Dividing Fractions: Can You Picture It?

We’ve looked at what it means to multiply fractions, including whole and mixed numbers; now it’s time for division of fractions. We’ll start here with pictures, similar to what we did for multiplication, but a little more complicated. Then next time, we’ll see additional ways to understand why we “invert and multiply”.

Dividing a whole by a fraction: 6 ÷ 1/4

We’ll start with a question from 1999, about the larger topic we examined a few weeks ago, why division by a fraction results in a larger number:

Dividing by a Fraction

When you divide, your answer should be smaller. Twenty-four divided by six is four. Four is lower than twenty-four. 

When you divide fractions, like 1/4 divided by 1/4, the answer is a larger number than 1/4. Why is this? I try telling myself that fractions are less than 1 and therefore the math may be different. One strategy we use in teaching children is that their answer must make sense. Is your answer smaller when you divide? I can't answer this.

When we divide \(\frac{1}{4}\) by \(\frac{1}{4}\), the result, 1, is greater than the number we started with, \(\frac{1}{4}\). One way to make sense of it will be to see it in terms of a physical model of the division. I answered:

Hi, Dorothy.

Yes, it's important for our math to make sense. If our number sense disagrees with the answer, either the answer or our sense has to be fixed. If we leave them both as they are, we'll be in trouble. In this case, what seems natural and obvious when we're used to working with whole numbers has to be modified when we introduce fractions.

I have seen students who seem to have learned long ago not to expect math to make sense; so unexpected answers don’t cause them to check their work, and they express no curiosity; it’s just magic. That is a tragedy. We need to either correct our answer, or correct our intuition!

A good way to do the latter is to connect the unexpected new idea to our comfortable old ideas, looking behind our intuition for an underlying reason that may explain both.

Let's think about why dividing by a whole number produces a smaller result. Dividing 24 by 4 means I want to find out how many 4's it takes to make 24. If I have 24 sticks

     ||||||||||||||||||||||||
                24

I can divide 24 by 4 by counting out 6 groups of 4 sticks:

     ||||  ||||  ||||  ||||  ||||  ||||
       1     2     3     4     5     6

Since each group contains more than one stick, there are fewer groups than sticks.

This is one of several ways to think of division; it asks, “How many groups can we make of this size?” We could also think more abstractly, “What number times 4 equals 24?”

Now let's divide 6 by 1/4. That means I want to find out how many 1/4's it takes to make 6. If I have 6 sticks

     | | | | | |
          6

and I break each of them into 4 pieces:

     ||||  ||||  ||||  ||||  ||||  ||||  
       1     2     3     4     5     6

I find that there are 24 quarter-sticks. Since each stick has been turned into 4 pieces, there are more pieces than sticks - the result of the division is greater than the original number of sticks. (By the way, this also helps to explain why dividing by a fraction is the same as multiplying by the reciprocal.)

This example shows that division by a fraction of the form \(\frac{1}{n}\) is the same as multiplying by its reciprocal, n.

You're exactly right that when you divide by a number less than one, things turn upside down, and the quotient is now larger than the dividend. In fact, you're multiplying by the reciprocal, which is a number greater than one, and you therefore increase the number. Similarly, multiplying by a number greater than one increases a number, and multiplying by a number less than one decreases it:

     24 / 4 = 6   smaller       6 * 4 = 24  bigger

    6 / 1/4 = 24  bigger     24 * 1/4 = 6   smaller

But dividing by a fraction doesn’t always mean dividing by a number less than one! The next example reverses it.

Dividing by a mixed number: 1 ÷ 1 1/2

Drawing pictures becomes harder as things get more complicated – which is why we don’t often do it! Here is another question from 1999:

Fraction Division Diagrams

How do you diagram the following problem:  1 divided by 1 1/2?

This was asked after explanations and diagramming of division problems where there were whole numbers divided by fractions less than 1. Mixed number divisors were not part of the diagramming examples but were part of the practice on the page.

I answered, starting with the easier case, dividing by a fraction less than 1:

Hi, Lanette.

I'm not sure exactly what kind of diagram you are using; if mine is too far off, maybe you can write back with an example for whole numbers and proper fractions.

