The difference between CI and SI on a certain sum of money for 3 years at 5% p.c.p.a. is ₹ 122. Find the sum invested.

### Concept & Difference Between CI and SI for 3 Years For a period of exactly 3 years, the difference between compound interest and simple interest on a principal $P$ at rate $R\%$ is given by a specific derived formula. $$Difference = P\left(\frac{R}{100}\right)^2\left(\frac{300 + R}{100}\right)$$ ### Step-by-Step Solution * **Given:** Difference $D = 122$, Rate $R = 5\%$, Time = 3 years. * Substitute the given values directly into the 3-year difference formula: * $122 = P \times \left(\frac{5}{100}\right)^2 \times \left(\frac{300 + 5}{100}\right)$ * $122 = P \times \left(\frac{1}{20}\right)^2 \times \left(\frac{305}{100}\right)$ * $122 = P \times \left(\frac{1}{400}\right) \times \left(\frac{61}{20}\right)$ * $122 = P \times \frac{61}{8000}$ * Solve for $P$: * $P = \frac{122 \times 8000}{61}$ * Since $61 \times 2 = 122$, the equation simplifies: * $P = 2 \times 8000 = 16000$ ### Exam Strategy & Shortcut Memorize the ratio fraction for 3 years at 5%. For $R = 5\%$, the fractional rate is $1/20$. The multiplier for the difference is $\frac{3(1/20) + (1/20)^2}{1} = \frac{61}{8000}$ of the principal. Knowing the multiplier $\frac{61}{8000}$ allows you to immediately write $122 = P \times \frac{61}{8000}$ and solve it in seconds. ### Common Pitfall Using the 2-year difference formula $P(R/100)^2$ instead of the 3-year formula, which would yield a completely incorrect principal. ### Final Answer Therefore, the correct answer is **₹ 16000**.