A sum of ₹ 10 is lent to be returned in 11 monthly instalments of ₹ 1 each, interest being simple. The rate of interest is :

### Concept & Formula The interest in an instalment plan is calculated on the outstanding principal for each month. The sum of these monthly outstanding balances acts as the equivalent principal for one month. $$I = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution * **Given:** Total loan amount = ₹ 10. It is repaid in 11 instalments of ₹ 1 each. * Total amount paid = $11 \times 1$ = ₹ 11. * Total simple interest paid = Total Amount - Principal = $11 - 10$ = ₹ 1. * Let the rate of interest be $R\%$ per annum. * **Calculation:** The principal outstanding each month is: * 1st month: ₹ 10 * 2nd month: ₹ 9 * ... * 11th month: ₹ 0 (Wait, principal remaining is 0 only after the 10th instalment reduces the core principal. But standard instalment outstanding principal is simply $10 + 9 + 8 + ... + 1$). * Sum of outstanding principal = $\frac{10 \times 11}{2}$ = ₹ 55 for one month. * Interest = $\frac{55 \times R \times 1}{100 \times 12} = 1$ * $R = \frac{1200}{55} = \frac{240}{11} = 21 \frac{9}{11}\%$ ### Exam Strategy & Shortcut For such instalment problems without down payments, calculate the total interest directly (Total Paid - Loan). Then, find the sum of an arithmetic progression for the outstanding principal and apply the basic SI formula for 1 month ($T = \frac{1}{12}$). ### Common Pitfall A common mistake is using $T = \frac{11}{12}$ on the total sum ₹ 55, which double-counts the time. The sum ₹ 55 is already aggregated into a 1-month equivalent. ### Final Answer Therefore, the correct answer is **$21 \frac{9}{11}\%$**.