What if two fractions in the list are equal?

Equivalent fractions such as $\frac{2}{4}$ and $\frac{3}{6}$ take the same place in the order. Write them next to each other with an equals sign: $\frac{1}{4} < \frac{2}{4} = \frac{3}{6} < \frac{5}{6}$.

Do I have to use the least common denominator?

No, any common denominator works. Multiplying all the denominators together always gives one. The least common denominator just keeps the numbers small, which means fewer mistakes.

How can I check my order?

Turn every fraction into a decimal with a calculator. In ascending order each decimal must be larger than the one before it, in descending order smaller.

Teacher Resources

Ordering Fractions

Q: Do I have to use the least common denominator?

No, any common denominator works. Multiplying all the denominators together always gives one. The least common denominator just keeps the numbers small, which means fewer mistakes.

Q: How can I check my order?

Turn every fraction into a decimal with a calculator. In ascending order each decimal must be larger than the one before it, in descending order smaller.

Learn to put three or more fractions in order from least to greatest or greatest to least, and to pick the fastest method every time.

Intermediate25 minLesson

Printable worksheet

Definition

Ordering fractions means putting three or more fractions in a row by size.

Ascending order (least to greatest): $\frac{1}{4} < \frac{1}{2} < \frac{3}{4}$
Descending order (greatest to least): $\frac{3}{4} > \frac{1}{2} > \frac{1}{4}$

You already know how to compare two fractions. To order several, pick the method that fits the whole list:

1.Same denominator: order the numerators. $\frac{2}{9} < \frac{4}{9} < \frac{7}{9}$
2.Same numerator: the larger the denominator, the smaller the fraction. $\frac{3}{10} < \frac{3}{8} < \frac{3}{5}$
3.Common denominator: rewrite every fraction with the least common denominator (LCD), then order the numerators.
4.Decimals: divide numerator by denominator and order the decimals.
5.Benchmarks $0$ , $\frac{1}{2}$ and $1$ : sort the fractions into groups first, then order inside each group.

Cross multiplication works for two fractions at a time, but four fractions already need six pairs. A common denominator puts the whole list on one scale in one go.

Try it now

Which list shows

\frac{4}{7}

\frac{1}{7}

\frac{6}{7}

\frac{3}{7}

from least to greatest?

Worked Examples

Order from least to greatest: $\frac{5}{9}$ , $\frac{2}{9}$ , $\frac{7}{9}$ , $\frac{4}{9}$

Check the denominators

All four fractions are ninths → Same denominator: compare the numerators

Order the numerators

$2 < 4 < 5 < 7$ → $2, 4, 5, 7$

Write the fractions in that order

Keep the denominator $9$ → $\frac{2}{9} < \frac{4}{9} < \frac{5}{9} < \frac{7}{9}$

Answer: $\frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}$

Common Mistakes

Ordering unit fractions by their denominators: writing $\frac{1}{3}, \frac{1}{4}, \frac{1}{6}$ as least to greatest because $3 < 4 < 6$

Why it's wrong: With the same numerator, a larger denominator means the whole is cut into more, smaller pieces.

Correct: Least to greatest is $\frac{1}{6} < \frac{1}{4} < \frac{1}{3}$ : the largest denominator gives the smallest fraction.

Comparing only the numerators when the denominators differ, e.g. putting $\frac{5}{12}$ after $\frac{2}{3}$ because $5 > 2$

Why it's wrong: Numerators can only be compared directly when the pieces have the same size.

Correct: Rewrite first: $\frac{2}{3} = \frac{8}{12}$ , and $\frac{5}{12} < \frac{8}{12}$ , so $\frac{5}{12} < \frac{2}{3}$ .

Changing only the denominator when rewriting: $\frac{3}{4} = \frac{3}{12}$

Why it's wrong: Multiplying only the denominator makes the pieces smaller without adding more of them, so the value changes.

Correct: Multiply the numerator and the denominator by the same number: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ .

Mixing up ascending and descending order

Why it's wrong: Students order correctly from least to greatest, but the question asked for greatest to least (or the other way round).

Correct: Underline the words in the question first. Ascending starts with the smallest fraction ( $<$ ), descending starts with the greatest ( $>$ ).

Interactive Visual

Fraction Number Line

Denominator

Select two fractions to compare them.

Numerator:

Denominator:

3/4= 75%

Click on the bar to change the fraction

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems

Problem 1 of 18

Easy

Which list shows $\frac{4}{7}$ , $\frac{1}{7}$ , $\frac{6}{7}$ , $\frac{3}{7}$ from least to greatest?

Why It Matters

Ordering fractions is something you do whenever several amounts have to be ranked:

Cooking: a recipe lists $\frac{3}{4}$ cup of flour, $\frac{2}{3}$ cup of sugar and $\frac{1}{2}$ cup of milk. Which ingredient do you need most?
Sports: runners cover $\frac{7}{10}$ km, $\frac{3}{4}$ km and $\frac{4}{5}$ km in training. Who ran farthest?
Surveys: $\frac{3}{5}$ of one class and $\frac{5}{8}$ of another walk to school. Which class walks more?
Measuring: drill bits and wrenches are labeled $\frac{3}{16}$ , $\frac{1}{4}$ and $\frac{5}{32}$ inch, and you must find them in size order.

