Math Arguments

Dividing by a fraction

You probably memorized it, but never got a good explanation why. Can you come up with an explanation?

Here's one explanation, from my other blog:

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 1 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 1: \(\dfrac{\frac{6}{5}}{\frac{6}{5}}\)

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.

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