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.

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:

Verification / Alternative check:

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.