Part of this shape is shaded and part of it is not. In fact, we can see that the shape is made of five equal parts, and only three of them have been shaded in. We can call this amount \dfrac{3}{5}.

The point marked on this number line is halfway between 0 and 1. That is, the point represents the number that is as far from 0 as it is from 1. We can call this number \dfrac{1}{2}.

These numbers are examples of fractions.

Parts of fractions

The top part of a fraction is called the numerator. This tells us how many parts are in the fraction. The bottom part is called the denominator. This tells us how much of a whole each part is. The line in the middle is called the vinculum.

The fraction 4 fifths where 4 is the numerator, 5 is the denominator and the horizontal line in the middle is the vinculum.

For example, consider \dfrac{4}{5}. In this fraction, the numerator is 4 and the denominator is 5. We can call this fraction 4 on 5, 4 over 5, 4 out of 5, or 4 fifths.

Examples

Example 1

What fraction of the hexagon below is shaded?

A hexagon divided into 6 parts, 5 of which are shaded

Worked Solution

Create a strategy

Write the fraction as the number of shaded parts over the total number of parts.

Apply the idea

The hexagon is divided into 6 parts. The number of shaded parts is 5. This means that 5 is the numerator and 6 is the denominator.

So the fraction is \dfrac{5}{6}.

Idea summary

A fraction is a number which can be made of equal parts of a whole number.

A fraction is made up of:

A top number called the numerator which says how many equal parts are in the fraction
A bottom number called the denominator which says how much of a whole each part is
A line between the two numbers called the vinculum

Improper fractions and mixed numbers

Fractions like \dfrac{4}{5} make up less than a whole. We can tell because the numerator is less than the denominator. We call these proper fractions.

What about a fraction like \dfrac{8}{5}? Notice that the numerator is greater than the denominator. This means that the fraction is greater than a whole. We call these improper fractions.

A rectangle divided into 10 equal parts with 8 parts shaded.

Each row in this grid has been split into five equal parts, and eight parts of the whole have been shaded. We can think about this as eight fifths of one row being shaded. This means that we have a complete row and then three more parts shaded.

We can write this number as 1\,\dfrac{3}{5}, which we call one and three fifths.

We call numbers like this mixed numbers or mixed numerals.

Examples

Example 2

What number is plotted on the number line? Give your answer as a mixed number.

Worked Solution

Create a strategy

For the fraction part, count the number of equal spaces between the two whole numbers then count how many spaces after the previous whole number the point is located.

Apply the idea

The point is located between 3 and 4. It is greater than 3 but less than 4. So the whole number part is 3.

There 10 equal spaces between 3 and 4, so each space represents \dfrac{1}{10}. The point is 9 spaces to the right of 3, so the fraction is \dfrac{9}{10}.

The number plotted on the number line is 3\,\dfrac{9}{10}.

Idea summary

Fractions where the numerator is less than the denominator are called proper fractions. Fractions where the numerator is more than the denominator are called improper fractions.

Numbers which are made up of a whole number and a fraction are called mixed numbers or mixed numerals.

Compare fractions

Which fraction is bigger out of \dfrac{3}{8} and \dfrac{5}{8}? The first thing we can do is make a visual model for each fraction.

Two rectangles both divided into 8 equal parts. The top rectangle has 3 shaded parts. The bottom rectangle has 5 shaded parts

We can see that more of the \dfrac{5}{8} fraction bar has been shaded than the \dfrac{3}{8} fraction bar. Try creating fraction bars for other fractions with a denominator of 8. Notice that the smaller the numerator, the smaller the fraction. This works for any two fractions with the same denominator.

Which fraction is bigger out of \dfrac{4}{6} and \dfrac{4}{10}? Again, we can make a visual model for each fraction.

One rectangle is divided into 6 equal parts with 4 shaded. The other rectangle is divided into 10 equal parts with 4 shaded

Here, more of \dfrac{4}{6} and has been shaded than \dfrac{4}{10}. Try creating fraction bars for other fractions with a numerator of 4. Notice that the smaller the denominator, the bigger the fraction. This works for any two fractions with the same denominator.

Exploration

Use the following applet to practise comparing fractions. Sort the fractions by dragging them until they are inside the black boxes.

Loading interactive...

Fraction bars can help us compare fractions.

Examples

Example 3

Which inequality symbol completes the sentence: \dfrac{1}{5} ⬚ \dfrac{1}{6}?

\lt

\gt

Worked Solution

Create a strategy

Since the numerators are the same, the fraction with the larger denominator is the smaller fraction.

Apply the idea

The numerators of the fraction are the same. The denominator of \dfrac{1}{5} is smaller than the denominator of \dfrac{1}{6}. So we have \dfrac{1}{5} \gt \dfrac{1}{6}

The answer is option B.

Idea summary

Fraction bars can help us compare fractions.

For any two fractions with the same denominator: the smaller the numerator, the smaller the fraction.

For any two fractions with the same numerator: the larger the denominator, the smaller the fraction.

2.01 Name, identify, order, and compare fractions

Introduction

Ideas

Parts of fractions

Examples

Example 1

Improper fractions and mixed numbers

Examples

Example 2

Compare fractions

Exploration

Examples

Example 3

Outcomes

MA4-5NA

What is Mathspace

About Mathspace