Dividing Fractions

I recently came across this question from Prakash Sahoo on LinkedIn:

I am really struggling with Division of fractions, (Specifically 2/3 divided by 5/6) … Why do we reciprocate the second fraction?

One week and 28 answers later, I’m still looking for an explanation that’s simple, clear, and intuitive. Let’s see if we can create one by starting with the basics.

What is Division?

We know (from Piles of Somethings) that we can arrange any group of objects into smaller groups, so that each group has the same number of objects (with the possible exception of the last group, which may contain fewer). For example, if we have 5 socks, we can make 2 pair with one sock left over. Division is simply math’s way of helping us figure out how many groups we can make, how many objects will be in each group, and how many will be left over.

A Tasty Example

There’s probably a Law of Division somewhere that states, “Every discussion about division must mention pizza.” So let’s open some boxes of fresh pizza and count the slices:  you’ll usually find each pizza cut into 8 slices, and they’re always so hot that, if you eat them right away, they’ll burn the roof of your mouth and leave those annoying “gum strings” that hang down and maddeningly distract you from all other bodily functions until you’ve rolled them around with your tongue and broken them off … but I digress.

Let’s make the pizza cool faster by separating the slices. When we’re done, we count the slices and find that we have 24. How many whole pizzas did we start with? Division gives us the answer: If we arrange the 24 slices into groups of 8, we’ll find that we have exactly 3 groups of 8 slices.

  

24 slices / 8 slices per pizza = 3 whole pizzas

Same Slices, Different Words

Here’s the fun part:  let’s say the same thing using slightly different language. Remember, we’re going to say exactly the same thing, using exactly the same pizza. We know that each pizza has been cut into 8 equal parts. We’ve been calling those parts slices, but a mathematician would scowl at us and complain that the word slice isn’t specific enough. After all, there’s no law that says we have to cut our pizza into 8 slices. We could just as well have cut it into 4 larger slices, or into 32 tiny slices, or even into 13 equal slices. (Try that someday: 13 equal slices. Good luck.) The point is that the word slice doesn’t tell us how big the piece is, or how many of them make up one pizza. What word would a mathematician use instead of slice, to make it clear that 8 slices make one pizza?  How about eighth? Each slice is exactly one eighth of a pizza. We write the fraction one eighth as 1/8.

Now we can write our pizza equation again, but let’s use the word eighth instead of slice.

We started with 24 slices … that is, 24 eighths … and wanted to know how many pizzas we could make. We found the answer by arranging the 24 slices into groups of 8 slices. That is, we divided 24 eighths by 8 eighths, and we saw that we had exactly 3 pizzas. Let’s write that down in a few different ways. Remember, I’m going to write exactly the same idea each time; I’m just going to change the words a little.

  

24 slices ÷ 8 slices = 3 whole pizzas.
24 eighths ÷ 8 eighths = 3 whole pizzas.
24/8 ÷ 8/8 = 3 whole pizzas.

The Most Important Rule of Division

That was easy, but I slipped something past you. It’s subtle, and it’s probably obvious, but it’s so absolutely critically important that I want to call your attention to it:  When we divide (in this case, 24 slices by 8 slices), we cannot divide objects of different types. That makes sense, doesn’t it?  We can divide slices by slices, or we can divide pizzas by pizzas, but we can’t divide shoelaces by turtles.

But wait! I hear you cry. We just divided pizzas into slices! Those aren’t the same things! Great observation. That’s exactly the subtlety I’m getting at. You see, in order to divide pizzas into slices, we had to think about the pizzas in terms of slices. In other words, first we changed the pizzas into slices, and then we could do the division.

Alternatively, we could have changed the slices into pizzas and done the division in terms of pizzas. As far as the math is concerned, it doesn’t matter which objects you use, as long as they’re the same. 1 pizza is exactly the same as 8 slices, and 1 slice is exactly the same as 1/8 of a pizza. But it’s easier to think about whole slices than fractions of pizzas. The Most Important Rule is simply that you need to use the same word to describe the things you’re dividing!

Why is that so important? Well, it’s actually the fundamental idea behind Prakash’s dilemma. Stick with me, and we’ll get there.

Enter the Children

Getting back to our pizza, we’ve polished off 2 entire pies when the kids come in and want to share our cheesy, sausage-y goodness. But a whole slice is too much for them to eat, so we cut all of the remaining pieces into thirds. One of the kids asks, “How much of a pizza am I eating?” Let’s try the math. Remember that the original slices were each 1/8 of a pizza, and we’ve divided each one into 3 pieces.

