Question

A sum of money amounts to Rs. 46,305 in 1 year and to Rs. 48,620.25 in $1\frac{1}{2}$ years at compound interest, compounded semi-annually. Find the sum and the rate of interest per annum.

Answer 1

Let the principle amount be Rs. $x$ and rate be $y\%$ per annum

Formula of amount in compound interest: $A=P{(1+\frac{r}{n})}^{nt}$

where $A=$ Amount

$P=$Principle

$n=$ number of times that interest is compounded per year

$t=$ number of years

$r=$rate(in decimals) per annum

Amount after $1$ year$=$Rs. $46305$

$n=2-[$since interest is compounded semi-annually]

$\therefore 46305=P{(1+\frac{y/100}{2})}^{2\times 1}$ ......$\left(1\right)$

Amount after $1\frac{1}{2}$year$=$Rs. $48620.25$

$48620.25=P{(1+\frac{y/100}{2})}^{2\times \frac{13}{2}}$ .......$\left(2\right)$

Dividing equation $\left(2\right)$ by $\left(1\right)$, we get

$\frac{48620.25}{46305}=(1+\frac{y}{200})$

On solving we get, $y=10\%$ p.a.

Substituting the value of y in equation $\left(1\right)$

$\therefore 46305=P{(1+\frac{10/100}{2})}^2$

On solving the above equation, we get

Principle(P)$=$Rs. $42,000$.