The present worth of a bill due 7 months hence is Rs. 1200. If the bill were due at the end of 2.5 years, its present worth would be Rs. 1016. Find the annual simple interest rate and the face value (sum) of the bill.

Introduction / Context:
Present worth under simple interest uses P = F / (1 + r * t). Two present-worth conditions at two different times allow solving for both r and F simultaneously.

Given Data / Assumptions:

• P1 = 1200 at t1 = 7/12 years
• P2 = 1016 at t2 = 2.5 years
• Simple interest, same rate r, same face value F

Concept / Approach:
Use the pair of equations: 1200 = F / (1 + r * 7/12) 1016 = F / (1 + 2.5 * r) Solve for r and F.

Step-by-Step Solution:
From first: F = 1200(1 + 7r/12). From second: F = 1016(1 + 2.5r). Equate: 1200(1 + 7r/12) = 1016(1 + 2.5r). Solve to get r = 0.10 (10%). Substitute to get F = 1270.

Verification / Alternative check:
P1 = 1270/(1 + 0.10 * 7/12) = 1200; P2 = 1270/(1 + 0.10 * 2.5) = 1016. Both confirm.

Why Other Options Are Wrong:
Other rate–sum pairs do not satisfy both present-worth equations simultaneously.

Common Pitfalls:
Mixing simple with compound interest; failing to convert months to years; algebraic mistakes when equating expressions for F.

Final Answer:
10%, Rs. 1270