A person lends Rs. 5000 to B for 2 years and Rs. 3000 to C for 4 years at the same annual rate of simple interest, and receives a total of Rs. 2200 as interest from both borrowers together. What is the rate of interest per annum?

Introduction / Context:
This question involves lending different amounts of money to two different people at the same rate of simple interest for different time periods. The combined interest from both loans is given, and you must determine the common annual rate of interest.

Given Data / Assumptions:

• Amount lent to B (P1) = Rs. 5000 for 2 years.
• Amount lent to C (P2) = Rs. 3000 for 4 years.
• The rate of simple interest per annum is the same for both loans and is denoted by R%.
• Total simple interest from both borrowers = Rs. 2200.
• All interest is calculated using simple interest formula with time measured in years.

Concept / Approach:
Simple interest is calculated using the formula SI = (P * R * T) / 100. Compute the interest from each loan in terms of R, then add them and equate the sum to the given total interest of Rs. 2200. This leads to a simple linear equation in R that can be solved directly.

Step-by-Step Solution:
Step 1: Let R% be the common annual rate of simple interest. Step 2: Interest from B, I1 = (P1 * R * T1) / 100 = (5000 * R * 2) / 100. Step 3: Simplify I1: (5000 * 2) / 100 = 100, so I1 = 100 * R. Step 4: Interest from C, I2 = (P2 * R * T2) / 100 = (3000 * R * 4) / 100. Step 5: Simplify I2: (3000 * 4) / 100 = 120, so I2 = 120 * R. Step 6: Total interest I = I1 + I2 = 100 * R + 120 * R = 220 * R. Step 7: Given that total interest is Rs. 2200, so 220 * R = 2200. Step 8: Solve for R: R = 2200 / 220 = 10% per annum.

Verification / Alternative check:
Check the correctness by computing each interest with R = 10%. For B: I1 = (5000 * 10 * 2) / 100 = Rs. 1000. For C: I2 = (3000 * 10 * 4) / 100 = Rs. 1200. Total interest = 1000 + 1200 = Rs. 2200, which matches the given value, confirming that the rate is 10% per annum.

Why Other Options Are Wrong:
5% would give I1 = 500 and I2 = 600, total only Rs. 1100, which is too low. 7% would give I1 = 700 and I2 = 840, total Rs. 1540, still less than 2200. 7.5% would give I1 = 750 and I2 = 900, total Rs. 1650, which is not equal to the required Rs. 2200.

Common Pitfalls:
Many learners mistakenly divide the total interest by the total principal, ignoring the different time periods. Others forget to separately compute the interest contributed by each loan. It is essential to correctly apply the simple interest formula to each loan and then add the results.

Final Answer:
The correct rate of simple interest per annum is 10%.