Dividing Fractions: How and Why

Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.

Note: In this post, I am going to replace the slash (/), which we had to use on the old site, with the obelus (÷) when referring to division (as opposed to fractions), to make it easier to follow what we say.

How do I divide fractions?

Here is a basic question, from 1999:

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 Olivia’s question, starting with pictures of fractions in order to make the ideas concrete:

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 ÷ 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.

So, dividing by a fraction with 1 in the numerator amounts to multiplying by that numerator.

Next problem: 

    2 ÷ 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 ÷ 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.

Here, in addition to multiplying by the denominator, 5, we had to divide by the numerator, 2. That amounts to multiplying by the reciprocal, 5/2. And that is the standard way to divide by a fraction.

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 ÷ 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

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

     5/7 ÷ 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.

We have given many other explanations of this, both the reason for it and the process. But before I get back to those, there is another question:

Do I have to multiply by the reciprocal?

First, a question (also from 1999) asking “Why can’t we?” triggers the answer, “Actually, we can!”:

Dividing Fractions

I know how to divide fractions. I want to know WHY you cannot divide them. Can you please help me?

What Alex meant, of course, was that you can’t just divide them as they stand, but have to change it to a division. But is that really true? I played around with alternatives just to see what I could do, and replied:

There are several ways to divide fractions, some more direct than others. The method of multiplying the reciprocal is usually the easiest.

Suppose we want to divide 4/9 by 2/3. We COULD do it just by dividing the numerators and denominators:

      4     2    4 ÷ 2    2
     --- ÷ --- = ----- = ---
      9     3    9 ÷ 3    3

You can see that it works.

Why do we usually not do this? Because I chose that problem carefully so the divisions would work out. More typically you would have trouble:

      3     5    3 ÷ 5   3/5
     --- ÷ --- = ----- = ---
      4     6    4 ÷ 6   2/3

You have not gained anything, since you still have one fraction divided by another. We have to fix this up; we can do this by multiplying numerator and denominator by the least common denominator (LCD) of the two fractions:

      3     5    3/5   3/5 * 15    9
     --- ÷ --- = --- = -------- = --
      4     6    2/3   2/3 * 15   10

There’s a surprise! I usually tell students you don’t need a common denominator to divide, only to add! But here, we ended up having to simplify a “complex fraction”, and that is another place where common denominators are used. We can also use common denominators first, before dividing numerator and denominator:

Or, we could first convert both fractions to use the same denominator:

      3     5     9   10    9 ÷ 10    9
     --- ÷ --- = -- ÷ -- = ------- = --
      4     6    12   12   12 ÷ 12   10

Since I made the denominators the same, the "denominator" of the quotient works out to 1.

But there is really no division in that method, is there? It is still multiplication.

Then I rewrote that division in fraction form, and found something interesting:

A variation on this method is:

      3       9
     ---     --
      4      12     9
    ----- = ---- = --
      5      10    10
     ---     --
      6      12

You just multiply numerator and denominator by 12 in the last step to eliminate the fractions.

Do you see what I did there? Let us write it out more carefully and you will see where "multiplying by the reciprocal" is hidden:

      3      3*6
     ---     ---
      4      4*6    3*6   3*3    9
    ----- = ----- = --- = --- = --
      5      5*4    5*4   5*2   10
     ---     ---
      6      6*4

Here I have ignored LEAST common denominators and just used the simplest common denominator, the product of the denominators 4 and 6. The product is simply the product of 3/4 and 6/5.

(These days, I sometimes call the product of the two denominators the OCD: Obvious Common Denominator. Sometimes you don’t really need the least. In fact, schools today often downplay the LCD, because it tends to turn what could be an open playing field into a rule-studded  minefield. Just find something useful to multiply by, without worrying whether it is the best choice; eventually you will learn how to save effort.)

The idea is that in converting both fractions to use this common denominator, and then canceling that denominator, I ultimately just multiplied the numerator of the dividend by the denominator of the divisor, and vice versa, 3*6 and 4*5.

So in my exploration of ways to divide without multiplying by the reciprocal, I ended up actually doing that.

To answer your question simply, we divide fractions by multiplying, not because we cannot divide, but because multiplication is easier than division, and because division by itself does not always produce whole numbers. All these methods just make you multiply the numerator of the dividend by the denominator of the divisor, and vice versa, with some division to simplify the fraction; the difference is only in how much you have to think about what you are doing.

So multiplying by the reciprocal is just the best way to make the process automatic. But sometime not being automatic means you are thinking more, which can be a good thing.

A more complicated example

Our next question, probably from a student in a “math for elementary teachers” course, called for a combination of the two ideas above: illustrating a division of fractions, and showing more than one way to think about it. This time, it was a complicated division:

(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?

The phrasing of the question interested me!

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.

(In effect, one could say that what I just did was to repeatedly subtract 3/4 from 9/2; that is one way to think of division.)

Before getting to the invert-and-multiply rule (which is perhaps not the most natural thing to do when actually working with wood), I pointed out an alternative:

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.)

Once again, although we usually tell students not to use a common denominator for multiplication and division, it is a perfectly reasonable thing to do in many cases. We don’t usually think of using a smaller unit as a common denominator, but what I did here was to rewrite 9/2 as 54/12, and 3/4 as 9/12, so that the problem became 54/12 ÷ 9/12; but I never had to think of it that way when I used inches.

