Let <em>x</em> and <em>y</em> be the unit rates at which one large pump and one small pump works, respectively.
Two large/one small operate at a unit rate of
(1 pool)/(4 hours) = 0.25 pool/hour
so that
2<em>x</em> + <em>y</em> = 0.25
One large/three small operate at the same rate,
(1 pool)/(4 hours) = 0.25 pool/hour
<em>x</em> + 3<em>y</em> = 0.25
Solve for <em>x</em> and <em>y</em>. We have
<em>y</em> = 0.25 - 2<em>x</em> ==> <em>x</em> + 3 (0.25 - 2<em>x</em>) = 0.25
==> <em>x</em> + 0.75 - 6<em>x</em> = 0.25
==> 5<em>x</em> = 0.5
==> <em>x</em> = 0.1
==> <em>y</em> = 0.25 - 2 (0.1) = 0.25 - 0.2 = 0.05
In other words, one large pump alone can fill a 1/10 of a pool in one hour, while one small pump alone can fill 1/20 of a pool in one hour.
Now, if you have four each of the large and small pumps, they will work at a rate of
4<em>x</em> + 4<em>y</em> = 4 (0.1) + 4 (0.05) = 0.6
meaning they can fill 3/5 of a pool in one hour. If it takes time <em>t</em> to fill one pool, we have
(3/5 pool/hour) (<em>t</em> hours) = 1 pool
==> <em>t</em> = (1 pool) / (3/5 pool/hour) = 5/3 hours
So it would take 5/3 hours, or 100 minutes, for this arrangement of pumps to fill one pool.
