The banker's gain on a bill due 1 year hence at 12% per annum simple interest is Rs 6. What is the true discount on this bill?

Introduction:
In this question, the banker's gain (BG) is given and we are asked to find the true discount (TD). Using the relationship between BG and TD in banker's discount problems, we can solve this quickly without knowing the face value explicitly.

Given Data / Assumptions:
Banker's gain BG = Rs 6. Rate r = 12% per annum. Time t = 1 year. We must find the true discount TD.

Concept / Approach:
For a bill with face value P at rate r for time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). BG = BD − TD. There is a useful relationship: BG = TD * r * t / 100. Given BG, and knowing r and t, we can solve directly for TD.

Step-by-Step Solution:
Step 1: Compute r * t. r * t = 12 * 1 = 12. Step 2: Use BG = TD * r * t / 100. 6 = TD * 12 / 100. TD = 6 * 100 / 12. TD = 600 / 12 = Rs 50.

Verification / Alternative check:
If TD = Rs 50 and r * t = 12, then the banker's gain should be: BG = TD * r * t / 100 = 50 * 12 / 100 = 600 / 100 = Rs 6, which matches the given value, confirming our answer.

Why Other Options Are Wrong:
Rs 62, Rs 58, Rs 47, Rs 60: None of these values satisfy the exact relation BG = TD * 12 / 100 = 6. Any other TD would either give a larger or smaller banker's gain than Rs 6.

Common Pitfalls:
A common error is to assume BG is simply BD * (some fraction) without recalling the precise formula. Another mistake is trying to find the face value first, which is unnecessary and leads to more complicated algebra.

Final Answer:
The true discount on the bill is Rs 50.