Tuesday, December 27, 2011

Dividing by a Fraction

We "invert and multiply", "multiply by the reciprocal" or insist on using the fraction key because we can't remember or were never really taught the reasons or the algorithm. Is there a simple explanation for the method we old farts memorized years ago in third or fourth grade? Why does it work?
Let's start with a problem: $\frac{3}{4} \div \frac{5}{6}$ and change to a compound fraction: $\dfrac{\frac{3}{4}}{\frac{5}{6}}$

Now what? Dividing by a fraction is confusing, but dividing by one is obvious. So we turn $\frac{5}{6}$ into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by $\frac{6}{5}$ without changing its value, so we multiply by one: $\dfrac{\frac{6}{5}}{\frac{6}{5}}$

All in one image: $\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}$

Divide by one. Seems simple to me.


  1. Or, another way to put this is that division and multiplication are inverse operations, so dividing by any number is the same as multiplying by its inverse. Understanding inverses and identities is important.

    Your picture is nice.

    I kept pounding into my kids heads: x/x = 1. I called x/x "a special form of one" even though it isn't really special. It was just a memory device.

  2. I once saw what I think is an interesting way (Sherman Stein book?): convert both fractions to equivalents with a common denominator, and then "divide straight across" the way we multiply fractions. The denominator of the solution will always be one, so the numerator becomes the answer.

    It has the advantage of using only skills students had already learned prior to dividing fractions.