Deepinder lends Rs 8,200 to Jairaj for 16 years and Rs 4,900 to Karna for 15 years, both at the same annual rate of simple interest. If he receives Rs 19,446.50 in total as interest from both of them, what is the rate of interest per annum?

Introduction / Context:
This problem tests your ability to handle multiple simple interest loans at the same rate but with different principals and time periods. You must combine the interest from both loans and then deduce the common rate of interest from the total interest received.

Given Data / Assumptions:

• Loan to Jairaj: Principal P1 = Rs 8,200, Time T1 = 16 years.
• Loan to Karna: Principal P2 = Rs 4,900, Time T2 = 15 years.
• Common rate of interest per annum, R%, to be found.
• Total simple interest received from both loans = Rs 19,446.50.

Concept / Approach:
The simple interest formula for each loan is:
SI1 = (P1 * R * T1) / 100SI2 = (P2 * R * T2) / 100Because the rate R is the same for both, we can add the interests:
Total SI = SI1 + SI2 = (R / 100) * (P1 * T1 + P2 * T2)Using the total interest, we can solve for R by treating the quantity (P1 * T1 + P2 * T2) as a combined principal time factor.

Step-by-Step Solution:
Step 1: Compute the product P1 * T1.P1 * T1 = 8200 * 16 = 131,200Step 2: Compute the product P2 * T2.P2 * T2 = 4900 * 15 = 73,500Step 3: Add these to get the combined factor.P1 * T1 + P2 * T2 = 131,200 + 73,500 = 204,700Step 4: Use the total simple interest formula.Total SI = (R / 100) * 204,700 = 19,446.50Step 5: Rearrange to solve for R.R = (19,446.50 * 100) / 204,700R = 1,944,650 / 204,700R = 9.5So, the rate of interest is 9.5% per annum.

Verification / Alternative check:
We can verify by computing the interest separately. For Jairaj: SI1 = (8200 * 9.5 * 16) / 100. For Karna: SI2 = (4900 * 9.5 * 15) / 100. Using the combined factor approach we already know these will sum to 19,446.50, but checking with a calculator or careful multiplication reassures that 9.5% is correct.

Why Other Options Are Wrong:

• 10%, 10.5%, and 11% would each produce a total interest larger than Rs 19,446.50 when applied to the combined factor of 204,700.
• Only 9.5% yields the exact given total interest.

Common Pitfalls:
Common mistakes include adding principals or times incorrectly, or failing to multiply principal and time before summing. Some learners also try to split the total interest arbitrarily between the two loans rather than working with the combined expression that contains R as a common factor. Using the compact formula Total SI = (R / 100) * (P1 * T1 + P2 * T2) keeps the algebra neat and reduces errors.

Final Answer:
The rate of simple interest charged to both Jairaj and Karna is 9.5% per annum.