The first approach that comes to mind is something like this, for 3 divided by 1/2:

    +---------------+---------------+---------------+
    |                       3                       |
    +---------------+---------------+---------------+
    +-------+-------+-------+-------+-------+-------+
    |  1/2  |  1/2  |  1/2  |  1/2  |  1/2  |  1/2  |
    +-------+-------+-------+-------+-------+-------+

There are 6 1/2's in 3.

This is very much like the stick model above, but with a single strip with length 3. To find how many pieces we can get with length \(\frac{1}{2}\), we split each unit into 2 halves, so there are \(3\times 2 = 6\) pieces. Again, we’ve multiplied by a reciprocal.

The problem we are working on will be harder in large part because the answer will be a fraction; so we need to work up to that while sticking with whole numbers. Possibly Lanette hasn’t seen such an example yet:

When the result is a fraction, it's a little more awkward; for 3 divided by 2, the best I can do is:

    +---------------+---------------+---------------+
    |                       3                       |
    +---------------+---------------+---------------+
    +---------------+---------------+---------------+---------------+
    |               2               |               2               |
    +---------------+---------------+---------------+---------------+

There are 1 1/2 2's in 1.

Notice how we have to see that the extra unit following the first whole 2 amounts to half of a second 2. We are using the 2 as a new unit to measure length, and 3 consists of \(1\frac{1}{2}\) of these new units.

Now we’re ready for the hard case:

For 1 divided by 1 1/2, it would look like

    +-------------------------------+
    |               1               |
    +-------------------------------+
    +-------------------------------+---------------+
    |                      1 1/2                    |
    +-------------------------------+---------------+

There are 2/3 of a 1 1/2 in 1 (since each 1/2 is 1/3 of 1 1/2).

Again we have to think of the \(1\frac{1}{2}\) bar (which is \(\frac{3}{2}\)) as the unit, of which 1 is \(\frac{2}{3}\). We can see there are 3 \(\frac{1}{2}\)’s in \(\frac{3}{2}\), so that each \(\frac{1}{2}\) is \(\frac{1}{3}\) of the new unit; 1 consists of 2 of these, making \(\frac{2}{3}\) of the new unit.

Whole divided by fraction again: 2 ÷ 1/5, 2 ÷ 2/5, …

Here is yet another from 1999, just asking how to divide:

Dividing Fractions

How do you divide fractions? I love math and I am good at it. This is the only thing slowing me down. I've tried and cannot find a way.

Doctor Rick answered, using a picture just to illustrate the ideas, and working up to the general rule:

Hi, Olivia.

Let's think about a few simple examples. Once you understand them, you will be able to put them together to make harder problems.

What is 2 divided by 1/5? Here are 2 blocks:

     +---------+---------+
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     +---------+---------+

Here is 1/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

To divide 2 by 1/5, we divide 2 blocks into pieces the size of the 1/5 block.

     +---------+---------+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +---------+---------+

There are 10 pieces this size in the 2 blocks. In other words,

     2 divided by 1/5 = 10

That's the same as

     2 x 5 = 10

Do you see why? To make 1/5 size blocks, we had to make each whole block into 5 pieces. In other words, we multiplied the number of pieces by 5.

Again, we are asking, how many \(\frac{1}{5}\)’s are there in 2; so we’ve counted them and found 10. In the process, we saw that dividing by \(\frac{1}{5}\) is the same as multiplying by 5.

Next problem: 

    2 divided by 2/5 = ?

Here is 2/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

Let's divide the 2 blocks into pieces this size.

     +---------+---------+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |                   |
     +---------+---------+

The bottom block is a different shape, but it's the same size, because it's made of 2 1/5-size pieces. Since the pieces we want are twice as big, there are only 1/2 as many of them. The answer is

     2 divided by 2/5 = 10/2 = 5

This is the same as

     2 x 5/2 = 5

Do you see how the two equations are related? You take the fraction, 2/5, and "turn it upside down" (in math words, we say "invert the fraction," or "take its reciprocal"). Then replace the division sign by a multiplication sign.

We counted the number of \(\frac{2}{5}\)’s that fit in 2, which in itself is not a very mathematical thing to do. But on thinking about it, we realized that the answer is just half of \(2\div\frac{1}{5}\). So we can think of the division as multiplying by 5 and then dividing by 2; or, putting it together, as multiplying 2 by the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\).

