The compound interest on a certain sum for 2 years at 12.5% per annum is Rs. 510. What is the simple interest on the same sum at the same rate for the same 2 years?

Introduction / Context:
This question connects compound interest and simple interest for the same principal, rate, and time. We are told the compound interest over 2 years and asked to deduce the simple interest over the same period. To do this, we must first find the principal using the compound interest formula and then apply the simple interest formula.

Given Data / Assumptions:

• Compound interest (CI) for 2 years = Rs. 510
• Rate of interest (R) = 12.5% per annum
• Time (T) = 2 years
• Same principal, rate and time for both CI and SI

Concept / Approach:
Let the principal be P. The amount after 2 years at compound interest is:
A = P * (1 + R / 100) ^ 2The compound interest is:
CI = A - PWe are given CI and must find P. After finding P, we compute simple interest for 2 years at the same rate using:
SI = (P * R * T) / 100

Step-by-Step Solution:
Rate R = 12.5% = 1 / 8So 1 + R / 100 = 1 + 1 / 8 = 9 / 8Amount after 2 years A = P * (9 / 8) ^ 2 = P * 81 / 64CI = A - P = P * (81 / 64 - 1) = P * (17 / 64)Given CI = 510, so P * (17 / 64) = 510P = 510 * 64 / 17510 / 17 = 30, so P = 30 * 64 = 1920Thus principal P = Rs. 1,920Now compute simple interest for 2 years at 12.5%SI = (1920 * 12.5 * 2) / 10012.5 * 2 = 25, so SI = (1920 * 25) / 100SI = 1920 * 0.25 = Rs. 480
Verification / Alternative check:
We can quickly recheck CI from this principal: CI = 1920 * 17 / 64. Dividing 1920 by 64 gives 30, so CI = 30 * 17 = Rs. 510, which is exactly the given CI. This confirms that our principal and subsequent SI calculation are consistent.

Why Other Options Are Wrong:
Values like Rs. 580, Rs. 550 or Rs. 470 do not match the simple interest produced by a principal of Rs. 1,920 at 12.5% per annum for 2 years. Only Rs. 480 fits the correct principal found from the compound interest expression.

Common Pitfalls:
Some students mistakenly treat the given Rs. 510 as simple interest directly and try to adjust it, or they use the simple interest formula to back calculate the principal. Others mis-handle the fractional rate 12.5%. It is best to convert it into a fraction like 1 / 8 to simplify calculations.

Final Answer:
The simple interest on the same sum at the same rate for 2 years is Rs. 480.