In compound interest calculations, what is the total compound interest earned on an investment of Rs 80,000 at a rate of 40% per annum for 1 full year, if the interest is compounded quarterly (four times in a year)?

Introduction:
In this compound interest aptitude question, we are given a principal amount, a relatively high annual interest rate, and the information that interest is compounded quarterly. The goal is to find the total compound interest earned in one year on Rs 80,000 at 40% per annum with quarterly compounding. Understanding how to convert annual rates to periodic rates and apply the compound interest formula is essential for bank exams and placement tests.

Given Data / Assumptions:

• Principal P = Rs 80,000
• Annual rate of interest r = 40% per annum
• Compounding frequency = quarterly (4 times a year)
• Time t = 1 year
• Interest is compounded at the same rate throughout the year

Concept / Approach:
For compound interest, the amount is calculated using:
A = P * (1 + r_periodic)^(number_of_periods)Here r_periodic is the rate per compounding period. For quarterly compounding, the annual rate is divided by 4, and the number of periods in one year is 4. The compound interest is then:
CI = A - P

Step-by-Step Solution:
Step 1: Convert the annual rate to a quarterly rate.r_annual = 40% = 40 / 100 = 0.40r_quarterly = r_annual / 4 = 0.40 / 4 = 0.10 (10% per quarter)Step 2: Determine the number of compounding periods in one year.n = 4 quarters in 1 yearStep 3: Compute the final amount A.A = 80,000 * (1 + 0.10)^4A = 80,000 * (1.10)^4(1.10)^2 = 1.21 and (1.10)^4 = 1.21 * 1.21 = 1.4641A = 80,000 * 1.4641 = Rs 117,128 (approximately)Step 4: Compute the compound interest.CI = A - P = 117,128 - 80,000 = Rs 37,128

Verification / Alternative check:
We can quickly approximate by noting that 40% simple interest on 80,000 for 1 year would be Rs 32,000. Since quarterly compounding slightly increases the effective rate, the compound interest must be a bit higher than Rs 32,000. The value Rs 37,128 is reasonably higher, and the calculation using the exact formula confirms it.

Why Other Options Are Wrong:

• Rs 28,317: This is lower than simple interest and cannot be correct for compound interest at 40%.
• Rs 18,732: Far too small compared to 40% of 80,000.
• Rs 21,387: Also significantly lower than even simple interest.
• Rs 32,400: Close to simple interest, but ignores the effect of quarterly compounding.

Common Pitfalls:
Candidates often forget to divide the annual rate by the number of compounding periods, or they wrongly use simple interest formula. Another frequent mistake is to multiply by 4 instead of using the power for compounding, which leads to underestimation of the amount.

Final Answer:
The total compound interest earned on Rs 80,000 at 40% per annum compounded quarterly for one year is Rs 37,128.