Question

At what rate of interest per annum will a sum of $Rs.62500$ earn a compound interest of $Rs.5100$ in one year? The interest is to be compounded half yearly.

Answer 1

Hint: This question is the direct application of the formula of compound interest. We just have to put the values of different variables in the formula to find the required rate of interest. One should also remember that the total amount will be equal to the sum of compound interest and principle.
Formula used:
$A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$
A $ = $ Final amount
P $ = $ Sum or initial principle
r $ = $ interest rate
n $ = $ number of times interest applied per time period
t $ = $ time period

Complete step-by-step solution:
As we know,
Total amount=compound interest + principle
A $ = $ C.I +P
A $ - $ P $ = $ C.I
From the given formula,
$P{\left( {1 + \dfrac{r}{n}} \right)^{nt}} - \,P\, = \,C.I$
Taking P common,
$P\left[ {{{\left( {1 + \dfrac{r}{n}} \right)}^{nt}} - 1} \right] = \,C.I$
From our given question:
$P\, = \,Rs.\,62500$
$n = \dfrac{\text{Number of times applied}}{\text{Time period}} = \,\dfrac{2}{1}\, = \,2$
r $ = $ interest rate
t $ = $ time period
C.I $ = $ Compound interest
C.I $ = $ $Rs.\,5100$
Putting the values,
$62500\left[ {{{\left( {1 + \dfrac{r}{{2 \times 100}}} \right)}^{2 \times 1}} - 1} \right] = \,5100$
In this question, we have to find the percentage. So, we had multiplied r by $100$.
Multiply $62500$ in left-hand side,
$62500{\left( {1 + \dfrac{r}{{200}}} \right)^2} - \,62500\, = \,5100$
$\Rightarrow 62500{\left( {1 + \dfrac{r}{{200}}} \right)^2} = \,5100 + 62500$
$\Rightarrow 62500{\left( {1 + \dfrac{r}{{200}}} \right)^2} = \,67600$
$\Rightarrow {\left( {1 + \dfrac{r}{{200}}} \right)^2} = \,\dfrac{{67600}}{{62500}}$
Taking square root both sides,
${\sqrt {{{\left( {1 + \dfrac{r}{{200}}} \right)}^2}} ^{}} = \,\sqrt {\dfrac{{67600}}{{62500}}} $
$\Rightarrow 1 + \dfrac{r}{{200}} = \,\dfrac{{26}}{{25}}$
Neglecting the negative value as rate of interest is always positive.
$\dfrac{r}{{200}} = \,\dfrac{1}{{25}}$
$\Rightarrow r\, = \,8$
Hence the rate of interest is $8\,\% $.

Note: put the values of different variables in the formula carefully. The value of rate of interest is always positive. So, while taking the root, consider only positive values. Take the value of n carefully, it might be confusing. The meaning of half-years is $2$ times a year and the meaning of quarterly is $4$ times a year.