On a certain principal, the simple interest for 2 years is Rs 3000 and the compound interest for the same period with annual compounding is Rs 3225. What is the annual rate of interest, in percent?

Introduction / Context:
This question combines simple interest and compound interest information for the same principal and time span to determine the annual rate. The simple interest and compound interest amounts for 2 years are given separately. Using the relationship between them over 2 years, we can form equations that lead to the rate. This is a classic style of aptitude question.

Given Data / Assumptions:
- Simple interest SI over 2 years is Rs 3000.
- Compound interest CI over 2 years (annual compounding) is Rs 3225.
- The principal P is the same in both cases.
- We must determine the annual rate of interest R in percent.

Concept / Approach:
First, we use the simple interest formula to relate P, R, and SI:
SI = (P * R * T) / 100Here T = 2 years, so SI = (P * R * 2) / 100. From this we get a relation between P and R. Second, we use the known relationship for the difference between CI and SI for 2 years:
Difference D = CI − SI = P * (R/100)^2We can compute D from the given CI and SI, then combine these two equations to solve for R.

Step-by-Step Solution:
Let R be the annual rate in percent and P be the principal.Given SI for 2 years: SI = 3000.Use SI formula: 3000 = (P * R * 2) / 100.Therefore, P * R = 3000 * 100 / 2 = 150000.Given CI for 2 years: CI = 3225.Difference D = CI − SI = 3225 − 3000 = 225.For 2 years, D = P * (R/100)^2.So, 225 = P * (R/100)^2.Write (R/100)^2 as R^2 / 10000, so 225 = P * R^2 / 10000.From P * R = 150000, multiply both sides by R to get P * R^2 = 150000 * R.Thus, 225 * 10000 = 150000 * R.Compute left side: 225 * 10000 = 2250000.So, 2250000 = 150000 * R, which gives R = 2250000 / 150000 = 15.Therefore, the annual rate of interest is 15 percent.

Verification / Alternative check:
We can check by finding P. From P * R = 150000 and R = 15, we get P = 150000 / 15 = 10000. Simple interest for 2 years at 15 %: SI = (10000 * 15 * 2) / 100 = 3000, which matches the given SI. For compound interest, amount A = 10000 * (1.15)^2 = 10000 * 1.3225 = 13225. Then CI = 13225 − 10000 = 3225, which matches the given CI. This confirms the rate is correct.

Why Other Options Are Wrong:
At 7.5 percent, both SI and CI would be much smaller and could not reach 3000 and 3225. A rate of 30 percent would produce SI and CI values far larger than given. 22.5 percent would also lead to larger interests. A rate of 10 percent would not produce a difference of 225 between CI and SI. Only 15 percent satisfies both the given SI and CI simultaneously.

Common Pitfalls:
Some candidates attempt to solve directly with the compound interest formula without using the helpful difference relation. Others forget to use the time as 2 years when applying the simple interest formula. Mismanaging algebra, especially with R^2 and the factor 10000, also causes mistakes. Carefully setting up and solving the two equations in P and R avoids these problems.

Final Answer:
The annual rate of interest is 15 percent.