Filling a Pool

Date: 07/28/97 at 21:28:15
From: moms address) (Jennifer Mann
Subject: Word Problem - fractions?

It takes 2 hours to fill a pool with the green hose, but 5 hours with 
the black hose - how long does it take to fill it with both hoses at 
same time ? 

My solution: after one hour the green hose fills 1/2 of the pool and 
the black hose fills 1/5 of the pool. After 1 hour the pool is 
1/2 + 1/5 full, or 7/10 full.  

How much longer would it be for the remaining 3/10 ?  I come up with 
3/7 of an hour or 3/7(60 minutes) or 25 minutes with 5/7 minute.  

My answer does not quite make sense to me (or my mom). One hour 25 
minutes and 5/7 of a minute ?! 

Date: 07/29/97 at 07:42:39
From: Doctor Anthony
Subject: Re: Word Problem - fractions ?

You are thinking along the right lines.  In 1 hour, with both hoses, 
you fill 1/2 + 1/5 = 7/10 of the pool.

So to fill the pool you require  10/7 =  1 hour 25 mins 42.86 seconds

Note that to get the total time, you simply find the reciprocal of 
7/10. It is not necessary to consider the remaining 3/10 in the way 
you did, although you will of course get the right answer, but only 
after a few extra steps.

To understand why 10/7 gives the total time, consider an easier 
situation. Suppose we fill 1/3 of the pool in 1 hour: then it is 
obvious that it will require 3 hours to fill the pool. Or suppose we 
fill 1/2 of the pool in 1 hour: then it would take 2 hours to fill the 
pool. In each case the total time in hours is the reciprocal of the 
fraction filled in 1 hour.

In arithmetic notation,  fraction per hour x number of hours = 1

                         number of hours = 1/fraction per hour

-Doctor Anthony,  The Math Forum
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