Dividing Fractions

Dividing means making equal groups.

Let's look at the meaning of 24 ÷ 6:

As you can see, division has two closely related meanings: either you know how many groups you will make, or you know how many go in each group, so you can find how many groups you get.

How many 23 are in 4?

Imagine we have 4 wholes, and let's split them into groups of 23.

We took 3 wholes and split them into 25 pieces: we got 7 whole 25 groups, but the white part left over is not a 25 piece. So how can we name the white part?

7 groups of 25 + half a 25 = 712 groups of 25

How many 34 are in 5?

We took 5 wholes and split them into groups of 34. Right now we have 6 groups of 34, and there are still 2 uncolored parts. I need to find how much of a group this is.

If the shape above, with 3 parts, is 33, then 2 parts are 23. So the parts marked ??? are a 23 group.

So I have 6 whole groups and a 23 group.

First, let's see what the division means:

It cannot mean "How many 4s are in 23?" (because there aren't any), so let's look at the other meaning of division.

If we split 23 into 4 equal groups, we can find how much is in one group.

Imagine you have 23 of a pizza, and you will share this pizza with 4 friends. How much pizza does each of you get? That is the question we are answering.

First, let's build 23:

Now let's split 23 into 4 equal groups:

The circled piece is one of the equal groups. So how much does this piece show?

I split it vertically with dashed lines; you could also split it horizontally, which is even easier.

So let's cut the whole into equal parts:

The whole is now cut into 12 equal parts. My goal is to find what one group (that is, 2 parts) is equal to: 2 of the 12 equal parts.

If we split 35 into 3 equal groups, how much / how many is in one group?

If you have learned unit fractions well, you can do this right away with logic.

35 is made of three 15 unit fractions:

Let's also show it with a model:

When it is split into 3 groups, each group is a 15 piece.

When we split 35 into 5 equal groups, how much / how many is in one group?

First, let's show 35:

Let's split the shaded part into 5 equal parts:

So let's cut the whole into equal parts:

My whole is now cut into 25 equal parts, and my group has 3 parts. So the result is:

First, let's see what it can mean:

How many 14 are in 12?

Let's split 12 into groups of 14.

First, let's see what the operation means:

I have 15 pieces of 120, and I must put them into groups of 8. Each group must be made of 8 pieces of 120.

How many groups could I make?

I made 1 full group, but the 2nd group is not complete.

Each group is made of 8 pieces, and the 2nd group has 7 pieces. If it had 8 pieces, my 2nd group would be complete. So what should I call it?

If 8 pieces are 1 group, then 1 piece is a 18 group, and 7 pieces are a 78 group.

I have 1 whole group and a 78 group, so the result is:

First, let's match the unit fractions so that we can group more easily.

The operation became:

What does it mean?

How many / how much 4024 is in 1824?

When I split 1824 into groups of 4024, how many groups can I get?

You might say right away: "But there is no 4024 in 1824!"

Actually, you are partly right: there is not a whole 4024 inside 1824. So you cannot make even 1 group, which means the result must be less than 1, right?

Imagine we have 18 little pieces, each 124 long, and 40 of them make one group. Our goal is to find how much of a group the 18 pieces make.

The length of the pieces is not really important. What matters is that all the pieces are equal and that 40 of them count as 1 group.

We can take each piece as a 140 group: 40 pieces of 140, that is 4040, make 1 whole group.

18 groups of 140 make 1840 of a group.

So the result is:

There is no special operation defined for dividing fractions: we divide fractions by using multiplication.

Let's start with an example:

Dividing any number by 2 means multiplying it by 12.

Solution: