Compare & order fractions
Theory
Comparing and ordering fractions is a Year 4 maths skill. When two fractions have the same denominator you just count the parts, and when they have the same numerator the one with the bigger denominator is smaller. This page shows how to use fraction bars, number lines, equivalent fractions and the benchmark one half to compare and order fractions of the same whole, and how to tell which fraction is closer to 1. Comparing fractions with related denominators, such as quarters and eighths, goes a little beyond Year 4 and continues in Year 5 in both curricula. In Western Australia this is taught in Year 5. Curriculum: VC2M4N03 / AC9M4N03.
A fraction has a numerator (the top number, how many parts you have) and a denominator (the bottom number, how many equal parts the whole is cut into). To compare fractions fairly, the wholes must be the same size.
There are two easy cases. When two fractions have the same denominator, the parts are the same size, so the fraction with more parts is larger: \(\dfrac{5}{6} > \dfrac{3}{6}\). When two fractions have the same numerator, a bigger denominator cuts the whole into more, thinner parts, so it is the smaller fraction: \(\dfrac{1}{3} > \dfrac{1}{6}\).
Equivalent fractions cover the same amount even though the numbers look different, like \(\dfrac{1}{2} = \dfrac{2}{4}\). To compare fractions that do not match, rename one so both have the same denominator, then count the parts.
Two more tools help. One half is a handy benchmark: a fraction is more than one half when its top number is more than half of its bottom number, like \(\dfrac{4}{6}\). And a fraction is closer to 1 when the part still missing from the whole is smaller.
These comparison facts are worth remembering. They work whenever the fractions are parts of the same whole.
How to compare two fractions
- Check the wholes. Make sure both fractions are parts of the same size whole, or use bars and number lines of equal length.
- Look at the numbers. If the denominators are the same, compare the numerators. If the numerators are the same, the bigger denominator is the smaller fraction.
- Rename if needed. If neither matches, rename one fraction so both have the same denominator, then compare the parts and write \(>\), \(<\) or \(=\).
- Or use one half. If one fraction is less than one half and the other is more, the one more than one half is larger.
How to find which fraction is closer to 1
- Find the part missing. Work out how much each fraction needs to make one whole. \(\dfrac{4}{5}\) needs \(\dfrac{1}{5}\) more.
- Compare the missing parts. The fraction with the smaller missing part is closer to 1, and it is the larger fraction.
How to order three or more fractions
- Rename to the same parts where you can, such as eighths or tenths.
- Order the top numbers from smallest to largest, then write the fractions in their original form.
| \(\text{both count sixths}\) | \(\to\) | \(\text{same size parts}\) |
| \(5\text{ parts}\) | \(>\) | \(3\text{ parts}\) |
| \(\dfrac{5}{6}\) | \(>\) | \(\dfrac{3}{6}\) |
The larger fraction is \(\dfrac{5}{6}\).
| \(\text{same top number}\) | \(\to\) | \(1\) |
| \(\text{thirds}\) | \(>\) | \(\text{sixths}\) |
| \(\dfrac{1}{3}\) | \(>\) | \(\dfrac{1}{6}\) |
The answer is \(>\).
| \(\dfrac{2}{4}\) | \(=\) | \(\dfrac{4}{8}\) |
| \(\dfrac{4}{8}\) | \(<\) | \(\dfrac{5}{8}\) |
| \(\dfrac{2}{4}\) | \(<\) | \(\dfrac{5}{8}\) |
The larger fraction is \(\dfrac{5}{8}\).
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{3}{6}\) |
| \(\text{in sixths}\) | \(:\) | \(\dfrac{1}{6},\ \dfrac{3}{6},\ \dfrac{5}{6}\) |
In order: \(\dfrac{1}{6},\ \dfrac{1}{2},\ \dfrac{5}{6}\).
| \(\dfrac{4}{5}\) | \(\to\) | \(\text{needs }\dfrac{1}{5}\text{ more}\) |
| \(\dfrac{7}{8}\) | \(\to\) | \(\text{needs }\dfrac{1}{8}\text{ more}\) |
| \(\dfrac{1}{8}\) | \(<\) | \(\dfrac{1}{5}\) |
\(\dfrac{7}{8}\) is missing the smaller part, so it is closer to \(1\) (and larger than \(\dfrac{4}{5}\)).
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{5}{10}\) |
| \(3\text{ tenths}\) | \(<\) | \(5\text{ tenths}\) |
| \(\dfrac{3}{10}\) | \(<\) | \(\dfrac{1}{2}\) |
The answer is \(<\).
Common pitfalls
Frequently asked questions
Which is bigger, 1/6 or 1/10?
One sixth is bigger. When the top numbers are the same, the fraction with the smaller bottom number has bigger parts. Sixths cut the whole into 6 parts, but tenths cut it into 10, so each sixth is longer than each tenth.
How do you compare fractions with the same denominator?
When the bottom numbers are the same, the parts are the same size, so just compare the top numbers. More parts means a larger fraction, so \(\dfrac{5}{6} > \dfrac{3}{6}\).
How do you compare fractions with the same numerator?
When the top numbers are the same, look at the bottom numbers. A bigger bottom number cuts the whole into more, thinner parts, so it is the smaller fraction. For example \(\dfrac{1}{3} > \dfrac{1}{6}\).
Why does a bigger denominator make a smaller fraction?
The denominator tells how many equal parts the whole is cut into. The more pieces you cut, the smaller each piece becomes, so one part out of 6 is smaller than one part out of 3.
What does it mean if two fractions are equivalent?
Equivalent fractions cover the same amount of the whole even though the numbers look different. For example, \(\dfrac{2}{5}\) and \(\dfrac{4}{10}\) shade the same length of a strip, so \(\dfrac{2}{5} = \dfrac{4}{10}\).
How do you put fractions in order from smallest to largest?
Rename them so they share the same bottom number, then order them by their top numbers. You can also line them up on a number line or with fraction bars: fractions further to the right, or with more shading, are larger.