest is 10%.

Note: Here, the formula of compound interest is mainly used to find the total amount of money at the certain compound interest. Also, remember the formula of simple interest (SI = P x R x N) which can be useful. in simple interest, the principal amount is to be constant while in compound interest it keeps on varying during the entire borrowing period.