Dividing Fractions: Why Invert and Multiply?

Last week, we looked at how to visualize division of fractions; in the process, we saw that you can multiply the first fraction (dividend) by the reciprocal of the second (divisor): “invert and multiply”. Here I want to look at a few of the many times we have been asked how to do it or why this works, when we did not answer with a picture.

How do you divide fractions?

We can start with this question from Karen in 1996:

Dividing Fractions

Hi! I'm in grade 6 going on to grade 7 next year. I always get stuck on dividing fractions. Please help me!

Doctor Anthony answered, starting with the rule:

A simple rule to remember when dividing fractions is that you take the fraction on the bottom line (denominator), turn it upside down and multiply.

So 5/(1/2) is the same as 5*(2/1) = 10

You will notice that the (1/2) on the bottom line has been turned upside down to 2/1, or simply 2, and then multiplied.

It’s worth noting that he is treating the division \(5\div\frac{1}{2}\) as a fraction, \(\frac{5}{\frac{1}{2}}\). So the “top line” and “bottom line” refer to the numerator and denominator, which are the same as the dividend and divisor. This is touched upon in the posts How to Convert a Fraction to a Decimal – and Why.

$$5\div\frac{1}{2} = \frac{5}{\frac{1}{2}} = 5\times\frac{2}{1} = \frac{5}{1}\times\frac{2}{1} = \frac{5\times 2}{1\times 1} = \frac{10}{1} = 10$$

Because in the old days we had to use “/” for division, it was often hard to tell which was intended; since they ultimately mean the same thing (as we’ll be seeing), it doesn’t really make a difference, but to students who are not yet familiar with fractions, it can be confusing. I will be replacing the slash “/” with the obelus “÷” in what follows, when it means division explicitly.

But we don’t like to just state rules to follow blindly; there has to be a “why”:

The reason why this method works is that multiplying both top line and bottom line by the bottom line turned upside down causes the bottom line to become 1, and we need only then consider what happens on the top line.

Example   Simplify   5/8     (5/8)(4/3)    5/6
                     ---  =  ---------- = ----- =  5/6
                     3/4     (3/4)(4/3)     1

The idea here is that a fraction is unchanged when you multiply both parts by the same quantity. This is covered in the post How Do Equivalent Fractions Work?.

If we multiply both numbers by the reciprocal of the divisor, the divisor becomes 1 and has no effect; while the new dividend is the product with that reciprocal. We’ll be seeing the same explanation repeated several times here with different details.

It’s worth noting, too, that we use the same idea when we divide by a decimal: To divide \(12.3\) by \(4.56\), we “move the decimal point” by multiplying both numbers by 100 (the implied denominator of the divisor, \(4.56 = \frac{456}{100}\)), and get \(1230\div 456\), which we can do more easily by hand: $$12.3\div 4.56 = (12.3\times 100)\div (4.56\times 100) = 1230\div 456 = 2.697…$$

In practice we don't need to carry out all the steps shown above. We just do the 'invert and multiply' that I described at the start.

    5/8
    --- =   5/8 * 4/3 = 5/6
    3/4

That is, $$\require{cancel}\frac{5}{8}\div\frac{3}{4} = \frac{5}{8}\times\frac{4}{3} = \frac{5\times \cancel{4}}{\underset{2}{\cancel{8}}\times 3} = \frac{5}{6}$$

Just the “how”, please

Karen wrote back,

Hi, again! I still don't understand what you meant. Please send me more details about dividing fractions.

Doctor Robert answered this time, keeping it simple by focusing on the “how” rather than the “why”, and starting with multiplication:

If you know how to multiply fractions, dividing is pretty easy.  You multiply fractions by multiplying the numerators together to get the numerator of the answer, and multiplying the denominators together to get the denominator of the answer.  

For example 2/3 times 4/5 = 8/15.

That is, $$\frac{2}{3}\times\frac{4}{5}=\frac{2\times 4}{3\times 5}=\frac{8}{15}$$

To divide one fraction by another, you simply invert the divisor (invert means make the numerator the denominator and make the denominator the numerator) and multiply. 

For example 2/3 divided by 4/5 is the same as 2/3 times 5/4 which is 10/12 or 5/6 when reduced.

That is, $$\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\times\frac{5}{4}=\frac{2\times 5}{3\times 4}=\frac{10}{12}=\frac{5}{6}$$

Again, this could have been simplified before multiplying: $$\frac{2}{3}\div\frac{4}{5}=\frac{\cancel{2}}{3}\times\frac{5}{\underset{2}{\cancel{4}}}=\frac{1\times 5}{3\times 2}=\frac{5}{6}$$

Keep – Change – Flip?

