Writing, Simplifying and Ordering Fractions

What you'll learn

How to write a fraction from a group of objects or an amount.
How to simplify fractions into their simplest form.
How to spot equivalent fractions.
How to order fractions and decide which fraction is closer to a given value.

What a fraction means

A fraction shows part of a whole or part of a group. For example, if 3 out of 8 equal parts are shaded, the fraction shaded is $38\frac{3}{8}$ .

Definition

Numerator and denominator

The numerator is the top number in a fraction. It tells you how many parts you have.
The denominator is the bottom number in a fraction. It tells you how many equal parts there are in total.

The diagram shows $38\frac{3}{8}$ because 3 equal parts are shaded out of 8 equal parts altogether.

Fraction diagram showing 3 shaded parts out of 8, labelled numerator and denominator

Writing a fraction from a group

When a question asks, “What fraction are red?” or “What fraction are blue?”, the total number usually goes on the bottom.

Key Idea

Fraction of a group

If you are choosing some objects from a group, use:

numerator = number of chosen objects
denominator = total number of objects

Example

Writing a fraction from a group

A bag contains 20 counters. 6 of the counters are yellow. What fraction of the counters are yellow?

The total number of counters is 20, so the denominator is 20.
The number of yellow counters is 6, so the numerator is 6.
The fraction that are yellow is $620\frac{6}{20}$ .

Finding “the rest”

Sometimes the question tells you how many objects are one colour, then asks about the remaining objects. The rest means what is left after you subtract.

Example

Writing a fraction for the remaining objects

There are 13 pencils in a pot. 8 pencils are black. The rest are orange. What fraction of the pencils are orange?

Start with the total number of pencils: 13.
Find how many are orange by subtracting: 13 - 8 = 5.
The numerator is 5 because 5 pencils are orange.
The denominator is 13 because there are 13 pencils altogether.
The fraction that are orange is $513\frac{5}{13}$ .

Common Mistake

Forgetting the total

Do not put the number “left over” on the bottom. The denominator is the total number in the group, not the number that are not chosen.

Equivalent fractions

Equivalent fractions have the same value, even though they look different. For example, $12\frac{1}{2}$ and $24\frac{2}{4}$ are equivalent because they both represent the same amount.

You make equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

Example

Spotting the fraction that is not equivalent

One of these fractions is not equivalent to $35\frac{3}{5}$ : $610\frac{6}{10}$ , $915\frac{9}{15}$ , $1220\frac{12}{20}$ , $1530\frac{15}{30}$ , $1830\frac{18}{30}$ . Find it.

Simplify $610\frac{6}{10}$ by dividing top and bottom by 2 to get $35\frac{3}{5}$ .
Simplify $915\frac{9}{15}$ by dividing top and bottom by 3 to get $35\frac{3}{5}$ .
Simplify $1220\frac{12}{20}$ by dividing top and bottom by 4 to get $35\frac{3}{5}$ .
Simplify $1530\frac{15}{30}$ by dividing top and bottom by 15 to get $12\frac{1}{2}$ .
Simplify $1830\frac{18}{30}$ by dividing top and bottom by 6 to get $35\frac{3}{5}$ .
The fraction that is not equivalent to $35\frac{3}{5}$ is $1530\frac{15}{30}$ .

Simplifying fractions

To simplify a fraction, divide the numerator and denominator by the same number. The value of the fraction stays the same, but the numbers become smaller.

Definition

Simplest form

A fraction is in simplest form when the numerator and denominator have no common factor apart from 1.

A factor is a whole number that divides exactly into another number. For example, 6 is a factor of 24 because 24 can be divided by 6 with no remainder.

Example

Simplifying a fraction

Write $3648\frac{36}{48}$ in its simplest form.

Look for a number that divides exactly into both 36 and 48.
12 divides into both numbers, so divide the numerator and denominator by 12.
The numerator becomes 3 because 36 divided by 12 is 3.
The denominator becomes 4 because 48 divided by 12 is 4.
So $3648\frac{36}{48}$ simplifies to $34\frac{3}{4}$ .

Tip

If you cannot spot the biggest factor

You can simplify in stages. For example, divide by 2, then by 2 again, then by 3 if possible. As long as you divide the top and bottom by the same number each time, the fraction stays equivalent.

Common Mistake

Dividing only one part

Never divide just the top or just the bottom. To keep a fraction equivalent, whatever you do to the numerator must also be done to the denominator.

Writing an increase as a fraction

If a question says “write the increase as a fraction of last year’s cost”, the denominator is the original amount.

Example

Increase as a fraction of the original amount

A bus pass cost £30 last year. This year it costs £42. Write the increase as a fraction of last year’s cost.

