Question

It can take 12 hours to fill a swimming pool using two pipes if the pipe with larger diameter is used for 4 hours and pipe of smaller diameter for 9 hours only half the pool can be filled .how long would it take for each pipe fill the pool separately?

( At least please answer the equation thus formed)

Answer 1

Solution :

Let P and Q be two pipes

• Water filled by Pipe P in in 1 hour = $\sf \: \dfrac{1}{P}$

• Water filled by Pipe Q in in 1 hour = $\sf \: \dfrac{1}{Q}$

Total Quantity of Water filled by both the pipes :

$\implies \sf \: \dfrac{1}{P } + \dfrac{1}{ Q}$

Given It can take 12 hours to fill the swimming pool Using both the pipes i.e.

$\implies \sf \: \dfrac{1}{P } + \dfrac{1}{ Q} = \dfrac{1}{12} ....(i)$

Quantity of Water filled in 4 hours by Pipe P = $\sf \: \dfrac{4}{P}$

Quantity of Water filled in 9 hours by Pipe Q = $\sf \: \dfrac{9}{Q}$

We Have,

$\implies \sf \: \dfrac{4}{P } + \dfrac{9}{Q} = \dfrac{1}{2} ....(ii)$

$\star \: \sf \: Let \: \dfrac{1}{P} \: be \: x \: and \: \dfrac{1}{Q} \: be \: y$

Rewriting the equation i) and ii)

$\implies \sf \: x + y = \dfrac{1}{12} ....(iii)$

$\implies \sf \: 4x + 9y = \dfrac{1}{2}.... iv)$

Multiply equation iii) by 4

$\longrightarrow \sf \: 4(x + y) = 4( \dfrac{1}{12} )....(iii)$

$\longrightarrow \sf \: 4x +4 y = \cancel\dfrac{4}{12}$

$\longrightarrow \sf \: 4x +4 y = \dfrac{1}{3}$

Subtracting from Equation iii),Thus we have :

$\sf \longrightarrow( 4x + 4y = \dfrac{1}{3} ) - (4x + 9y = \dfrac{1}{2} )$

$\sf \longrightarrow4x - 4x + 9y - 4y = \dfrac{1}{2} - \dfrac{1}{3}$

$\longrightarrow \sf \: 5y = \dfrac{1}{2} - \dfrac{1}{3}$

$\longrightarrow \sf \: 5y = \dfrac{3 - 2}{6}$

$\longrightarrow \sf \: 5y = \dfrac{1}{6}$

$\longrightarrow \sf \: y = \dfrac{1}{ 30}$

$\longrightarrow \sf \: \dfrac{1}{Q} \: = \dfrac{1}{ 30}$

$\longrightarrow \boxed{\purple{\tt \: Q \: = \: 30}}$

Pipe Q alone can fill the tank in 30 hours.

Putting the value of y = 1/30 in equation iii)

$\longrightarrow\sf \: x + \dfrac{1}{30} = \dfrac{1}{12}$

$\longrightarrow\sf \: x = \dfrac{1}{12} - \dfrac{1}{30}$

$\longrightarrow\sf \: x = \dfrac{5 - 2}{60}$

$\longrightarrow\sf \: x = \cancel\dfrac{ 3}{60}$

$\longrightarrow\sf \: x = \dfrac{ 1}{20}$

$\longrightarrow \sf \: \dfrac{1}{P} \: = \dfrac{1}{ 20}$

$\longrightarrow \boxed{\purple{\tt \: P \: = \: 20}}$

Pipe P alone can fill the tank in 20 hours.