While we’re looking at “how”, here is a 2007 question about different ways to say what we are doing:

Comments on Dividing Fractions

I was reading your posts on how to divide fractions and it took me a while to understand.  It's basically multiply and find the reciprocal right?  My teacher told me you can use the KCF to help. KCF stands for "Keep it. Change it. Flip it."

ex. 4/5 divided by 1/5

 K   C    F
4/5  ÷   1/5

4/5  x   5/1  =  4

We can still use that too, right?

The idea here is that we change a division to a multiplication by Keeping the first number as is, Changing the operation from multiplication to division, and Flipping the second number over (making its reciprocal). It’s a way to avoid the big word “reciprocal”, just as “invert” and “upside down” have been used above! They all mean the same thing.

I answered this:

Hi, Tiffany.

You didn't say which page(s) you read; sometimes we explain things the long way in order to talk about why we do something, rather than just tell you quickly what to do.  We usually avoid just giving mnemonics like your KCF, but there's nothing wrong with it.  It's merely a very short way to say the same thing we say.

I chose not to say anything negative about what Tiffany was taught (which is taught even in remedial Arithmetic classes in my college, whose students need things kept simple). But we prefer understanding over rote methods, and teaching useful words rather than avoiding them, where feasible.

My own "quick version", which is more in line with the way mathematicians like to think, is this: "division is defined as multiplication by the reciprocal".  That is,

  a ÷ b = a * 1/b

So, to divide a by b, we multiply a by the reciprocal of b, which means exactly what your teacher says: we "change" the operation from division to multiplication, and we "flip" the divisor to use its reciprocal.

Beginning students may not recognize that when b is a fraction, “\(\frac{1}{b}\)” means to flip the fraction over; I’m taking a more advanced perspective here. It really means to divide 1 by b. And why does that produce the reciprocal? Because any fraction times its reciprocal is 1, so 1 divided by any fraction flips it over.

Similarly, I've heard subtraction explained as KCC, or "Keep Change Change", meaning that you change the subtraction to an addition, and change the sign of the second number (the subtrahend).  The mathematician's version of that one is that subtraction is defined as addition of the negative (additive inverse).  That is,

  a - b = a + -b

So really, subtraction and division are just addition and multiplication with the second operand "inverted" in an appropriate way.  I rarely hear anyone point out the similarity of the two rules (or definitions), but I think it's very useful to see it.

Both division and subtraction are inverse functions, and both are accomplished by applying the direct operation to the appropriate inverse of the second operand.

Why do we do it that way?

Here is a question from 1997 focusing on the “why”:

Dividing Fractions

I was reading over the explanation on dividing fractions and why we have to flip the second fraction, and I am still confused. I understand that we have to do it in order to do the problem, but I want to know the reason why we have to flip it. I want a simpler explanation.

Doctor Rob answered, starting with an illustration quite different from our pictures last week:

In a fraction its value represents the number of things of size measured by the denominator you add up to get one thing of size measured by the numerator. For example, 31/11 is the number of 11-pound objects you put together to to get one 31-pound object. When the denominator is itself a fraction, as in your situation, this does not change.

Although he is writing this as a fraction and calling it a fraction, in terms of the question it is a division: How many 11-pound gold bars can be melted down to make a 31-pound statue? That’s asking, “What number, times 11, gives 31?” \(31\div 11 = \frac{31}{11}\), because \(\frac{31}{11}\times 11 = 31\).

Now we divide by a fraction:

For example, (14/3)÷(2/5) is the number of objects, each weighing 2/5 pound, which are put together to get a weight of 14/3 pounds. 

How many does it take to make one pound? Answer: 5/2, each weighing 2/5 pound, will make one pound. How did we get 5/2?  By inverting 2/5, or, in other words, finding its "reciprocal."  Why is this the right answer?  Because (5/2)*(2/5) = (5*2)/(2*5) = 10/10 = 1.  

Then to get 14/3 pounds it will take (14/3)*(5/2) objects, each weighing 2/5 pound. (Of course this equals 35/3, so you'll need 11 and 2/3 objects each weighing 2/5 pound to make 14/3 pounds.)

There is nothing special about 14, 3, 2, and 5. They could be replaced by any four numbers - except zero: remember, you can't divide by zero!

