At what annual rate of compound interest per annum will a sum of Rs 20,000 grow to Rs 23,328 in 2 years, assuming interest is compounded annually?

Introduction / Context:
In this question, the principal, the amount after a specified time, and the time period are given, and we are asked to find the annual compound interest rate. This is a reverse application of the compound interest formula. Instead of computing the amount from the rate, we must infer the rate from the known principal and amount. This kind of problem is very common in aptitude tests and develops skill in algebraic manipulation of exponential equations.

Given Data / Assumptions:

• Principal (P) = Rs 20,000.
• Amount after 2 years (A) = Rs 23,328.
• Time (T) = 2 years.
• Interest is compounded annually.
• We must find the annual rate of interest R (in percent).

Concept / Approach:
For annual compounding, the compound interest amount formula is:
A = P * (1 + R / 100)^TWe know A, P, and T, so we can solve for (1 + R / 100) and then for R. For T = 2 years, this becomes:
23328 = 20000 * (1 + R / 100)^2Rearranging gives:
(1 + R / 100)^2 = 23328 / 20000Taking the square root yields 1 + R / 100, from which R can be determined.

Step-by-Step Solution:
Step 1: Write the basic equation: 23328 = 20000 * (1 + R / 100)^2.Step 2: Divide both sides by 20000 to isolate the squared factor: (1 + R / 100)^2 = 23328 / 20000.Step 3: Simplify the ratio 23328 / 20000 = 1.1664.Step 4: So (1 + R / 100)^2 = 1.1664.Step 5: Take the positive square root (since 1 + R / 100 must be positive): 1 + R / 100 = sqrt(1.1664).Step 6: The square root of 1.1664 is 1.08 (since 1.08^2 = 1.1664).Step 7: Therefore, 1 + R / 100 = 1.08, so R / 100 = 0.08.Step 8: Multiply by 100 to find R: R = 8.Step 9: The required annual compound interest rate is 8% per annum.

Verification / Alternative check:
We can verify by computing the amount using R = 8%. For 2 years, A = 20000 * (1.08)^2. First 1.08^2 = 1.1664. Then A = 20000 * 1.1664 = 23328, which matches the given amount in the problem. This confirms that our calculated rate of 8% per annum is correct.

Why Other Options Are Wrong:
A rate of 16% would produce (1.16)^2 = 1.3456, giving an amount of 20000 * 1.3456 = 26912, which is too high. A rate of 24% would give (1.24)^2 = 1.5376, and amount 30752, which is much too high. A 12% rate yields (1.12)^2 = 1.2544, giving 25088, again higher than 23328. A 10% rate gives (1.10)^2 = 1.21, amount 24200. None of these match the target amount; only 8% reproduces exactly Rs 23,328.

Common Pitfalls:
Students sometimes treat the increase from 20000 to 23328 as 16.64% over 2 years and then incorrectly divide by 2 to get about 8.32%, forgetting that compound interest is multiplicative, not additive. Others may take the square root incorrectly or use approximate decimal values without checking. Writing out the steps clearly and verifying by substituting back into the original formula helps avoid these issues.

Final Answer:
The annual rate of compound interest that turns Rs 20,000 into Rs 23,328 in 2 years is 8 percent per annum.