Find the annual rate from CI − SI over 2 years: The difference between compound interest and simple interest on ₹ 5000 for 2 years is ₹ 72. What is the annual rate of interest (compounded yearly)?

Introduction / Context:
For 2 years at a fixed rate with annual compounding, the excess of CI over SI equals P * r^2, where r is expressed as a decimal. This comes from the second-year “interest on interest.”

Given Data / Assumptions:

• P = ₹ 5000
• CI − SI = ₹ 72
• t = 2 years, annual compounding

Concept / Approach:
Use CI − SI = P * r^2. Solve for r: r = sqrt((CI − SI)/P). Convert decimal r to percent by multiplying by 100.

Step-by-Step Solution:
P * r^2 = 72 → r^2 = 72 / 5000 = 0.0144r = sqrt(0.0144) = 0.12Annual rate = 0.12 × 100% = 12%

Verification / Alternative check:
At 12%, CI − SI should be 5000 * 0.12^2 = 5000 * 0.0144 = ₹ 72, exactly as given.

Why Other Options Are Wrong:
10%, 8%, 6%, or 9% do not satisfy P * r^2 = 72 for P = 5000.

Common Pitfalls:
Using 2 * r or r * t in the difference; the 2-year CI − SI shortcut works only with P * r^2 for annual compounding at a fixed r.

Final Answer:
12%