The banker's gain on a sum due 3 years hence at 12% per annum simple interest is Rs 270. What is the banker's discount on this sum?

Introduction:
Here we are given the banker's gain over a 3-year period at 12% simple interest and must determine the banker's discount. This involves using the link between banker's gain and true discount, and then connecting true discount to banker's discount.

Given Data / Assumptions:
Banker's gain BG = Rs 270. Rate r = 12% per annum. Time t = 3 years. We must find the banker's discount BD.

Concept / Approach:
For a bill with face value P at rate r and time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). BG = BD − TD. The key relation: BG = TD * r * t / 100. Hence: TD = BG * 100 / (r * t). Once TD is known, we use: BD / TD = (100 + r * t) / 100, or simply: BD = TD * (100 + r * t) / 100.

Step-by-Step Solution:
Step 1: Compute r * t. r * t = 12 * 3 = 36. Step 2: Find TD from BG. BG = TD * r * t / 100 ⇒ 270 = TD * 36 / 100. TD = 270 * 100 / 36 = 27000 / 36 = Rs 750. Step 3: Compute BD from TD. BD = TD * (100 + r * t) / 100. BD = 750 * (100 + 36) / 100 = 750 * 136 / 100. BD = 750 * 1.36 = Rs 1020.

Verification / Alternative check:
We can confirm BG: BG = BD − TD = 1020 − 750 = Rs 270, which is exactly the given banker's gain. This check confirms that our BD value is correct.

Why Other Options Are Wrong:
Rs 1315, Rs 1150, Rs 980, Rs 900: None of these values, when used as BD with TD = 750, produce a banker's gain of exactly Rs 270, nor do they satisfy the exact rate-time relationships.

Common Pitfalls:
Many learners try to guess BD directly or incorrectly assume BG is a certain percentage of BD without calculation. It is safer to compute TD from BG first using BG = TD * r * t / 100 and then compute BD using BD = TD * (100 + r * t) / 100.

Final Answer:
The banker's discount on the sum is Rs 1020.