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

Introduction / Context:
This problem asks us to determine the banker's discount when the banker's gain is given for a loan or bill due after 3 years at a rate of 12 percent per annum. We use the relationships among banker's discount, true discount and banker's gain for a fixed rate and time. Once we express banker's gain as a fraction of the face value, we can find the face value and then compute the banker's discount.

Given Data / Assumptions:
- Time until the sum is due t = 3 years
- Rate of simple interest r = 12 percent per annum
- Banker's gain BG = Rs 270
- Let face value of the bill be F
- Banker's discount BD and true discount TD are both based on simple interest at this rate and time

Concept / Approach:
For a bill with face value F at rate r and time t, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Banker's gain BG equals BD − TD. For fixed r and t, BG simplifies to a fixed fraction of F. With r = 12 and t = 3, we can compute r t and express BD and TD accordingly, then solve for F from BG, and finally find BD from its formula.

Step-by-Step Solution:
Step 1: Compute r × t = 12 × 3 = 36.Step 2: Banker's discount BD = F × 36 / 100 = 9F / 25.Step 3: True discount TD = F × 36 / (100 + 36) = F × 36 / 136 = 9F / 34.Step 4: Banker's gain BG = BD − TD = 9F / 25 − 9F / 34.Step 5: Simplify BG = 9F × (1 / 25 − 1 / 34) = 9F × (34 − 25) / (25 × 34) = 9F × 9 / 850 = 81F / 850.Step 6: Given BG = 270, so 81F / 850 = 270 and therefore F = 270 × 850 / 81 = 8500 / 3.Step 7: Banker's discount BD = 9F / 25 = 9 × (8500 / 3) / 25 = 8500 / 25 = Rs 1020.

Verification / Alternative check:
We can verify by computing TD from F. True discount TD = 9F / 34 = 9 × (8500 / 3) / 34 = 8500 / 34 = Rs 250. Then BD − TD = 1020 − 250 = Rs 770 which is not equal to 270, so we must be careful. Let us recalculate TD carefully: 36 / 136 simplifies to 9 / 34, so TD = F × 9 / 34. With F = 8500 / 3, TD = (8500 / 3) × 9 / 34 = 8500 × 3 / 34 = 25500 / 34 ≈ 750. Therefore BD = 1020, TD ≈ 750, and BG = BD − TD = 1020 − 750 = Rs 270, which matches the given banker's gain. This confirms BD = 1020 as correct.

Why Other Options Are Wrong:
Values like 980 or 1150 correspond to different combinations of BD and TD that would not yield a banker's gain of 270 at 12 percent over 3 years. The option 1315 is much too large and would make BG far greater than the stated 270. Only 1020 is consistent with both the formulas and the given banker's gain.

Common Pitfalls:
One common error is simplifying fractions incorrectly when computing BD and TD, which leads to a wrong expression for BG. Another mistake is forgetting that BG is BD minus TD and not the other way round. Working systematically with r t, expressing all quantities in terms of F, and checking calculations avoids such slips.

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