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.

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, again, is what it means to multiply by the reciprocal, 4/3.

More practical examples

Here is one more question from 1999:

Explaining Division of Fractions

My wife and I are helping some neighbor kids prepare for 6th grade. Since they will be entering an advanced program in science and math, those are the areas we're concentrating on. In a review of fractions, I stumbled when I could not come up with a practical explanation of how division of fractions works. Other than stating the rule about inverting the second fraction and multiplying to get the correct answer, I could not give a concrete example to help them understand the concept. For example, in the simple problem of 1/2 divided by 1/2, the correct answer is 1. However, I was unable to explain why division using whole numbers results in a smaller amount, but dividing by fractions produces an answer greater than what you started with. I hope I am clear. The kids left a few moments ago and my brain still feels like mush.

There are really several questions here: Why does dividing by a fraction yield a larger number, contrary to experience with whole numbers; where does invert-and-multiply come from; and how can we illustrate such divisions? Doctor Twe took it on, starting with an abstract view:

You're right, dividing fractions is confusing and seems counterintuitive. The logic, of course, is that division is the inverse function of multiplication. So if 1 * 1/2 = 1/2, then 1/2 ÷ 1/2 = 1. But that is distinctly unsatisfying. The "real world" doesn't provide much help, either. There simply aren't many examples in the real world of dividing by a fraction. When we visualize division, we picture splitting something into more than one part - not less than one part.

Note that his first attempt at an example used what is called a “partitive” model of division, splitting into a specified number of equal parts and asking how big each is. This can be difficult to picture with fractions. After giving a relatively unsuccessful example, he came back with some more. Not surprisingly, these are based on “quotative” models, where we split something into parts of a specified size, and ask how many there are (like my board cutting above):

I came up with a few more (and I hope better) examples. The first one is nice because it correlates directly with what we do with integers. Here they are:

Integer example:

I went to a dairy farm and bought a 10-gallon canister of milk. The canister won't fit in my refrigerator, so I want to pour it into several 2-gallon jugs. How many jugs do I need?

Solution:

     10
     --  =  5 jugs needed.
      2

Fraction example:

I went to the store and bought 1/2 gallon of milk. The container won't fit in my refrigerator (I have a "mini-fridge"), so I want to pour it into several 1/8-gallon (one pint) containers. How many containers do I need?

Solution:

     1/2
     ---  =  4 containers needed.
     1/8

In both cases, we divide the total quantity of milk by the capacity of the containers. This shows that dividing a fraction by a smaller fraction produces a value larger than one (you need more than one of the smaller containers).

This nicely answers the question about division increasing a number. Then, as I did in one answer above, he observed that this example motivates a different method of division:

This also demonstrates an alternative way to solve dividing fractions. The "real world" problem can be solved using integers by converting the quantities to pints. 1/2 gallon = 4 pints, 1/8 gallon = 1 pint. Then 4/1 = 4 containers needed. The equivalent mathematical operation is called "eliminating the fraction," and is accomplished by multiplying both the dividend and divisor by a number that will eliminate the denominators. The most efficient value to use is the Least Common Multiple (LCM) of the denominators of the two fractions - in this example 8.

     8   1/2     8*1/2     4 pints
     - * ---  =  -----  =  -------  =  4 containers.
     8   1/8     8*1/8     1 pint

But if you look carefully, you see that what he has done can also be described as multiplying 1/2 by 8, the reciprocal of the divisor.

A second example:
(A) How many 1/2-hour 'Simpsons' episodes can you watch in 1/2 hour?  
(B) How many 1/4-hour 'Rugrats' episodes can you watch in 1/2 hour?

Solution:
(A)
     1/2     1   2
     ---  =  - * -  =  1 'Simpsons' episode.
     1/2     2   1

and

(B)
     1/2     1   4
     ---  =  - * -  =  2 'Rugrats' episodes.
     1/4     2   1

Here again, we can solve the problem by converting to a smaller unit (minutes). 1/2 hour = 30 minutes, 1/4 hour = 15 minutes. So 30/30 = 1 'Simpsons' episode, and 30/15 = 2 'Rugrats' episodes. Eliminating the fraction can be accomplished by multiplying both the dividend and divisor by 30. (Note that 30 is _not_ the LCM, but it works because it is a common multiple of 2 and 4).

(A)
     60   1/2     60*1/2     30 min.
     -- * ---  =  ------  =  -------  =  1 'Simpsons' episode.
     60   1/2     60*1/2     30 min.

(B)
     60   1/2     60*1/2     30 min.
     -- * ---  =  ------  =  -------  =  2 'Rugrats' episodes.
     60   1/4     60*1/4     15 min.
A final thought:

We eliminate the fraction in the "real world" situations by converting to a smaller unit (pints instead of gallons, or minutes instead of hours). We can think of the mathematical method as converting to a smaller base unit. We are counting in 1/8ths (or /60ths, etc.) instead of 1's.

In closing, here are three more of the many explanations we have given for dividing fractions; all of these use a similar method that is more algebraic in flavor than the visual approaches I have focused on here:

Dividing Fractions [Doctor Rob, 1999]

Dividing Fractions: Why Invert? [Doctor Greenie, 2001]

Multiplying by the Reciprocal to Divide Two Fractions [Doctor Ian, 2008]

Leave a Comment

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.