Now we can tackle harder problems. The same method always works, even if the answer isn't a whole number as in the simple examples. For instance,

     5 divided by 3/8 = ?

Invert the fraction 3/8; you get 8/3. Then multiply by this fraction:

     5 x 8/3 = 40/3 = 13 1/3

It isn’t necessary to keep making pictures (though we’ll do so later just as a demonstration) once we have the general process: To divide any number, whole or not, by a fraction, we just multiply by its reciprocal:

We can even replace the whole number in the problem with a fraction, like this:

     5/7 divided by 3/8 = ?

     5/7 x 8/3 = ?

      5     8     40       19
     --- x --- = ---- = 1 ----
      7     3     21       21

If you are comfortable with multiplying fractions, then you will have no trouble dividing fractions. Just remember: invert the fraction you are dividing by (the divisor), and multiply instead of dividing.

Pictures are a good way to get the idea initially, but sooner or later you need to generalize the idea and make it an abstract concept and a regular process.

Yet people still tend to want physical or visual examples …

Dividing a fraction by a fraction: 2/3 ÷ 1/2

A teacher in 2001 explicitly asked for a picture:

Picturing Dividing Fractions

Dr. Math,

My fifth graders can divide fractions without a problem. They use the reciprocal of a given fraction without any trouble. Our problem is that we can't draw the division problems out to prove our answers.  

For instance, please draw 2/3 divided by 1/2. We know the answer is 1 1/3. We're working with fraction pieces from circles. First we took 2/3 and taped them together. Next we place a 1/2 fraction piece on top of the 2/3 section. We could see that it took 1 piece that was 1/2 in size to cover up part of the 2/3. However, the part of the 2/3 piece that was not covered up by the 1/2 section did not equal 1/3. Help!

Thanks,
Mrs. S.

To mathematician, “proof” means starting from formal definitions and axioms. To the rest of the world, “proof” means showing that something fits their understanding of what things mean (informal definitions) and how things work (informal axioms). That is, they want a visual demonstration just to confirm the validity of the abstract work they are doing. And the difficulty they are having is just what I mentioned in an earlier example, where the answer is a fraction. It is very easy to misinterpret it!

I took up the challenge:

Hi, Katherine.

I'll use rectangles rather than circles, since that's easier to draw in text form. Here's 2/3:

    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|         |
    +---------+---------+---------+

Now I'll lay several 1/2's next to it, so we can find out how many 1/2's it takes to make 2/3:

    0                  2/3        1
    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|         |
    +---------+---------+---------+
    +--------------+--------------+
    |11111111111111|22222222222222|
    +--------------+--------------+
     <------1-----> <-?->

Now, that extra piece after the first half is 1/6 of the original bar (or circle in your case). But the question is not what number is left, but HOW MANY HALVES IS IT?

Since 1/6 is 1/3 of 1/2, that extra piece is 1/3 of a half, and the whole 2/3 is 1 1/3 halves. That's the answer you're looking for.

It's a little tricky, isn't it? When we divide, we're looking for how many of the things we're dividing by it takes to make the thing we're dividing; but when we work with fractions it's easy to get mixed up and count units rather than divisors. To demonstrate it, you'll want to cover the second 1/2 with three 1/6's, and explain that they are thirds of the 1/2.

To make this last bit more visible, we might mark out fractions using the common denominator, 6:

    0                  2/3        1
    +----+----+----+----+----+----+
    |XXXXXXXXX|XXXXXXXXX|         |
    +----+----+----+----+----+----+
    +----+----+----+----+----+----+
    |11111111111111|22222222222222|
    +----+----+----+----+----+----+
     <------1-----> <-?->

Now we can see that the extra bit is \(\frac{1}{6}\), and that this is \(\frac{1}{3}\) of that second half. We’ll be seeing the use of a common denominator again below.

The process is exactly what they described using the circles, but it’s a little easier to measure this way. And just as I did above, I suggested starting with a simpler case to get the students used to seeing the result this way:

It might help to work up to this problem with an intermediate one, where we get a fractional answer, but are dividing whole numbers. Try dividing 3 by 2:

    0         1         2         3
    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|XXXXXXXXX|
    +---------+---------+---------+
    +-------------------+-------------------+
    |1111111111111111111|2222222222222222222|
    +-------------------+-------------------+
     <--------1--------> <---?--->

This time we have 1 1/2 2's: one 2, and a 1 left over, which is 1/2 of a 2. You can probably figure out a better way to say that.

