Learn about ordering fractions – KS3 Maths

Key points

It can be useful to order a group of numbers by their value. Fractions can be arranged in or order.
When fractions have the same , the are compared. The greater the numerator, the greater the fraction.
When fractions have the same numerators, the denominators are compared. The greater the denominator, the smaller the fraction.
When fractions have different numerators and denominators, they are rewritten as with . The denominators then match. The numerators are compared to order the fractions.
Learning about equivalent fractions is a useful skill for ordering fractions.

Ordering fractions with the same denominator

When fractions have the same , they are ordered by the size of their numerators:

The fraction with the smallest numerator is the smallest fraction.
The fraction with the greatest numerator is the greatest fraction.
Fractions can be arranged in or order. Ascending order starts with the smallest value and ascends to the greatest value. Descending order starts with the greatest value and descends to the smallest value.
Two fractions can be compared in size using

Examples

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$Example 1: nineteen over twenty six written as a fraction. Fifteen over twenty six written as a fraction. Five over twenty six written as a fraction. Seventeen over twenty six written as a fraction. Twenty-five over twenty six written as a fraction.$
Image caption,
Arrange these fractions in ascending and descending order.
$Five fractions - the denominator is highlighted orange. Nineteen over twenty six written as a fraction. Fifteen over twenty six written as a fraction. Five over twenty six written as a fraction. Seventeen over twenty six written as a fraction. Twenty-five over twenty six written as a fraction.$
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Each fraction has the same denominator.
$Five fractions: twenty five over twenty six. Nineteen over twenty six. Seventeen over twenty six. Fifteen over twenty six. Five over twenty six. Underneath an arrow labelled descending order points from left to right. The left is labelled greatest on the left and smallest on the right.$
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Use the numerators to arrange the fractions from highest to lowest value. The fractions are in descending order.
$Five fractions. Five over twenty six. Fifteen over twenty six. Seventeen over twenty six. Nineteen over twenty six. Twenty five over twenty six. Underneath an arrow labelled ascending order points from left to right. The left is labelled smallest on the left and greatest on the right.$
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These fractions are in ascending order. The numerators are ordered from lowest to highest value.
$Example 2. Four fractions in a line. Twenty five over twenty six. Fifteen over twenty six. Five over twenty six. Nineteen over twenty six.$
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Use inequality signs to compare the fractions.
$Two pairs of fractions in a line. Twenty five over twenty six next to Fifteen over twenty six. A greater than symbol points towards twenty five over twenty six. Five over twenty six next to Nineteen over twenty six. A less than symbol points towards five over twenty six.$
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25 is greater than 15 so the first fraction is greater than the second fraction. 5 is less than 19 so the first fraction is less than the second fraction.

Question

Arrange the fractions in ascending order. You might need a pen and paper to do this.

$ \frac{5}{8} \frac{1}{8} \frac{7}{8} \frac{3}{8} $

Ordering fractions with the same numerators

To order fractions with the same , the fractions are ordered by the size of their denominators.

The fraction with the smallest denominator is the greatest fraction.
The fraction with the greatest denominator is the smallest fraction.
A has a numerator of 1. Unit fractions are also ordered using their denominators.
Fractions may be arranged in or order.

Examples

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$Example 1: Four over nine. Four over eleven. Both written as fractions.$
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Which is smaller - 4⁄9 or 4⁄11?
$A rectangle divided into nine parts. One over nine is written as a fraction in the first part. Another rectangle divided into eleven parts. One over eleven is written as a fraction in the first part.$
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The larger the denominator, the smaller each part. A larger denominator splits a shape into more parts. Each part will be smaller. The bar model shows that 1⁄11 is smaller than 1⁄9
$A rectangle divided into nine parts. The first four parts are shaded green and labelled with fractions. One over nine. Two over nine. Three over nine. Four over nine. Another rectangle underneath is divided into eleven parts. The first four parts are coloured purple labelled with fractions. One over eleven. Two over eleven. Three over eleven. Four over eleven.$
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The bar models show that 4⁄11 is less than 4⁄9. When numerators are the same, the greater the denominator, the smaller the fraction.
$Example 2: Five fractions in a horizontal line. One over five. One over three. One over eleven. One over four. One over eight.$
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Arrange the unit fractions in ascending order.
$Five fractions in a line. Each denominator is highlighted orange. One over eleven. One over eight. One over five. One over four. One over three. An arrow underneath goes form left to right labelled smallest on the left and greatest on the right. The arrow is labelled ascending order.$
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The numerators are all the same. Use the denominators to order the unit fractions. The greater the denominator, the smaller the fraction. Start with the largest denominator - this is the smallest fraction. The fractions are in ascending order.
$Example three: Five fractions in a horizontal line. Five over ninety one. Five over thirty two. Five over eight. Five over one hundred. Five over nine.$
Image caption,
Arrange these fractions in descending order.
$Five fractions in a line. Each denominator is highlighted orange. Five over eight. Five over nine. Five over thirty two. Five over ninety one. Five over one hundred. An arrow below points from left to right with the left side labelled as greatest and the right side labelled as smallest. The arrow is labelled as descending order.$
Image caption,
The numerators are all the same. Use the denominators to order the fractions. The greater the denominator, the smaller the fraction. Start with the smallest denominator – this is the largest fraction. The fractions are in descending order.

