Ordering fractions

There are many methods used to order fractions, including:

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 lowest common multiple 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 order:

\frac{1}{2}, \frac{2}{3}, \frac{7}{12}, \frac{5}{6}, \frac{1}{4}

First, consider all of the denominators 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 multiples 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 order:

\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.

0 in units column

\frac{1}{2} = 1 \div 2 = 0.5

Example of long division (1 / 2)

\frac{4}{5} = 4 \div 5 = 0.8

Example of long division (4 / 5)

\frac{3}{8} = 3 \div 8 = 0.375

Example of long division (3 / 8)

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.

FractionDecimalPercentage
\frac{1}{10}0.110%
\frac{1}{5}0.220%
\frac{1}{4}0.2525%
\frac{1}{2}0.550%
\frac{1}{3} 0. \dot{3} 33. \dot{3} \%