The key idea here is that a number times its reciprocal is 1. If each bar weighs \(\frac{2}{5}\) pounds, then \(\frac{5}{2}\) bars will weigh \(\frac{5}{2}\times\frac{2}{5} = \frac{10}{10} = 1\) pound, and \(\frac{14}{3}\) pounds is \(\frac{14}{3}\) times that many bars. So we multiply by the reciprocal: $$\frac{14}{3}\div\frac{2}{5} = \frac{14}{3}\times\frac{5}{2} = \frac{\overset{7}{\cancel{14}}\times 5}{3\times \cancel{2}} = \frac{35}{3} = 11\frac{2}{3}$$

We can do the same thinking abstractly, using the idea of multiplying numerator and denominator by the same number, namely the reciprocal of the denominator:

Another way to look at this is to start with your original compound fraction, and multiply the numerator and denominator of the fraction by 5/2.  You get:

   14     14   5     14   5
   --     -- * -     -- * -
    3      3   2      3   2    14   5
  ---- = -------- = -------- = -- * -
    2      2   5        1       3   2
    -      - * -
    5      5   2

Why did we pick 5/2?  Because 5/2 is the reciprocal of 2/5, the denominator, and when you multiply any number by its reciprocal, you get 1, which is what we want to create in the denominator.

Making an equivalent fraction

We’ll head toward the finish with a 2008 question about “how”, that got a “why” answer:

Multiplying by the Reciprocal to Divide Two Fractions

I can not figure out how you divide fractions, like 2/9 by 7/45.

I do not get reciprocals and how you change it from dividing to multiplying.

Doctor Ian answered:

Hi Amanda,

The "rule" is that you invert the bottom fraction and multiply by it instead.  Here's an explanation of why that works:

  Multiplying and Dividing Fractions
  http://mathforum.org/library/drmath/view/58080.html

We’ll look at this long answer soon. For now, though, he gave the same kind of answer we’ve been seeing with a slightly different twist, treating the division as a big fraction:

A shorter explanation is that you can just use equivalent fractions to get the result.  That is, if we have a fraction like 

  2
  -
  9

we can multiply it by n/n, where n is any number but zero; and we get a fraction with the same value, e.g., 

  2   5   10
  - * - = --          2/9 and 10/45 have the same value
  9   5   45

That’s the general principle. Now we apply it to division of two fractions, rather than to a single fraction:

Does that look familiar?  If so, consider that we can start with this,

    2
    -
    9
  ---- 
    7
   --
   45

and multiply both the top and bottom by 45/7, i.e., the reciprocal of the bottom fraction:


    2   45
    - * --
    9    7
  ----------
    7   45
   -- * --
   45    7

Now, what's the bottom going to equal?  It has to be 1, right?  So the whole thing is equal to 

    2   45
    - * --
    9    7
  ----------

      1

or just

    2   45
    - * --
    9    7

Which is to say, it's equal to the numerator times the reciprocal of the denominator.

Amanda replied,

THANK YOU.....I hope my grade goes up I do understand it NOW.....THANK YOU SOOOOOOO MUCH.

We’ve had some students find this explanation fully satisfying, and others not. That’s why interaction is essential, and multiple answers are good.

Fractions as two operations combined

Here is the 2001 question Doctor Ian referred to, which deals with both multiplication and division. I considered including it last week, but didn’t because it uses pictures only for the multiplication part. It seems to fit here, because he takes a totally different perspective:

Multiplying and Dividing Fractions

We were just trying to figure out how to understand division of fractions and multiplication of fractions. It is weird because when I divided 1/2 by 1/2 on the calculator, I got 1, but when I multiplied them, I got 1/4.... 

I want to be able to explain in a drawing. HELP!
Adom and Jamie

It is not uncommon that students discover facts like these while playing with a calculator, and become curious. That is one of the good roles calculators can play in education!

Multiplying by a fraction

Doctor Ian answered, starting with pictures of multiplication:

Hi guys,

Let's start from the beginning.  When you multiply by a whole number, you replicate something some number of times:

    * * *  x  4  =  * * * 
                    * * * 
                    * * * 
                    * * *

And when you divide by a whole number, you cut something into some number of pieces, and throw away all but one of them:

    * * *  ÷  4  =  * * *
    * * *
    * * *
    * * *

When you multiply by a fraction, you do BOTH of these things.  For example, to multiply by 3/4, you divide by 4 and then multiply by 3:

  * * * * *  x (3/4)  =  * * * * *  x  3  =  * * * * *
  * * * * *                                  * * * * *
  * * * * *                                  * * * * *
  * * * * *

or you multiply by 3 and then divide by 4:                            

  * * * * *  x (3/4)  =  * * * * *  ÷  4  =  * * * * *
  * * * * *              * * * * *           * * * * *
  * * * * *              * * * * *           * * * * *
  * * * * *              * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *

Either way you end up with the same result.

