The Dean’s Column:
A Juicy Math Problem
When students are challenged by a problem, they often set it aside if it involves too much reading, for fear that the concentration required would be too exhausting to make the problem pleasurable. Although this problem does require a fair bit of reading it is rather easy to explain to a class, and could even be dramatized. Once past the statement of the problem, it is very easy to understand, but quite difficult to solve by conventional means.
This is where the beauty of the problem comes it. The solution—as unexpected as it is—almost makes the problem trivial. That is, the problem and its conventional solution will not get much of an enthusiastic reaction from students, but after having struggled with a solution attempt, the novel approach we will present here will gain you much favor with the class.
So let’s state the problem:
We have two one-gallon bottles. One contains a quart of grape juice and the other, a quart of apple juice. We take a tablespoonful of grape juice and pour it into the apple juice. Then we take a tablespoon of this new mixture (apple juice and grape juice) and pour it into the bottle of grape juice. Is there more grape juice in the apple juice bottle, or more apple juice in the grape juice bottle?
To solve the problem, we can figure this out in any of the usual ways—often referred to as “mixture problems”—or we can use some clever logical reasoning and look at the problem’s solution as follows:
With the first “transport” of juice there is only grape juice on the tablespoon. On the second “transport” of juice, there is as much apple juice on the spoon as there is grape juice in the “apple juice bottle.” This may require students to think a bit, but most should “get it” soon.
The simplest solution to understand and the one that demonstrates a very powerful strategy is that of using extremes. We use this kind of reasoning in everyday life when we resort to the option: “ such and such would occur in a worst case scenario….”
Let us now employ this strategy for the above problem. To do this, we will consider the tablespoonful quantity to be a bit larger. Clearly the outcome of this problem is independent of the quantity transported. So we will use an extremely large quantity. We’ll let this quantity actually be the entire one quart. That is, following the instructions given the problem statement, we will take the entire amount (one quart of grape juice), and pour it into the apple juice bottle. This mixture is now 50% apple juice and 50% grape juice. We then pour one quart of this mixture back into the grape juice bottle…The mixture is now the same in both bottles. Therefore, there is as much apple juice in the grape juice bottle as there is grape juice in the apple juice bottle!
We can consider another form of an extreme case, where the spoon doing the juice transporting has a zero quantity. In this case the conclusion follows immediately: There is as much grape juice in the apple juice bottle as there is apple juice in the grape juice bottle, that is, zero!
Carefully presented, this solution can be very significant in the way students approach future mathematics problems and even how they may analyze everyday decision-making.#
You may find other such examples in Math Wonders: To Inspire Teachers and Students, by Alfred S. Posamentier (ASCD, 2003) see: www.ascd.org. or Math Charmers: Tantalizing Tidbits for the Mind. By Alfred S. Posamentier (Prometheus Books, 2003) see: www.prometheusbooks.com. If you wish to learn more about p, see: p: A Biography of the world’s Most Mysterious Number, By Alfred S. Posamentier (Prometheus Books, 2004) see: www.prometheusbooks.com.
Dr. Alfred S. Posamentier is Dean of the School of Education at City College of NY, author of over 35 books on math, and member of the NYS Standards Committee on Math.