If 40 men or 60 women or 80 children can do a piece of work in 6 months, then 10 men, 10 women and 10 children together do half of the work in

Let the amount of work be 360 units

So it is given that it takes 40 men, 6 months or 180 days to complete 360 units of work which means

1 men 1 day work = $$\frac{1}{20}$$ units

it is given that it takes 60 women, 6 months or 180 days to complete 360 units of work which means

1 women 1 day work = $$\frac{1}{30}$$ units

it is given that it takes 80 children, 6 months or 180 days to complete 360 units of work which means

1 children 1 day work = $$\frac{1}{40}$$

Now, (1 men +1 women+ 1 children)'s 1 day work = $$(\frac{1}{20} + \frac{1}{30} + \frac{1}{40} )$$ = $$\frac{13}{120}$$ units

So to complete $$\frac{360}{2}$$=180 units of work (10 men +10 women+ 10 children) will take = $$\frac{180 \times 120}{130}$$ days = $$\frac{180 \times 12}{13 \times 30}$$

= $$\frac{6 \times 12}{13 \times 1}$$

= $$\frac{72}{13}$$

= 5 $$\frac{7}{13}$$ months