Mr Duggal invested ₹ 20000 with rate of interest @ 20 p.c.p.a. The interest was compounded half-yearly for first one year and in the next year it was compounded yearly. What will be the total interest earned at the end of 2 years?

### Concept & Variable Compounding Terms When compounding frequencies change during the investment period, calculate the amount phase by phase, using the end amount of the first phase as the principal for the next. ### Step-by-Step Solution * **Given:** Principal ($P$) = ₹ 20000, Rate ($R$) = 20% p.a. * **Step 1:** Phase 1 (First year, compounded half-yearly). The rate is halved to 10% per half-year. The time is 2 half-years. Amount after 1st year ($A_1$) = $20000 \times (1 + \frac{10}{100})^2$ $A_1 = 20000 \times (1.10)^2 = 20000 \times 1.21 = 24200$ * **Step 2:** Phase 2 (Second year, compounded yearly). The new principal is ₹ 24200. The rate is 20% p.a. for 1 year. Amount after 2nd year ($A_2$) = $24200 \times (1 + \frac{20}{100})^1$ $A_2 = 24200 \times 1.20 = 29040$ * **Step 3:** Calculate Total Interest. Total Interest = Final Amount - Initial Principal Total Interest = $29040 - 20000 = 9040$ ### Exam Strategy & Shortcut Use successive percentages. First year effective rate (two 10% cycles) = $10 + 10 + \frac{10 \times 10}{100} = 21\%$. Second year rate = 20%. Total effective rate for 2 years = $21 + 20 + \frac{21 \times 20}{100} = 41 + 4.2 = 45.2\%$. Total Interest = $45.2\%$ of 20000 = $45.2 \times 200 = 9040$. ### Common Pitfall Applying the 20% rate for both years without adjusting for the half-yearly compounding in the first year, which would mistakenly yield $20000 \times 1.44 = 28800$, leading to an interest of 8800. ### Final Answer Therefore, the correct answer is **₹ 9040**.