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The difference between the simple interest on a certain principal at 10% per annum for 2 years and the compound interest on the same principal for 2 years, compounded half yearly, is Rs. 124.05. What is the principal sum?

Difficulty: Medium

Correct Answer: Rs. 8000

Explanation:


Introduction / Context:
This problem compares simple interest and compound interest on the same principal, time, and nominal rate, but with compounding taking place every 6 months. The key idea is to use known formulas for simple interest and compound interest and then use the given difference to find the unknown principal.


Given Data / Assumptions:

  • Rate of interest = 10% per annum.
  • Time period = 2 years.
  • For simple interest, interest is calculated yearly.
  • For compound interest, interest is compounded every half year, so there are 4 compounding periods in 2 years.
  • Difference between compound interest and simple interest = Rs. 124.05.
  • Principal is the same in both cases and is to be determined.


Concept / Approach:

Simple interest is given by SI = P * r * t / 100. For compound interest with half yearly compounding, we use the periodic rate r/2 and number of periods 2t. The difference CI - SI becomes a small additional amount due to compounding more frequently than once per year. We express the difference in terms of the principal and then solve for the principal.


Step-by-Step Solution:

Step 1: Let principal be P. Step 2: Simple interest for 2 years at 10% per year is SI = P * 10 * 2 / 100 = 0.2P. Step 3: For compound interest, half yearly rate = 10 / 2 = 5% per half year and number of half yearly periods = 2 * 2 = 4. Step 4: Amount under compound interest is A = P * (1 + 0.05)^4. Step 5: Compute (1.05)^2 = 1.1025 and then (1.05)^4 = 1.1025^2 = 1.21550625. Step 6: Compound interest for 2 years is CI = P * (1.21550625 - 1) = P * 0.21550625. Step 7: The difference is CI - SI = P * (0.21550625 - 0.2) = P * 0.01550625. Step 8: Given that CI - SI = 124.05, so P * 0.01550625 = 124.05. Hence P = 124.05 / 0.01550625 = 8000.


Verification / Alternative check:

Check: For P = 8000, SI = 0.2 * 8000 = 1600. CI = 8000 * 0.21550625 = 1724.05. The difference is 1724.05 - 1600 = 124.05, which matches the given value, confirming the principal.


Why Other Options Are Wrong:

Rs. 6000 gives a smaller difference than Rs. 124.05, while Rs. 12000 gives a larger difference. The option None of these is incorrect because a valid principal of Rs. 8000 exactly fits the condition. Therefore only Rs. 8000 satisfies the given data.


Common Pitfalls:

Students often forget to adjust the rate and the number of periods when interest is compounded half yearly or quarterly. Another common mistake is to treat the nominal annual rate directly as the periodic rate. It is also easy to confuse the difference formula and attempt to compute CI and SI numerically without expressing them algebraically in terms of P, which can lead to unnecessary complexity.


Final Answer:

The principal sum is Rs. 8000.

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