Question

A tank is fitted with 22 pipes. some of these are fill pipes that fill the tank and others waste pipes that drain the tank. if each fill pipe can fill the tank in 24 hours and each drain pipe can drain the tank in 16 hours, and if all the pipes are kept open simultaneously when the tank is empty, the tank overflows in 12 hours. how many of these are waste pipes?

Answer 1

let volume of tank = LCM(24, 16,12)= 48l
then each pipe fills 48/24=2l in one hour
and each can drain 48/16=3 litres in 1 hour
suppose there are x pipes that fill the tank
then there are 22-x pipes that drains the tank
then total water filled in tank when all the pipes are opened = 2(x)-3(22-x) litre water
but it takes 12hours to overflow the tank when all taps are opened , IE water filled in 1 hour =48/12=4
thus according to questio
$2x - 3(22 - x) = 4 \\ 2x - 66 + 3x = 4 \\ 5x = 4 + 66 \\ 5x = 70 \\ x = \frac{70}{5} = 14$
there are 14 pipes that fill the tank and 22-14=8 pipes that drain the tank
.

Answer 2

Consider the volume of tank to be v.
let x and y be the numbers of fill and drain pipes hence x+y = 22.
rate of filling of a fill pipe = v/24.
rate of draining of a drain pipe = v/16.
It is said that if you keep all of them open together it would take 12 hours to reach the point of overflow which means,
((x × v/24) -(y×v/16))×12=v
i.e. (x/24 )-(y/16)=1/12
but y=22-x
Therefore
$\frac{x}{24} - \frac{22 - x}{16} = \frac{1}{12} \\ 40x - 22 \times 24 = 32 \\ x = 14 \\ y = 22 - 14 = 8$
There will be 14 fill pipes and 8 drain pipes
Hope this helps.