Ready-money settlement of mutual dues at a true-discount rate: A owes B Rs. 456.75 payable 4 1/2 months hence; B owes A Rs. 455.51 payable 3 months hence. If they settle now at a true-discount rate of 4% per annum (simple), who pays whom and how much?

Introduction / Context:
To settle mutual future-dated dues immediately, compute the present worth (true discount method) of each obligation at the stated simple-interest rate and net the results. The party with the higher present obligation pays the difference to the other party.

Given Data / Assumptions:

• A → B: 456.75 due in 4.5 months (t1 = 4.5/12 years).
• B → A: 455.51 due in 3 months (t2 = 3/12 years).
• True-discount rate r = 4% p.a. simple.
• PW = A / (1 + r * t).

Concept / Approach:
Compute PW for each amount using its own time to maturity. Compare PWs to decide direction and amount of settlement.

Step-by-Step Solution:
PW(A’s liability) = 456.75 / (1 + 0.04 * 4.5/12) = 456.75 / (1 + 0.015) = 456.75 / 1.015 = 450.PW(B’s liability) = 455.51 / (1 + 0.04 * 3/12) = 455.51 / (1 + 0.01) = 455.51 / 1.01 = 451.Net = 451 − 450 = 1, in favor of A. Therefore B must pay A Re. 1, or equivalently A receives Re. 1.

Verification / Alternative check:
Accruing both PWs to either 3 months or 4.5 months at 4% gives equalized future values that differ by exactly Re. 1 accumulated appropriately, confirming fairness.

Why Other Options Are Wrong:

• “Rs. 2, B” and “Rs. 2, A” double the correct difference.
• “Re. 1, B” reverses the direction (A is the receiver).

Common Pitfalls:

• Using months directly without converting to years in r * t.
• Applying banker’s discount A * r * t instead of dividing by 1 + r * t.

Final Answer:
Re.1, A