# The meaning of division

To understand division of fractions, first, one must understand what division means;
Division is basically separating numbers into groups of other numbers.

In other words,

The group number X The number of elements in each group = The divided number

Let’s look at the meaning of 24:6

Let’s say we have 24 marbles, and they are placed inside these 6 different buckets one by one…

Note that, by the end of the procedure, you will obtain 6 buckets with 4 marbles each inside.

As a result, when you divide 24 marbles into 6 groups, there are 4 marbles in each group.

2nd Meaning: When we divide 24 into 6, how many groups are created? (How many 6’s are there in 24 is the answer of the question)

When we divide 24 marbles into groups of 6, 4 equal groups were created…

As you can see, division has two equivalent meanings. Either you must know
how many groups you will be splitting into or how many you want in each group; so that, we can
see how many total groups are formed.

The fundamental logic for division is same when it comes to
the division of fractions.

## Dividing Integers to Fractions

$$\displaystyle 4:\frac{2}{3}$$

How many $$\displaystyle \frac{2}{3}$$ are there in a 4?
Let’s imagine we have complete 4 and let’s divide it into $$\displaystyle \frac{2}{3}$$ groups.

There are 6 groups of $$\displaystyle \frac{2}{3}$$

## The result is 6

$$3:\frac{2}{5}=?$$

How many $$\displaystyle \frac{2}{5}$$'s are there in 3?
When we divide 3 , into $$\displaystyle \frac{2}{5}$$ groups, how many groups do I have?

We took 3 as a whole and divided it into $$\displaystyle \frac{2}{5}$$ parts, there are 7 complete $$\displaystyle \frac{2}{5}$$ 's . However, one cannot fit another $$\displaystyle \frac{2}{5}$$  into remaining white area. How can we express what the white part is? Above  you see a $$\displaystyle \frac{2}{5}$$ th part.It consists of 2 equal parts. If two parts are equal to $$\displaystyle \frac{2}{5}$$ th ,a single part is a half of $$\displaystyle \frac{2}{5}$$

7 , $$\displaystyle \frac{2}{5}$$ th + A half of $$\displaystyle \frac{2}{5}$$ th =  $$\displaystyle 7\frac{1}{2}$$ of $$\displaystyle \frac{2}{5}$$ th.

So the result is $$\displaystyle 7\frac{1}{2}$$

$$\displaystyle 5:\frac{3}{4}$$

How many $$\displaystyle \frac{3}{4}$$ 's are there in 5 ?

We took a whole 5 and divided into $$\displaystyle \frac{3}{4}$$ groups . Now there are 6 , $$\displaystyle \frac{3}{4}$$ groups. And we still have 2 empty white parts. We need to find a way to evaluate these regions .

We had said the $$\displaystyle \frac{3}{4}$$ piece  was one group.  One group consists of 3 parts.

If one group consists of 3 parts, what is a group that consists of 2 parts? If the above shape of 3 parts represents  $$\displaystyle \frac{3}{3}$$  then 2 parts represent $$\displaystyle \frac{2}{3}$$ So the groups with ??? are a $$\displaystyle \frac{2}{3}$$ 'rd group . So we have 6 whole groups and one $$\displaystyle \frac{2}{3}$$ 'rd group .

So the result is ;
$$\displaystyle 6\frac{2}{3}$$

## Dividing fractions by whole (integers)

$$\displaystyle \frac{2}{3}:4=?$$

Let’s have a look at the meaning of what the division will be:

How many 4 are there in $$\displaystyle \frac{2}{3}$$  rds ? We can't do that  (because it is not possible to do so), so let us have a  look at the other meaning of division.

If we divide $$\displaystyle \frac{2}{3}$$ rds into 4 equal parts, we can see how many parts are there  in each group.

Imagine you have $$\displaystyle \frac{2}{3}$$  rd of a piece of pizza in your hand and you want to divide this among 4 friends. How much of a piece of pizza will each of you have?

The solution to the question is below.

