Fractions describe parts of a whole, parts of a group and numbers between whole numbers.
You use fraction ideas whenever you divide food equally, measure part of a quantity, share something between people or describe how much of something is left.
This guide explains fractions using practical examples and simple number reasoning.
What is a fraction?
A fraction can describe one or more equal parts of a whole.
For example:
1/2
means one of two equal parts.
And:
3/4
means three of four equal parts.
The important word is equal.
The parts must be equal
Imagine a pizza shared between four people.
If it is divided into four equal pieces, each person can receive:
1/4 of the pizza
If one piece is much larger than the others, the four pieces do not each represent one quarter.
Fractions depend on equal partitioning of the same whole.
Numerator and denominator
Consider:
3/5
The bottom number is called the denominator.
It tells us that the whole has been divided into five equal parts.
The top number is called the numerator.
It tells us that we are considering three of those parts.
So:
3/5 means three parts, each of size 1/5.
Unit fractions
A unit fraction has numerator 1.
Examples include:
- 1/2
- 1/3
- 1/4
- 1/5
- 1/10
Each describes one equal part of a whole.
More parts means smaller pieces
Suppose two identical cakes are divided equally.
One cake is divided into two equal pieces.
The other is divided into eight equal pieces.
One half of the first cake is larger than one eighth of the second cake.
So:
1/2 is greater than 1/8
When the whole stays the same, dividing it into more equal pieces makes each piece smaller.
The whole must be the same when comparing parts
One half of a small pizza may be smaller than one quarter of a very large pizza.
When comparing fractions by size, make sure the fractions refer to the same whole or to equal sized wholes.
This prevents misleading comparisons.
Fractions are numbers
A fraction is not only a picture of a sliced object.
Fractions are numbers that can be located on a number line.
For example:
0 1/4 1/2 3/4 1
Each fraction has its own position between zero and one.
Fractions can also be greater than one.
Fractions greater than one
Consider:
5/4
Four quarters make one whole.
Five quarters therefore means one whole and one additional quarter.
So:
5/4 = 1 1/4
A fraction can represent a quantity greater than one whole.
Proper fractions and mixed numbers
A fraction such as:
3/4
is less than one whole.
A mixed number combines a whole number and a fraction, such as:
1 1/2
This represents one whole plus one half.
Everyday example: sharing food
Four friends share two identical pizzas equally.
Each pizza can be divided into four equal quarters.
There are eight quarters altogether.
If eight quarters are shared equally among four people, each person receives two quarters.
Two quarters are equivalent to one half.
So each person receives:
1/2 of a pizza
Equivalent fractions
Different fractions can represent the same quantity.
For example:
1/2 = 2/4
Imagine the same rectangle.
If it is divided into two equal parts and one part is shaded, half is shaded.
If each of those parts is divided again, there are four equal parts and two are shaded.
The shaded amount has not changed.
Only the way the whole is divided has changed.
More equivalent fractions
For the same whole:
1/2 = 2/4 = 3/6 = 4/8
These fractions occupy the same position on a number line.
This idea becomes important when comparing, adding and subtracting fractions.
Why multiplying top and bottom by the same number works
Start with:
1/2
Multiply both numerator and denominator by 2:
2/4
The number of selected parts doubled, but the total number of equal parts also doubled.
The proportion did not change.
That is why the two fractions are equivalent.
Simplifying a fraction
Sometimes a fraction can be written using smaller numbers without changing its value.
For example:
6/8
Both 6 and 8 can be divided by 2.
So:
6/8 = 3/4
Both forms represent the same quantity.
Compare fractions with the same denominator
Consider:
3/8 and 5/8
The pieces are the same size because both fractions use eighths.
Five eighths contains more of those pieces than three eighths.
Therefore:
5/8 is greater than 3/8
Compare unit fractions
Consider:
1/3 and 1/6
For the same whole, thirds are larger pieces than sixths.
Therefore:
1/3 is greater than 1/6
Use one half as a benchmark
One half is a useful reference point.
Consider:
3/8
Half of eight eighths is four eighths.
Since three eighths is less than four eighths:
3/8 is less than 1/2
Now consider:
5/8
Five eighths is greater than four eighths, so:
5/8 is greater than 1/2
Finding a fraction of a quantity
Suppose you want to find:
1/4 of 20
One quarter means dividing the quantity into four equal groups.
So:
20 divided by 4 = 5
Therefore:
1/4 of 20 = 5
Finding several fractional parts
Now find:
3/4 of 20
First find one quarter:
20 divided by 4 = 5
Then take three of those parts:
3 × 5 = 15
Therefore:
3/4 of 20 = 15
Everyday example: a class group
A group contains 24 learners.
One third of the group joins one activity.
To find one third:
24 divided by 3 = 8
So eight learners join that activity.
Everyday example: time
One hour contains 60 minutes.
