Eighteen men can complete a piece of work in 64 days. 9 women can complete it in 108 days, whereas 7 children can finish it in 216 days. How many days will 16 men, 9 women and 21 children together take to complete the same work?

18 men can complete the work in 64 days.

9 women can complete the work in 108 days.

7 children can complete the work in 216 days.

Work done is inversely proportional to the number of days taken.

​$$\text{Work Rate of one person} = \frac{1}{\text{Number of days taken to complete the work by that person}}$$​​

18 men can complete the work in 64 days, so the work rate of 1 man is:

​$$\text{Rate of one man} = \frac{1}{18 \times 64} = \frac{1}{1152} \text{ (work per day)}$$​​

9 women can complete the work in 108 days, so the work rate of 1 woman is:

​$$\text{Rate of one woman} = \frac{1}{9 \times 108} = \frac{1}{972} \text{ (work per day)}$$​​

7 children can complete the work in 216 days, so the work rate of 1 child is:

​$$\text{Rate of one child} = \frac{1}{7 \times 216} = \frac{1}{1512} \text{ (work per day)}$$​​

Work rate of 16 men:

​$$16 \times \frac{1}{1152} = \frac{16}{1152} = \frac{1}{72} \text{ (work per day)}$$​​

Work rate of 9 women:

​$$9 \times \frac{1}{972} = \frac{9}{972} = \frac{1}{108} \text{ (work per day)}$$​​

Work rate of 21 children:

​$$21 \times \frac{1}{1512} = \frac{21}{1512} = \frac{1}{72} \text{ (work per day)}$$​​

Now,

​$$\text{Total work rate} = \frac{1}{72} + \frac{1}{108} + \frac{1}{72} ​\\ \ \\= \frac{1}{72} + \frac{1}{108} + \frac{1}{72} \\\ \\= \frac{3}{216} + \frac{2}{216} + \frac{3}{216} = \frac{8}{216} =\frac{1}{27}$$​​

Since the total work rate is $$\frac{1}{27}$$​​ work per day, the time to complete the work is:

$$\text{Time} = \frac{1}{\frac{1}{27}} = 27 \text{ days}$$​

Thus, 16 men, 9 women, and 21 children together will take 27​ days to complete the work.