A sum of Rs. 2600 is invested at two different rates of simple interest. The difference between the simple interest amounts earned after 4 years at these two rates is Rs. 402.80. What is the difference between the two rates of interest (in percentage points)?

Introduction / Context:
This problem asks you to determine how far apart two simple interest rates are, given the common principal, the common time, and the difference in the amounts of interest earned under those rates.

Given Data / Assumptions:

• Principal (P) = Rs. 2600.
• Time (T) = 4 years.
• Difference in simple interest after 4 years at two different rates = Rs. 402.80.
• Let the two rates be R1% and R2%, with difference (R1 - R2) in percentage points.
• Interest is simple interest in both cases.

Concept / Approach:
For simple interest, SI = (P * R * T) / 100. If the same principal and time are used, the difference between the interest amounts from two rates is proportional to the difference between the rates. Thus, Difference in interest = (P * (R1 - R2) * T) / 100. From this equation, you can solve for the difference in rates directly.

Step-by-Step Solution:
Step 1: Let the difference in rates be d = R1 - R2. Step 2: Use the formula for difference in simple interest: Difference = (P * d * T) / 100. Step 3: Substitute the known values: 402.80 = (2600 * d * 4) / 100. Step 4: Simplify the denominator: (2600 * 4) / 100 = 104. Step 5: So 402.80 = 104 * d. Step 6: Solve for d: d = 402.80 / 104. Step 7: Compute 402.80 / 104 = 3.873076923..., which rounds to approximately 3.87 percentage points.

Verification / Alternative check:
To verify, you can check that a difference of about 3.87% on Rs. 2600 over 4 years yields about Rs. 402.80 in extra interest. Difference in interest = (2600 * 3.87 * 4) / 100 = 2600 * 0.1548 = approximately Rs. 402.48, and allowing for slight rounding, this matches the given Rs. 402.80 closely, confirming the choice.

Why Other Options Are Wrong:
1.58%, 2.63%, and 4.02% lead to differences in interest far from Rs. 402.80 when plugged back into the formula, so they do not satisfy the condition.

Common Pitfalls:
One common mistake is to treat Rs. 402.80 as the interest from a single rate, instead of the difference between two interests. Another error is failing to apply the time factor correctly, forgetting to multiply by 4 years when using the formula for simple interest. Keeping track of P, T, and the difference in rates clearly avoids confusion.

Final Answer:
The difference between the two rates of simple interest is approximately 3.87%.