Simple Interest — A sum is lent for a fixed time. At 10% per annum simple interest it amounts to ₹ 400, and at 4% per annum for the same time it amounts to ₹ 200. Find the original principal (sum).

Introduction / Context:
When the same principal is invested for the same time under simple interest at two different rates, the two final amounts give two linear equations in the principal and time. Solving those equations yields the principal directly.

Given Data / Assumptions:

• Let principal be P and time be t years (same in both cases).
• At 10% p.a., amount A1 = ₹ 400.
• At 4% p.a., amount A2 = ₹ 200.
• Simple interest applies (linear growth).

Concept / Approach:
Under simple interest: A = P * (1 + r * t), with r as a decimal per year. Form two equations and solve for P and t.

Step-by-Step Solution:
P * (1 + 0.10 t) = 400 … (1)P * (1 + 0.04 t) = 200 … (2)Divide (1) by (2): (1 + 0.10 t) / (1 + 0.04 t) = 2.1 + 0.10 t = 2 + 0.08 t ⇒ 0.02 t = 1 ⇒ t = 50 years.From (2): P = 200 / (1 + 0.04 * 50) = 200 / 3 = ₹ 200/3.

Verification / Alternative check:
At 10% for 50 years: A1 = (200/3) * (1 + 5) = 200/3 * 6 = 400. At 4%: A2 = (200/3) * (1 + 2) = 200/3 * 3 = 200. Both match.

Why Other Options Are Wrong:
₹ 100, ₹ 400/3, ₹ 500/3 do not satisfy both amounts simultaneously with a single t under simple interest.

Common Pitfalls:
Confusing simple interest with compound interest or assuming different times; both scenarios share the same t here.

Final Answer:
₹ 200/3