3 men or 4 women can build a wall in 43 days. The number of days that 7 men…

Concept: When a problem says that "A men OR B women can do a job in D days", it means each group ALONE completes the SAME job in D days, so their total effort in worker-days is equal: A × D man-days = B × D woman-days = one full job. This lets us find each individual's daily work rate, and then combine any mix of men and women using those rates.

Application:

1. Total work in man-days: 3 men complete the wall in 43 days, so total work = 3 × 43 = 129 man-days.

2. Total work in woman-days: 4 women complete the wall in 43 days, so total work = 4 × 43 = 172 woman-days.

3. One man's daily rate = 1/129 of the wall, since 129 man-days make up 1 wall.

4. One woman's daily rate = 1/172 of the wall, since 172 woman-days make up 1 wall.

5. Combined daily rate of 7 men and 5 women = 7 × (1/129) + 5 × (1/172) = 7/129 + 5/172.

6. Put both fractions over the LCM of 129 and 172, which is 516: 7/129 = 28/516 and 5/172 = 15/516.

7. Add the fractions: 28/516 + 15/516 = 43/516 = 1/12 of the wall per day.

8. At a combined rate of 1/12 of the wall per day, the whole wall is finished in 12 days.

Cross-check: Convert everyone into 'man-equivalent' workers using the given exchange rate 3 men = 4 women, i.e. 1 woman = 3/4 man in daily output. Then 7 men and 5 women together equal 7 + 5 × (3/4) = 7 + 3.75 = 10.75 man-equivalents. Since 3 men finish the wall in 43 days, one man-equivalent's daily rate is 1/(3 × 43) = 1/129 of the wall, matching the rate found in step 3, so 10.75 man-equivalents' daily rate is 10.75/129 = 43/516 = 1/12 of the wall per day — the same rate found above, confirming the day count.

Result: 7 men and 5 women together complete the wall in 12 days.