Simple Interest — Same duration at two rates; recover P and time: A certain sum becomes ₹ 600 at 6% p.a. SI in a certain time and ₹ 200 at 1% p.a. SI in the same time. Find the principal and the time.
Aptitude
Simple Interest
Difficulty: Hard
Choose an option
Answer
Correct Answer: ₹ 120 and 66 2/3 yr
Explanation
Introduction / Context:Two simple-interest amounts with the same time but different rates provide two linear equations in principal P and time t. Solving them simultaneously yields both the initial sum and the duration, even when the time is unusually long.
Given Data / Assumptions:
- A1 = P * (1 + 0.06 * t) = ₹ 600
- A2 = P * (1 + 0.01 * t) = ₹ 200
- Same t in both cases
Concept / Approach:Divide equations or subtract to eliminate P or t. One direct approach is to solve as a 2×2 system for P and t in real numbers under SI linearity.
Step-by-Step Solution:
From A2: P = 200 / (1 + 0.01 t).Substitute in A1: 200 * (1 + 0.06 t) / (1 + 0.01 t) = 600 ⇒ (1 + 0.06 t) / (1 + 0.01 t) = 3.Cross-multiply: 1 + 0.06 t = 3 + 0.03 t ⇒ 0.03 t = 2 ⇒ t = 66.666... years.Then P = 200 / (1 + 0.6666...) = 200 / 1.6666... = ₹ 120.Verification / Alternative check:
Check A1: 120 * (1 + 0.06 * 66.666...) = 120 * 5 = 600; A2: 120 * (1 + 0.01 * 66.666...) = 120 * 1.6666... = 200.Why Other Options Are Wrong:
- Other principals do not satisfy both amounts simultaneously for one common t.
Common Pitfalls:
- Assuming realistic times; math permits long durations under SI.
Final Answer:₹ 120 and 66 2/3 years.