2 men or 8 women can do a job in 3 days. 2 men work for 1 day and leave. The…

When two groups of workers (such as men and women) each complete the same amount of work in the same time, the two groups' total work rates are equal. Comparing headcounts then converts one worker's rate into an equivalent count of the other group's workers (for example, if 2 men produce the same total rate as 8 women, one man's rate equals 4 women's rate). The total work can then be expressed in a single common unit (such as "woman-days"), completed work subtracted from it to get the remaining work, and the remaining work divided by the new workforce's daily rate to find the required count or time.

1. Since 2 men and 8 women each complete the same job in 3 days, their work rates are equal: 2 x (one man's rate) = 8 x (one woman's rate). This simplifies to 1 man's rate = 4 women's rate.

2. Express the total work in woman-day units: since 8 women complete the job in 3 days, the total work equals 8 x 3 = 24 woman-days.

3. Find the work done by the 2 men in 1 day: 2 men working for 1 day is equivalent (using 1 man = 4 women) to 2 x 4 = 8 women working for 1 day, i.e., 8 woman-days of work.

4. Find the remaining work: 24 woman-days minus 8 woman-days = 16 woman-days.

5. Find the number of women required in 8 days: let x be the number of women. x women working for 8 days complete x x 8 woman-days of work. Setting x x 8 = 16 gives x = 2.

Cross-check using man-day units instead: the total work is 2 men x 3 days = 6 man-days. The 2 men worked for 1 day, using 2 man-days, so 6 minus 2 = 4 man-days remain. Since 1 man-day equals 4 woman-days, the remaining work equals 4 x 4 = 16 woman-days, matching the earlier figure and confirming the required count of women.