Here we are thinking of a fraction as a two-step process, a multiplication and a division combined. So multiplication by 3/4 means multiplying by 3 and dividing by 4. That’s one benefit of the notation: \(\frac{3}{4} = 3\times\frac{1}{4}\).

So there are no new ideas here, just a couple of old ideas bunched together. 

If the fraction is less than 1, you lose more in the division than you gain in the multiplication. (For example, when you multiply a dollar by 3/4, you break the dollar into 4 quarters, and triple one of them, leaving you with 3 quarters.)

If the fraction is greater than 1, you gain more in the multiplication than you lose in the division. (For example, when you multiply a dollar by 5/4, you break the dollar into 4 quarters, and quintuple one of them, leaving you with 5 quarters.)

If the fraction is equal to 1, the multiplication and the division cancel each other out, and you end up where you started.

And we can think of the fraction itself as a multiplication and a division starting with 1:

If it helps, you can think of 

  1/4 * 1/4

as being the same thing as

  1 * 1/4 * 1/4

So, you start with a whole something; cut it into four pieces, and keep one; then cut _that_ into four pieces, and keep one.  Think about doing that with a pizza. How many pieces of the final size would you need to make a whole pizza? You'd need 16 of them, right? 
Conveniently, 

  1/4 * 1/4 is 1/16.

We’re dividing by 4 twice, which is the same as dividing by 16.

How about 2/3 * 3/2?  Again, you start with a whole something; cut it into three pieces, and double one of them. Then you take those two pieces, cut each one in half, and triple each one of them.  What do you end up with?  Six pieces, each of which is 1/6th of the original thing. That is, you end up where you started. 

You can actually do this with paper and scissors, and that's not a bad idea if it isn't yet clear how this works. If you don't absolutely understand what it means to multiply something by a fraction, you are going to have great difficulty in every math class that you take from now on.

He didn’t say so, but the example he chose here demonstrated the idea of a reciprocal, which is essential for division: \(\frac{2}{3}\times\frac{3}{2} = 1\) because it both multiplies and divides by 2, and both divides and multiplies by 3.

Dividing

Okay, so what about dividing by a fraction? Well, there is no good way that I know of to illustrate that with the kinds of pictures that you can use for multiplication. But perhaps that's not such a big deal, because division is primarily just another way of looking at multiplication.  That is, once we know something like

  24 = 6 * 4

this is really exactly the same piece of information as

  24 ÷ 6 = 4     and    24 ÷ 4 = 6

isn't it?  In fact, we DEFINE division this way.  We say that

  a ÷ b = c     WHENEVER   c * b = a

This is the general definition of division, which works even when reciprocals don’t exist (as when you are restricted to whole numbers only). Division is the inverse operation to multiplication; it undoes it.

That's the definition of division. That's what division MEANS. So how does this apply to fractions? Well, now that you know how to multiply fractions, you understand why

  9 * 2/3 = 6

right?  We divide 9 by 3 to get 3, and multiply that by 2 to get 6; or we multiply 9 by 2 to get 18, and divide by 3 to get 6.  Either way, this isn't a surprising fact.  

Well, because of how we've DEFINED division, if we say

  9 * 2/3 = 6

this is really exactly the same piece of information as

   6 ÷ 9 = 2/3       and      6 ÷ (2/3) = 9

So knowing a product of two numbers also tells us about a couple divisions, namely the product divided by either of the factors gives the other factor. In particular, since \(9\times\frac{2}{3} = 6\), we know that \(6\div\frac{2}{3} = 9\). Hold that thought …

Make sure you understand why this is true.  If it helps, look at these patterns again, and match up the letters with the numbers:

  b * c = a      <------->     a ÷ b = c, and a ÷ c = b
            
So, what do we have to do to get 6 from 9?  Well, we could cut it in half, and then triple what we get; or we could triple it, and take half of what we get. In other words, we get from 6 to 9 in this way:

  6 * (3/2) = 9

But we also know that

  6 ÷ (2/3) = 9

And when two things are equal to the same thing, they must be equal to each other, right?  That means

  6 * (3/2) = 6 ÷ (2/3)

So when you want to divide by a fraction, you invert the fraction and multiply instead.

Since division undoes multiplication, division by a fraction divides by the numerator and multiplies by the denominator; that is, it multiplies by the reciprocal.

This is really all that's going on.  Note that I've used some particular numbers: 6, 9, 2/3, etc. - but you can work it out using only letters, and you'll see that so long as we accept the DEFINITION of division, we have to invert and multiply in order to divide by a fraction. If we did anything else, we'd get crazy results.  

In a sense, this goofy rule for dividing by fractions is the price we pay to keep the rest of math running smoothly.

This is the essence of a proof, which could be stated algebraically, but has been done instead by example.