On a certain sum of money, the simple interest for 2 years is Rs. 660, while the compound interest for the same period and at the same annual rate is Rs. 696.30. What is the rate of interest per annum?

Introduction / Context:
This problem compares simple and compound interest for the same principal, time, and rate, over a 2 year period. The difference between the two interest amounts helps us determine the rate of interest per annum, without directly knowing the principal.

Given Data / Assumptions:

• Simple interest for 2 years, SI2 = Rs. 660.
• Compound interest for 2 years, CI2 = Rs. 696.30.
• Time period = 2 years.
• The principal P and rate r% per annum are the same for both cases.
• Interest in the compound case is compounded annually.

Concept / Approach:

For 2 years, simple interest is SI2 = P * r * 2 / 100. Compound interest over 2 years is CI2 = P * [(1 + r / 100)^2 - 1]. The difference D = CI2 - SI2 has a useful simplified form: D = P * r^2 / 100^2. With SI2 and CI2 known, we can first find D, then express P * r and P * r^2 and finally solve for r using algebra.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

If r = 10% or 12% or 13%, the difference CI2 - SI2 would not equal Rs. 36.30. Each of those rates leads to a different algebraic relationship between P * r and P * r^2. Only r = 11% satisfies both the simple and compound interest conditions at the same time.

Common Pitfalls:

A common error is to try to guess the rate by trial instead of using the compact formula for the difference between compound and simple interest. Another mistake is to miscalculate the difference or to forget to multiply by 10000 when using the P * r^2 relation, which leads to an incorrect rate.

Final Answer:

The rate of interest is 11% per annum.