#### Question

To fill a swimming pool two pipes are to be used. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.

#### Solution

Let the time taken by the pipe of larger diameter to fill the pool completely be *x* hours and the pipe of smaller diameter be *y* hours.

In one hour,

Part of the pool filled by the pipe of larger diameter = `1/x`

Part of the pool filled by the pipe of smaller diameter = `1/y`

According to question,

`4/x+9/y=1/2`

y−x=10 .....(ii)

Substituting the value of *y* from (ii) in (i), we get

`4/x+9/(x+10)=1/2`

`(4(x+10)+9x)/((x+10)x)=1/2`

`(4x+40+9x)/(x^2+10x)=1/2`

`(13x+40)/(x^2+10x)=1/2`

26x+80=x^{2}+10x

x^{2}−16x−80=0

x^{2}−20x+4x−80=0

x(x−20)+4(x−20)=0

(x+4)(x−20)=0

x=20

Putting the value of *x *in (ii), we get

y−20=10

y=30

Therefore, the time taken by the pipe of larger diameter to fill the pool is 20 hours and the time taken by the pipe of smaller diameter to fill the pool is 30 hours