Find the increase: £42 - £30 = £12.
The increase is 12, so the numerator is 12.
The original cost was £30, so the denominator is 30.
The fraction is $1230\frac{12}{30}$ .
Simplify by dividing the numerator and denominator by 6.
The increase as a fraction of last year’s cost is $25\frac{2}{5}$ .

Ordering fractions

To put fractions in order, it is usually easiest to give them a common denominator.

Definition

Common denominator

A common denominator is a denominator that several fractions can all be changed to.

Once the denominators are the same, compare the numerators. The smaller numerator gives the smaller fraction.

The number line below shows how fractions can be compared once they are written with the same denominator. Fractions further to the right are larger.

$Number line showing fractions ordered from one fifth to one half using twentieths$

Example

Ordering fractions from smallest to largest

Put these fractions in order, starting with the smallest: $14\frac{1}{4}$ , $25\frac{2}{5}$ , $310\frac{3}{10}$ , $12\frac{1}{2}$ , $15\frac{1}{5}$ .

Choose a common denominator. A good choice is 20 because 4, 5, 10 and 2 all divide into 20.
Change each fraction into twentieths.
The fractions become $14=520\frac{1}{4}=\frac{5}{20}$ , $25=820\frac{2}{5}=\frac{8}{20}$ , $310=620\frac{3}{10}=\frac{6}{20}$ , $12=1020\frac{1}{2}=\frac{10}{20}$ and $15=420\frac{1}{5}=\frac{4}{20}$ .
Order the twentieths from smallest to largest: $420\frac{4}{20}$ , $520\frac{5}{20}$ , $620\frac{6}{20}$ , $820\frac{8}{20}$ , $1020\frac{10}{20}$ .
Convert back to the original fractions: $15\frac{1}{5}$ , $14\frac{1}{4}$ , $310\frac{3}{10}$ , $25\frac{2}{5}$ , $12\frac{1}{2}$ .

Tip

Same denominator shortcut

If two fractions have the same denominator, just compare the numerators. For example, $720\frac{7}{20}$ is bigger than $320\frac{3}{20}$ because 7 parts is more than 3 parts.

Which fraction is closer?

To decide which fraction is closer to a number, find the distance or gap between each fraction and the target.

For example, a fraction just below 1 might be closer than a fraction just above 1. You need to compare the gaps, not just guess by looking.

Example

Which fraction is closer to 1?

Which fraction is closer to 1: $89\frac{8}{9}$ or $109\frac{10}{9}$ ?

Find the gap between $89\frac{8}{9}$ and 1. Since 1 is the same as $99\frac{9}{9}$ , the gap is $19\frac{1}{9}$ .
Find the gap between $109\frac{10}{9}$ and 1. Since 1 is the same as $99\frac{9}{9}$ , the gap is $19\frac{1}{9}$ .
The gaps are the same, so both fractions are equally close to 1.

Example

Which fraction is closer to one half?

Which fraction is closer to $12\frac{1}{2}$ : $25\frac{2}{5}$ or $58\frac{5}{8}$ ?

Compare $25\frac{2}{5}$ with $12\frac{1}{2}$ . Use denominator 10: $25=410\frac{2}{5}=\frac{4}{10}$ and $12=510\frac{1}{2}=\frac{5}{10}$ , so the gap is $110\frac{1}{10}$ .
Compare $58\frac{5}{8}$ with $12\frac{1}{2}$ . Use denominator 8: $12=48\frac{1}{2}=\frac{4}{8}$ , so the gap is $18\frac{1}{8}$ .
Compare the gaps $110\frac{1}{10}$ and $18\frac{1}{8}$ . Since tenths are smaller parts than eighths, $110\frac{1}{10}$ is the smaller gap.
Therefore $25\frac{2}{5}$ is closer to $12\frac{1}{2}$ .

Exam technique

In the exam

For “fraction of” questions, put the chosen amount on top and the total amount on the bottom.
For simplifying, divide the numerator and denominator by the same common factor.
For ordering fractions, change them to a common denominator before comparing.
For “closer to” questions, show the gap from the target for each fraction.

Self review

Check yourself

Can you explain what the numerator and denominator mean?
Can you simplify a fraction by dividing the top and bottom by the same number?
Can you order fractions by changing them to a common denominator?

Writing, Simplifying and Ordering Fractions

What you'll learn

How to write a fraction from a group of objects or an amount.
How to simplify fractions into their simplest form.
How to spot equivalent fractions.
How to order fractions and decide which fraction is closer to a given value.

What a fraction means

A fraction shows part of a whole or part of a group. For example, if 3 out of 8 equal parts are shaded, the fraction shaded is $38\frac{3}{8}$ .

Definition

Numerator and denominator

The numerator is the top number in a fraction. It tells you how many parts you have.
The denominator is the bottom number in a fraction. It tells you how many equal parts there are in total.

