A person deposited ₹ 400 for 2 years, ₹ 550 for 4 years and ₹ 1200 for 6 years. He received the total simple interest of ₹ 1020. The rate of interest per annum is (S.S.C., 2007)

### Concept & Formula The total simple interest from multiple deposits is the sum of the individual simple interests calculated for each deposit using the common rate of interest. $$Total SI = \frac{P_1 \times R \times T_1}{100} + \frac{P_2 \times R \times T_2}{100} + \frac{P_3 \times R \times T_3}{100}$$ ### Step-by-Step Solution 1. **Given:** - Deposit 1: $P_1 = 400$, $T_1 = 2$ years - Deposit 2: $P_2 = 550$, $T_2 = 4$ years - Deposit 3: $P_3 = 1200$, $T_3 = 6$ years - Total Interest = ₹ 1020 - Let the uniform rate of interest be $R\%$ p.a. 2. Set up the equation for total simple interest: $\frac{400 \times R \times 2}{100} + \frac{550 \times R \times 4}{100} + \frac{1200 \times R \times 6}{100} = 1020$ 3. Simplify the terms by cancelling out the hundreds: $(4 \times 2 \times R) + (5.5 \times 4 \times R) + (12 \times 6 \times R) = 1020$ 4. Multiply out the components: $8R + 22R + 72R = 1020$ 5. Add the coefficients: $102R = 1020$ 6. Solve for $R$: $R = \frac{1020}{102} = 10$ ### Exam Strategy & Shortcut Factor out $R/100$ right from the start to save writing time. Total Interest = $\frac{R}{100} \times [ (400 \times 2) + (550 \times 4) + (1200 \times 6) ]$ $1020 = \frac{R}{100} \times [800 + 2200 + 7200]$ $1020 = \frac{R}{100} \times 10200$ $1020 = 102R \implies R = 10\%$. ### Common Pitfall A common error is miscalculating the products before adding them up, such as incorrectly evaluating $550 \times 4$ as 2000 instead of 2200. ### Final Answer Therefore, the correct answer is **10%**.