Ordering fractions
There are many methods used to order fractions, including:
- using A common denominator exists when the denominator (the number at the bottom of a fraction) of two or more fractions is the same. Common denominators help to compare or add/subtract two or more fractions. Common denominators are made by using equivalent fractions, eg a common denominator for 1/4 and 1/3 would be twelfths.
- converting fractions to decimals and then ordering
Ordering fractions using common denominators
Fractions can be compared by finding equivalent fractions with the same denominator. Common denominators are made using common multiples of the two numbers, for example 24 is the The smallest positive number that is a multiple of two or more numbers. of 8 and 12 (\(8 \times 3 = 24\) and \(12 \times 2 = 24\)). There are many other common multiples of 8 and 12 but 24 is the lowest.
Example
Place the following fractions in Ascending means starting at the bottom and going up, for example, 0, 1, 2, 3... or A, B, C, D...:
\(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{7}{12}\), \(\frac{5}{6}\), \(\frac{1}{4}\)
First, consider all of the The bottom part of a fraction. For â…ť, the denominator is 8, which represents 'eighths'. of the fractions. In this case, these are 2, 3, 12, 6 and 4. Find the lowest common multiple of these numbers. One way to do this is to look at the The multiple of a number is the number multiplied by an integer. The first four multiples of 5 are 5, 10, 15 and 20. of one of the numbers and consider each time whether the other numbers will also go into this multiple.
The lowest common multiple of 2, 3, 12, 6 and 4 is 12 (\(2 \times 6 = 12\), \(3 \times 4 = 12\), \(12 \times 1 = 12\), \(6 \times 2 = 12\) and \(4 \times 3 = 12\)) so the common denominator of all of the fractions needs to be 12.
Re-write each fraction as an equivalent fraction with 12 as the denominator:
\(\frac{1}{2} \times \frac{6}{6} = \frac{6}{12}\)
\(\frac{2}{3} \times \frac{4}{4} = \frac{8}{12} \)
\(\frac{7}{12} \times \frac{1}{1} = \frac{7}{12}\)
\(\frac{5}{6} \times \frac{2}{2} = \frac{10}{12}\)
\(\frac{1}{4} \times \frac{3}{3} = \frac{3}{12}\)
Now all the fractions have the same denominator, it is much easier to compare them. Use the numerators to place them in ascending order:
\(\frac{3}{12}\), \(\frac{6}{12}\), \(\frac{7}{12}\), \(\frac{8}{12}\), \(\frac{10}{12}\)
Lastly, write the final answer as the fractions appeared in the question.
\(\frac{3}{12} = \frac{1}{4}\)
\(\frac{6}{12} = \frac{1}{2}\)
\(\frac{7}{12} = \frac{7}{12}\)
\(\frac{8}{12} = \frac{2}{3}\)
\(\frac{10}{12} = \frac{5}{6}\)
The final answer is: \(\frac{1}{4}\), \(\frac{1}{2}\), \(\frac{7}{12}\), \(\frac{2}{3}\), \(\frac{5}{6}\).
Converting fractions to decimals and then ordering
Another method for ordering fractions is to convert fractions to decimals.
Example
Place the following fractions in Descending means starting at the top and going down.:
\(\frac{3}{4}\), \(\frac{1}{2}\), \(\frac{4}{5}\), \(\frac{3}{8}\)
First, convert each fraction to decimals.
\(\frac{3}{4}\) is worked out below.
1 of 6
\(\frac{1}{2} = 1 \div 2 = 0.5\)
\(\frac{4}{5} = 4 \div 5 = 0.8\)
\(\frac{3}{8} = 3 \div 8 = 0.375\)
Now order the decimals in descending order (as the question asked for) and use this to order the fractions:
0.8, 0.75, 0.5, 0.375
\(\frac{4}{5}\), \(\frac{3}{4}\), \(\frac{1}{2}\:\), \(\frac{3}{8}\:\)
The table shows common conversions of fractions, decimals and percentages.
| Fraction | Decimal | Percentage |
| \(\frac{1}{10}\) | 0.1 | 10% |
| \(\frac{1}{5}\) | 0.2 | 20% |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{3}\) | \(0. \dot{3}\) | \(33. \dot{3} \%\) |
| Fraction | \(\frac{1}{10}\) |
|---|---|
| Decimal | 0.1 |
| Percentage | 10% |
| Fraction | \(\frac{1}{5}\) |
|---|---|
| Decimal | 0.2 |
| Percentage | 20% |
| Fraction | \(\frac{1}{4}\) |
|---|---|
| Decimal | 0.25 |
| Percentage | 25% |
| Fraction | \(\frac{1}{2}\) |
|---|---|
| Decimal | 0.5 |
| Percentage | 50% |
| Fraction | \(\frac{1}{3}\) |
|---|---|
| Decimal | \(0. \dot{3}\) |
| Percentage | \(33. \dot{3} \%\) |