4 men and 6 women complete a work in 8 days, while 3 men and 7 women complete it in 10 days. In how many days will 10 women alone complete the work?

Problem restatementGiven two linear combinations of men and women producing 1 unit of work, find the rate of women alone and then the days for 10 women.

Given data

• (4m + 6w) × 8 = 1 ⇒ 32m + 48w = 1
• (3m + 7w) × 10 = 1 ⇒ 30m + 70w = 1

Concept/ApproachSolve the linear system for w (woman’s daily rate). Then 10 women rate is 10w; invert to get days.

Step-by-step calculationFrom 32m + 48w = 1 ⇒ m = (1 − 48w) ÷ 32Substitute in 30m + 70w = 1:30(1 − 48w)÷32 + 70w = 1 ⇒ (15÷16)(1 − 48w) + 70w = 115÷16 − 45w + 70w = 1 ⇒ 15÷16 + 25w = 1 ⇒ 25w = 1÷16 ⇒ w = 1÷40010 women rate = 10w = 1÷40 per day ⇒ days = 40

Verification/AlternativeBack-substitute to confirm both original equations hold.

Common pitfallsMixing “days to finish” with “rate per day.” Always convert to rates first.

Final Answer40 days