Question 746812
>>...pipe A can fill the pool in 8 hours...<<
<pre>
Therefore pipe A can fill 1 pool in 8 hours.  Therefore
A's pool-filling rate is 1 pool per 8 hours, which can be
written as {{{1_pool/8_hr}}} or {{{1/8}}}{{{pool/hr}}}.

Suppose pipe B can fill the pool in x hours.

Then pipe B can fill 1 pool in x hours.  Therefore
B's pool-filling rate is 1 pool per x hours, which can be
written as {{{1_pool/x_hr}}} or {{{1/x}}}{{{pool/hr}}}.

Suppose pipe C can fill the pool in y hours.

Then pipe C can fill 1 pool in y hours.  Therefore
C's pool-filling rate is 1 pool per y hours, which can be
written as {{{1_pool/y_hr}}} or {{{1/y}}}{{{pool/hr}}}. 
</pre>
>>...pipes A and C can fill the pool in 6 hours...<<
<pre>
So A and C's combined pool-filling rate is 1 pool per 6 hours, 
which can be written as {{{1_pool/6_hr}}} or {{{1/6}}}{{{pool/hr}}}.

Now we use the fact that A and C's combined pool-filling rate
is the sum of their individual pool-filling rates to get this
equation:

{{{(matrix(5,1,
"A's",pool-filling,rate,1/8,pool/hr))}}}{{{""+""}}}{{{(matrix(5,1,
"C's",pool-filling,rate,1/y,pool/hr))}}}{{{""=""}}}{{{(matrix(6,1,
"A_and_C's",combined,pool-filling,rate,1/6,pool/hr))}}}

So

{{{1/8}}}{{{""+""}}}{{{1/y}}}{{{""=""}}}{{{1/6}}}

Above we have C's pool-filling rate as {{{1/y}}}{{{pool/hr}}},
so we solve that for {{{1/y}}}

     {{{1/y}}}{{{""=""}}}{{{1/6}}}{{{""-""}}}{{{1/8}}}
     {{{1/y}}}{{{""=""}}}{{{4/24}}}{{{""-""}}}{{{3/24}}}
     {{{1/y}}}{{{""=""}}}{{{1/24}}}

so C's pool-filling rate is {{{1/24}}}{{{pool/hr}}},

</pre>
>>...pipes B and C can fill the pool in 10 hours...<<
<pre>
So B and C's combined pool-filling rate is 1 pool per 10 hours, 
which can be written as {{{1_pool/10_hr}}} or {{{1/10}}}{{{pool/hr}}}.

Now we use the fact that B and C's combined pool-filling rate
is the sum of their individual pool-filling rates to get this
equation:

{{{(matrix(5,1,
"B's",pool-filling,rate,1/x,pool/hr))}}}{{{""+""}}}{{{(matrix(5,1,
"C's",pool-filling,rate,1/24,pool/hr))}}}{{{""=""}}}{{{(matrix(6,1,
"A_and_C's",combined,pool-filling,rate,1/10,pool/hr))}}}

So

{{{1/x}}}{{{""+""}}}{{{1/24}}}{{{""=""}}}{{{1/10}}}

Above we have B's pool-filling rate as {{{1/x}}}{{{pool/hr}}},
so we solve that for {{{1/x}}}



     {{{1/x}}}{{{""=""}}}{{{1/10}}}{{{""-""}}}{{{1/24}}}
     {{{1/x}}}{{{""=""}}}{{{12/120}}}{{{""-""}}}{{{5/120}}}
     {{{1/x}}}{{{""=""}}}{{{7/120}}} 

so B's pool-filling rate is {{{7/120}}}{{{pool/hr}}}.
</pre>
>>...How long will it take to fill the pool if all three pipes are used together?...<<
<pre>

Suppose all three pipes can fill the pool in z hours.

So the combined pool-filling rate of all three is 1 pool per z hours, 
which can be written as {{{1_pool/z_hr}}} or {{{1/z}}}{{{pool/hr}}}.  

Then the equation comes from

{{{(matrix(5,1,
"A's",pool-filling,rate,1/8,pool/hr))}}}{{{""+""}}}{{{(matrix(5,1,
"B's",pool-filling,rate,7/120,pool/hr))}}}{{{""+""}}}{{{(matrix(5,1,
"C's",pool-filling,rate,1/24,pool/hr))}}}{{{""=""}}}{{{(matrix(6,1,
"A,_B_and_C's",combined,pool-filling,rate,1/z,pool/hr))}}}

{{{1/8}}}{{{""+""}}}{{{7/120}}}{{{""+""}}}{{{1/24}}}{{{""=""}}}{{{1/z}}}

Multiply through by LCD = 120z

15z + 7z + 5z = 120
           27z = 120
             z = {{{120/27}}}
             z = {{{40/9}}} or {{{4&4/9}}} hour

or 4 hours, 26 minutes, 40 seconds.

Edwin</pre>