What annual payment will discharge a debt of ₹ 1025 due in 2 years at the rate of 5% compound interest?

### Concept & Instalments for Compound Interest The present value of a debt paid in equal instalments is calculated using the formula for compound interest present worth. $$P = \frac{x}{\left(1 + \frac{R}{100}\right)} + \frac{x}{\left(1 + \frac{R}{100}\right)^2}$$ where $P$ is the principal borrowed, $x$ is the annual instalment, and $R$ is the rate of interest. ### Step-by-Step Solution * **Given:** Debt (Principal) $P = 1025$, Rate $R = 5\%$, Time $T = 2$ years. * Let the annual instalment be $x$. * $1025 = \frac{x}{1 + \frac{5}{100}} + \frac{x}{\left(1 + \frac{5}{100}\right)^2}$ * $1025 = \frac{x}{1.05} + \frac{x}{(1.05)^2} = \frac{x}{\frac{21}{20}} + \frac{x}{\left(\frac{21}{20}\right)^2}$ * $1025 = x\left(\frac{20}{21} + \frac{400}{441}\right)$ * $1025 = x\left(\frac{420 + 400}{441}\right) = x\left(\frac{820}{441}\right)$ * $x = \frac{1025 \times 441}{820}$ * $x = \frac{205 \times 441}{164} = \frac{5 \times 441}{4} = \frac{2205}{4} = 551.25$ ### Exam Strategy & Shortcut Recognize that the sum of the series for 2 years at 5% is $\frac{820}{441}$. The instalment is simply $P \times \frac{441}{820}$. Notice that 820 and 1025 are both multiples of 205 (which is $41 \times 5$), simplifying the calculation quickly to $5 \times \frac{441}{4}$. ### Common Pitfall A common mistake is treating the "debt due in 2 years" as the future amount rather than the present principal, leading to incorrect applications of the instalment formula. ### Final Answer Therefore, the correct answer is **₹ 551.25**.