The difference between the compound interest and the simple interest on a certain sum at 12% per annum for 2 years is Rs. 90. What will be the total amount (principal plus interest) at the end of 3 years at the same rate of compound interest?

Introduction / Context:
This question combines both simple interest and compound interest ideas and asks you to use the known difference between them to deduce the principal. After finding the principal, you must compute the compound amount for a longer period. It is a classic multi-step aptitude question that checks your conceptual understanding of interest formulas.

Given Data / Assumptions:

• Rate of interest r = 12% per annum
• Time T1 = 2 years
• Difference between compound interest and simple interest for 2 years = Rs. 90
• Same principal P in both cases
• We need the amount at compound interest after T2 = 3 years

Concept / Approach:
For the same principal P, rate r and time 2 years, the difference between compound interest and simple interest is:
Difference = P * (r / 100)^2This comes from expanding the compound interest expression for 2 years. Once P is found, the amount after 3 years at compound interest is:
A3 = P * (1 + r / 100)^3

Step-by-Step Solution:
Step 1: Use the difference formula for 2 years.Difference = P * (r / 100)^290 = P * (12 / 100)^2 = P * (144 / 10000)P = 90 * 10000 / 144 = 6250Step 2: Use compound interest formula for 3 years.A3 = 6250 * (1 + 12 / 100)^3A3 = 6250 * (1.12)^3(1.12)^3 = 1.404928A3 = 6250 * 1.404928 = Rs. 8780.80

Verification / Alternative check:
You can double-check the principal by computing simple and compound interest for 2 years. Simple interest for 2 years:
SI2 = P * r * 2 / 100 = 6250 * 12 * 2 / 100 = Rs. 1500Compound amount after 2 years:
A2 = 6250 * (1.12)^2 = 6250 * 1.2544 = 7840CI2 = A2 - P = 7840 - 6250 = Rs. 1590Difference = CI2 - SI2 = 1590 - 1500 = Rs. 90This matches the given difference.

Why Other Options Are Wrong:
Rs. 8560, Rs. 8673, Rs. 8746, Rs. 8620: These values correspond to using incorrect principal, rate factor, or computing only 2 years instead of 3 years of compounding.

Common Pitfalls:
One common error is to treat Rs. 90 as the interest for a third year, which is incorrect. Another mistake is forgetting that the difference formula applies specifically for 2 years and involves the square of the rate fraction. Some students also wrongly use simple interest formulas for 3 years instead of compound interest. Keeping track of formulas and time periods avoids these errors.

Final Answer:
The value of the amount at the end of 3 years at 12% compound interest is Rs. 8780.80.