A bill is due 4 months hence at 15% p.a. The banker’s discount (BD) is ₹420. Find the true discount (TD).

Introduction / Context:
When BD and x = r * t are known, TD follows from TD = BD / (1 + x). This is because BD = (1 + x) * TD under simple-interest bill discounting.

Given Data / Assumptions:

• BD = ₹420.
• t = 4 months = 1/3 year; r = 15% p.a.

Concept / Approach:
Compute x = r * t; then TD = BD / (1 + x). No need to compute the face value explicitly.

Step-by-Step Solution:

Verification / Alternative check:
BG = BD − TD = 420 − 400 = ₹20. Also BG = x * TD = 0.05 * 400 = ₹20, matches.

Why Other Options Are Wrong:
₹380, ₹390, and ₹410 are off the exact ratio division by 1.05.

Common Pitfalls:
Using months directly without converting to years; confusing TD with PW.

Final Answer:
Rs. 400