Dividing Fractions

This video explains how to divide fractions by looking at the concepts involved.

Transcript for Dividing Fractions

We are first going to do a quick review of division, so we can apply the concepts to fractions.

So let’s look at the problem 12 divided by 3. We can think of this as dividing 12 items, such as these pigs, into groups of 3 and seeing how many groups can form. We can see that we have 4 groups of 3, and we all know that 12 divided by 3 is 4, so there are no surprises here.

So what would it look like if we had the problem 5 divided by a $\frac{1}{3}$. How many groups of $\frac{1}{3}$ can I make? First, we need to divide all the circles into thirds and put $\frac{1}{3}$ into each pile, giving us 15 piles.

What if we change the problem to 5 divided by $\frac{2}{3}$? Well, we would treat it the same way. We would divide 5 into thirds and put $\frac{2}{3}$ in each pile, giving us $7\frac{1}{2}$ piles.

Coming up with Rules

We can see how to find the answer using fraction pieces, but we need a rule we can follow. We can’t always have fraction pieces handy so let’s start by looking at two problems that you probably already know the answer to.

I’m going start with 4 times $\frac{1}{2}$. Did you get two? And how did you get it? Did you multiply it out or did you just divide by 2? What about 12 times $\frac{1}{3}$? I got 4 but didn’t actually multiply by $\frac{1}{3}$. I divided by 3. In both situations, I got the same answer so there must be a connection between multiplying and dividing.

In the first example, we saw that 4 times $\frac{1}{2}$ was equivalent to 4 divided by 2 and in the second that 12 times $\frac{1}{3}$ is equivalent to 12 divided by 3. I noticed that the number 2 and the fraction $\frac{1}{2}$ are reciprocals and I also notice that 3 and $\frac{1}{3}$ are reciprocals. So that would suggest that multiplying is the same thing as dividing by the reciprocal and, perhaps even more useful, that dividing is the same a multiplying by the reciprocal.

Let’s see if this process works for the problems we just did with the fraction tiles. 5 divided by $\frac{1}{3}$. Well, if I took the reciprocal of $\frac{1}{3}$ I get 3 so that would be 5 times 3 and I see in both cases I got 15. Now, let’s look at 5 divided by $\frac{2}{3}$. Well, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ and when I multiply that I get $\frac{15}{2}$ or 7 and $\frac{1}{2}$. So, let’s practice dividing fractions by multiplying by the reciprocal. Let’s say I had $\frac{5}{8}$ divided by $\frac{2}{3}$. Well, that would be the same thing as $\frac{5}{8}$ times the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$ and when I multiply across I get $\frac{15}{16}$. Notice when I divide by a number less than 1, I get a solution that’s actually greater than what I started with.

Mixed Numbers

Now, let’s look at the problem involving mixed numbers $1\frac{1}{2}$ divided by $2\frac{1}{3}$. Just like a multiplication, the first thing I need to do is to change my mixed numbers into fractions. $1\frac{1}{2}$ becomes $\frac{3}{2}$ and $2\frac{1}{3}$ becomes $\frac{7}{3}$. Now, I need to switch my division to multiplying by the reciprocal so that would be $\frac{3}{2}$ times $\frac{3}{7}$ and when I multiply across I get $\frac{9}{14}$. Now notice that when I divide by a number that’s larger than the one I was dividing into I’m going to get something smaller. I’m going to get a fraction less than 1.

A little confused?

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