It is also the basis for ordering decimals, percentages and negative numbers later on.

Real World Applications

Recipes

Recipes often give ingredients as fractions of a cup, and you may want to know which ingredient you need the most or the least of.

Example:

A cake uses $\frac{3}{4}$ cup of flour, $\frac{1}{3}$ cup of sugar and $\frac{1}{2}$ cup of milk. With benchmarks: $\frac{1}{3} < \frac{1}{2} < \frac{3}{4}$ , so flour is the largest amount.

1Try It Yourself

A granola recipe uses $\frac{3}{4}$ cup of flour, $\frac{2}{3}$ cup of sugar, $\frac{1}{2}$ cup of milk and $\frac{5}{6}$ cup of oats.

Put the amounts in order from least to greatest.

Step 1: Choose the right answer

Rewrite all four amounts in twelfths and pick the right order:

Race Distances

Coaches compare training distances that are often given as fractions of a kilometer or a mile.

Example:

Three runners covered $\frac{3}{5}$ km, $\frac{3}{4}$ km and $\frac{3}{10}$ km. The numerators are the same, so $\frac{3}{10} < \frac{3}{5} < \frac{3}{4}$ .

2Try It Yourself

In training, runner A ran $\frac{7}{10}$ km, runner B ran $\frac{3}{4}$ km, runner C ran $\frac{4}{5}$ km and runner D ran $\frac{13}{20}$ km.

Which distance is the second shortest?

Step 1: Write the mathematical expression

Rewrite every distance in twentieths and give the second shortest one as a fraction:

Survey Results

Survey results from groups of different sizes are often reported as fractions, and ranking them shows which group answered yes most often.

Example:

In a survey, $\frac{1}{4}$ of the students chose music, $\frac{1}{2}$ chose sports and $\frac{1}{8}$ chose art. Same numerators: $\frac{1}{8} < \frac{1}{4} < \frac{1}{2}$ .

3Try It Yourself

Four classes were asked whether they walk to school. Yes answers: class A $\frac{3}{5}$ , class B $\frac{5}{8}$ , class C $\frac{11}{20}$ , class D $\frac{16}{25}$ .

Which class has the largest share of students who walk?

Step 1: Write the mathematical expression

Write each fraction as a decimal and give the largest one as a fraction:

Key Takeaways

1Ascending order goes from least to greatest ( $<$ ), descending order from greatest to least ( $>$ )
2Same denominator: order the numerators
3Same numerator: the larger the denominator, the smaller the fraction
4Different denominators: rewrite every fraction with the least common denominator, order the numerators, then write the original fractions
5Decimals are fastest when the LCD is large and the denominators divide easily (2, 4, 5, 8, 10, 20, 25)
6Benchmarks $0$ , $\frac{1}{2}$ and $1$ split the list into groups, so you only compare fractions inside each group

Frequently Asked Questions

Look at the whole list. Same denominators or same numerators? Use that rule. Do the benchmarks

0

\frac{1}{2}

and

1

split the fractions into clear groups? Sort first. Otherwise use the LCD when it is small (like 12 or 24), or decimals when the LCD is large or you have a calculator.

Glossary

Ascending order: Arranged from least to greatest, written with $<$
Descending order: Arranged from greatest to least, written with $>$
Common denominator: A denominator shared by several fractions after they are rewritten as equivalent fractions
Least common denominator (LCD): The smallest common denominator, equal to the least common multiple of the denominators
Benchmark fraction: A familiar reference value such as $0$ , $\frac{1}{2}$ or $1$ used to sort fractions quickly
Equivalent fractions: Fractions that name the same amount, like $\frac{3}{4}$ and $\frac{9}{12}$

Formula Card

Same denominator

\frac{a}{d} < \frac{b}{d} \text{ when } a < b

Order the numerators

Same numerator

\frac{n}{a} < \frac{n}{b} \text{ when } a > b

The larger denominator gives the smaller fraction

Rewriting with a common denominator

\frac{a}{b} = \frac{a \times k}{b \times k}

Multiply numerator and denominator by the same number k until every fraction has the LCD

Fraction to decimal

\frac{a}{b} = a \div b

Divide the numerator by the denominator, then order the decimals

Ordering Fractions

Definition

Worked Examples

Common Mistakes

Interactive Visual

Fraction Number Line

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Recipes

Race Distances

Survey Results

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Same denominator

Same numerator

Rewriting with a common denominator

Fraction to decimal

More in This Topic

Comparing Fractions with the Same Denominator

Comparing Fractions with the Same Numerator

Fractions on a Number Line

Comparing Mixed Numbers