A bill for Rs 6000 is drawn on July 14 at 5 months. It is discounted on October 5 at 10% per annum simple interest. Find the banker's discount, the true discount, the banker's gain, and the net amount (money) that the holder of the bill receives.

Introduction:
This is a comprehensive banker's discount problem involving a dated bill with a term in months, a discount date, and a given annual interest rate. We must carefully determine the time from the date of discount to the bill's due date (including days of grace), then calculate the banker's discount (BD), true discount (TD), banker's gain (BG), and the net amount received by the holder.

Given Data / Assumptions:
Face value of the bill P = Rs 6000. Bill drawn on = July 14. Term of the bill = 5 months. Rate of interest r = 10% per annum (simple interest). Date of discount = October 5. Bills allow 3 days of grace, so due date is term end + 3 days.

Concept / Approach:
Steps required: 1. Find the nominal due date: add 5 months to July 14. 2. Add 3 days of grace to get the final due date. 3. Find the number of days between the discount date and the final due date. 4. Convert this into years for simple interest calculations. 5. Compute banker's discount using BD = P * r * t / 100. 6. Compute true discount using TD = P * r * t / (100 + r * t). 7. Banker's gain BG = BD − TD. 8. Amount received = P − BD.

Step-by-Step Solution:
Step 1: Nominal due date. 5 months after July 14 is December 14. Step 2: Add 3 days of grace. Final due date = December 17. Step 3: Time from discount date (October 5) to final due date (December 17). From October 5 to October 31 = 26 days. November = 30 days. December 1 to 17 = 17 days. Total days = 26 + 30 + 17 = 73 days. Step 4: Convert to years. t = 73 / 365 = 1/5 year (since 73 * 5 = 365). Step 5: Banker's discount. BD = P * r * t / 100 = 6000 * 10 * (1/5) / 100. BD = 6000 * 2 / 100 = Rs 120. Step 6: True discount. Compute r * t: r * t = 10 * (1/5) = 2. TD = P * r * t / (100 + r * t) = 6000 * 2 / (100 + 2). TD = 12000 / 102 ≈ Rs 117.65. Step 7: Banker's gain. BG = BD − TD ≈ 120 − 117.65 ≈ Rs 2.35. Step 8: Amount received by the holder. Net amount received = P − BD = 6000 − 120 = Rs 5880.

Verification / Alternative check:
We can check the relation BG = TD * r * t / 100: BG ≈ 117.65 * 2 / 100 ≈ 2.353 ≈ Rs 2.35. This is consistent with BD − TD, supporting the calculations.

Why Other Options Are Wrong:
Option B, C, D, E: All use incorrect values for BD, TD, BG, or the final amount. They do not respect the precise time interval of 73 days, nor the exact formulas for true discount and banker's discount, and thus lead to incorrect net proceeds.

Common Pitfalls:
Common mistakes include forgetting the 3 days of grace, miscounting the days between the discount date and the due date, or using months as 1/12 year instead of the exact number of days. Another error is assuming TD ≈ BD for short periods, which ignores the subtle difference captured in the formula TD = P * r * t / (100 + r * t).

Final Answer:
The correct set of values is Banker's discount = Rs 120; true discount ≈ Rs 117.65; banker's gain ≈ Rs 2.35; amount received = Rs 5880.