3 women and 18 children together take 2 days to complete a piece of work. How many days will 9 children alone take to complete the piece of work if 6 women alone can complete the piece of work in 3 days?

6 women complete a piece of work in 3 days.

Hence, one woman can complete $$\frac{1}{18}$$ piece of work in one day.

3 women and 18 children take 2 days to complete the work.

3 women in 2 days complete $$3*2*\frac{1}{18}=\frac{1}{3}$$

Hence, in two days 18 children complete $$1-\frac{1}{3}=\frac{2}{3}$$ of the work.

So, amount of work done by 1 child in a day is $$\frac{2}{3}*\frac{1}{36}=\frac{1}{54}$$

So, 9 children complete $$9*\frac{1}{54}=\frac{1}{6}$$ part of the work in a day.

Hence, the number of days in which 9 children alone complete the work is $$6$$