On a certain sum of money, the simple interest for 2 years is Rs 660, while the compound interest for the same sum and period is Rs 696.30, the rate of interest being the same in both cases. Find the rate of interest per annum.

Introduction:
Here you are given both simple interest and compound interest for the same principal and the same time period. This information allows you to determine the annual rate of interest by forming equations and solving for the unknown rate, which is a common type of aptitude question.

Given Data / Assumptions:

• Simple interest SI2 for 2 years = Rs 660.
• Compound interest CI2 for 2 years = Rs 696.30.
• Same principal P and rate r for both types of interest.
• Interest under compound interest is compounded annually.
• Time t = 2 years.

Concept / Approach:
Simple interest is SI = P * r * t / 100, which yields one relation between P and r. Compound interest for 2 years is CI2 = P * (1 + r/100)^2 − P. Knowing both SI and CI gives two equations in P and r. Solving them simultaneously yields the rate r. This is an algebraic manipulation exercise combined with interest formulas.

Step-by-Step Solution:
From simple interest: SI2 = P * r * 2 / 100 = 660So P * r / 100 = 330. Call this equation (1).From compound interest: CI2 = P * (1 + r/100)^2 − P = 696.30Let x = r/100. Then CI2 = P * [(1 + x)^2 − 1] = P * (2x + x^2) = 696.30But from (1), P * x = 330.Thus P * (2x + x^2) = 2 * P * x + P * x^2 = 2 * 330 + P * x^2 = 696.30So 660 + P * x^2 = 696.30, giving P * x^2 = 36.30Now divide P * x^2 by P * x: (P * x^2) / (P * x) = x = 36.30 / 330 = 0.11Hence r/100 = 0.11, so r = 11% per annum.

Verification / Alternative Check:
Using r = 11%, we can find P from P * r / 100 = 330. So P * 0.11 = 330, which gives P = 3000. Simple interest for 2 years at 11%: 3000 * 11 * 2 / 100 = 660 (checks). Compound amount: 3000 * (1.11)^2 = 3000 * 1.2321 = 3696.30. CI2 = 3696.30 − 3000 = 696.30 (checks again).

Why Other Options Are Wrong:
10%, 12% and 13%: Each of these rates, when used with an appropriate principal, does not simultaneously produce SI2 = 660 and CI2 = 696.30.9%: Gives even smaller values of CI and cannot match the given compound interest.

Common Pitfalls:
Common mistakes include treating the difference CI − SI as directly equal to P * (r/100)^2 without verifying the assumptions or mismanaging the algebra when solving for r. Working step by step with clear substitutions helps keep the work accurate.

Final Answer:
The rate of interest per annum is 11%.