A swimming pool can be filled by a large pipe in 4 hours and by a small pipe in 6 hours. A third pipe can empty the pool in 3 hours. How long would it take to fill the pool if all three pipes were open at the same time? Can you help me to get to the starting equation?
Make this chart:
number of number of rate in
pools filled hours pools/hour
Large pipe
Small pipe
Third pipe
All three
-----------------------------------------------------------------
Let x = the time it would take to fill the pool if all three were open. So
put x for the number of hours for all three, and since we want to know how
long it would take to fill 1 pool, we put 1 for the number of pools filled.
number of number of rate in
pools filled hours pools/hour
Large pipe
Small pipe
Third pipe
All three 1 x
-----------------------------------------------------------------
>>...A swimming pool can be filled by a large pipe in 4 hours...<<
Since it would take the large pipe 4 hours to fill 1 pool, put 1 for
the large pipe's number of pools and 4 for its number of hours.
number of number of rate in
pools filled hours pools/hour
Large pipe 1 4
Small pipe
Third pipe
All three 1 x
-----------------------------------------------------------------
>>...and by a small pipe in 6 hours...<<
Since it would take the small pipe 6 hours to fill 1 pool, put 1 for
the small pipe's number of pools and 6 for its number of hours.
number of number of rate in
pools filled hours pools/hour
Large pipe 1 4
Small pipe 1 6
Third pipe
All three 1 x
-----------------------------------------------------------------
>>...A third pipe can empty the pool in 3 hours...<<
Since it would take the third pipe 3 hours to EMPTY 1 pool, put -1 for
the small pipe's number of pools, because that is a LOSS of one pool!
and put 3 for its number of hours to "fill negative one pool" which means
to EMPTY 1 pool.
number of number of rate in
pools filled hours pools/hour
Large pipe 1 4
Small pipe 1 6
Third pipe -1 3
All three 1 x
-----------------------------------------------------------------
Next we put in the rates in pools/hour by dividing the number of
pools by the number of hours:
number of number of rate in
pools filled hours pools/hour
Large pipe 1 4 1/4
Small pipe 1 6 1/6
Third pipe -1 3 -1/3
All three 1 x 1/x
-----------------------------------------------------------------
Now the equation comes from:
The sum of the three rates = their combined rate:
Solve that and get x = 12 hours. Hint: multiply through by LCD of 12x
to clear of fractions.
Edwin