Saturday, March 8, 2014

51: Dividing by a fraction

Why is dividing by a fraction equivalent to multiplying by the reciprocal ... 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.


  1. I like your way of doing it, and I have helped some students do it this way. I personally usually prefer the technique of multiplying by the LCM of the denominators. So in your example, I would multiply both fractions by 12, so you would get (3/4)*12 divided by (5/6)*12 = 9 divided by 10.

    For those who struggle with finding the LCM (i.e., most of my students), I tell them to cancel each fraction, one-by-one. So maybe multiply by 4/4 to cancel the top fraction: (3/4)*4 divided by (5/6)*4 = 3 divided by (20/6). Assuming you don't simplify the 20/6 (though you could), then multiply by 6/6 to cancel the bottom fraction: 3*6 divided by (20/6)*6 = 18 divided by 20 = 9 divided by 10.

    I use this latter method when working with students to determine the derivative of a rational function using the limit process. For kids who struggle with fractions, they tend to be able to deal with the fractions in a fraction well.

    1. Ignore the "usually" in the second sentence.

  2. I always think of it slightly differently. Helps to divide a whole # by a fraction. So 9 ÷ (2/3). Since that's asking how many times we can find (2/3) in 9. So first we have to find how many thirds are in 9. (9*3=27 thirds) and then of all those thirds, how many times can I make groups of 2. (27 ÷ 2).

  3. Here's a possibly useful analogy you can present to students:

    Dividing by a/b is the same as multiplying by b/a

    Subtracting a-b is the same as adding b-a