Mixture Applications

Mixture problems involve mixing two things … usually chemicals.

I use a diagram to help set up the equation.  The diagram looks like this.

% (amount)     +     % (amount)    =    % (total amount)

A chemist need to mix 20 liters of 40% acid solution with some 70% solution to get a mixture that is 50% acid.  How many liters of the 70% solution should be used?

[Note:  I have colored coded some of the key elements of this problem to help you set up the problem “with me”.]

First always identify, what we are trying to find.  The question asks … How many liters of 70% solution … so we begin with ... Let x = the amount of 70%

The first pieces of information that I look for in the problem are the percentages.  You need to fill these in the diagram in the appropriate places.  You really shouldn’t have too much trouble doing this especially if you remember that the “middle” value percentage has to be on the right-hand side of the equation (resultant mixture).  In other words, in our problem we have percentages of 40, 50, and 70 … 50 is the (numerical) median and therefore is on the “right-hand” side of the equation.

Fill in the percentages now.

                                    % (amount)     +     % (amount)    =    % (total amount)

                                    40 (amount)     +     70 (amount)    =    50 (total amount)

Next try and determine any given amounts.  The problem links the amounts to the percentages by use of the word of.  In math, of may often be interpreted as multiplication.  This particular problem states that we have 20 liters of 40%, so 20 is the amount linked to 40% by multiplication.  Also, the question asks How many liters of 70% (in the problem it also says we have “some” 70% … this means we have an unknown.) and therefore the variable x is the amount of 70%.

Fill in this information.

                                    40 (amount)     +     70 (amount)    =    50 (total amount)

                                    40 (20)              +     70 (x)               =    50 (??????)

Now, we still need one more piece of information … the total amount.  The total amount is the sum of the two amounts we mixed … 20 + x is the total amount.

                                    40 (20)     +     70 (x)    =    50 (20 + x)

Now we need to solve the equation. 

                                    % (amount)     +     % (amount)    =    % (total amount)

                                    40 (20)              +     70 (x)               =    50 (20 + x)

                                    800                   + 70x                       =    1000   +  50x
                                  –800                   – 50x                             – 800   – 50x

                                                                20x                      =      200
                                                                20                                  20

                                                                             x         =     10 Liters

So the amount of 70% needed is 10 Liters.

Let’s try another problem similar to that problem.

How many gallons of a 12% solution must be mixed with a 20% solution to get 10 gallons of a 14% solution?

If we let x = amount of 12%

                                    % (amount)     +     % (amount)    =    % (total amount)

                                    12  (x)               +      20 (10-x)         =    14(10)

Notice that we identified the amount of 12% as an unknown “x” and we were given 10 gallons of 14%… the total amount of the mixture.  How do we figure out the amount of 20%?   If we are given the total amount then the sum of the two amounts on the left must equal the total sum on the right.  We cannot have 2 unknowns with 2 different variables and we cannot use x to represent both amounts.  So, if we let x = amount of 12% we must let the amount of 20% be the total amount – x.

[Think about it … IF I had 2 gallons of 12% that would mean that I have (10-2) = 8 gallons of 20% to total 10 gallons of 14% … it will always work this way when you are given the total amount!]

Now, to solve the equation:

                                    12  (x)               +      20 (10-x)         =    14(10)

                                    12x                    + 200     – 20x         =   140

                                                            – 8x                           =   – 60

                                                                        x   =  7.5 gallons of 20%