P completes a work in 12 days working 8 hours per day. Q completes the same work in 8 days working 10 hours per day. If both work together for 8 hours per day, in how many days will they complete the work?

Problem restatementNormalize each worker’s productivity to an hourly rate, add them, then convert back to days at 8 hours per day.

Given data

• P: 12 days × 8 h/day = 96 h for full work ⇒ P rate = 1÷96 per hour
• Q: 8 days × 10 h/day = 80 h for full work ⇒ Q rate = 1÷80 per hour
• Joint schedule: 8 h/day

Concept/ApproachUse hourly rates to combine fairly, since daily hours differ in the solo statements.

Step-by-step calculationCombined hourly rate = 1÷96 + 1÷80 = (5 + 6)÷480 = 11÷480 per hourCombined daily rate at 8 h/day = 8 × 11÷480 = 11÷60 per dayDays required = 1 ÷ (11÷60) = 60÷11 days ≈ 5.45 days

Verification/AlternativeIn 5.45 days at 8 h/day ≈ 43.6 h total; at 11÷480 per hour this gives about 1 unit of work.

Common pitfallsDo not average 12 and 8 days; different daily hours make that invalid.

Final Answer60÷11 days (≈ 5.45 days)