Question 107438
Here's one way of looking at the problem:
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First, since Jim can fill the pool in 30 minutes, this means that for each minute he fills
{{{1/30}}} of the pool.
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Next, since Sue can fill the pool in 45 minutes, this means that in one minute she fills
{{{1/45}}} of the pool.
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Finally, since Tony can fill the pool in 90 minutes (90 minutes = 1 1/2 hours) for each minute
he works, he fills {{{1/90}}} of the pool.
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Then for each minute that they work together their total contribution to filling the pool is 
the sum of their individual contributions ... {{{(1/30 + 1/45 + 1/90)}}}
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90 is the least common denominator. Convert the fractions so they all have 90 as the denominator.
When you do, the sum becomes:
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{{{1/30 + 1/45 + 1/90 = 3/90 + 2/90 + 1/90 = 6/90}}}
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and 6 is a common factor of both the numerator and denominator of {{{6/90}}}. Therefore,
{{{6/90 = (6*1)/(6*15) = (cross(6)*1)/(cross(6)*15) = 1/15}}}
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This shows that when Jim, Sue, and Tony work together that each minute they fill {{{1/15}}} of the
pool. Therefore, in 15 minutes they will fill the pool.
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In equation form this becomes:
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{{{(1/30 + 1/45+1/90)*T = 1}}} 
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where the 1 on the right side represents 1 pool full and T is the unknown number of minutes
that it takes to fill 1 pool.
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When you combine the three fractions as we did previously this equation becomes:
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{{{(1/15)*T = 1}}}
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Solve for T by either dividing both sides of this equation by {{{1/15}}} or by multiplying
both sides by 15 to get:
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{{{T = 15}}} 
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and since we were working in units of minutes, the answer for filling the pool when all 
three persons work together is T = 15 minutes.
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Another way of looking at the problem is to find a common number of minutes that is divisible
by 30, 45, and 90. That number is 90. Then look at it this way ... in 90 minutes Jim would
fill the pool 3 times, Sue would fill it twice, and Tony would fill it once. Therefore, 
if all three worked together, they would fill the pool a total of 6 times in 90 minutes. So
to find how long it takes to fill the pool once, divide 90 minutes by 6 times and you
get 15 minutes ... the same answer as we got previously.
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Hope this helps you to understand how to do the problem.
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