What is the compound interest earned on Rs 80,000 invested for 1 year at 40% per annum, when the interest is compounded quarterly (every 3 months)?

Introduction / Context:
Here we apply the standard compound interest formula with a nominal annual rate and quarterly compounding. The principal is Rs 80,000, the nominal rate is 40% per annum, and the compounding is done four times a year. The question asks for the compound interest earned in 1 year, not the final amount, so after finding the amount we must subtract the principal to obtain the interest. This is a straightforward calculation that checks a student's comfort with handling non-annual compounding.

Given Data / Assumptions:

• Principal P = Rs 80,000.
• Nominal annual rate r = 40% per annum.
• Compounding frequency: quarterly (4 times per year).
• Time t = 1 year.
• We must find the compound interest (CI) earned in this period.

Concept / Approach:
For compound interest with nominal rate r and n compounding periods per year, the amount after t years is A = P * (1 + r/(100n))^(n*t). Here, r = 40, n = 4 and t = 1. This gives a periodic rate of 40/4 = 10% per quarter, and there are 4 quarters in 1 year. Once we compute A, the compound interest is CI = A - P. Because the rate is high, the growth over the year is significant, and accurate multiplication is important to avoid arithmetic mistakes.

Step-by-Step Solution:
Periodic rate per quarter = 40% / 4 = 10% = 0.10. Number of compounding periods in 1 year = 4 * 1 = 4. Amount A = P * (1 + 0.10)^4 = 80,000 * (1.10)^4. Compute (1.10)^2 = 1.21; then (1.10)^4 = (1.21)^2 = 1.4641. So A = 80,000 * 1.4641 = 117,128 (approximately). Compound interest CI = A - P = 117,128 - 80,000 = 37,128. Thus, the interest earned is Rs 37,128.

Verification / Alternative check:
We can also build the amount step by step each quarter. After 1st quarter: 80,000 * 1.10 = 88,000. After 2nd quarter: 88,000 * 1.10 = 96,800. After 3rd quarter: 96,800 * 1.10 = 106,480. After 4th quarter: 106,480 * 1.10 = 117,128. Subtracting the original principal 80,000 from 117,128 again yields 37,128. This stepwise approach confirms the result without relying on the shortcut of squaring 1.21.

Why Other Options Are Wrong:
Rs 28,317 and Rs 18,732 are both significantly lower than the correct interest and would correspond to a much lower effective annual growth than 40% nominal with quarterly compounding. Rs 21,387 is still too small, and Rs 40,512 is too large for one year on Rs 80,000 at the given rate and frequency. Only Rs 37,128 matches the arithmetic of the compound interest formula.

Common Pitfalls:
Some students mistakenly use simple interest, computing 80,000 * 40/100 = 32,000, which ignores the effect of quarterly compounding. Others incorrectly use 40% directly in the formula as if it were the quarterly rate, which leads to an exaggerated result. Always divide the nominal rate by the number of compounding periods, and be careful to raise the expression to the correct power n * t.

Final Answer:
The compound interest earned in 1 year is Rs 37,128.