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, what is the simple interest rate per annum?

Introduction / Context:
This question combines two different simple interest loans made at the same rate but for different principals and time periods. The total interest from both loans over their entire durations is given, and the task is to find the common annual rate of interest. This type of problem checks whether learners can sum the interests and treat the common rate as a single unknown for the entire situation.

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. - Annual simple interest rate R percent is the same for both loans. - Total interest from both loans SI total = Rs 19,446.50. - Interest is calculated using simple interest, without compounding.

Concept / Approach:
Simple interest on each loan is SI1 = (P1 * R * T1) / 100 and SI2 = (P2 * R * T2) / 100. The total interest is SI total = SI1 + SI2. Since the rate R is common, factor R out and express SI total as R * (P1 * T1 + P2 * T2) / 100. Then solve for R using the known total interest and the sum P1 * T1 + P2 * T2. This gives the annual rate in percent per annum.

Step-by-Step Solution:
Step 1: Write total interest as SI total = (P1 * R * T1) / 100 + (P2 * R * T2) / 100. Step 2: Factor R / 100: SI total = (R / 100) * (P1 * T1 + P2 * T2). Step 3: Compute P1 * T1 = 8,200 * 16. Step 4: 8,200 * 16 = 1,31,200. Step 5: Compute P2 * T2 = 4,900 * 15 = 73,500. Step 6: Sum them: P1 * T1 + P2 * T2 = 1,31,200 + 73,500 = 2,04,700. Step 7: Use SI total = 19,446.50, so 19,446.50 = (R / 100) * 2,04,700. Step 8: Rearrange for R: R = (19,446.50 * 100) / 2,04,700. Step 9: Compute numerator: 19,446.50 * 100 = 19,44,650. Step 10: Divide: 19,44,650 / 2,04,700 = 9.5. Step 11: Thus R = 9.5 percent per annum.

Verification / Alternative check:
Check the interest on each loan with R = 9.5 percent. SI1 = (8,200 * 9.5 * 16) / 100. First 8,200 * 9.5 = 77,900, then multiply by 16 to get 12,46,400, divide by 100 to get Rs 12,464. SI2 = (4,900 * 9.5 * 15) / 100. Here 4,900 * 9.5 = 46,550, multiply by 15 to get 6,98,250 and divide by 100 to get Rs 6,982.50. Total interest = 12,464 + 6,982.50 = Rs 19,446.50, which matches the given total.

Why Other Options Are Wrong:
At 10 percent, the total interest would be higher than 19,446.50. At 10.5 percent or 11 percent, the total interest becomes even larger. Only the rate of 9.5 percent yields a combined interest equal to Rs 19,446.50, so the other options do not fit the total interest condition.

Common Pitfalls:
Some learners incorrectly average the rates or times instead of using the weighted sum P1 * T1 + P2 * T2. Others may forget to include one of the loans in the total interest calculation. Treating the total interest expression systematically and solving for R ensures an accurate answer.

Final Answer:
The simple interest rate per annum is 9.5% per year.