BG is 9/25 of BD over 2.5 years: find the annual rate: For a certain sum due 2.5 years hence, the banker’s gain equals 9/25 of the banker’s discount. Find the simple annual rate of interest.

Introduction / Context:
For any bill, BG/BD = (r * t) / (1 + r * t). This comes from BD = S r t and TD = BD / (1 + r t), hence BG = BD − TD = BD * (r * t)/(1 + r * t).

Given Data / Assumptions:
BG/BD = 9/25; t = 2.5 years.

Concept / Approach:
Solve r * t / (1 + r * t) = 9/25. Let y = r * t. Then y/(1 + y) = 9/25 → 25y = 9 + 9y → 16y = 9 → y = 9/16 = 0.5625. Hence r = y / t.

Step-by-Step Solution:

Verification / Alternative check:
Check: r t = 0.5625 → BG/BD = 0.5625 / 1.5625 = 9/25 ✔️.

Why Other Options Are Wrong:
18.12%, 24.13%, 22.22% do not satisfy the exact ratio for t = 2.5 years.

Common Pitfalls:
Solving for r t but forgetting to divide by t; confusing BG/BD with BD/TD.

Final Answer:
22.5% per annum