Multiplying Fractions

This video shows how to multiply fractions 3 different ways: focusing on the concepts, using the box method, and using the traditional method.

Now that we have the basics, let’s try the mixed numbers $1 \frac{1}{2}$ times $2 \frac{1}{3}$.
Multiplying mixed numbers follows the same rules as multiplying fractions, in most cases you just need to change the mixed number into a fraction. In this video we look at how to multiply a fraction 3 different ways: conceptually, using the box method, and using the traditional algorithm.

Multipling Fractions

We will start by looking at the problem $\tfrac{1}{2}$ of 4.

So, what do we mean by that. One way to think of it is to look at it on a number line. We can think of it as $\tfrac{1}{2}$, 4 times.

Now lets look at the problem $\tfrac{3}{4}$ of $\tfrac{1}{5}$

We will start with $\tfrac{3}{4}$ of a square and then take $\tfrac{1}{5}$ of that.
Remember when we are dealing with fractions all the pieces need to be the same size, so we will be cutting all the sections in the square into fifths, giving us 20 pieces
Therefore, $\tfrac{3}{4}$ of $\tfrac{1}{5}$ is $\tfrac{3}{20}$

Now let’s look at what happens when a mixed number is involved.

Let’s say we are trying to find $\tfrac{1}{3}$ of $2\tfrac{2}{3}$ .

Visually:

First we can shade $2\tfrac{2}{3}$ squares in yellow, and then we will take $\tfrac{1}{3}$ of each piece.

A fourth of 1 is $\tfrac{1}{4}$ and when we divide $\tfrac{1}{2}$ into fourths we get $\tfrac{1}{8}$
Before we can combine those numbers, we need to be working with equal size pieces so we will need to divide the whole into 8th and you can see we have $\tfrac{2}{8}$ and when we combine that with the $\tfrac{1}{8}$ we get a total of $\tfrac{3}{8}$.
Well I don’t want to draw out every problem I encounter, so let’s look at 2 other ways we can solve the problem

First let revisit the multiplication array:

2 and $\tfrac{2}{3}$ is the same as $2 + \tfrac{2}{3}$ times $ \tfrac{1}{3}$
$2$ times $ \tfrac{1}{3}$ is $\tfrac{2}{3}$
and $\tfrac{1}{3} * \tfrac{2}{3}$ is $\tfrac{2}{9}$ and when we combine those we get $\tfrac{8}{9}$.

Now let’s look at the traditional algorithm.

First, we need to convert the mixed number into a fraction
So we would write $2\tfrac{2}{3}$ as $\tfrac{8}{3}$.

\[\begin{align*}
2\frac{2}{3} *\frac{1}{3} &= \\
\\
\frac{8}{3} *\frac{1}{3} &= \frac{8}{9}
\end{align*}\]

Transcript for Multiplying Mixed Numbers

Let’s look at multiplying a mixed number times another mixed number.

We will use the example of 1½ times $2\tfrac{1}{3}$.
Or we could say 1½ of $2\tfrac{1}{3}$.
We are going to show you how to solve this in 3 different ways.
We are going to start off by using drawings.
First, we will draw $2\tfrac{1}{3}$ — we need that one time.
Now, we are going to draw it a second time, but this time we are going to take ½ of everything.
So, we have $2\tfrac{1}{3}$ + $1\tfrac{1}{6}$ and when we combing those we get $3\tfrac{3}{6}$ and when we simplify that we get 3½.

Even though we can solve this by drawing, sometimes it is easier to just multiply it out.
We are going to demonstrate how to do this using the multiplication array.
We have $1\tfrac{1}{2}$ times $2\tfrac{1}{3}$.
So when we multiply that out we get 2 + 1/3+1+1/6.

We notice that my common denominator would be 6 and when we add those we get $3\tfrac{3}{6}$
or simplified that would be 3½
Using the traditional algorithm we would first convert the mixed numbers into fractions.
So 1½ would become $\tfrac{3}{2}$ and $2\tfrac{1}{3}$ would become $\tfrac{7}{3}$
getting $\tfrac{3}{2}$ * $\tfrac{7}{3}$ which equals $\tfrac{21}{6}$, to make this into a mixed number we need to see how many times 6 goes into 21 and we get $3\tfrac{3}{6}$, or 3½ again.

When multiplying fractions most people use the traditional algorithm, but by looking at the other methods we can more fully understand what is going on.