First of all let’s create the  $$\displaystyle \frac{2}{3}$$ ‘rds piece;

Now let’s divide $$\displaystyle \frac{2}{3}$$ ’rds into a group of 4,

I have preferred to divide vertically with white lines, you can do it horizontally – in fact, and it is easier.

The main rule of Fractions: For something to be a fraction, the whole needs to be made up of equal parts.

So let’s divide the whole into equal parts:

The whole has been divided into 12 equal parts. Our goal is to calculate how much one group (two pieces) actually is.

So there are 2 within 12 equal parts;

$$\displaystyle \frac{2}{12}$$

$$\frac{3}{5}:3$$

If we divide $$\displaystyle \frac{3}{5}$$ ’ths into 3 groups, how many will there be in the group?
If you have learnt unit fractions well we can calculate this easily using the fallowing logic.

$$\displaystyle \frac{3}{5}$$ ’ths are made up of three $$\displaystyle \frac{1}{5}$$  ’ths .

$$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{3}{5}$$

So the result is ; $$\frac{3}{5}$$

Let’s show this with modelling it:

When we divide the groups into three each group is made up of $$\displaystyle \frac{1}{5}$$ th.

$$\displaystyle \frac{3}{5}:5$$

When we divide $$\displaystyle \frac{3}{5}$$  ths into 5 equal groups, how many are there in a group?

First let’s show  $$\displaystyle \frac{3}{5}$$  ’ths ;

Let’s first divide the coloured part into 5 equal parts.

The main rule of fractions: For something to be a fraction, the whole needs to be made
up of equal parts.

So let’s divide the whole into equal parts:

The whole has been divided into 25 parts, and in this group there are 3 parts.
So the result is;

$$\displaystyle \frac{3}{25}$$

## Dividing Fractions by Fractions

$$\frac{1}{2}: \frac{1}{4}$$

First of all, -let’s look at the meaning,

How many $$\displaystyle \frac{1}{4}$$  ths , in $$\displaystyle \frac{1}{2}$$

First let’s form our $$\displaystyle \frac{1}{2}$$

Let’s divide $$\displaystyle \frac{1}{2}$$ th's into $$\displaystyle \frac{1}{4}$$ ths .

There are two $$\displaystyle \frac{1}{4}$$ ths in $$\displaystyle \frac{1}{2}$$
so the result is ;
$$\displaystyle 2$$

$$\frac{3}{4}: \frac{2}{5}$$

First let’s look at what this means. How many $$\displaystyle \frac{2}{5}$$ ’ths are there in $$\displaystyle \frac{3}{4}$$ ’s?

When we divide $$\displaystyle \frac{3}{4}$$ ’s so that there are  $$\displaystyle \frac{2}{5}$$’ths in each group, how many groups do we have?

To easily group, -let us pair the unit fractions. The objective is to make equal fractions so we can group them. Imagine pears in a bag, how can we group these? If there were only apples, for example, I could group them together as 3’s or 5’s. . It is actually quite easy to understand when we actually pair them.

$$\displaystyle \frac{3}{4}.\frac{5}{5}=\frac{15}{20}$$
15 units are   $$\displaystyle \frac{1}{5}$$
$$\displaystyle \frac{2}{5}.\frac{4}{4}=\frac{8}{20}$$
8 units are $$\displaystyle \frac{1}{20}$$

We have equal unit fractions at $$\displaystyle \frac{1}{20}$$

So now the division is ;

$$\displaystyle \frac{15}{20}: \frac{8}{20}$$

How many $$\displaystyle \frac{8}{20}$$ 's in $$\displaystyle \frac{15}{20}$$ ?

When we group $$\displaystyle \frac{15}{20}$$’s into $$\displaystyle \frac{8}{20}$$ how many groups are formed?

Each group must consist of 8 , $$\displaystyle \frac{1}{20}$$ .

How many groups have been formed?

