### Mixture Problems

My daughter has been asking for help on mixture problems in which they use little tables to arrange the available information. I used to teach this method myself, in "Intermediate Algebra" at the college, with great enthusiasm, I might add. Now I have my doubts.

In case you are not familiar with this method, here is an example.

Suppose I have a solution of 50% alcohol and another solution of 25% alcohol. How much of each of these solutions should I mix to make 2 liters of 40% alcohol?

The table is arranged with an equation across the top that relates the quantities in the mixture, and rows underneath for each component of the solution and the final result. The student is instructed to fill in all the known information, then assign a variable based on the question.

Next, the student will apply the equation across the top to calculate the last column, whose first two entries must add up to the entry in the result, thus generating the algebraic equation to solve:

Thus .50x + .50 - .25x = .80.25x = .30x = 1.20

Therefore we should use 1.2L of 50% alcohol and .8L of 25% alcohol.

This all works out beautifully. It seems very elegant. Unfortunately, it's also a gimmick. My daughter doesn't understand why the table is set up the way it is, nor does she really understand why this table results in an equation that gives the solution to the problem. I think this is fairly typical of students who are taught this method. Furthermore, it applies to only a limited set of problems, although it's true that they do vary somewhat from this example. Especially interesting are the variations that involve replacement of part of the mixture (how much weak antifreeze to drain out of your radiator and replace with pure antifreeze to get it right). The method can also be used with Distance=Rate x Time problems. These seem to be even more confusing, though.

My belief is that this type of problem is best approached as a system of equations in two unknowns. Let's see how this works with the example. I have to mix two solutions, let's say x of the 50% solution and y of the 25% solution and get 2 liters. Then x+y=2. The key feature of mixture problems is that there are two things that you can keep track of. In this example, one is the total amount of solution, the other is the amount of the pure stuff (alcohol) in the solution. We have an equation for the first, now we need one for the second. That will be (.50)x+(.25)y=.80. This system of equations is easily solved by substitution, resulting in an equation just like the one generated by the table method. However, the conceptual differences could be significant. If we use the system of equations, the focus is on reasoning about the relationships in the problem, while with the table the focus is, well, how to fill out the table. Teachers can stress the concepts underlying the table and perhaps bring the students to the same understanding, but they may not always do that. Too often, there is more of a focus on "learning the procedure/pattern" than on the essential "why."

One reason that this is a far superior method of teaching these types of problems is that it requires students to think about the sources of their equations. When students get to real applications in science or engineering classes, for example, it will be a VERY common thing to solve a problem by saying something like, "I have two unknowns here. I know one equation, where can I get another?" They should be practicing this kind of thinking in algebra.

The tables are now standard fare in high school algebra, though I used to be surprised when some of my college students said they hadn't seen them before. Now, I just think they are a gimmick that we would be better off doing without.

In case you are not familiar with this method, here is an example.

Suppose I have a solution of 50% alcohol and another solution of 25% alcohol. How much of each of these solutions should I mix to make 2 liters of 40% alcohol?

The table is arranged with an equation across the top that relates the quantities in the mixture, and rows underneath for each component of the solution and the final result. The student is instructed to fill in all the known information, then assign a variable based on the question.

Amt Sol | %Purity | Amt Pure | |

First Sol | x | .50 | |

Second Sol | 2-x | .25 | |

Final Sol | 2 | .40 |

Next, the student will apply the equation across the top to calculate the last column, whose first two entries must add up to the entry in the result, thus generating the algebraic equation to solve:

Amt Sol | %Purity | Amt Pure | |

First Sol | x | .50 | .50x |

Second Sol | 2-x | .25 | .50-.25x |

Final Sol | 2 | .40 | .80 |

Therefore we should use 1.2L of 50% alcohol and .8L of 25% alcohol.

This all works out beautifully. It seems very elegant. Unfortunately, it's also a gimmick. My daughter doesn't understand why the table is set up the way it is, nor does she really understand why this table results in an equation that gives the solution to the problem. I think this is fairly typical of students who are taught this method. Furthermore, it applies to only a limited set of problems, although it's true that they do vary somewhat from this example. Especially interesting are the variations that involve replacement of part of the mixture (how much weak antifreeze to drain out of your radiator and replace with pure antifreeze to get it right). The method can also be used with Distance=Rate x Time problems. These seem to be even more confusing, though.

My belief is that this type of problem is best approached as a system of equations in two unknowns. Let's see how this works with the example. I have to mix two solutions, let's say x of the 50% solution and y of the 25% solution and get 2 liters. Then x+y=2. The key feature of mixture problems is that there are two things that you can keep track of. In this example, one is the total amount of solution, the other is the amount of the pure stuff (alcohol) in the solution. We have an equation for the first, now we need one for the second. That will be (.50)x+(.25)y=.80. This system of equations is easily solved by substitution, resulting in an equation just like the one generated by the table method. However, the conceptual differences could be significant. If we use the system of equations, the focus is on reasoning about the relationships in the problem, while with the table the focus is, well, how to fill out the table. Teachers can stress the concepts underlying the table and perhaps bring the students to the same understanding, but they may not always do that. Too often, there is more of a focus on "learning the procedure/pattern" than on the essential "why."

One reason that this is a far superior method of teaching these types of problems is that it requires students to think about the sources of their equations. When students get to real applications in science or engineering classes, for example, it will be a VERY common thing to solve a problem by saying something like, "I have two unknowns here. I know one equation, where can I get another?" They should be practicing this kind of thinking in algebra.

The tables are now standard fare in high school algebra, though I used to be surprised when some of my college students said they hadn't seen them before. Now, I just think they are a gimmick that we would be better off doing without.

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