Simple interest on a certain sum of money at 9% is ₹ 450 in 2 years. Find the compound interest, on the same sum, at the same rate for 1 year, if the interest is reckoned half yearly.

Given, R = 9 %, S.I. = ₹ 450, T = 2 years

Let P be the principal amount.

$\text{S.I.} = \dfrac{P \times R \times T}{100}\\[1em] ⇒ ₹ 450 = \dfrac{P \times 9 \times 2}{100}\\[1em] ⇒ ₹ 450 = \dfrac{18P}{100}\\[1em] ⇒ P = ₹ \dfrac{450 \times 100}{18}\\[1em] ⇒ P = ₹ \dfrac{45,000}{18}\\[1em] ⇒ P = ₹ 2,500$

For 1st $\dfrac{1}{2}$ year :

P = ₹ 2,500, R = 9 % , T = $\dfrac{1}{2}$ year

$\text{Interest for first year} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{2,500 \times 9 \times \dfrac{1}{2}}{100}\\[1em] = \dfrac{2,500 \times 9 \times \dfrac{1}{2}}{100}\\[1em] = \dfrac{22,500}{200}\\[1em] = ₹ 112.5$

A = P + I

= ₹ 2,500 + 112.5

= ₹ 2,612.5

For 2nd $\dfrac{1}{2}$ year :

P = ₹ 2,612.5, R = 9 %, T = $\dfrac{1}{2}$ year

$\text{Interest for second year} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{2,612.5 \times 9 \times \dfrac{1}{2}}{100}\\[1em] = \dfrac{23,512.5}{200}\\[1em] = ₹ 117.56$

A = P + I

= ₹ 2,612.5 + 117.56

= ₹ 2,730.06

Compound Interest = Final amount - Initial principal

= ₹ 2,730.06 - 2,500

= ₹ 230.06

Hence, the compound interest = ₹ 230.06.