We have formed one complete group, but the second one is not complete. Each group consists of 8 parts; where there are 7 parts in the second group. If it was 8 parts then we would have 2 complete groups, so how can we call this?

If 8 parts is one group, one piece is $$\displaystyle \frac{1}{8}$$’th group, 7 pieces are a $$\displaystyle \frac{7}{8}$$ ’th group.

We have 1 whole group and one $$\displaystyle \frac{7}{8}$$ th group, so the result is ;

$$\displaystyle 1 \frac{7}{8}$$

$$\displaystyle 1 \frac{3}{4}:\frac{10}{6}$$

First lets pair the unit fractions, so we can group them easily.

$$\displaystyle \frac{3}{4}.\frac{6}{6}=\frac{18}{24}$$
$$\displaystyle \frac{10}{6}.\frac{4}{4}=\frac{40}{24}$$
We have paired the unit fractions at  $$\displaystyle \frac{1}{24}$$ .

The equation has been reduced to;

$$\displaystyle \frac{18}{24}: \frac{40}{24}$$

What does this mean?

How many $$\displaystyle \frac{40}{24}$$’s are there in $$\displaystyle \frac{18}{24}$$ .?

You can say tell straight away; there are no $$\displaystyle \frac{40}{24}$$ 's in $$\displaystyle \frac{18}{24}$$ ?

You are actually right, there are no $$\displaystyle \frac{40}{24}$$ 's in $$\displaystyle \frac{18}{24}$$  . So we cannot form 1 group. So the result must be less than 1, right?

Let there be 18 pieces   that are formed of  $$\displaystyle \frac{1}{24}$$ lengths.
40 of these make a group. Our goal is to see how many groups are made out of 18 of these.

We can take each piece as a $$\displaystyle \frac{1}{40}$$ , 40 pieces $$\displaystyle \frac{1}{40}$$'s so $$\displaystyle \frac{40}{40}$$ is 1 group.
A group of 18 $$\displaystyle \frac{1}{40}$$'s make a $$\displaystyle \frac{18}{40}$$ group.
So the result is : $$\displaystyle \frac{18}{40}$$

## Dividing Fractions with an Equation

There is not a certain equation for dividing fractions so we use multiplication.
Let’s start by giving an example;
If we divide any number by 2, this also means multiplying by  $$\displaystyle \frac{1}{2}$$

We will use cross multiplication for dividing fractions.

$$\displaystyle \frac{3}{4}: \frac{2}{5}$$

The first fraction remains the same, the second fraction flips and then the fractions are multiplied. This leads to cross multiplication.

$$\frac{3}{4}. \frac{5}{2}=\frac{15}{8}$$    $$\displaystyle \frac{3}{4}. \frac{5}{2}=\frac{15}{8}$$

Things to watch out for when dividing fractions:

- By writing 1 underneath a whole number you can show this number as a fraction.

Turn the whole fractions into compound fractions – Make sure the whole number is added to the top of fraction .

Don’t forget that you will always carry out your calculation after turning it into
the following form:    $$\displaystyle \frac{a}{b}: \frac{c}{d}$$

$$\displaystyle 5: \frac{7}{8}$$
So, write 1 below the whole number to show it as a fraction;

$$\displaystyle \frac{5}{1}: \frac{7}{8}$$

Let’s now use the rule and flip the second fraction on its head.
$$\displaystyle \frac{5}{1}. \frac{8}{7}=\frac{40}{7}$$
so the result is :
$$\displaystyle \frac{40}{7}$$

$$\displaystyle 2 \frac{3}{5}: \frac{4}{6}$$

If the fractions are a whole number then according to the rule we add 2  and we get:
$$\displaystyle \frac{3}{5}$$ .

$$\displaystyle \frac{13}{5}:\frac{4}{6}$$
According to the rule the first fraction remains the same and we flip the second.
$$\displaystyle \frac{13}{5}. \frac{6}{4} = \frac{78}{20}$$
so the result is :   $$\displaystyle \frac{78}{20}$$