A lent ₹ 5000 to B for 2 years and ₹ 3000 to C for 4 years on simple interest at the same rate of interest and received ₹ 2200 in all from both of them as interest. The rate of interest per annum is

### Concept & Simple Interest To find the rate of interest, we can set up an equation summing the simple interest earned from both B and C and equate it to the total interest received. Formula for Simple Interest: $$SI = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution - **Given:** - Loan to B: Principal $P_1 = 5000$, Time $T_1 = 2$ years. - Loan to C: Principal $P_2 = 3000$, Time $T_2 = 4$ years. - Total Interest $SI = 2200$. - Let the common rate of interest be $R$\%. - **Calculation:** - Interest from B = $\frac{5000 \times R \times 2}{100} = 100R$ - Interest from C = $\frac{3000 \times R \times 4}{100} = 120R$ - Total Interest = $100R + 120R = 220R$ - Set the total interest equal to the given amount: $220R = 2200$ - Solving for $R$: $R = \frac{2200}{220} = 10$ ### Exam Strategy & Shortcut Instead of writing out the full fraction for both, just multiply the principal by the time and think of it as "total principal-years". Total principal-years = $(5000 \times 2) + (3000 \times 4) = 10000 + 12000 = 22000$. Interest on $22000$ for 1 year at $R$\% is $2200$. $R = \frac{2200 \times 100}{22000} = 10\%$. ### Common Pitfall A common error is confusing the time periods or trying to average the amounts instead of finding the sum of the simple interests directly. ### Final Answer Therefore, the correct answer is **10%**.