The difference between C.I. and S.I. on ₹ 7500 for two years is ₹ 12 at the same rate of interest per annum. Find the rate of interest.

Given,

P = ₹ 7500

Time = 2 years

Let rate of interest be r%.

By formula,

S.I. = $\dfrac{P \times R \times T}{100}$

Substituting values we get :

$S.I. = \dfrac{7500 \times r \times 2}{100} \\[1em] = 150r.$

By formula,

C.I. = A - P

$C.I. = P\Big(1 + \dfrac{r}{100}\Big)^n - P \\[1em] = 7500 \times \Big(1 + \dfrac{r}{100}\Big)^2 - 7500 \\[1em] = 7500 \times \Big(\dfrac{100 + r}{100}\Big)^2 - 7500 \\[1em] = 7500 \times \Big(\dfrac{10000 + r^2 + 200r}{10000}\Big) - 7500 \\[1em] = \dfrac{3}{4} \times (10000 + r^2 + 200r) - 7500 \\[1em] = 7500 + \dfrac{3r^2}{4} + 150r - 7500 \\[1em] = \dfrac{3r^2}{4} + 150r$

The difference between C.I. and S.I. on ₹ 7500 for two years is ₹ 12.

∴ C.I. - S.I. = ₹ 12

$\therefore \dfrac{3r^2}{4} + 150r - 150r = 12 \\[1em] \Rightarrow \dfrac{3r^2}{4} = 12 \\[1em] \Rightarrow r^2 = \dfrac{12 \times 4}{3} \\[1em] \Rightarrow r^2 = 16 \\[1em] \Rightarrow r = \sqrt{16} = 4\%.$

Hence, rate of interest = 4%.