CI–SI Comparison — Find rate from 2-year difference and SI total: For a sum, the difference between compound and simple interest for 2 years is ₹ 160. The simple interest for 2 years is ₹ 2880. Find the annual rate (percent per annum).
Aptitude
Compound Interest
Difficulty: Medium
Choose an option
Answer
Correct Answer: 11 1/9%
Explanation
Introduction / Context:Two years at the same annual rate produces neat identities: SI(2y) = 2 P r and CI − SI = P r^2, where r is the annual rate in decimal and P is the principal. Having both the 2-year SI and the 2-year difference allows elimination of P to solve directly for r, then convert it to a percent value.
Given Data / Assumptions:
- SI over 2 years = ₹ 2880 ⇒ 2 P r = 2880
- CI − SI over 2 years = ₹ 160 ⇒ P r^2 = 160
- r is an annual rate (decimal), P > 0
Concept / Approach:Divide the two equations to eliminate P: (P r^2)/(2 P r) = r/2 = 160/2880. This quickly yields r. Finally, express r as a mixed fraction percentage for clarity.
Step-by-Step Solution:
From 2 P r = 2880, we have P r = 1440.From P r^2 = 160, compute r = (P r^2)/(P r) = 160 / 1440 = 1/9.Therefore r = 1/9 ≈ 0.111… ⇒ 11 1/9% per annum.Verification / Alternative check:
Check: With r = 1/9, CI − SI (2y) = P r^2 = P/81. Also, SI(2y) = 2 P r = 2P/9. Ratio (difference)/(SI) = (P/81)/(2P/9) = 1/18 = 160/2880 (consistent).Why Other Options Are Wrong:
- 9%, 10%, 12 1/2%, 55/9% do not satisfy both equations simultaneously; only 11 1/9% works.
Common Pitfalls:
- Treating r as “11.1” instead of 0.111…; always convert properly between decimal and percent.
Final Answer:11 1/9% per annum.