Question 451472
one pipe 2 feet in diameter takes 12 hours to fill a reservoir. The second pipe 3 feet in diameter takes 6 hours to fill the reservoir. How long would it take for both pipes working at the same time to fill the reservoir?
<pre>
Two ways to solve it:

1. In your head
2. By algebra.

First way: In your head

Ignore the diameters.  Pretend there were several 
identical reservoirs that could be filled.

In 12 hours the smaller pipe would fill 1 reservoir and the
larger pipe could fill 2 reservoirs (since it fills 1 in 6 hours)
So in 12 hours they both could fill 2+1 or 3 reservoirs.
So it would only take 1/3 as long for them both to fill just 1 
reservoir,

so the answer is 1/3 of 12 hours, or 4 hours.

Second way: by algebra:

Make this chart

            reservoirs filled   time   rate in reservoirs/hour
small pipe        
large pipe                     
both pipes                     

Let the answer be t. So put t for the time it takes both
pipes to fill 1 reservoir when both are open. So put t
in the bottom middle slot and 1 for the number of reservoirs
filled by both pipes.

            reservoirs filled   time   rate in reservoirs/hour
small pipe        
large pipe          
both pipes          1             t           

Fill in the two times, 12 hours and 6 hours, for the two pipes'
times and 1 for the number of reservoirs they can fill in those
times

            reservoirs filled   time   rate in reservoirs/hour
small pipe          1            12           
large pipe          1             6           
both pipes          1             t           

Next put in the rates in reservoirs/hour by dividing the reservoirs
filled by the number of hours: 

            reservoirs filled   time   rate in reservoirs/hour
small pipe          1            12           1/12
large pipe          1             6           1/6
both pipes          1             t           1/t

The equation comes from:

Small pipe's rate + Large pipe's rate = Their combined rate

                           1/12 + 1/6 = 1/t

Multiply through by LCD of 12t

                 12t(1/12) + 12t(1/6) = 12t(1/t)        
                               t + 2t = 12
                                   3t = 12
                                    t = 4

answer: 4 hours.

Edwin</pre>