On a certain sum, the banker's discount is Rs. 200 and the true discount is Rs. 100 for the same time and rate of interest. What is the sum due (face value of the bill)?

Introduction / Context:
This question asks you to determine the face value of a bill when both the banker's discount and true discount are known. It is a direct application of the relationships between banker's discount, true discount, banker's gain and the principal (sum due). Problems like this are frequently used in aptitude tests to assess understanding of commercial discounting.

Given Data / Assumptions:

• Banker's discount BD = Rs. 200.
• True discount TD = Rs. 100.
• Both apply to the same sum S, for the same time and rate.
• We assume simple interest.

Concept / Approach:
We use the standard relations:

• Banker's gain BG = BD − TD.
• Present worth P = S − TD.
• Banker's gain can also be written as BG = TD^2 / P.

However, in this particular case, there is a quicker way using the relation between BD, TD and S: BD = (S * TD) / (S − TD). We can solve this directly for S using the given BD and TD.

Step-by-Step Solution:
Step 1: Use the relation BD = (S * TD) / (S − TD).Step 2: Substitute BD = 200 and TD = 100 into the formula: 200 = (S * 100) / (S − 100).Step 3: Cross multiply: 200(S − 100) = 100S.Step 4: Expand left side: 200S − 20000 = 100S.Step 5: Bring terms together: 200S − 100S = 20000.Step 6: So 100S = 20000, giving S = 200.Step 7: Therefore, the sum due (face value) is Rs. 200.

Verification / Alternative check:
We can verify by computing present worth P and banker's gain BG. Present worth P = S − TD = 200 − 100 = 100. Banker's gain BG should be BD − TD = 200 − 100 = 100. From the formula BG = TD^2 / P, we get TD^2 / P = 100^2 / 100 = 100, which matches. Also, using BD = S * TD / P gives BD = 200 * 100 / 100 = 200, confirming all relationships are satisfied.

Why Other Options Are Wrong:
If S were Rs. 400 or Rs. 300, substituting in BD = (S * TD) / (S − TD) with TD = 100 would not yield BD = 200. Rs. 100 as S is impossible because the true discount cannot equal the sum itself. Rs. 250 similarly fails to satisfy the algebraic relations. Only S = 200 produces BD = 200, TD = 100 and the correct banker's gain.

Common Pitfalls:

• Assuming S must be BD + TD without checking formulas.
• Confusing present worth with the sum due.
• Not using the key relation BD = (S * TD) / (S − TD) and instead trying to guess.
• Ignoring banker's gain and its relation to true discount and present worth.

Final Answer:
The sum due (face value of the bill) is Rs. 200.