Why invert and multiply? 9/2 ÷ 3/4

We’ll close with a 2003 question, focused on illustrating  the general rule with a concrete problem:

(9/2)/(3/4): A Picture

When dividing fractions, why do you have to invert and multiply?

In my college class we were trying to find a division fraction question and everything we came up with turned into a multiplication fraction problem. Why is that?

I answered again:

Hi, Julie.

I'd like to know how your division questions "turned into" multiplications; it sounds as if you may have come close to seeing why we invert and multiply. Division problems naturally turn into multiplication problems when you see them the right way.

Let's take a simple, practical division question: I want to cut a board that is 4 1/2 feet long (that is, 9/2 foot) into pieces that are 3/4 foot long. How many pieces will I get?

Here's the board:

    +-----------+-----------+-----------+-----------+-----+
    |                                                     |
    +-----------+-----------+-----------+-----------+-----+

Here's one piece:

    +--------+
    |        |
    +--------+

I could, of course, just cut until I use up the board:

    +--------+--------+--------+--------+--------+--------+
    |        |        |        |        |        |        |
    +--------+--------+--------+--------+--------+--------+

But I'd like to think first and find the answer mathematically.

What I described there was what is called “division by repeated subtraction”: Just keep subtracting \(\frac{3}{4}\) until there is nothing left; or, equivalently, keep adding \(\frac{3}{4}\) until you reach the goal of \(\frac{9}{2}\).

It turned out that the first, and perhaps most obvious, ways to do this division visually was not the standard invert-and-multiply algorithm, but one we don’t teach often (because it is not always useful):

One way (believe it or not!) is to use a common denominator. That means choosing a different unit to measure in. Since both 4 1/2 feet and 3/4 foot can be easily expressed in inches, I'll use that (which amounts to using 12 as a common denominator). So I want to divide 4 1/2 * 12 = 54 inches by 3/4 * 12 = 9 inches; the answer is 6. (In fact, that's how I drew my picture; each character is one inch.)

This is one of the reasons we have different units. The Romans, for example, invented the inch essentially as a way to avoid having to invent fractions. But this is also a valid way to do division on paper. Rather than using 12 as a default denominator (1 inch = \(\frac{1}{12}\) foot), we can use the common denominator 4, by rewriting \(\frac{9}{2}\) as \(\frac{18}{4}\), then seeing that \(\frac{18}{4}\) divided by \(\frac{3}{4}\) is the same as 18 divided by 3, which is 6. This is sometimes a good trick for mental arithmetic.

But that's not the way we usually teach division of fractions. We want to divide in a way that uses the numerator and denominator of our fractions. So let's take those one at a time. Rather than work with the 3/4 foot boards, let's think about a 1/4 foot piece, paying attention to the denominator alone for now. How many 1/4 foot pieces will fit into our 4 1/2 foot board? Well, there will be 4 pieces in each foot, so we want to MULTIPLY 4 1/2 feet by 4, which is the DENOMINATOR of our divisor. We find that 18 of these smaller pieces will fit in our board.

But we don't want 1/4 foot pieces; we want 3/4 foot pieces. So we have to take those 1/4 foot pieces and glue each three of them back together! How many will we end up with? Well, for every three of the 18 pieces we have, we will end up with only one piece; so we will DIVIDE 18 by 3, the NUMERATOR of the divisor. That gives 6, which is our answer.

This is the same idea we’ve seen earlier: We can divide \(\frac{9}{2}\) by \(\frac{3}{4}\) by first multiplying \(\frac{9}{2}\) by 4 (getting 18), and then dividing by 3 (getting 6).

Here's a picture of what we did:

    +-----------------------------------------------------+
    |                                                     | 4 1/2 ft
    +-----------------------------------------------------+
    times 4 fourths per foot
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 18
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    divided by 3 pieces per fourth
    +--------+--------+--------+--------+--------+--------+
    |        |        |        |        |        |        | 6
    +--------+--------+--------+--------+--------+--------+

So we multiplied by the denominator, 4, and then divided by the numerator, 3, to get the answer.

And that is the same as multiplying by \(\frac{4}{3}\).

We’ll be looking at other ways to carry out the division of fractions, and other explanations, next time.