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 interest per annum?

Introduction / Context:
This question is a simpler variant of multiple loan problems under simple interest. Two loans are given at the same rate but different principals and time periods, and you must find the common annual interest rate from the total interest earned.

Given Data / Assumptions:

• Loan 1: Principal P1 = Rs 400, Time T1 = 3 years.
• Loan 2: Principal P2 = Rs 500, Time T2 = 4 years.
• Total simple interest from both loans = Rs 160.
• Common annual simple interest rate, R%, is unknown.

Concept / Approach:
Simple interest on each loan is given by:
SI1 = (P1 * R * T1) / 100SI2 = (P2 * R * T2) / 100Since the rate is the same, we can add these to express the total interest in terms of R:
Total SI = (R / 100) * (P1 * T1 + P2 * T2)We then solve for R using the given total interest.

Step-by-Step Solution:
Step 1: Compute P1 * T1.P1 * T1 = 400 * 3 = 1,200Step 2: Compute P2 * T2.P2 * T2 = 500 * 4 = 2,000Step 3: Add them for the combined factor.P1 * T1 + P2 * T2 = 1,200 + 2,000 = 3,200Step 4: Use total simple interest expression.Total SI = (R / 100) * 3,200 = 160Step 5: Rearrange to find R.(R / 100) * 3,200 = 160R * 32 = 160R = 160 / 32 = 5Thus, the annual simple interest rate is 5%.

Verification / Alternative check:
Compute interest for each loan separately using R = 5%. For the first loan: SI1 = (400 * 5 * 3) / 100 = 60. For the second loan: SI2 = (500 * 5 * 4) / 100 = 100. Total SI = 60 + 100 = 160, matching the given total interest, so the rate is confirmed as 5%.

Why Other Options Are Wrong:

• 7%, 9%, and 10% all produce total interest values larger than Rs 160 when applied to the combined factor of 3,200.
• Only 5% gives exactly Rs 160 as total simple interest.

Common Pitfalls:
Some students mistakenly average the two principals or two times instead of using the correct product method. Others may miscalculate the combined factor P1 * T1 + P2 * T2. Remember that under a common rate, adding interests is equivalent to applying the rate to the sum of the principal time products.

Final Answer:
The rate of simple interest charged by the moneylender is 5% per annum.