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

Introduction / Context:
This question provides both the simple interest (SI) and compound interest (CI) for 2 years on the same principal and asks for the annual rate of interest. By combining the basic SI formula with the standard CI–SI difference relationship, we can find the rate without ever knowing the principal explicitly. This is a neat algebraic question that combines two closely related interest concepts.

Given Data / Assumptions:

• Simple interest SI over 2 years = Rs. 3000.
• Compound interest CI over 2 years, compounded annually = Rs. 3225.
• Principal P is the same in both cases.
• Rate of interest r percent per annum is the same in both calculations.
• Time t = 2 years.

Concept / Approach:
Under simple interest: SI = (P * r * t) / 100. For t = 2 years, SI = (P * r * 2) / 100. Under compound interest over 2 years, the difference between CI and SI is D₂ = P * (r/100)^2. We know SI and CI, so D₂ = CI − SI = 3225 − 3000 = 225. Using SI, we find P * r. Using the difference formula, we find P * r^2. Then, dividing (P * r^2) by (P * r) gives r directly.

Step-by-Step Solution:
Step 1: Compute the difference between CI and SI for 2 years: D₂ = CI − SI = 3225 − 3000 = 225. Step 2: From simple interest, SI = (P * r * 2) / 100 = 3000. Step 3: Rearranging gives P * r = (3000 * 100) / 2 = 150000. Step 4: From the difference formula, D₂ = P * (r/100)^2 = 225. Step 5: Multiply both sides by 10000 to eliminate the denominator: P * r^2 = 225 * 10000 = 2250000. Step 6: We now have P * r = 150000 and P * r^2 = 2250000. Step 7: Divide P * r^2 by P * r to get r: (P * r^2) / (P * r) = r = 2250000 / 150000. Step 8: Compute 2250000 / 150000 = 15. Step 9: Hence, the annual rate of interest is 15 percent per annum.

Verification / Alternative check:
We can verify by picking P from the equation P * r = 150000. With r = 15, P = 150000 / 15 = 10000. For simple interest: SI = (10000 * 15 * 2) / 100 = 3000, which matches the given SI. For compound interest: amount A = 10000 * (1.15)^2. Compute (1.15)^2 = 1.3225, so A = 13225. CI = A − P = 13225 − 10000 = 3225, which matches the given CI. This confirms that r = 15 percent is correct.

Why Other Options Are Wrong:
At 7.5 percent, the SI and CI values would be much smaller and their difference would not match 225. At 10 percent, the difference CI − SI for 2 years is P * (0.10)^2 = 0.01P, which would not align with SI = 3000. Higher rates such as 22.5 percent or 30 percent would produce much larger differences and inconsistent SI values. Only 15 percent satisfies both the simple interest and compound interest conditions simultaneously.

Common Pitfalls:
Some learners try to solve directly from the CI formula without leveraging the neat ratio trick of using P * r and P * r^2. Others confuse the formulas for CI and SI or forget that the difference formula D₂ = P * (r/100)^2 applies for 2 years. Another frequent pitfall is mixing up the units when moving between r and r/100. Following a structured, algebraic approach helps avoid these errors.

Final Answer:
The annual rate of interest that makes the simple interest Rs. 3000 and the compound interest Rs. 3225 in 2 years on the same principal is 15 percent per annum.