3 men can finish a work in 10 days, 4 women can finish it in 12 days and 10 qualified workers can finish it in 6 days. In how many days is the work finished by 4 men, 4 women and 4 qualified workers, working together every day?

Given:

3 men can finish a work in 10 days.

4 women can finish it in 12 days.

10 qualified workers can finish it in 6 days.

Formula Used:

Work done per day by a person = Total work / Number of days

Combined work done per day = Sum of individual work done per day

Total days to finish the work = Total work / Combined work done per day

Calculation:

Let the total work be W.

Work done per day by 3 men = W / 10

Work done per day by 1 man = (W / 10) / 3 = W / 30

Work done per day by 4 women = W / 12

Work done per day by 1 woman = (W / 12) / 4 = W / 48

Work done per day by 10 qualified workers = W / 6

Work done per day by 1 qualified worker = (W / 6) / 10 = W / 60

Combined work done per day by 4 men, 4 women, and 4 qualified workers:

Work done per day by 4 men = 4 × (W / 30) = 4W / 30 = 2W / 15

Work done per day by 4 women = 4 × (W / 48) = 4W / 48 = W / 12

Work done per day by 4 qualified workers = 4 × (W / 60) = 4W / 60 = W / 15

Combined work done per day = (2W / 15) + (W / 12) + (W / 15)

Finding a common denominator for the fractions:

Combined work done per day = (2W / 15) + (W / 12) + (W / 15)

⇒ Combined work done per day = (8W / 60) + (5W / 60) + (4W / 60)

⇒ Combined work done per day = (8W + 5W + 4W) / 60

⇒ Combined work done per day = 17W / 60

Total days to finish the work = W / (17W / 60)

⇒ Total days = 60 / 17

The work is finished by 4 men, 4 women, and 4 qualified workers in \( \frac{60}{17} \) days.