If the true discount on a certain sum due 6 months hence at 15% per annum is Rs. 120, then for the same sum, same time period and same rate, what will be the banker's discount?

Introduction / Context:
This question tests your knowledge of commercial arithmetic concepts related to bills, including true discount (T.D.), banker's discount (B.D.) and their relationship with the sum due. It focuses on a case where the true discount is known and you are asked to compute the banker's discount for the same bill, time and rate of interest. Such problems are common in banking and competitive exams dealing with bills of exchange and present worth.

Given Data / Assumptions:

• True discount (T.D.) = Rs. 120.
• Time = 6 months = 0.5 years.
• Rate of interest = 15% per annum.
• The sum is due after 6 months.
• We assume simple interest and standard formulas for discount.

Concept / Approach:
For a bill of face value S, rate R% per annum and time T years, the formulas are:

• True discount: TD = S * (R * T) / (100 + R * T).
• Banker's discount: BD = S * R * T / 100.

We first use the given true discount to find the sum due S. Once S is known, we substitute back into the banker's discount formula to get BD directly.

Step-by-Step Solution:
Step 1: Here R = 15% and T = 0.5 years, so R * T = 15 * 0.5 = 7.5.Step 2: Use TD formula: TD = S * 7.5 / (100 + 7.5) = S * 7.5 / 107.5.Step 3: Given TD = 120, so 120 = S * 7.5 / 107.5.Step 4: Rearranging, S = 120 * 107.5 / 7.5 = 1720.Step 5: Now compute banker's discount: BD = S * R * T / 100 = 1720 * 15 * 0.5 / 100.Step 6: 15 * 0.5 = 7.5, so BD = 1720 * 7.5 / 100 = 129.Step 7: Therefore the banker's discount is Rs. 129.

Verification / Alternative check:
Once S = 1720 is found, you can recompute the true discount as TD = 1720 * 7.5 / (100 + 7.5) = 1720 * 7.5 / 107.5 = 120, confirming consistency with the given data. Then, BD = S * 7.5 / 100 must be larger than TD because banker's discount is calculated on the full face value, whereas true discount is based on the present worth. The value 129 is reasonably slightly higher than 120, which fits the theory.

Why Other Options Are Wrong:
Rs. 109 and Rs. 119 are both less than the true discount, which contradicts the fact that banker's discount is always greater than true discount for a positive rate and time. Rs. 139 and Rs. 149 are too large compared to the difference between face value and present worth implied by the true discount; they do not satisfy the algebraic relationship between TD and BD for the given rate and time.

Common Pitfalls:

• Confusing the formulas of true discount and banker's discount.
• Forgetting to convert 6 months into 0.5 years before using R * T.
• Trying to guess BD directly without first finding the sum due S.
• Assuming BD and TD are equal, which is never true except in trivial cases.

Final Answer:
The banker's discount for the given bill is Rs. 129.