At 10% p.a. simple interest, the banker’s discount (BD) and true discount (TD) on the same bill for the same time are Rs 100 and Rs 80, respectively. Find the face value (sum due at maturity) and the time.

Introduction / Context:
Banker’s discount (BD) is simple interest on the face value for the full term; true discount (TD) is the difference between face value and its present worth. The pair (BD, TD) at the same rate/time pins down both the time and the face value.

Given Data / Assumptions:

• Rate r = 10% p.a. (simple interest).
• BD = Rs 100, TD = Rs 80, same time t.
• Let face value (amount due at maturity) be A.

Concept / Approach:
BD = A · r · t / 100. True discount TD = A · k/(1 + k) with k = r t / 100. Equivalently TD = BD/(1 + k). Two equations in k and A solve the problem.

Step-by-Step Solution:

Verification / Alternative check:
k = r t/100 = 0.25; BD = A k = 400 × 0.25 = 100; TD = A k/(1 + k) = 400 × 0.25/1.25 = 80 (consistent).

Why Other Options Are Wrong:
Other pairs mismatch BD/TD relationships with r = 10%.

Common Pitfalls:
Confusing present worth discounting (TD) with interest on face value (BD).

Final Answer:
Sum = Rs 400; Time = 2.5 years