A and B can finish a work together in 30 days. B and C can finish the same work together in 24 days and A and C can finish the work together in 40 days. If all three work together, how long will it take them to complete the work?

Let A's work be a units/day , B's work be b units/ day and C's work be c units/day
Now total work will always remain same 
Let it be W 
so we get 30a+30b =24b+24c=40a+40c
Now we get
30a+6b-24c=0
5a+b-4c =0    (1) and
10a-30b+40c =0
a-3b+4c=0    (2)
From 1 and 2
we get
$$\frac{a}{8}=\frac{b}{24}=\frac{c}{16}$$
we get
$$a=\frac{b}{3}=\frac{c}{2}$$
so let a=k,b=3k and c=2k
Now W =120k
Together time will be 120k/6k = 20 days