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

Introduction:
This question asks for the annual compound interest rate that grows a principal of Rs. 20,000 to Rs. 25,088 in 2 years with annual compounding. It is a classic reverse compound interest calculation, similar to finding the rate from a known amount and principal.

Given Data / Assumptions:

• Principal (P) = Rs. 20,000.
• Final amount (A) after 2 years = Rs. 25,088.
• Time (t) = 2 years.
• Compounding frequency = annually.

Concept / Approach:
The fundamental compound interest amount formula is:
A = P * (1 + r/100)^tWe know P, A, and t, and need to solve for r. Rearranging the equation for t = 2 years allows us to take a square root to find 1 + r/100.

Step-by-Step Solution:
Step 1: Write the equation: 25,088 = 20,000 * (1 + r/100)^2.Step 2: Divide both sides by 20,000: (1 + r/100)^2 = 25,088 / 20,000.Step 3: Compute the ratio: 25,088 / 20,000 = 1.2544.Step 4: So (1 + r/100)^2 = 1.2544.Step 5: Take the square root of both sides: 1 + r/100 = sqrt(1.2544).Step 6: Evaluate the square root: sqrt(1.2544) = 1.12.Step 7: Therefore, 1 + r/100 = 1.12 which implies r/100 = 0.12.Step 8: Multiply by 100 to get r = 12%.Thus the required compound interest rate is 12 percent per annum.

Verification / Alternative check:
Check by calculating the amount at 12% for 2 years. Amount after first year: 20,000 * 1.12 = 22,400. Amount after second year: 22,400 * 1.12 = 25,088. The computed final amount matches the given figure exactly, confirming that 12 percent is correct.

Why Other Options Are Wrong:
At 8 percent or 10 percent, the amount after 2 years would be less than 25,088 because their squared growth factors are smaller than 1.2544. At 16 percent or 24 percent, the growth factors are much larger than 1.2544, leading to amounts significantly higher than 25,088. Only a 12 percent annual compound rate fits the data.

Common Pitfalls:
Some learners attempt to use simple interest methods, averaging yearly increases, which do not account for compounding. Others forget to take the square root when solving for a squared factor. It is crucial to manipulate the compound interest formula correctly: divide by principal, then take the appropriate root, and only then interpret the result as 1 + r/100.

Final Answer:
The sum of Rs. 20,000 will amount to Rs. 25,088 in 2 years at an annual compound interest rate of 12 percent.