A moneylender lends Rs 400 to one person for 3 years and Rs 500 to another person for 4 years at the same annual rate of simple interest; if the total interest received from both loans together is Rs 160, what is the rate of simple interest per annum?

Introduction / Context:
This problem involves two small loans made at the same simple interest rate but with different principals and durations. The total interest earned from both loans is known, and the objective is to find the common annual interest rate. It is another example of combining multiple simple interest calculations into one equation with a single unknown rate.

Given Data / Assumptions:
- Loan 1: principal P1 = Rs 400, time T1 = 3 years. - Loan 2: principal P2 = Rs 500, time T2 = 4 years. - Rate of simple interest R percent per annum is the same for both loans. - Total simple interest from both loans SI total = Rs 160. - Interest is calculated using the simple interest formula, without compounding.

Concept / Approach:
Simple interest on each loan is SI1 = (P1 * R * T1) / 100 and SI2 = (P2 * R * T2) / 100. The sum SI1 + SI2 equals the total interest of Rs 160. By factoring R and using SI total = (R / 100) * (P1 * T1 + P2 * T2), we can solve for R. This method is efficient because it reduces the two separate equations to one combined expression.

Step-by-Step Solution:
Step 1: Write SI total = (P1 * R * T1) / 100 + (P2 * R * T2) / 100. Step 2: Factor R / 100 to get SI total = (R / 100) * (P1 * T1 + P2 * T2). Step 3: Compute P1 * T1 = 400 * 3 = 1,200. Step 4: Compute P2 * T2 = 500 * 4 = 2,000. Step 5: Sum them: P1 * T1 + P2 * T2 = 1,200 + 2,000 = 3,200. Step 6: Use SI total = 160, so 160 = (R / 100) * 3,200. Step 7: Rearrange to solve for R: R = (160 * 100) / 3,200. Step 8: Compute numerator: 160 * 100 = 16,000. Step 9: Divide: 16,000 / 3,200 = 5. Step 10: Therefore, R = 5 percent per annum.

Verification / Alternative check:
Verify by computing interest on each loan at 5 percent. For the Rs 400 loan over 3 years, SI1 = (400 * 5 * 3) / 100 = (400 * 15) / 100 = 6,000 / 100 = Rs 60. For the Rs 500 loan over 4 years, SI2 = (500 * 5 * 4) / 100 = (500 * 20) / 100 = 10,000 / 100 = Rs 100. Total interest SI1 + SI2 = 60 + 100 = Rs 160, which matches the given total.

Why Other Options Are Wrong:
At 7 percent, total interest would be larger than Rs 160. At 9 percent or 10 percent, the interest amounts would be even higher. None of these match the total interest of Rs 160. Only 5 percent per annum produces exactly Rs 160 in combined simple interest.

Common Pitfalls:
A common mistake is to average the time periods or principals instead of using the weighted sum P1 * T1 + P2 * T2. Some learners also forget to divide by 100 when handling percentages. Writing the combined equation clearly and solving step by step avoids these errors.

Final Answer:
The simple interest rate per annum is 5% per year.