To solve this problem, we need to break it down into calculable parts and follow a systematic approach.
### Step 1: Determine the work rate of one man and one woman
1. Work rate of one man:
- 16 men can complete the project in 20 days.
- Therefore, the amount of work one man can complete in one day is [tex]\( \frac{1}{16 \times 20} \)[/tex].
- This simplifies to [tex]\( \frac{1}{320} \)[/tex] of the work per day.
2. Work rate of one woman:
- 14 women can complete the project in 30 days.
- Therefore, the amount of work one woman can complete in one day is [tex]\( \frac{1}{14 \times 30} \)[/tex].
- This simplifies to [tex]\( \frac{1}{420} \)[/tex] of the work per day.
### Step 2: Calculate the work done by 40 men in 5 days
1. Work rate for 40 men:
- Work rate of one man is [tex]\( \frac{1}{320} \)[/tex].
- Work rate of 40 men is [tex]\( 40 \times \frac{1}{320} \)[/tex].
- This simplifies to [tex]\( \frac{40}{320} = \frac{1}{8} \)[/tex] of the work per day.
2. Work completed by 40 men in 5 days:
- [tex]\( \frac{1}{8} \)[/tex] of the work per day.
- In 5 days, the work completed is [tex]\( 5 \times \frac{1}{8} = \frac{5}{8} \)[/tex] of the work.
### Step 3: Determine the remaining work
1. Total work needed to complete the project is assumed to be 1 unit.
2. Work completed by 40 men in 5 days is [tex]\( \frac{5}{8} \)[/tex].
3. Remaining work:
- [tex]\( 1 - \frac{5}{8} = \frac{3}{8} \)[/tex] of the work is still remaining.
### Step 4: Calculate the number of days 21 women need to complete the remaining work
1. Work rate for 21 women:
- Work rate of one woman is [tex]\( \frac{1}{420} \)[/tex].
- Work rate of 21 women is [tex]\( 21 \times \frac{1}{420} \)[/tex].
- This simplifies to [tex]\( \frac{21}{420} = \frac{1}{20} \)[/tex] of the work per day.
2. Days needed for 21 women to complete the remaining [tex]\( \frac{3}{8} \)[/tex] of the work:
- [tex]\( \frac{3}{8} \)[/tex] of the work remaining and the work rate of 21 women is [tex]\( \frac{1}{20} \)[/tex] per day.
- Number of days = [tex]\( \frac{\frac{3}{8}}{\frac{1}{20}} = \frac{3}{8} \times 20 = \frac{60}{8} = 7.5 \)[/tex] days.
### Conclusion
The number of days in which the 21 women can finish the remaining work is 7.5 days.
