Banker’s Gain → Present Worth (simple interest): The banker's gain on a certain sum due 2 years hence at 5% per annum is Rs. 80. What is the present worth (i.e., the discounted value today) of the bill?

Introduction / Context:
Banker's Discount (BD), True Discount (TD), Banker's Gain (BG), Face Value (F), and Present Worth (P) are classic simple-interest notions used for short-term bill discounting. The banker usually deducts BD = F * r * t upfront, while the mathematically correct reduction is TD = F * r * t / (1 + r * t). The difference BG = BD − TD quantifies the banker's extra gain relative to true discounting.

Given Data / Assumptions:

• Rate r = 5% per annum; time t = 2 years ⇒ r * t = 0.10.
• Banker's Gain BG = Rs. 80.
• Simple interest framework; P = present worth; F = face value at maturity.

Concept / Approach:
For simple interest, a key identity is BG = F * (r * t)^2 / (1 + r * t). Also, P = F / (1 + r * t). We first compute F from BG, then obtain P.

Step-by-Step Solution:
Let x = r * t = 0.10.BG = F * x^2 / (1 + x) = F * 0.01 / 1.10 = F / 110.Given BG = 80 ⇒ F = 80 * 110 = Rs. 8800.Present Worth P = F / (1 + x) = 8800 / 1.10 = Rs. 8000.

Verification / Alternative check:
BD = F * x = 8800 * 0.10 = 880. TD = F * x / (1 + x) = 880 / 1.10 = 800/1.0? Actually 880/1.10 = 800. Then BG = 880 − 800 = 80, matching the given.

Why Other Options Are Wrong:
Rs. 8800 is the face value (F), not the present worth. Rs. 1600 and Rs. 1200 are far too small; Rs. 880 is the banker's discount in this case.

Common Pitfalls:
Confusing BD with TD or P; using compound interest instead of simple interest; forgetting that BG = BD − TD.

Final Answer:
Rs. 8000