Compare fractions with related denominators
Theory
When one denominator is a multiple of the other, rename the smaller-denominator fraction to the larger denominator, then compare the numerators. For \(\dfrac{2}{3}\) and \(\dfrac{5}{6}\): \(\dfrac{2}{3}=\dfrac{4}{6}\), and \(\dfrac{4}{6}<\dfrac{5}{6}\).
Denominators are related when the larger one is a multiple of the smaller, like \(3\) and \(6\).
Rename the smaller-denominator fraction so its denominator matches the larger one.
With matching denominators, the fraction with the bigger numerator is larger: \(\dfrac{2}{3}=\dfrac{4}{6}<\dfrac{5}{6}\).
Rename the smaller-denominator fraction to the larger denominator, then compare the numerators.
| Compare | Rename | Result |
|---|---|---|
| \(\dfrac{2}{3}\) and \(\dfrac{5}{6}\) | \(\dfrac{2}{3}=\dfrac{4}{6}\) | \(\dfrac{2}{3}<\dfrac{5}{6}\) |
| \(\dfrac{3}{4}\) and \(\dfrac{5}{8}\) | \(\dfrac{3}{4}=\dfrac{6}{8}\) | \(\dfrac{3}{4}>\dfrac{5}{8}\) |
| \(\dfrac{1}{2}\) and \(\dfrac{3}{8}\) | \(\dfrac{1}{2}=\dfrac{4}{8}\) | \(\dfrac{1}{2}>\dfrac{3}{8}\) |
How to compare fractions with related denominators
- Check the larger denominator is a multiple of the smaller.
- Rename the smaller-denominator fraction to the larger denominator.
- Compare the numerators.
| \(\dfrac{2}{3}\) | \(=\) | \(\dfrac{4}{6}\) |
| \(=\) | \(\dfrac{4}{6}<\dfrac{5}{6}\) |
So \(\dfrac{2}{3}<\dfrac{5}{6}\).
| \(\dfrac{3}{4}\) | \(=\) | \(\dfrac{6}{8}\) |
| \(=\) | \(\dfrac{6}{8}>\dfrac{5}{8}\) |
So \(\dfrac{3}{4}>\dfrac{5}{8}\).
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{4}{8}\) |
| \(=\) | \(\dfrac{4}{8}>\dfrac{3}{8}\) |
So \(\dfrac{1}{2}>\dfrac{3}{8}\).
| \(\dfrac{2}{5}\) | \(=\) | \(\dfrac{4}{10}\) |
| \(=\) | \(\dfrac{4}{10}<\dfrac{7}{10}\) |
So \(\dfrac{2}{5}<\dfrac{7}{10}\).
Common pitfalls
Frequently asked questions
What are related denominators?
Denominators are related when the larger is a multiple of the smaller, like \(3\) and \(6\) or \(4\) and \(8\).
How do you compare fractions with related denominators?
Rename the smaller-denominator fraction to the larger denominator, then compare the numerators. \(\dfrac{2}{3}=\dfrac{4}{6}<\dfrac{5}{6}\).
Which fraction do you rename?
The one with the smaller denominator, so its denominator matches the larger one.
Why does this method work?
Because the larger denominator is a multiple of the smaller, the smaller-denominator fraction can be split into the same-sized parts.