Alternate-day working and fair split: Total wages for completing a job is ₹ 1280. A alone can finish in 8 days, B alone in 12 days. If they work on alternate days starting with A and stop exactly when the work finishes, what is A’s share of the money?

Introduction / Context:
When workers alternate days, the fair split should still reflect actual work done. We compute the cumulative work over cycles until completion, including any partial last day, then use those fractions to divide the fixed ₹ 1280. The alternation begins with A.

Given Data / Assumptions:

• A's rate = 1/8 job per day
• B's rate = 1/12 job per day
• Work alternates daily: A then B, and so on, until job completes
• Total wage = ₹ 1280

Concept / Approach:
One 2-day cycle produces (1/8 + 1/12) = 5/24 of the job. After 4 full cycles (8 days), they complete 20/24 = 5/6. Day 9 (A's turn) adds 1/8 = 3/24, reaching 23/24. Only 1/24 remains, which B completes in half a day (since B does 1/12 per full day). Split money by total work each has completed.

Step-by-Step Solution:
A's work: 4 full days in cycles + day 9 = 5 days total ⇒ 5*(1/8) = 5/8B's work: 4 full days + partial last day = 4*(1/12) + 1/24 = 8/24 + 1/24 = 9/24 = 3/8Check: 5/8 + 3/8 = 1A's share = 5/8 * 1280 = ₹ 800

Verification / Alternative check:
Cycle method is robust; alternatively integrate until cumulative sum hits 1, accounting for partial-day completion on the final day for B.

Why Other Options Are Wrong:
₹ 500, ₹ 600, ₹ 700 do not correspond to the exact 5/8 fraction of the total payment.

Common Pitfalls:
Dividing by number of days worked instead of work done; missing that B only works a fraction of the last day; rounding prematurely.

Final Answer:
₹ 800