6 men and 8 women complete a task in the same time as 10 men and 2 women do.…

Concept: When two teams complete the exact same task in the exact same length of time, their combined work rates must be equal - this lets you find the work-rate ratio between different types of workers (here, men and women) and treat the whole task as a fixed number of "work units" equal to that common rate.

1. Let 1 woman's work rate = 1 unit per unit time, and let 1 man's rate = m units (in the same unit).

2. Since 6 men + 8 women finish the task in the same time as 10 men + 2 women, the two teams' combined rates are equal: 6m + 8 = 10m + 2.

3. Solve for m: 8 - 2 = 10m - 6m, so 6 = 4m, giving m = 1.5. So 1 man's rate equals 1.5 women's rate.

4. The total task (in work units) equals either team's combined rate, since each finishes it in the same reference time: Total work = 6m + 8 = 6(1.5) + 8 = 9 + 8 = 17 units.

5. The new team of 4 men and 6 women has a combined rate of 4m + 6 = 4(1.5) + 6 = 6 + 6 = 12 units, in that same reference time.

6. Fraction of the task finished by 4 men and 6 women = 12/17.

Cross-check: Using the other original team, 10 men + 2 women = 10(1.5) + 2 = 15 + 2 = 17 units, which matches the total work of 17 units found above - confirming m = 1.5 is consistent with both given teams.

So 4 men and 6 women finish 12/17 of the task in the same time.