Answer:
76 mL of 8% and 304 mL of 13%
Step-by-step explanation:
Set up a table like this (this is the same table for any mixture problem):
                    mL total     *           % vinegar       =              % vinegar/mL
8%
13%
total %
What you do with this table is multiply the first 2 columns and they equal the last column.  We already know the middle column's entries from what we are given.  But remember that a percent needs to be expressed as a decimal:
                        mL total     *       % vinegar      =        % vinegar/mL
8%                                                 .08
13%                                                .13
total %                                   
Now we also know that the total he wants to make is 380 mL that is 12%, so all that info goes into the last row:
                       mL total       *       % vinegar        =        % vinegar/mL
8%                                                   .08
13%                                                  .13
total %            380                          .12
If he wants this total mixture to come from the 8% and the 13% vinegars, he is mixing them together, or adding them.  We need to know how much 8% + how much 13% will give us 380 mL of 12%.  So we can fill in the amounts accordingly.  If he has x mL of 8%, then he will have 380 - x of 13%.  The table tells us that we are multiplying the first 2 columns to get the third, so we will do that at the same time:
                        mL total         *          % vinegar        =        % vinegar/mL
8%                        x                *             .08                =          .08x
13%                 380 - x            *             .13                 =      .13(380 - x)
total %               380              *             .12                 =          45.6
Since we are adding the different vinegars together to get the total, we can do that with the last column as well.  This is our equation:
.08x + .13(380 - x) = 45.6  Multiply everything through by 100 to get rid of the decimals now:
8x + 13(380 - x) = 4560 and
8x + 4940 - 13x = 4560 and
-5x = -380 so
x = 76
This means that there is 76 mL of 8% and 380 - 76 = 304 mL of 13%