1 slice ÷ 3
1 eighth ÷ 3
1/8 ÷ 3

This is probably when you start to wish you’d made soup for dinner. But then you remember the picture of 3 pizzas that had serendipitously been cut into 24 slices, and you realize that 3 pizzas = 24 slices = 24 eighths = 24/8. Even better, you gleefully recognize that you’ve described the pizzas and slices in terms of eighths, and you remember that the Most Important Rule tells us we can divide objects of the same type:

1/8 ÷ 3
1/8 ÷ 24/8
1 eighth ÷ 24 eighths
1 ÷ 24
1/24

Each new slice is 1/24 of a whole pizza!

More Pizza, More Questions

Never satisfied, the little miscreants decide they each want not one, but 2 of the smaller slices. Each of the smaller slices is 1/24 of a pizza, so each child will eat 2/24.

Concerned that they haven’t confused you enough, the kids want to invite their friends over for dinner. By now, only 5 of the original slices (5/8 of an entire pizza) remain uneaten. And your little mathematicians-in-the-making want to know how many servings are left.

How can we figure this out? Well, we can cheat and use the picture: there are 5×3 = 15 small slices, and each child will eat 2. If we arrange the 15 slices into groups of 2, we’ll have 15/2 = 7 groups with one slice left over. So if we do it the hard way–by dividing fractions–the answer had better be 5×3÷2 = 7 1/2.

Will You Please Answer Prakash’s Question?

Ok, back to the math. We’re solving this problem:

5/8 ÷ 2/24
5/8 ÷ (2 × 1/24)

Note that the parentheses mean we must divide by the product of 2 and 1/24. This is the same as dividing by 2 and also dividing by 1/24. Dividing by 1/24 is the same as multiplying by 24:

5/8 ÷ 2 × 24

We’re allowed to rearrange the order of multiplications and divisions, so let’s rewrite this as

5/8 × 24 ÷ 2
5/8 × 24/2

This is precisely what Prakash asked:  Why do we reciprocate the second fraction; in other words, why is 5/8 ÷ 2/24 the same as 5/8 × 24/2?  The answer is that division by any number is the same as multiplying by its reciprocal.  For example, division by 2 is the same as multiplication by 1/2.  And conversely, division by 1/2 is the same as multiplication by 2. This hold true for both whole numbers and fractions, so division by 3/16 is the same as multiplication by 16/3: you’re dividing by (3 × 1/16), which is the same as dividing by 3 and multiplying by 16.

That makes sense! It’s simple! So why complicate things with pizza, children, and turtles? Because I like math to be intuitive, and I needed a real-world example.  Sure, we can memorize rules, but it’s so much more satisfying when the numbers represent real objects, isn’t it? And, besides, we still haven’t solved the equation.

English, Not Numbers, Please

Let’s take this one step further and rearrange the multiplications and divisions once more. Two equations ago, we wrote

5/8 × 24 ÷ 2

This time, let’s swap the division by 8 and the multiplication by 24:

5 × 24 ÷ 8 ÷ 2

Now, let’s try to figure out how this equation relates to pizza, slices, and children. We started with 5 leftover slices. We cut each of those slices into 24/8 = 3 smaller pieces. (Remember, we started with 8 slices per pizza, but we cut them so each pizza would have 24 kid-sized slices.) So the quantity

5 × ( 24 ÷ 8 ) = 5 × 3 = 15

represents the 15 smaller pieces we have left. Finally, we divide by 2 because each child will eat 2 of the smaller slices, and we want to know how many portions of 2 slices we can make.

15/2
7 1/2, or 7 with 1 left over.

So we can serve 2 slices to each of 7 children, with one slice left over.

Putting it All Together

Prakash originally asked:

I am really struggling with Division of fractions, (Specifically 2/3 divided by 5/6) … Why do we reciprocate the second fraction?

Understanding with Your Head

We can answer mathematically by recognizing that

2/3 ÷ 5/6

can be written as

2/3 ÷ (5 × 1/6)

which is the same as

2/3 ÷ 5 × 6

or, by rearranging the division and multiplication

2/3 × 6  ÷ 5

which is

2/3 × 6/5
(2 × 6) / (3 × 5)
12/15
4/5

Understanding with Your Gut

Alternatively, we can rearrange the original equation

2/3 ÷ 5/6

like this

2 × (6/3)  ÷ 5

We can interpret this to mean, “We originally had 2 pieces of something. We divided them into (6/3) smaller parts, and then we arranged the smaller parts into equal groups of 5.”

Indeed, if we start with 2 objects and divide each in half, we’ll have 4 smaller objects. If we arrange the 4 smaller objects into 5 equal groups, then each group will have 2/3 ÷ 5/6 = 4/5 of an original object.

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