Placing Fractions on the Number Line

In this article, we'll learn how to order fractions on the number line with ease, speed, and without any trouble.
Let's first look at the number line and analyze it.

Here we see a number line from $0$ to $5$.
We notice that the line is divided into $5$ sections, and each mark indicates an increase of $1$.
However, if we look at this number line:

We'll see that each mark indicates an increase of $2$.

To place fractions on the number line, the first thing we need to understand is how much we increase from one segment to the next.
The lines that mark the end of each segment on the number line are called points. It is also common to say that – represents the space between the two points.

Complete the missing numbers on the number line:

Solution:
To fill in the numbers, first let's mark, with an arc, the spaces that are between the dots of the two given numbers.

We will notice that there are $3$ arcs between the number $20$ and $80$.
Remember, all the spaces are identical.
Let's subtract $80-20=60$.
$60$ is the total space.
Since there are $3$ arcs, we will divide the $60$ by $3$ to find the measure of each arc.
We will find that:
$60 ÷ 3 = 20$
Therefore, each arc measures $20$.
We will place marks on the number line so we can easily fill in the remaining numbers:

Now that we know a bit more about the number line and have learned how to find the distance between two points, we can move forward.

Finding the Measure of the Arc Between Two Points in the Following Way:
Subtract the two given numbers and keep the difference.
Count the number of arcs that are between the numbers.
Divide the result of the subtraction by the number of arcs to find out the measure of each arc.

Placing the Numbers: Reduction and expansion can be applied if necessary.

Put the fractions $\frac{1}{2}$ and $1\frac{1}{2}$ on the number line.

Let's ask ourselves, what is the difference between the $2$ given numbers?
For example, between $1$ and $2$?
The difference is $1$.
How many arcs are there between $1$ and $2$?
$4$ arcs.
We will divide the difference we calculated by the number of arcs and we will get:
$1:4=\frac{1}{4}$
Each arc measures $1 \over 4$.
Now let's think about where to place the $1 \over 2$.
We can scale up the $1 \over 2$ by $2$ and we get $2 \over 4$.
Therefore, $1 \over 2$ will be placed here:

Now let's move on to $1 \frac{1}{2}$
We see that there is a whole number 1 in the fraction, so it will be located between $1$ and $2$
We saw that each segment measures $1 \over 4$ and deduced that $1 \over 2$ is equivalent to $2 \over 4$.
Therefore, $1 \frac{1}{2}$ will be located here:

Find the number that the arrow points to:

Solution:
First, let’s find out how much each segment represents.
Take the two given numbers $1$ and $2$
and count how many segments there are between them.
There are $7$ segments between $1$ and $2$.
Subtract:
$2-1=1$
Now divide by $7$
$1 \div 7=\frac{1}{7}$
Each segment represents $1 \over 7$.
Now, let's see how many segments we need to jump to reach the arrow:
The answer is $4$, so the number will be $1 \frac{4}{7}$.
Make sure not to forget the whole number $1$.

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