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2 taps A and B can fill a swimming pool in 3 hours. If turned on alone, it takes tap A 5 hours less than tap B to fill the same pool. How many hours does it take tap A to fill the pool?

Guest May 20, 2015

Best Answer 

 #1
avatar+92744 
+10

Let x be the time (in hours) that B takes to fill the pool.....then( x - 5)  is the time it takes for A to fill the pool.

 

And every hour, B fills 1/x of the pool and A fills 1/(x -5) of the pool 

 

And we know that :

 

Rate per hour x time  = amount of job done....so.......

 

[(1/x) * 3]  + [1/(x -5) * 3]  = 1

 

3/x + 3/(x -5)  = 1

 

[ 3(x - 5) + 3(x)] = x(x - 5)

 

[6x - 15 ] = x^2 - 5x     simplify

 

x^2  - 11x + 15  = 0       and using the onsite solver, we have

 

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{11}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,-\,}}{\mathtt{11}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1.594\: \!875\: \!162\: \!046\: \!672\: \!8}}\\
{\mathtt{x}} = {\mathtt{9.405\: \!124\: \!837\: \!953\: \!327\: \!2}}\\
\end{array} \right\}$$

 

Reject 1.59......so A takes about (x - 5)= (9.405 - 5) = 4.405 hours to fill the pool working alone

 

Check....  (3/9.405) + (3/4.405) ≈ 1

 

CPhill  May 20, 2015
 #1
avatar+92744 
+10
Best Answer

Let x be the time (in hours) that B takes to fill the pool.....then( x - 5)  is the time it takes for A to fill the pool.

 

And every hour, B fills 1/x of the pool and A fills 1/(x -5) of the pool 

 

And we know that :

 

Rate per hour x time  = amount of job done....so.......

 

[(1/x) * 3]  + [1/(x -5) * 3]  = 1

 

3/x + 3/(x -5)  = 1

 

[ 3(x - 5) + 3(x)] = x(x - 5)

 

[6x - 15 ] = x^2 - 5x     simplify

 

x^2  - 11x + 15  = 0       and using the onsite solver, we have

 

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{11}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,-\,}}{\mathtt{11}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{11}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1.594\: \!875\: \!162\: \!046\: \!672\: \!8}}\\
{\mathtt{x}} = {\mathtt{9.405\: \!124\: \!837\: \!953\: \!327\: \!2}}\\
\end{array} \right\}$$

 

Reject 1.59......so A takes about (x - 5)= (9.405 - 5) = 4.405 hours to fill the pool working alone

 

Check....  (3/9.405) + (3/4.405) ≈ 1

 

CPhill  May 20, 2015
 #2
avatar+94110 
+5

2 taps A and B can fill a swimming pool in 3 hours. If turned on alone, it takes tap A 5 hours less than tap B to fill the same pool. How many hours does it take tap A to fill the pool?

 

A is the fast one:   let tap A take X hours to fill the pool

A fills 1 pool in X hours

that is 3 pools in 3X hours   OR      3/X pools in 3 hours

B fills 1 pool in X+5 hours

that is 3 pools in 3(X+5) hours    OR        3/(X+5) pools in 3 hours

So together in 3 hours they will fill

 

$$\frac{3}{X}+\frac{3}{X+5} \;\;pools\\\\
=\frac{3(X+5)+3X}{X(X+5)}\;\;pools\; in\; 3 \;hours\\\\
$But in 3 hours they fill 1 pool so $\\\\
\frac{3(X+5)+3X}{X(X+5)}=1\\\\
3(X+5)+3X=X(X+5)\\\\
3X+15+3X=X^2+5X\\\\
6X+15=X^2+5X\\\\
X^2+5X-6X-15=0\\\\
X^2-X-15=0\\\$$

 

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{15}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{61}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{3.405\: \!124\: \!837\: \!953\: \!327\: \!2}}\\
{\mathtt{x}} = {\mathtt{4.405\: \!124\: \!837\: \!953\: \!327\: \!2}}\\
\end{array} \right\}$$

 

Obviously the negative answer is invalid  so X = 4.4051 hours to fill the pool

 

$${\mathtt{60}}{\mathtt{\,\times\,}}{\mathtt{0.405\: \!1}} = {\mathtt{24.306}}$$

 

So that is near enough to 4 hours and 24 minutes

 

A will take 4 hours and 24 minutes to fill the pool

and

B will take 9 hours and 24 minutes to fill the pool.

Melody  May 20, 2015
 #3
avatar+94110 
0

Did you struggle with that one for as long as I did Chris?

I have done heaps of these but they turn into a saga EVERY time.    LOL

Oh well I got there in the end, CPhill and I did it a bit differently but we both got the same answer.

That is always a good sign   

Melody  May 20, 2015
 #4
avatar+92744 
0

It wasn't too much of a struggle.....but.......it does make me nervous when I get answers like 4.405....It defeats my "Nice Round Numbers"  theory........LOL!!!

 

CPhill  May 20, 2015
 #5
avatar+94110 
0

I have added this to our 'Great Answers to Learn From' Sticky thread.  :)

Melody  May 21, 2015

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