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Find the compound interest on Rs. 16,000 at 20% per annum for 9 months, when interest is compounded quarterly.

Difficulty: Medium

Correct Answer: Rs. 2522

Explanation:


Introduction / Context:
This question involves compound interest over a fractional year with quarterly compounding. We must correctly identify the periodic rate and the number of compounding periods for 9 months, then apply the standard compound interest formula to find the interest amount.


Given Data / Assumptions:

  • Principal P = Rs. 16,000.
  • Annual rate of interest r = 20% per annum.
  • Time period = 9 months.
  • Interest is compounded quarterly, that is, every 3 months.


Concept / Approach:

When interest is compounded quarterly, the annual rate is divided by 4 and the time in years is converted into the equivalent number of quarters. For each quarter, the amount is multiplied by (1 + r / 4 / 100). The general formula is A = P * (1 + r / (4 * 100))^(4t). Once we have the total amount A for 9 months, the compound interest is simply A minus the principal.


Step-by-Step Solution:

Step 1: Annual rate r = 20%, so quarterly rate = 20 / 4 = 5% per quarter. Step 2: Time = 9 months. Each quarter is 3 months, so number of quarters n = 9 / 3 = 3. Step 3: Amount A after 3 quarters is A = 16000 * (1 + 5 / 100)^3. Step 4: Compute 1 + 5 / 100 = 1.05. Step 5: Compute (1.05)^2 = 1.1025 and then (1.05)^3 = 1.1025 * 1.05 = 1.157625. Step 6: Therefore A = 16000 * 1.157625 = 18522. Step 7: Compound interest CI = A - P = 18522 - 16000 = 2522.


Verification / Alternative check:

An alternative is to compute the interest accumulated each quarter: first interest is 5% of 16000, second interest is 5% of the new amount after first quarter, and so on. Performing those three successive steps also leads to the same final amount of Rs. 18522 and interest of Rs. 2522, confirming the result.


Why Other Options Are Wrong:

Rs. 2422, Rs. 2622, and Rs. 2722 correspond to slightly different approximations or to using simple interest or an incorrect number of compounding periods. Only Rs. 2522 matches the precise calculation for quarterly compounding over 9 months at 20% per annum.


Common Pitfalls:

A very common mistake is to treat 9 months as 0.75 years and use annual compounding directly instead of finding the number of quarters. Another error is to forget that the rate per period must be 20% divided by 4, not 20% itself. Careful handling of time and rate conversions is crucial in compound interest problems with non annual compounding.


Final Answer:

The compound interest is Rs. 2522.

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