At what annual rate of compound interest (in percent per annum) will a sum of Rs. 15,000 grow to Rs. 18,150 in 2 years, if interest is compounded once every year?

Introduction / Context:
This problem asks for the annual compound interest rate that converts a known principal into a known amount over a fixed time. It is a classic reverse compound interest calculation, which is very useful in determining the rate of return on an investment or the implied rate of a loan.

Given Data / Assumptions:

• Principal P = Rs. 15,000.
• Amount after 2 years A = Rs. 18,150.
• Time period n = 2 years.
• Interest is compounded annually.
• We must find the annual rate of interest r (in percent per annum).

Concept / Approach:
Under annual compounding, the amount A is related to the principal P by the formula A = P * (1 + r)^n, where r is the annual interest rate in decimal form and n is the number of years. To find r, we rearrange the formula to (1 + r)^n = A / P and then take the nth root. In this case, because n is 2, we can take a square root and then subtract 1 to find r.

Step-by-Step Solution:
Step 1: Write the compound interest formula A = P * (1 + r)^n.Step 2: Substitute values: 18,150 = 15,000 * (1 + r)^2.Step 3: Divide both sides by 15,000 to isolate (1 + r)^2.Step 4: (1 + r)^2 = 18,150 / 15,000.Step 5: Compute 18,150 / 15,000 = 1.21.Step 6: Take the square root: 1 + r = square root of 1.21 = 1.1, so r = 1.1 - 1 = 0.1, which is 10% per annum.

Verification / Alternative check:
We can check the rate by applying it back to the principal. At 10% per annum compounded annually, after one year the amount is 15,000 * 1.1 = 16,500. After the second year, it becomes 16,500 * 1.1 = 18,150. This exactly matches the given amount, which confirms that the rate of 10% per annum is correct.

Why Other Options Are Wrong:

• 11: At 11% per annum, the two year factor would be (1.11)^2 = 1.2321, leading to an amount of about Rs. 18,481.5, which is larger than Rs. 18,150.
• 9: At 9% per annum, the two year factor would be (1.09)^2 = 1.1881, leading to an amount of about Rs. 17,821.5, which is smaller than Rs. 18,150.
• 12: At 12% per annum, the factor is (1.12)^2 = 1.2544, giving an amount around Rs. 18,816, again not equal to Rs. 18,150.

Common Pitfalls:
Some students mistakenly use the simple interest formula rather than the compound interest formula, which would give a different rate. Others divide the total increase by 2 and conclude that the rate is simply the average yearly gain, ignoring compounding. It is also easy to forget to take the square root when n = 2 and instead divide by 2, which is incorrect in a compound interest context. Always remember to use the appropriate power and its inverse when solving for the rate in compound situations.

Final Answer:
The annual rate of compound interest is 10% per annum.