Ordering Fractions

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 min18 problems
Practice Problems

Definition

Ordering fractions means putting three or more fractions in a row by size.
  • Ascending order (least to greatest): 14<12<34\frac{1}{4} < \frac{1}{2} < \frac{3}{4}
  • Descending order (greatest to least): 34>12>14\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. 1.Same denominator: order the numerators. 29<49<79\frac{2}{9} < \frac{4}{9} < \frac{7}{9}
  2. 2.Same numerator: the larger the denominator, the smaller the fraction. 310<38<35\frac{3}{10} < \frac{3}{8} < \frac{3}{5}
  3. 3.Common denominator: rewrite every fraction with the least common denominator (LCD), then order the numerators.
  4. 4.Decimals: divide numerator by denominator and order the decimals.
  5. 5.Benchmarks 00, 12\frac{1}{2} and 11: 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 47\frac{4}{7}, 17\frac{1}{7}, 67\frac{6}{7}, 37\frac{3}{7} from least to greatest?

Worked Examples

Order from least to greatest: 59\frac{5}{9}, 29\frac{2}{9}, 79\frac{7}{9}, 49\frac{4}{9}
1

Check the denominators

All four fractions are ninths → Same denominator: compare the numerators
2

Order the numerators

2<4<5<72 < 4 < 5 < 7 → 2,4,5,72, 4, 5, 7
3

Write the fractions in that order

Keep the denominator 99 → 29<49<59<79\frac{2}{9} < \frac{4}{9} < \frac{5}{9} < \frac{7}{9}

Common Mistakes

Ordering unit fractions by their denominators: writing 13,14,16\frac{1}{3}, \frac{1}{4}, \frac{1}{6} as least to greatest because 3<4<63 < 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 16<14<13\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 512\frac{5}{12} after 23\frac{2}{3} because 5>25 > 2

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

Correct: Rewrite first: 23=812\frac{2}{3} = \frac{8}{12}, and 512<812\frac{5}{12} < \frac{8}{12}, so 512<23\frac{5}{12} < \frac{2}{3}.

Changing only the denominator when rewriting: 34=312\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: 3×34×3=912\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

02/124/126/128/1210/121

Select two fractions to compare them.

3
3/4= 75%

Click on the bar to change the fraction

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which list shows 47\frac{4}{7}, 17\frac{1}{7}, 67\frac{6}{7}, 37\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 34\frac{3}{4} cup of flour, 23\frac{2}{3} cup of sugar and 12\frac{1}{2} cup of milk. Which ingredient do you need most?
  • Sports: runners cover 710\frac{7}{10} km, 34\frac{3}{4} km and 45\frac{4}{5} km in training. Who ran farthest?
  • Surveys: 35\frac{3}{5} of one class and 58\frac{5}{8} of another walk to school. Which class walks more?
  • Measuring: drill bits and wrenches are labeled 316\frac{3}{16}, 14\frac{1}{4} and 532\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 34\frac{3}{4} cup of flour, 13\frac{1}{3} cup of sugar and 12\frac{1}{2} cup of milk. With benchmarks: 13<12<34\frac{1}{3} < \frac{1}{2} < \frac{3}{4}, so flour is the largest amount.

1Try It Yourself

A granola recipe uses 34\frac{3}{4} cup of flour, 23\frac{2}{3} cup of sugar, 12\frac{1}{2} cup of milk and 56\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 35\frac{3}{5} km, 34\frac{3}{4} km and 310\frac{3}{10} km. The numerators are the same, so 310<35<34\frac{3}{10} < \frac{3}{5} < \frac{3}{4}.

2Try It Yourself

In training, runner A ran 710\frac{7}{10} km, runner B ran 34\frac{3}{4} km, runner C ran 45\frac{4}{5} km and runner D ran 1320\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, 14\frac{1}{4} of the students chose music, 12\frac{1}{2} chose sports and 18\frac{1}{8} chose art. Same numerators: 18<14<12\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 35\frac{3}{5}, class B 58\frac{5}{8}, class C 1120\frac{11}{20}, class D 1625\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 00, 12\frac{1}{2} and 11 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 00, 12\frac{1}{2} and 11 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.
Look at the whole list. Same denominators or same numerators? Use that rule. Do the benchmarks 00, 12\frac{1}{2} and 11 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.
Equivalent fractions such as 24\frac{2}{4} and 36\frac{3}{6} take the same place in the order. Write them next to each other with an equals sign: 14<24=36<56\frac{1}{4} < \frac{2}{4} = \frac{3}{6} < \frac{5}{6}.
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.
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.

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 00, 12\frac{1}{2} or 11 used to sort fractions quickly
Equivalent fractions
Fractions that name the same amount, like 34\frac{3}{4} and 912\frac{9}{12}

Formula Card

Same denominator

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

Order the numerators

Same numerator

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

The larger denominator gives the smaller fraction

Rewriting with a common denominator

ab=a×kb×k\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

ab=a÷b\frac{a}{b} = a \div b

Divide the numerator by the denominator, then order the decimals

Today's challengeOne problem, three tries

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