A sum of money amounts to ₹13230 in one year and to ₹13891.50 in $1\dfrac{1}{2}$ years at compound interest, compounded semi-annually. Find the sum and rate of interest per annum.

Let sum be ₹x and rate be r%.

Since C.I. is reckoned half-yearly, rate = $\dfrac{r\%}{2}.$

$A = P\Big(1 + \dfrac{r}{100}\Big)^n$

In one year,

n = 2

A = ₹13230

rate = $\dfrac{r\%}{2}$ as interest is calculated half-yearly.

Substituting values in formula,

$13230 = x\Big(1 + \dfrac{\dfrac{r}{2}}{100}\Big)^2 \\[1em] 13230 = x\Big(1 + \dfrac{r}{200}\Big)^2 ……..(i)$

In one and half year,

n = 3

A = ₹13891.50

rate = $\dfrac{r\%}{2}$ as interest is calculated half-yearly.

Substituting values in formula,

$13891.50 = x\Big(1 + \dfrac{\dfrac{r}{2}}{100}\Big)^3 \\[1em] 13891.50 = x\Big(1 + \dfrac{r}{200}\Big)^3 ……..(ii)$

Dividing eqn. (ii) by (i),

$\Rightarrow \dfrac{13891.50}{13230} = \dfrac{x\Big(1 + \dfrac{r}{200}\Big)^3}{x\Big(1 + \dfrac{r}{200}\Big)^2} \\[1em] \Rightarrow \dfrac{138915}{132300} = 1 + \dfrac{r}{200} \\[1em] \Rightarrow \dfrac{r}{200} = \dfrac{138915}{132300} - 1 \\[1em] \Rightarrow \dfrac{r}{200} = \dfrac{138915 - 132300}{132300} \\[1em] \Rightarrow \dfrac{r}{200} = \dfrac{6615}{132300} \\[1em] \Rightarrow r = \dfrac{6615 \times 200}{132300} \\[1em] \Rightarrow r = \dfrac{1323000}{132300} \\[1em] \Rightarrow r = 10\%.$

Putting value of r in eq. (i) we get,

$\Rightarrow 13230 = x\Big(1 + \dfrac{10}{200}\Big)^2 \\[1em] \Rightarrow 13230 = x\Big(1 + \dfrac{1}{20}\Big)^2 \\[1em] \Rightarrow 13230 = x\Big(\dfrac{21}{20}\Big)^2 \\[1em] \Rightarrow 13230 = x \times \dfrac{441}{400} \\[1em] \Rightarrow x = \dfrac{13230 \times 400}{441} \\[1em] \Rightarrow x = ₹12000.$

Hence, sum = ₹12000 and rate = 10% per annum.