In this post, we are going to learn how to **divide fractions**. In order to do this, we are going to take a look at two different methods:

## Method 1 for dividing fractions: Cross-multiplication

- This method consists of multiplying the numerator of the first fraction by the denominator of the second fraction and then writing the answer in the resulting fraction’s numerator.
- Next, we multiply the denominator of the first fraction by the numerator of the second fraction, and then write the answer in the resulting fraction’s denominator*.
- Lastly, we simplify the final fraction.

For example, in order to divide the fraction

We multiply the numerator of the first fraction (3) by the denominator of the second fraction (10). This gives us the numerator for the final fraction: 3 x 10 = 30.

Next, we multiply the denominator of the first fraction (4) by the numerator of the second fraction (6). This gives us the denominator of the final fraction: 4 x 6 = 24.

The last step is to simplify the fraction. Since both numbers are multiples of 6, we can divide the numerator and denominator by 6.

- 30 ÷ 6 = 5
- 24 ÷ 6 = 4
- Therefore, the result of the division is 5/4.

### Method 2 of dividing fractions: Inverting and multiplying

- Step 1: Invert the second fraction. That is, swap the numerator for the denominator.
- Step 2: Simplify any numerator with any denominator.
- Step 3: Multiply across.

For example, we are going to divide:

Step 1: We invert the second fraction 6/4. This becomes 4/6.

- Step 2: We simplify the numerators with the denominators.
- Numerators are:

12 = 2 x 2 x 3 - 4 = 2 × 2
- Denominators are:

5 = 5 - 6 = 2 × 3

We can simplify both from numerator and denominator a 2 and a 3. We call this process “cross canceling” if one numerator has a common factor with the other denominator.

And we multiply across:

## Multiplying and Dividing Fractions

Multiplying fractions is easy: you multiply the top numbers and multiply the bottom numbers. For instance:

When it's possible, you reduce the fraction by cancelling off common factors; that is, you cross out any factors from one side of the fraction line that are duplicated on the other side of the line. In the example above, however, nothing reduces, because 8 and 45 have no factors in common.

- If you're not sure whether anything can be cancelled off, you can always factor the numerator and denominator, and check for any duplicated factors:
- Nothing is duplicated between the top and the bottom, so nothing cancels.
- Often, though, something will cancel:

To do the multiplication, I multiply all the top numbers (the numerators) with each other, and multiply all the bottom numbers (the denominators) with each other. However, to make life a little easier for myself, I'll first cancel off any factors that are common to both the numerators and the denominators:

Then the simplified product is .

You can use the Mathway widget below to practice multiplying fractions. Try the entered exercise, type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

Please accept “preferences” cookies in order to enable this widget.

(Clicking on “Tap to view steps” on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

Dividing fractions is just about as easy as multiplying them; there's just one extra step. When you divide by a fraction, the first thing you do is “flip-n-multiply”. That is, you take the second fraction, flip it upside-down (that is, you “find the reciprocal”), and then you multiply the first fraction by this flipped fraction.

My first step will be to convert this to multiplication by flipping the 9/4 to get 4/9. Then I can proceed with the simple multiplication, cancelling off any factors that are duplicated:

Then my simplified answer is .

This one is a bit tricky, but I can deal with the whole-number 5 by converting it into a fraction. Remember that any whole number is a fraction, if you put it over “1”. So I convert the 5 into the fraction 5/1, and flip-n-multiply:

Then my simplified answer is .

For this exercise, I'll first have to convert the mixed numbers to (improper) fractional form. (Multiplying and dividing fractions are places where fractions are sooooo much nicer than mixed numbers!) Once I've got fractions, then I can flip-n-multiply.

Then my mixed-number answer is .

Note: When the inputs are mixed numbers, as in the last example above, the book (or instructor, or grader) usually expects a mixed number as the output, too. So, if your answer is an impropr fraction, you'll need to convert it back to mixed-number form. Don't forget this step!

You can use the Mathway widget below to practice dividing fractions. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

Please accept “preferences” cookies in order to enable this widget.

(Clicking on “Tap to view steps” on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

Next, we move on to the much-more-difficult addition and subtraction of fractions…

URL: https://www.purplemath.com/modules/fraction3.htm

## Making Sense of Invert and Multiply

As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it’s normally accompanied with Keep-Change-Flip or the saying “Yours is not the reason why, just invert and multiply.”

