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The compound interest on a sum of money for 2 years is Rs. 832 and the simple interest on the same sum for the same period is Rs. 800. What is the difference between compound interest and simple interest on the same sum for 3 years at the same rate of interest?

Difficulty: Hard

Correct Answer: Rs. 98.56

Explanation:


Introduction / Context:
This problem links simple interest and compound interest for the same principal, time, and rate, and then extends the result to a different time period. We are given the compound and simple interest for 2 years and must use that information to find both the rate and the principal. Then we can calculate the new difference between compound and simple interest for 3 years at the same rate.


Given Data / Assumptions:

  • Compound interest for 2 years, CI2 = Rs. 832.
  • Simple interest for 2 years, SI2 = Rs. 800.
  • Time period in both cases = 2 years.
  • Rate and principal are the same for all comparisons.
  • We must find CI3 - SI3 for 3 years.


Concept / Approach:

For 2 years, SI2 = P * r * 2 / 100 and CI2 = P * [(1 + r / 100)^2 - 1]. Their difference CI2 - SI2 has a compact form: CI2 - SI2 = P * r^2 / 100^2. From the two given interest values we can first find r and P. Then we compute SI3 and CI3 for 3 years and finally find CI3 - SI3.


Step-by-Step Solution:

Step 1: Let the principal be P and the annual rate be r%. Step 2: Simple interest for 2 years is SI2 = P * r * 2 / 100 = 800. Step 3: Compound interest for 2 years is CI2 = P * [(1 + r / 100)^2 - 1] = 832. Step 4: The difference for 2 years is CI2 - SI2 = 832 - 800 = 32. Step 5: For 2 years, CI2 - SI2 = P * r^2 / 100^2, so P * r^2 / 10000 = 32. Step 6: Also SI2 = 800 = P * r * 2 / 100, so P * r = 800 * 100 / 2 = 40000. Step 7: From P * r^2 = 32 * 10000 = 320000 and P * r = 40000, divide the first by the second to get r = 320000 / 40000 = 8% per annum. Step 8: Now find P from P * r = 40000: P = 40000 / 8 = Rs. 5000. Step 9: For 3 years, simple interest is SI3 = P * r * 3 / 100 = 5000 * 8 * 3 / 100 = Rs. 1200. Step 10: For compound interest over 3 years, A3 = P * (1 + r / 100)^3 = 5000 * (1.08)^3. Step 11: Compute (1.08)^2 = 1.1664 and (1.08)^3 = 1.1664 * 1.08 = 1.259712. Step 12: Therefore A3 = 5000 * 1.259712 = 6298.56 and CI3 = A3 - P = 6298.56 - 5000 = Rs. 1298.56. Step 13: The required difference for 3 years is CI3 - SI3 = 1298.56 - 1200 = Rs. 98.56.


Verification / Alternative check:

You can check that the 2 year values are consistent: SI2 = 2 * 1200 / 3 = 800 and CI2 can be recalculated for P = 5000 and r = 8% to verify that it equals 832. This confirms that the base parameters are correct and validates the 3 year calculations.


Why Other Options Are Wrong:

Rs. 48.00 and Rs. 66.56 both underestimate the additional compound interest over 3 years, while a significantly larger value than Rs. 98.56 would not match the precise calculation. The option None of these is invalid because one of the given numerical options matches the exact difference. Only Rs. 98.56 satisfies the detailed computation.


Common Pitfalls:

Many learners forget the compact form for CI2 - SI2 and instead try to calculate compound and simple interest separately without algebraic simplification, which can lead to errors. Another frequent mistake is to apply the 2 year difference directly to 3 years by simple proportion, which is not valid because compound interest grows non linearly with time.


Final Answer:

The difference between compound interest and simple interest for 3 years is Rs. 98.56.

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