The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years is Rs. 405. What is the rate of interest per cent per annum?

Introduction / Context:
This question uses an important shortcut involving the difference between compound interest and simple interest for 2 years on the same principal at the same rate. The difference is given and the principal is known, so we can use a known relation to directly find the rate. This kind of problem is often asked to test familiarity with special formulas and to avoid lengthy calculations.

Given Data / Assumptions:

• Principal P = Rs. 18,000.
• Time n = 2 years.
• Difference between compound interest and simple interest over 2 years is Rs. 405.
• Let the rate of interest be r percent per annum.
• Interest is compounded annually for the compound interest part.

Concept / Approach:
For 2 years on the same principal and rate, the difference between compound interest and simple interest is given by: Difference = P * (r/100)^2 This relation comes from expanding the 2 year compound interest expression and comparing it to the simple interest formula. Given the difference and P, we can solve for r by rearranging the formula and taking a square root.

Step-by-Step Solution:
Given Difference = 405 Formula: Difference = P * (r/100)^2 So 405 = 18000 * (r/100)^2 (r/100)^2 = 405 / 18000 405 / 18000 = 0.0225 Therefore, (r/100)^2 = 0.0225 r/100 = square root of 0.0225 = 0.15 r = 0.15 * 100 = 15 percent Hence, the rate of interest per annum is 15 percent

Verification / Alternative check:
We can verify with actual simple and compound interest calculations. For r = 15 percent and P = 18000: Simple interest for 2 years = P * r * n / 100 = 18000 * 15 * 2 / 100 = 5400 Compound amount after 2 years = 18000 * (1.15)^2 (1.15)^2 = 1.3225, so amount = 18000 * 1.3225 = 23805 Compound interest = 23805 - 18000 = 5805 Difference = 5805 - 5400 = 405 This matches the given difference exactly, confirming that r = 15 percent is correct.

Why Other Options Are Wrong:
Rates like 12 percent, 13 percent, or 14 percent do not produce a difference of 405 when plugged into the formula P * (r/100)^2 with P = 18000. For example, if r = 10 percent, the difference would be much smaller, equal to 18000 * 0.01 = 180. Only r = 15 percent gives the exact difference of 405 and also matches the full interest calculations.

Common Pitfalls:
Students sometimes forget the shortcut and try to compute both simple and compound interest separately from scratch, which is more time consuming and prone to arithmetic mistakes. Another error is to misapply the formula and use P * r * r instead of P * (r/100)^2. Mixing up the sign of the difference or forgetting that compound interest is larger than simple interest over multiple years can also cause confusion. Remembering this special relation for 2 years helps solve such questions quickly and accurately.

Final Answer:
The rate of interest per annum is 15 percent.