Half an hour is:
1/2 of 60 = 30 minutes
A quarter of an hour is:
1/4 of 60 = 15 minutes
Three quarters of an hour is:
3/4 of 60 = 45 minutes
Everyday example: measuring ingredients
A recipe may use quantities such as:
- 1/2 cup
- 1/4 cup
- 3/4 cup
If you have two measurements of 1/4 cup, together they make:
2/4 cup
which is equivalent to:
1/2 cup
Equivalent fractions can therefore help with practical measurement.
Everyday example: distance
Suppose a walking route is 8 kilometres long.
You complete one quarter of the route.
Calculate:
8 divided by 4 = 2
You have completed 2 kilometres.
Fractions and division are connected
The fraction:
3/4
can also represent:
3 divided by 4
This connection helps explain why fractions appear when quantities are shared equally.
For example, if three identical items are shared equally among four people, each person's share can be represented by 3/4 of one item.
Fractions and decimals can represent the same number
Some familiar relationships are:
1/2 = 0.5
1/4 = 0.25
3/4 = 0.75
Fractions and decimals are different ways of representing numbers.
You will connect these ideas to percentages in a later Practical Mathematics article.
Fractions can describe proportions
Suppose 3 of 12 books on a shelf are science books.
The fraction is:
3/12
This simplifies to:
1/4
So one quarter of the books are science books.
Add fractions with the same denominator
Consider:
2/7 + 3/7
Both fractions use pieces of size one seventh.
Two seventh sized parts plus three seventh sized parts make five seventh sized parts.
So:
2/7 + 3/7 = 5/7
The denominator remains seven because the size of each part has not changed.
Why we do not simply add denominators
Consider:
1/4 + 1/4
One quarter plus another quarter gives two quarters:
2/4 = 1/2
It does not give two eighths.
The denominator describes the size of each fractional unit.
Subtract fractions with the same denominator
Suppose you have:
7/8
and use:
2/8
The amount remaining is:
7/8 less 2/8 = 5/8
Estimating with fractions
Fraction sense can help you judge whether an answer is reasonable.
Suppose someone claims:
1/4 of 20 = 15
One quarter is much less than the whole and also less than one half.
Half of 20 is 10.
Therefore one quarter of 20 cannot be 15.
This quick comparison reveals that the result is unreasonable.
Practice 1: Identify numerator and denominator
For:
5/8
what are the numerator and denominator?
Answer
The numerator is 5.
The denominator is 8.
Practice 2: Which unit fraction is larger?
Which is larger?
1/4 or 1/10
Answer
1/4.
If the same whole is divided into four equal parts, each part is larger than if it is divided into ten equal parts.
Practice 3: Find an equivalent fraction
Complete:
1/3 = ___/6
Answer
2/6.
Both numerator and denominator were multiplied by 2.
Practice 4: Simplify
Simplify:
4/8
Answer
1/2.
Divide both numerator and denominator by 4.
Practice 5: Find a fraction of a quantity
Find:
1/5 of 30
Answer
Divide 30 into five equal groups:
30 divided by 5 = 6
So:
1/5 of 30 = 6
Practice 6: Find several parts
Find:
3/5 of 30
Answer
One fifth of 30 is 6.
Three fifths is:
3 × 6 = 18
Practice 7: Compare with one half
Is 7/10 greater than or less than one half?
Answer
Greater than one half.
One half is equivalent to 5/10, and 7/10 is greater than 5/10.
Practice 8: Add equal sized fractional parts
Calculate:
2/9 + 4/9
Answer
6/9
This can also be simplified to:
2/3
Common fraction mistakes
- forgetting that the parts of a whole must be equal
- comparing fractions that refer to different sized wholes without noticing the difference
- assuming a larger denominator always means a larger fraction
- adding denominators when adding fractions with the same denominator
- forgetting that fractions are numbers as well as parts of objects
- forgetting to simplify when a simpler equivalent form would help
A fraction sense checklist
When working with a fraction, ask:
- What is the whole?
- Into how many equal parts is the whole divided?
- How many of those parts are being considered?
- Is the fraction less than, equal to or greater than one?
- Can I compare it with a familiar benchmark such as one half?
- Is there an equivalent form that makes the problem easier?
- Does my final answer make sense for the original quantity?
The main idea
Fractions are numbers built from equal parts.
The denominator describes the size of the fractional parts, while the numerator tells how many of those parts are being considered.
Fractions can represent parts of objects, parts of groups, points on a number line, division and proportions.
Understanding equivalent fractions, benchmarks and fractions of quantities gives you a foundation for decimals, percentages and many everyday calculations.
Continue learning
Review Practical Number Sense for Everyday Mathematics if you want to strengthen estimation and number comparison skills.
The next Practical Mathematics article will connect these ideas to percentages and everyday percentage calculations.