Find the difference between the compound interest compounded yearly and half-yearly on ₹ 10000 for 18 months at 10% per annum.

Given,

P = ₹ 10000

n = 18 months or 1.5 years

r = 10%

When interest is compounded yearly :

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

For 1st year :

$A = 10000 \times \Big(1 + \dfrac{10}{100}\Big)^1 \\[1em] = 10000 \times \dfrac{110}{100} \\[1em] = ₹ 11000.$

For next half-year :

₹ 11000 is the principal.

A = $P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2}$

Substituting values we get :

$A = 11000 \times \Big(1 + \dfrac{10}{2 \times 100}\Big)^{\dfrac{1}{2} \times 2} \\[1em] = 11000 \times \Big(1 + \dfrac{1}{20}\Big)^1 \\[1em] = 11000 \times \dfrac{21}{20} \\[1em] = ₹ 11550.$

When rate of interest is compounded half-yearly :

A = $P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2}$

Substituting values we get :

$A = 10000 \times \Big(1 + \dfrac{10}{2 \times 100}\Big)^{1.5 \times 2} \\[1em] = 10000 \times \Big(1 + \dfrac{1}{20}\Big)^3\\[1em] = 10000 \times \Big(\dfrac{21}{20}\Big)^3 \\[1em] = 10000 \times \dfrac{9261}{8000} \\[1em] = ₹ 11576.25$

Difference in C.I. between two cases = ₹ 11576.25 - ₹ 11550 = ₹ 26.25

Hence, difference between C.I. in two cases = ₹ 26.25