It 10 men or 20 women or 40 children can do a piece of work in 7 months, then 5 men, 5 women and 5 children together can do half of the work in how many months :

Assume that every month has 30 days , so in 7 months = 7 x 30 = 210 days

Let assume the amount of work be 840 units (which is LCM of 10,20,40, 210)

It is given that 10 men can do 840 units of work in = 210 days

then 1 men's 1 day work = $$\frac{2}{5}$$ units

5 men's 1 day work = $$\frac{2}{5}$$ units x 5 = 2 units

20 women can do 840 units of work in = 210 days

1 women's 1 day work = $$\frac{1}{5}$$ units

5 women's 1 day work = $$\frac{1}{5}$$ units x 5 = 1 units

40 children's can do 840 units of work in = 210 days

1 children's 1 day work = $$\frac{1}{10}$$ units

5 children's 1 day work = $$\frac{1}{10}$$ units x 5 = 0.5 units

Now , (5 men + 5women + 5 children) 's 1 day work = 2 + 1 + 0.5 = 3.5 units

So, number of days required by (5 men + 5women + 5 children) to complete $$\frac{840}{2}$$ = 420 units of work = $$\frac{420}{3.5}$$ = 120 days

= $$\frac{120}{30}$$ = 4 months