The diagram shows $38\frac{3}{8}$ because 3 equal parts are shaded out of 8 equal parts altogether.

Fraction diagram showing 3 shaded parts out of 8, labelled numerator and denominator

Writing a fraction from a group

When a question asks, “What fraction are red?” or “What fraction are blue?”, the total number usually goes on the bottom.

Key Idea

Fraction of a group

If you are choosing some objects from a group, use:

numerator = number of chosen objects
denominator = total number of objects

Example

Writing a fraction from a group

A bag contains 20 counters. 6 of the counters are yellow. What fraction of the counters are yellow?

The total number of counters is 20, so the denominator is 20.
The number of yellow counters is 6, so the numerator is 6.
The fraction that are yellow is $620\frac{6}{20}$ .

Finding “the rest”

Sometimes the question tells you how many objects are one colour, then asks about the remaining objects. The rest means what is left after you subtract.

Example

Writing a fraction for the remaining objects

There are 13 pencils in a pot. 8 pencils are black. The rest are orange. What fraction of the pencils are orange?

Start with the total number of pencils: 13.
Find how many are orange by subtracting: 13 - 8 = 5.
The numerator is 5 because 5 pencils are orange.
The denominator is 13 because there are 13 pencils altogether.
The fraction that are orange is $513\frac{5}{13}$ .

Common Mistake

Forgetting the total

Do not put the number “left over” on the bottom. The denominator is the total number in the group, not the number that are not chosen.

Equivalent fractions

Equivalent fractions have the same value, even though they look different. For example, $12\frac{1}{2}$ and $24\frac{2}{4}$ are equivalent because they both represent the same amount.

You make equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

Example

Spotting the fraction that is not equivalent

One of these fractions is not equivalent to $35\frac{3}{5}$ : $610\frac{6}{10}$ , $915\frac{9}{15}$ , $1220\frac{12}{20}$ , $1530\frac{15}{30}$ , $1830\frac{18}{30}$ . Find it.

Simplify $610\frac{6}{10}$ by dividing top and bottom by 2 to get $35\frac{3}{5}$ .
Simplify $915\frac{9}{15}$ by dividing top and bottom by 3 to get $35\frac{3}{5}$ .
Simplify $1220\frac{12}{20}$ by dividing top and bottom by 4 to get $35\frac{3}{5}$ .
Simplify $1530\frac{15}{30}$ by dividing top and bottom by 15 to get $12\frac{1}{2}$ .
Simplify $1830\frac{18}{30}$ by dividing top and bottom by 6 to get $35\frac{3}{5}$ .
The fraction that is not equivalent to $35\frac{3}{5}$ is $1530\frac{15}{30}$ .

Simplifying fractions

To simplify a fraction, divide the numerator and denominator by the same number. The value of the fraction stays the same, but the numbers become smaller.

Definition

Simplest form

A fraction is in simplest form when the numerator and denominator have no common factor apart from 1.

A factor is a whole number that divides exactly into another number. For example, 6 is a factor of 24 because 24 can be divided by 6 with no remainder.

Example

Simplifying a fraction

Write $3648\frac{36}{48}$ in its simplest form.

Look for a number that divides exactly into both 36 and 48.
12 divides into both numbers, so divide the numerator and denominator by 12.
The numerator becomes 3 because 36 divided by 12 is 3.
The denominator becomes 4 because 48 divided by 12 is 4.
So $3648\frac{36}{48}$ simplifies to $34\frac{3}{4}$ .

Tip

If you cannot spot the biggest factor

You can simplify in stages. For example, divide by 2, then by 2 again, then by 3 if possible. As long as you divide the top and bottom by the same number each time, the fraction stays equivalent.

Common Mistake

Dividing only one part

Never divide just the top or just the bottom. To keep a fraction equivalent, whatever you do to the numerator must also be done to the denominator.

Writing an increase as a fraction

If a question says “write the increase as a fraction of last year’s cost”, the denominator is the original amount.

Example

Increase as a fraction of the original amount

A bus pass cost £30 last year. This year it costs £42. Write the increase as a fraction of last year’s cost.

Find the increase: £42 - £30 = £12.
The increase is 12, so the numerator is 12.
The original cost was £30, so the denominator is 30.
The fraction is $1230\frac{12}{30}$ .
Simplify by dividing the numerator and denominator by 6.
The increase as a fraction of last year’s cost is $25\frac{2}{5}$ .

Ordering fractions

To put fractions in order, it is usually easiest to give them a common denominator.

Definition

Common denominator

A common denominator is a denominator that several fractions can all be changed to.

Once the denominators are the same, compare the numerators. The smaller numerator gives the smaller fraction.

The number line below shows how fractions can be compared once they are written with the same denominator. Fractions further to the right are larger.

