At what annual rate of compound interest will a sum of Rs. 50000 amount to Rs. 73205 in 2 years, assuming annual compounding?

Introduction / Context:
In this question, the initial investment, final amount, and time are given, and we are asked to determine the annual compound interest rate. This is another inverse compound interest problem similar to the previous one, but with different numbers. The goal is to identify the growth factor over two years and then derive the rate that produces this factor under annual compounding.

Given Data / Assumptions:

• Principal P = Rs. 50000.
• Amount after 2 years A = Rs. 73205.
• Time period n = 2 years.
• Interest is compounded annually.
• We must find the annual compound interest rate r in percent.

Concept / Approach:
With annual compounding for two years, the amount is given by A = P * (1 + r) ^ 2. Dividing A by P gives (1 + r) ^ 2 = A / P. We then take the square root to get 1 + r and subtract 1 to isolate r in decimal form. Finally, we multiply by 100 to express r as a percentage. Because the numbers are chosen for a clean square, we expect a simple integer percentage rate as the final answer.

Step-by-Step Solution:
Step 1: Compute the ratio A / P = 73205 / 50000. Step 2: 73205 / 50000 equals 1.4641. Step 3: From the amount formula, (1 + r) ^ 2 = 1.4641. Step 4: Take the square root of 1.4641 to find 1 + r. Step 5: The square root of 1.4641 is 1.21. Step 6: Therefore, 1 + r = 1.21, which implies r = 1.21 - 1 = 0.21. Step 7: Convert r to a percentage: r = 0.21 * 100% = 21%. Step 8: Thus, the required annual rate of compound interest is 21 percent.

Verification / Alternative check:
We can check this by calculating the amount for two years at 21% per annum on 50000. After one year, the amount is 50000 * 1.21 = 60500. After the second year, it becomes 60500 * 1.21 = 73205. This matches the given final amount exactly, confirming that 21% is the correct rate. The neat square 1.21 squared equals 1.4641, which explains why the resulting numbers are so clean and typical of exam style questions.

Why Other Options Are Wrong:
A rate of 19% would give a two year growth factor of (1.19) ^ 2, which is about 1.4161, resulting in an amount around 70805, smaller than 73205. A rate of 17% would yield (1.17) ^ 2, about 1.3689, giving an amount near 68445. At 15%, the factor (1.15) ^ 2 equals 1.3225, resulting in only 66125. The option 23% would be even higher, giving (1.23) ^ 2 around 1.5129, and the amount would exceed 75000. None of these match the required 73205 except the 21% case.

Common Pitfalls:
Some learners mistakenly treat the two year growth ratio 1.4641 as the sum 1 + 2r and solve for r linearly, which would suggest r is around 23.2% in a simple interest mindset. This is incorrect for compound interest, where the rate is linked to the square root of the growth factor. Others may confuse square root with half and incorrectly take half of 1.4641 instead of its square root. It is important to remember that for n years with annual compounding, A / P equals (1 + r) raised to the power n, not multiplied by n.

Final Answer:
The annual compound interest rate that turns Rs. 50000 into Rs. 73205 in 2 years is 21 percent, which corresponds to option A.