Question 1184118
<pre>
This is mixing a weaker solution of OJ with a stronger solution of OJ and
ending up with a medium strength solution of OJ.

You can do this one in your head.  Since 60% is just as much above 50% as
40% is below 50%, you would mix the same amount of each.  So the answer
should be 9 gallons.

But that's not really working it the way your teacher expects.  Most mixture
problems can't be done in your head.  This one was special.  Let's do it as
your teacher expects.

Most teachers suggest that you should make a chart with such word problems,
at least until you get a little more proficient.  One column for the amounts
of liquid, one for the percents, and for the third we multiply what's in the
other two columns, to get the amount of pure substance contained in the two
liquids.

------------------------------------------------------------
                 |        | percent as |   gallons of      |
                 | gallons|  a decimal | pure OJ contained |
============================================================
weaker solution  |   x    |     0.6    |      0.6x         |  
------------------------------------------------------------
stronger solution|   9    |     0.4    |    0.4(9)         |
------------------------------------------------------------
medium-strength  |  x+9   |     0.5    |   0.5(x+9)        |
       solution  |        |            |                   |
------------------------------------------------------------

The equation comes from the last column

{{{0.6x+0.4(9)=0.5(x+9))}}}

Remove the decimals by multiplying through by 10:

{{{6x+4(9)=5(x+9))}}}

Distribute:

{{{6x+36=5x+45}}}
{{{x=9}}}

Just as we figured, 9 gallons of each.

Edwin</pre>