Recover bill amount and present worth from a given true discount: The true discount on a bill due 8 months hence at 12% per annum (simple) is Rs. 240. Find the amount of the bill and its present worth.

Introduction / Context:
Given a true discount (TD) at a stated rate and time, we can recover the due amount A and then obtain its present worth PW = A − TD. This is a direct inversion of the TD formula.

Given Data / Assumptions:

• TD = 240.
• t = 8 months = 2/3 year.
• r = 12% p.a. simple.

Concept / Approach:
TD = A * (r * t) / (1 + r * t). Compute r * t = 0.12 * 2/3 = 0.08. Then TD = A * 0.08 / 1.08. Solve for A and then PW = A − TD.

Step-by-Step Solution:
TD = A * 0.08 / 1.08 ⇒ 240 = A * (2/27) ⇒ A = 240 * (27/2) = 3240.PW = A − TD = 3240 − 240 = 3000.

Verification / Alternative check:
Check: PW = 3000; at 12% for 8 months, interest = 3000 * 0.12 * 2/3 = 240. PW + TD = 3240 = amount due, as required.

Why Other Options Are Wrong:

• The other pairs do not satisfy TD = 240 under 12% for 8 months.

Common Pitfalls:

• Using 8/10 of a year instead of 8/12.
• Applying banker’s discount A * r * t directly as TD.

Final Answer:
Rs. 3000, Rs . 3240