Question

Arrange the fractions in descending order. You might need a pen and paper to do this.

$ \frac{5}{11} \frac{5}{7} \frac{5}{6} \frac{5}{8} $

Ordering fractions with different denominators and numerators

To order fractions with different denominators and numerators, the denominators have to be the same. To do this:

Find the of the denominators. This will be the denominator of the equivalent fractions.
Create the by multiplying the numerator and denominator by the same value. The equivalent fractions should have the same denominators.
Compare the equivalent fractions using their numerators. The greater the numerator, the greater the fraction.
Write the original fractions in order, or compare the original fractions using symbols.

Examples

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Fractions with different denominators have to be rewritten as equivalent fractions to be able to compare them.
$Four fractions. Two values are missing and replaced with a blank box. Eight over fifteen equals a blank box over forty five. Five over nine equals a blank box over forty five.$
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Find the lowest common multiple (LCM) of the denominators. The LCM of 15 and 9 is 45
$Eight over fifteen equals twenty four over forty five. A small times three symbol is in between both fractions. Five over nine and twenty five over forty five. A small times by 5 symbol is seen in between both fractions.$
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Create equivalent fractions with a common denominator of 45. Multiply the numerator and denominator of each fraction by the same value. 8⁄15 is equivalent to 24⁄45. 5⁄9 is equivalent to 25⁄45
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Compare the equivalent fractions with the same denominators to decide which is greater than or less than. Compare the original fractions using the correct inequality sign (< or >). 8⁄15 is less than 5⁄9
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Write the fractions in ascending order.
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The lowest common multiple of 8, 5, 40 and 4 is 40. Write these denominators out. 26⁄40 already has a denominator of 40 so does not need to change.
$Five over eight equals twenty five over forty. A small multiply by five label is between both fractions. Four over five equals thirty two over forty. A small multiply by eight label is between both fractions. Twenty six over forty. Three over four equals thirty over forty. A small multiply by ten label is in between both fractions.$
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Create equivalent fractions with a common denominator of 40
$Four fractions in a line. Each numerator is highlight orange. Twenty five over forty. Twenty six over forty. Thirty over forty. Thirty two over forty. A label on the left says smallest. A label on the right says greatest. Four fractions below. Five over eight. Twenty six over forty. Three over four. Four over five. An arrow pointing from left to right says ascending order.$
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Compare the numerators of the equivalent fractions. Write the fractions in order from smallest to greatest. Then write the original set of fractions in ascending order.

Game - Comparing and ordering fractions

Practise your comparing and ordering fractions skills with these questions from our Divided Islands maths game.

Play the full Divided Islands game.

Practise ordering fractions

Activity

Arrange the fractions with the same numerators in ascending order.

Quiz

Practise what you've learned about ordering fractions with this quiz. You may need a pen and paper to help you work out the answers.

Real-world maths

Pots of paint on a mat in three colurs. Blue. Orange and yellow. A paintbrush spreads paint on the mat that the paint sits on. — Image caption,
Different shades of paint can be compared using fractions.

Shades of paint can be compared by ordering fractions. A paint colour that is $ \frac{1}{3} $ blue and $ \frac{2}{3} $ white will be darker than a paint that is $ \frac{1}{5} $ blue and $ \frac{4}{5} $ white. This is because the fraction of blue paint is greater.

When shopping online being able to compare discounts that involve fractions can be helpful. For example, a £15 jumper is cheaper if there is $ \frac{1}{3} $ off (£5 off) compared to a reduction of $ \frac{1}{4} $ (reduced by £3.75).