Ordering Fractions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Ordering Fractions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Ordering fractions means arranging a set of fractions from least to greatest (or greatest to least) by converting them to a common denominator or to decimals so their sizes can be directly compared.

Convert all fractions to a common denominator and then read off the order from the numerators.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Ordering fractions is repeated comparison using one shared scale.

Common stuck point: The procedure for ordering fractions is the easy part; the trap is ordering by denominator size alone. Asking "Can I place every fraction on the same scale?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I place every fraction on the same scale?

Worked Examples

Example 1

easy

Order

\frac{1}{2}

\frac{1}{3}

, and

\frac{1}{4}

from least to greatest.

Answer

\frac{1}{4} < \frac{1}{3} < \frac{1}{2}

First step

All fractions have numerator

1

(unit fractions). Larger denominator

\Rightarrow

smaller piece.

Full solution

2
Order of denominators from largest to smallest: $4 > 3 > 2$ .
3
So the fractions from least to greatest: $\frac{1}{4} < \frac{1}{3} < \frac{1}{2}$ .

For unit fractions (numerator = 1), the fraction with the largest denominator is the smallest because you are dividing a whole into more pieces. This is a useful shortcut that applies only when numerators are equal.

Example 2

medium

Order

\frac{5}{6}

\frac{3}{4}

\frac{7}{12}

, and

\frac{2}{3}

from least to greatest.

Four fractions located on the number line, least to greatest

Example 3

medium

Order least to greatest using a common denominator:

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

Example 4

medium

Use the benchmark

\frac{1}{2}

to order

\frac{4}{9}, \frac{6}{11}, \frac{3}{7}

least to greatest.

Example 5

hard

Order least to greatest using cross-multiplication:

\frac{5}{7}, \frac{7}{10}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Order

\frac{2}{5}

\frac{3}{10}

, and

\frac{1}{2}

from greatest to least.

Example 2

hard

Arrange

\frac{4}{9}

\frac{5}{12}

\frac{7}{18}

, and

\frac{11}{36}

from least to greatest.

Example 3

easy

Order from least to greatest:

\frac{1}{5}

\frac{3}{5}

\frac{2}{5}

Example 4

easy

Order from least to greatest:

\frac{1}{2}

\frac{1}{4}

\frac{1}{3}

Example 5

easy

Order from greatest to least:

\frac{2}{7}

\frac{5}{7}

\frac{4}{7}

Example 6

easy

Which fraction comes first when ordering

\frac{3}{4}

\frac{1}{4}

least to greatest?

Example 7

easy

Order using the benchmark

\frac{1}{2}

\frac{1}{3}

and

\frac{2}{3}

Example 8

easy

Order least to greatest:

0

\frac{1}{2}

1

The three benchmark values equally spaced on the number line

Example 9

easy

Order least to greatest:

\frac{3}{10}

\frac{7}{10}

\frac{1}{10}

Example 10

easy

Order greatest to least:

\frac{1}{6}

\frac{1}{2}

\frac{1}{3}

Example 11

medium

Order least to greatest:

\frac{2}{3}

\frac{3}{4}

\frac{5}{8}

Example 12

medium

Order least to greatest:

\frac{1}{2}

\frac{4}{9}

\frac{5}{9}

Example 13

medium

Order least to greatest:

\frac{3}{5}

\frac{5}{8}

\frac{1}{2}

Example 14

medium

Order greatest to least:

\frac{7}{8}

\frac{5}{6}

\frac{11}{12}

Example 15

medium

Three runners finished a race having run

\frac{3}{4}

\frac{5}{6}

, and

\frac{2}{3}

of a practice course. Order their distances least to greatest.

Example 16

medium

Convert to decimals and order least to greatest:

\frac{3}{8}

\frac{2}{5}

\frac{1}{3}

Example 17

challenge

Order least to greatest:

\frac{5}{7}

\frac{7}{9}

\frac{2}{3}

\frac{3}{4}

Example 18

challenge

Five fractions all lie between

\frac{1}{2}

and

\frac{2}{3}

. Could

\frac{7}{12}

be one of them? Justify.

7/12 located between the two boundary fractions on the number line

Example 19

challenge

Arrange least to greatest and identify the median:

\frac{1}{4}

\frac{2}{5}

\frac{1}{2}

\frac{3}{5}

\frac{7}{10}

Example 20

medium

Order least to greatest:

\frac{1}{3}

\frac{2}{5}

\frac{1}{2}

Example 21

medium

Order greatest to least:

\frac{2}{3}

\frac{3}{5}

\frac{4}{7}

Example 22

medium

Order least to greatest:

\frac{1}{4}

\frac{2}{7}

\frac{1}{3}

Example 23

easy

Order least to greatest:

\frac{2}{9}, \frac{5}{9}, \frac{8}{9}

Example 24

easy

Order least to greatest:

\frac{1}{5}, \frac{1}{8}, \frac{1}{2}

Example 25

easy

Order greatest to least:

\frac{3}{8}, \frac{5}{8}, \frac{1}{8}

Example 26

easy

Order least to greatest:

\frac{4}{6}, \frac{1}{6}, \frac{5}{6}

Example 27

easy

Order least to greatest:

\frac{2}{10}, \frac{2}{5}, \frac{2}{3}

Example 28

easy

Order greatest to least:

\frac{1}{2}, \frac{3}{4}, 1, \frac{1}{4}

Four values at equal intervals; greatest is at the right end

Example 29

medium

Order least to greatest:

\frac{5}{6}, \frac{7}{8}, \frac{11}{12}

Example 30

medium

Order greatest to least:

\frac{2}{3}, \frac{4}{5}, \frac{7}{10}

Example 31

medium

Order least to greatest:

\frac{3}{8}, \frac{1}{3}, \frac{1}{2}

Example 32

medium

Order least to greatest:

\frac{5}{8}, \frac{11}{16}, \frac{3}{4}

Example 33

medium

Order greatest to least:

\frac{3}{5}, \frac{7}{10}, \frac{13}{20}

Example 34

medium

Three pitchers hold

\frac{2}{3}

\frac{5}{8}

, and

\frac{3}{4}

of a liter. Order the volumes least to greatest.

Example 35

medium

Order least to greatest:

\frac{7}{12}, \frac{5}{8}, \frac{11}{24}

Example 36

hard

Order least to greatest:

\frac{7}{15}, \frac{4}{9}, \frac{1}{2}, \frac{8}{17}

Example 37

hard

Order greatest to least:

\frac{9}{20}, \frac{4}{9}, \frac{11}{25}

Example 38

hard

Insert

\frac{3}{7}

into the ordered list

\frac{2}{5}, \frac{1}{2}, \frac{5}{8}

keeping it least to greatest.

Example 39

hard

Order least to greatest:

\frac{17}{20}, \frac{4}{5}, \frac{9}{10}

Example 40

hard

Four students reported they had completed

\frac{3}{8}, \frac{2}{5}, \frac{7}{20}

, and

\frac{9}{25}

of their reading. Order least to greatest.

Example 41

challenge

Find a fraction with denominator

12

that lies strictly between

\frac{3}{5}

and

\frac{5}{7}

Example 42

challenge

Order least to greatest using any method:

\frac{99}{100}, \frac{100}{101}, \frac{101}{102}

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Related Concepts

Comparing Fractions Fraction on a Number Line

Background Knowledge

These ideas may be useful before you work through the harder examples.

fraction comparison