Both are conceptual cripplers.

I’ve been drafting the 4th installment of the Making Sense Series involving fractions and I’m sharing this post as more of a personal reference should K-C-F make its way round these parts again…and I’m sure it will.

**Side note**: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.

Let’s start with a model for 3/5 ÷ 1/4.

Modeling measurement division of fraction by a fraction.

At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.

From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of “1 whole” so they begin to purposefully leave it out. In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they’re seen as repetitive. We’ve even had one student that “invented” and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

## Working with Fractions: Dividing Fractions

For most of us dividing fraction is one of those things where we got the right answer in math, but we don’t really remember why.

Most of us memorized a procedure with a concept we don’t understand: “don’t ask why, just invert and multiply.” Maybe it’s time to ask why!!

When we look at dividing fractions with our students today, we can start with a lot of questions that challenge that memorized procedure: Is it possible to actually divide by fractions or can you only multiply by fractions? And why do we flip or invert fractions and then multiply??

I was in a classroom not too long ago, modeling the concepts we’re talking about today by using the pattern blocks to help students explore their knowledge, and I asked the students *why do we invert and multiply (or “keep, change, flip” in some classrooms)*?

A student raised his hand and said, *So you can get more of an accurate answer.* I responded by asking *Is it going to be inaccurate if you divide??*

In today’s blog, we’re going to explore how we should have learned about dividing fractions before we learned a trick to get us the right answer.

### A Foundation of Division

## Division of fractions part 3: why invert and multiply?

- We ended the previous post with a bit of a cliffhanger, with two possible diagrams to represent $1frac34 div frac12$:
- The first of these diagrams is more familiar to students because it reflects their past work, but the second is more productive for understanding “dividing by a unit fraction is the same as multiplying by its reciprocal.”

Why is the first one more familiar? In grades 3 and 4, students study both the “how many in each (or one) group?” and “how many groups?” interpretations for division with whole numbers (see our last blog post for examples). In grade 5, they study dividing whole numbers by unit fractions and unit fractions by whole numbers. But, as we mentioned in that post, in grade 5 the “how many groups?” interpretation is easier when dividing whole numbers by unit fractions because students do not have to worry about fractions of a group. Going from $3 div frac12$ to $1frac34 div frac12$ using this interpretation feels fairly natural:

The main intellectual work here is seeing that $frac14$ cup is $frac12$ of a container, but because the structure of the problem is the same and that structure can be easily seen in the diagrams, students can focus on that one new twist. The transition also helps students see that “how many groups” questions can be asked and answered when the numbers in the division are arbitrary fractions.

So the “how many groups” interpretation is useful for understanding important aspects of fraction division and has an important role in students’ learning trajectory. It enables students to see that dividing by $frac12$ gives a result that is 2 times as great. But it doesn’t give much insight into why this should be the case when the dividend is not a whole number.

The “how much in each group” interpretation shows why. Here are diagrams using that interpretation showing $3 div frac12 = 2 cdot 3$ and $1 frac34 div frac12 = 2 cdot 1 frac34$.In fact, the structure of this context is so powerful, we can see why dividing any number by $frac12$ would double that number: $$x div frac12 = 2 cdot x = x cdot frac21$$

This is true for dividing by any unit fraction, for example $frac15$:In the diagram above, we can see that $1frac34$ is $frac15$ of a container, so a full container is $1frac34 div frac15$. Looking at the diagram, we can see why it must be that the full container is $5 cdot 1 frac34 = 1 frac34 cdot frac51$.

- With a little more work to make sense of it, we can use this interpretation to see why we multiply by the reciprocal when we divide by any fraction, for example $frac25$:In the diagram above, we can see that $1frac34$ is $frac25$ of a container, so a full container is $1frac34 div frac25$. We can see in the diagram that $frac12$ of $1frac34$ is $frac15$ of the container, so our first step is to multiply by $frac12$: $$1frac34 cdot frac12$$
- Now, just as before, to find the full container, we multiply by 5:
- $left (1frac34 cdot frac12

ight) cdot 5 = 1frac34 cdot frac52$ - This shows that dividing by $frac25$ is the same as multiplying by $frac52$!

There is nothing special about these numbers, and a similar argument can be made for dividing any number by any fraction. Now students, instead of saying “ours is not to reason why, just invert and multiply,” can say “now I know the reason why, I’ll just invert and multiply.”

Next time: Beyond diagrams.

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