The simple interest on a sum of money at 12% per annum for 1 year is ₹ 900. Find :

(i) the sum of money and

(ii) the compound interest on this sum for 1 year, payable half-yearly at the same rate.

(i) Given,

I = ₹ 900

R = 12%

T = 1 year

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

$\Rightarrow 900 = \dfrac{P \times 12 \times 1}{100}\\[1em] \Rightarrow 900 = \dfrac{P \times 12}{100}\\[1em] \Rightarrow P = \dfrac{900 \times 100}{12} \\[1em] \Rightarrow P = ₹ 7,500.$

Hence, principal = ₹ 7,500.

(ii) When rate of interest is compounded half-yearly :

By formula,

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

Substituting values we get :

$\Rightarrow A = 7500 \Big(1 + \dfrac{12}{2 \times 100}\Big)^{2 \times 1} \\[1em] \Rightarrow A = 7500 \Big(\dfrac{200 + 12}{200}\Big)^2 \\[1em] \Rightarrow A = 7500 \Big(\dfrac{212}{200}\Big)^2 \\[1em] \Rightarrow A = 7500 \Big(1.06\Big)^2 \\[1em] \Rightarrow A = 7500 \times 1.1236 \\[1em] \Rightarrow A = ₹ 8,427$

By formula,

Compound interest = Amount - Principal = ₹ 8,427 - ₹ 7500 = ₹ 927.

Hence, compound interest = ₹ 927.