The banker’s gain (BG) on a bill due 6 years hence at 12% p.a. simple interest is Rs 540. Find the banker’s discount (BD).

Introduction / Context:
Banker’s gain BG = BD − TD quantifies the excess of banker’s discount over true discount for a given bill. With rate and time known, BG and BD are linked by a fixed factor.

Given Data / Assumptions:

• Rate r = 12% p.a., time t = 6 years.
• k = r t / 100 = 0.72.
• BG = Rs 540; we seek BD.

Concept / Approach:
For a face value A, BD = A k and TD = A k/(1 + k). Hence BG = BD − TD = A k − A k/(1 + k) = A k²/(1 + k). Also, TD = BD/(1 + k), so BG = BD · k/(1 + k). This directly relates BG and BD.

Step-by-Step Solution:

Verification / Alternative check:
Once BD is known, TD = BD/(1 + k) = 1290/1.72 = 750; BG = 1290 − 750 = 540 (checks).

Why Other Options Are Wrong:
They do not satisfy BG = BD · 0.72/1.72.

Common Pitfalls:
Using compound interest or mixing up the definitions of BD and TD.

Final Answer:
Rs 1290