$Number line showing fractions ordered from one fifth to one half using twentieths$

Example

Ordering fractions from smallest to largest

Put these fractions in order, starting with the smallest: $14\frac{1}{4}$ , $25\frac{2}{5}$ , $310\frac{3}{10}$ , $12\frac{1}{2}$ , $15\frac{1}{5}$ .

Choose a common denominator. A good choice is 20 because 4, 5, 10 and 2 all divide into 20.
Change each fraction into twentieths.
The fractions become $14=520\frac{1}{4}=\frac{5}{20}$ , $25=820\frac{2}{5}=\frac{8}{20}$ , $310=620\frac{3}{10}=\frac{6}{20}$ , $12=1020\frac{1}{2}=\frac{10}{20}$ and $15=420\frac{1}{5}=\frac{4}{20}$ .
Order the twentieths from smallest to largest: $420\frac{4}{20}$ , $520\frac{5}{20}$ , $620\frac{6}{20}$ , $820\frac{8}{20}$ , $1020\frac{10}{20}$ .
Convert back to the original fractions: $15\frac{1}{5}$ , $14\frac{1}{4}$ , $310\frac{3}{10}$ , $25\frac{2}{5}$ , $12\frac{1}{2}$ .

Tip

Same denominator shortcut

If two fractions have the same denominator, just compare the numerators. For example, $720\frac{7}{20}$ is bigger than $320\frac{3}{20}$ because 7 parts is more than 3 parts.

Which fraction is closer?

To decide which fraction is closer to a number, find the distance or gap between each fraction and the target.

For example, a fraction just below 1 might be closer than a fraction just above 1. You need to compare the gaps, not just guess by looking.

Example

Which fraction is closer to 1?

Which fraction is closer to 1: $89\frac{8}{9}$ or $109\frac{10}{9}$ ?

Find the gap between $89\frac{8}{9}$ and 1. Since 1 is the same as $99\frac{9}{9}$ , the gap is $19\frac{1}{9}$ .
Find the gap between $109\frac{10}{9}$ and 1. Since 1 is the same as $99\frac{9}{9}$ , the gap is $19\frac{1}{9}$ .
The gaps are the same, so both fractions are equally close to 1.

Example

Which fraction is closer to one half?

Which fraction is closer to $12\frac{1}{2}$ : $25\frac{2}{5}$ or $58\frac{5}{8}$ ?

Compare $25\frac{2}{5}$ with $12\frac{1}{2}$ . Use denominator 10: $25=410\frac{2}{5}=\frac{4}{10}$ and $12=510\frac{1}{2}=\frac{5}{10}$ , so the gap is $110\frac{1}{10}$ .
Compare $58\frac{5}{8}$ with $12\frac{1}{2}$ . Use denominator 8: $12=48\frac{1}{2}=\frac{4}{8}$ , so the gap is $18\frac{1}{8}$ .
Compare the gaps $110\frac{1}{10}$ and $18\frac{1}{8}$ . Since tenths are smaller parts than eighths, $110\frac{1}{10}$ is the smaller gap.
Therefore $25\frac{2}{5}$ is closer to $12\frac{1}{2}$ .

Exam technique

In the exam

For “fraction of” questions, put the chosen amount on top and the total amount on the bottom.
For simplifying, divide the numerator and denominator by the same common factor.
For ordering fractions, change them to a common denominator before comparing.
For “closer to” questions, show the gap from the target for each fraction.

Self review

Check yourself

Can you explain what the numerator and denominator mean?
Can you simplify a fraction by dividing the top and bottom by the same number?
Can you order fractions by changing them to a common denominator?

Writing, Simplifying and Ordering Fractions

What you'll learn

What a fraction means

Writing a fraction from a group

Finding “the rest”

Equivalent fractions

Simplifying fractions

Writing an increase as a fraction

Ordering fractions

Which fraction is closer?

Recap questions

You've reached the end

Writing, Simplifying and Ordering Fractions

What you'll learn

What a fraction means

Writing a fraction from a group

Finding “the rest”

Equivalent fractions

Simplifying fractions

Writing an increase as a fraction

Ordering fractions

Which fraction is closer?

Recap questions

You've reached the end

Writing, Simplifying and Ordering Fractions

What you'll learn

What a fraction means

Writing a fraction from a group

Finding “the rest”

Equivalent fractions

Simplifying fractions

Writing an increase as a fraction

Ordering fractions

Which fraction is closer?

Recap questions

You've reached the end

Writing, Simplifying and Ordering Fractions Revision Guide

Writing, Simplifying and Ordering Fractions

What you'll learn

What a fraction means

Writing a fraction from a group

Finding “the rest”

Equivalent fractions

Simplifying fractions

Writing an increase as a fraction

Ordering fractions

Which fraction is closer?

Recap questions

You've reached the end

Writing, Simplifying and Ordering Fractions Revision Guide