At what annual rate of compound interest (per annum) will a sum of Rs 20,000 amount to Rs 25,088 in 2 years, if interest is compounded annually?

Introduction / Context:
This question asks you to determine the annual compound interest rate when the principal, final amount, and time period are known. It is another direct application of the compound interest amount formula, where the unknown quantity is the rate of interest per annum.

Given Data / Assumptions:

• Principal P = Rs 20,000.
• Final amount after 2 years A = Rs 25,088.
• Time period n = 2 years.
• Interest is compounded annually at some rate r%.
• No additional investments or withdrawals occur during this period.

Concept / Approach:
Using the compound amount formula A = P * (1 + r/100)^n, we can isolate (1 + r/100)^2 as A / P. Taking the square root lets us find 1 + r/100, from which we can easily solve for r. This procedure is an example of reversing the compound interest formula when the final amount is known.

Step-by-Step Solution:
Step 1: Write the formula: A = P * (1 + r/100)^2. Step 2: Substitute the known values: 25088 = 20000 * (1 + r/100)^2. Step 3: Divide both sides by 20000: (1 + r/100)^2 = 25088 / 20000. Step 4: Compute the ratio: 25088 / 20000 = 1.2544. Step 5: Take the square root of both sides: 1 + r/100 = square root of 1.2544 = 1.12. Step 6: Therefore r/100 = 0.12, which gives r = 12.
Verification / Alternative check:
Check by recomputing A with r = 12%: A = 20000 * (1.12)^2 = 20000 * 1.2544 = 25,088, exactly the given amount, confirming the correctness of the rate.
Why Other Options Are Wrong:
8% would give 20000 * 1.08^2 = 20000 * 1.1664 = 23,328, which is too low. 16% would yield 20000 * 1.16^2 = 20000 * 1.3456 = 26,912, which is too high. 24% is far higher than needed and would overshoot the final amount significantly.
Common Pitfalls:
Some students attempt to use simple interest formulas or divide the total increase evenly over years, which does not account for compounding. Others forget to take the square root when solving a squared factor, leading to a rate that is either doubled or otherwise incorrect.
Final Answer:
The required annual rate of compound interest is 12 percent.