A swimming pool is fitted with 3 pipes A, B, C to fill the pool. A and B together can fill the pool in half the time that is required for C to fill the pool. B takes 20 hours more than the time required for A and 14 hours more than the time required for C to fill the pool. Then the time (in hours) required for all the 3 pipes together to fill the pool is

Let say, A,B,C require a,b and c hour to fill the pool.

According to question,

b=20+a  and  b=c+14.

So,$$\ 20+a=c+14.$$

or,$$c-a=6.$$..........................(1)

Again, A and B together can fill the pool in half the time that is required for C to fill the pool.

So,$$\ \frac{ab\ }{a+b}=\frac{c\ }{2}.$$

Putting value from (1) and b=20+a :

$$\ \frac{a\left(20+a\right)\ }{a+20+a}=\frac{a+6\ }{2}.$$

or,$$\ \frac{a\left(20+a\ \right)}{a+10}=\frac{a+6\ }{1}.$$

or,$$\ a^2+20a=\left(a+6\right)\left(a+10\right).$$

or,$$\ a^2+20a=a^2+6a+10a+60.$$

or,$$20a=16a+60.$$

or,$$4a=60.$$

or,$$a=15.$$

So,b=35 and c=21.

So,together they will do in 1 hour$$\left(\ \frac{\ 1}{15}+\frac{1\ }{35}+\frac{\ 1}{21}\right)$$ part

or $$\frac{\ 1}{15}+\frac{1\ }{35}+\frac{\ 1}{21}=\frac{\ 7+3+5}{105}=\frac{15\ }{105}=\frac{1}{7}$$part.

So,they together will complete in 7 hour.