Bill dated July 14, 5 months; discounted Oct 5 at 10%: A bill of ₹ 10,200 is drawn on July 14 at 5 months. It is discounted on Oct 5 at 10% per annum. Find (i) banker’s discount, (ii) true discount, (iii) banker’s gain, and (iv) the proceeds received.

Introduction / Context:
We compare banker’s discount (BD) and true discount (TD) for a bill discounted before maturity. BD is computed on the face value for the time from discount date to maturity; TD discounts the face value back at simple interest to obtain present worth; their difference is banker’s gain (BG).

Given Data / Assumptions:

• Face value S = ₹ 10,200.
• Drawn July 14 at 5 months → nominal due Dec 14; add 3 days grace → maturity Dec 17.
• Discount date = Oct 5; rate r = 10% p.a.

Concept / Approach:
Time between Oct 5 and Dec 17 is 73 days. Use t = 73/365 years. BD = S * r * t. TD = BD / (1 + r * t). Proceeds = S − BD. BG = BD − TD.

Step-by-Step Solution:

Verification / Alternative check:
BG = S * r^2 * t^2 / (1 + r t) = 10200 * 0.01 * 0.04 / 1.02 = ₹ 4 ✔️.

Why Other Options Are Wrong:
Other tuples do not satisfy both BD and TD identities together for t = 0.2 year.

Common Pitfalls:
Omitting 3 days of grace; using months as 30 days blindly; forgetting that TD = BD / (1 + r t).

Final Answer:
BD = ₹ 204, TD = ₹ 200, BG = ₹ 4, Proceeds = ₹ 9996