The banker's discount on a certain sum of money is Rs. 72 and the true discount on the same sum for the same time and rate is Rs. 60. What is the sum due (the face value of the bill)?

Introduction / Context:
This question illustrates how to determine the face value of a bill when both banker's discount and true discount are known for the same time period and rate. These two quantities are closely related through the concept of banker's gain and present worth, and being able to move between them is important in commercial arithmetic and banking-related questions.

Given Data / Assumptions:

• Banker's discount BD = Rs. 72.
• True discount TD = Rs. 60.
• Both are computed on the same sum S, at the same rate and for the same time.
• We assume simple interest and a single-payment bill.

Concept / Approach:
We use the concept of banker's gain and some standard relationships:

• Banker's gain (BG) = BD − TD.
• Present worth P = S − TD.
• A key relation: BG = TD^2 / P.

From BG and TD we can find P, and then add TD to get the sum due S. This avoids dealing directly with rate and time, which are not given.

Step-by-Step Solution:
Step 1: Compute banker's gain: BG = BD − TD = 72 − 60 = 12.Step 2: Let P be the present worth. Use BG = TD^2 / P.Step 3: Substitute values: 12 = 60^2 / P = 3600 / P.Step 4: Rearranging: P = 3600 / 12 = 300.Step 5: Present worth P = S − TD, so S − 60 = 300.Step 6: Therefore, S = 300 + 60 = 360.Step 7: The sum due (face value of the bill) is Rs. 360.

Verification / Alternative check:
We can verify using another relation BD = (S * TD) / P. With S = 360, TD = 60 and P = 300, BD = (360 * 60) / 300 = 21600 / 300 = 72, which matches the given banker's discount. The difference BD − TD = 72 − 60 = 12 is equal to TD^2 / P = 3600 / 300 = 12, confirming consistency of all formulas.

Why Other Options Are Wrong:
Values like Rs. 290, Rs. 420, Rs. 480 or Rs. 540 do not satisfy the relationships among BD, TD and P. If you substitute any of these as S into BD = (S * TD) / (S − TD), you do not obtain BD = 72. Only S = 360 correctly reproduces both the banker's discount and true discount values given in the question.

Common Pitfalls:

• Confusing present worth P with the face value S.
• Attempting to use simple interest formulas directly without using the TD–BD relations.
• Forgetting that banker's gain is BD − TD, not BD + TD.
• Ignoring the helpful relation BG = TD^2 / P, which simplifies the calculation significantly.

Final Answer:
The sum due (face value of the bill) is Rs. 360.