Compare & order fractions (denom 2,3,4,5,6,8,10)
Theory
To decide which fraction is bigger, look at how they are built. With the same denominator, more parts is bigger. With the same numerator, smaller parts (a bigger denominator) is smaller. Otherwise, compare each fraction with one half, or rewrite both with a common denominator. This Year 5 maths page shows all four ways and how to put fractions in order.
Same denominator: the parts are the same size, so compare the numerators. \(\dfrac{5}{6} > \dfrac{4}{6}\) because \(5\) sixths is more than \(4\) sixths.
Same numerator: the number of parts is the same, so compare the size of the parts. Sevenths are smaller than fifths, so \(\dfrac{2}{7} < \dfrac{2}{5}\).
Benchmark one half: a fraction is more than a half when its numerator is more than half its denominator. \(\dfrac{4}{7}\) is more than a half; \(\dfrac{2}{5}\) is less.
Common denominator: rewrite both fractions with the same denominator using equivalent fractions, then compare the numerators. \(\dfrac{2}{3} = \dfrac{8}{12}\) and \(\dfrac{3}{4} = \dfrac{9}{12}\), so \(\dfrac{3}{4}\) is larger.
The symbol \(>\) means is greater than and \(<\) means is less than. Ascending order is smallest to largest; descending order is largest to smallest.
Choose the quickest test:
How to compare two fractions
- Same denominator? The larger numerator is the larger fraction.
- Same numerator? The smaller denominator is the larger fraction.
- Neither? Compare each with one half, or rewrite both with a common denominator.
- Write the answer with \(>\) or \(<\).
How to order several fractions
- Rewrite them all with one common denominator.
- Order the numerators.
- Write the original fractions in that order.
| \(3\) | \(<\) | \(7\) |
\(\dfrac{3}{10} < \dfrac{7}{10}\).
Both have 3 parts; eighths are smaller than fifths.
| \(\dfrac{3}{8}\) | \(<\) | \(\dfrac{3}{5}\) |
\(\dfrac{3}{8}\) is smaller.
(i) \(\dfrac{4}{7}\)
(ii) \(\dfrac{2}{5}\)
Half of \(7\) is \(3\tfrac{1}{2}\); half of \(5\) is \(2\tfrac{1}{2}\).
| \(4\) | \(>\) | \(3\tfrac{1}{2}\) |
| \(2\) | \(<\) | \(2\tfrac{1}{2}\) |
(i) More than one half.
(ii) Less than one half.
\(\dfrac{3}{10}\) is less than \(\dfrac{5}{10} = \dfrac{1}{2}\), and \(\dfrac{2}{3}\) is more than a half.
| \(\dfrac{3}{10}\) | \(<\) | \(\dfrac{1}{2} < \dfrac{2}{3}\) |
\(\dfrac{2}{3}\) is larger.
| \(\dfrac{3}{4}\) | \(=\) | \(\dfrac{15}{20}\) |
| \(\dfrac{7}{10}\) | \(=\) | \(\dfrac{14}{20}\) |
\(15 > 14\), so \(\dfrac{3}{4} > \dfrac{7}{10}\).
Rewrite in tenths.
| \(\dfrac{4}{5}\) | \(=\) | \(\dfrac{8}{10}\) |
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{5}{10}\) |
| \(\dfrac{3}{5}\) | \(=\) | \(\dfrac{6}{10}\) |
\(\dfrac{3}{10},\ \dfrac{1}{2},\ \dfrac{3}{5},\ \dfrac{4}{5}\).
| \(\dfrac{5}{8}\) | \(=\) | \(\dfrac{15}{24}\) |
| \(\dfrac{2}{3}\) | \(=\) | \(\dfrac{16}{24}\) |
\(\dfrac{2}{3}\) is larger, but only by \(\dfrac{1}{24}\).
| \(\dfrac{1}{8}\) | \(<\) | \(\dfrac{1}{5}\) |
A whole cut into \(8\) parts has smaller parts than one cut into \(5\). So \(\dfrac{1}{5}\) is bigger.
Common pitfalls
Frequently asked questions
How do you compare fractions with the same denominator?
Compare the numerators. \(\dfrac{7}{10}\) is more than \(\dfrac{3}{10}\) because \(7\) tenths is more than \(3\) tenths.
How do you compare fractions with the same numerator?
The one with the smaller denominator is bigger, because its parts are bigger. \(\dfrac{3}{5} > \dfrac{3}{8}\).
How can one half help?
If one fraction is less than a half and the other is more, you know which is bigger. \(\dfrac{3}{10} < \dfrac{1}{2} < \dfrac{2}{3}\).
What is a common denominator?
A denominator that both fractions can be rewritten with. For thirds and fifths, \(15\) works: \(\dfrac{1}{3} = \dfrac{5}{15}\) and \(\dfrac{2}{5} = \dfrac{6}{15}\).
What is ascending order?
Smallest to largest. Descending order is largest to smallest.