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Year 6 Maths Fractions

Compare & order fractions (denom 2,3,4,5,6,8,10)

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Theory

To compare fractions, make them easy to match. With the same denominator, compare the numerators; with the same numerator, the smaller denominator gives bigger parts; otherwise compare with one half or rename both with a common denominator. This Year 6 maths page shows each method and how to order several fractions.

Same denominator: the parts are the same size, so more parts is bigger. \(\dfrac{4}{5} > \dfrac{2}{5}\).

Same numerator: the same number of parts, but a smaller denominator means bigger parts. \(\dfrac{2}{3} > \dfrac{2}{5}\).

Benchmark one half: a fraction is more than a half when its numerator is more than half its denominator. \(\dfrac{4}{6} > \dfrac{1}{2}\) and \(\dfrac{3}{10} < \dfrac{1}{2}\).

Common denominator: rename both fractions with the same denominator (a multiple of both), then compare numerators. \(\dfrac{3}{4} = \dfrac{9}{12}\) and \(\dfrac{4}{6} = \dfrac{8}{12}\).

Two thirds and two fifths Two bars the same length. The top bar is split into 3 equal parts with 2 shaded. The bottom bar is split into 5 equal parts with 2 shaded. The top shaded part is longer. 23 25
Same numerator: thirds are bigger than fifths, so \(\dfrac{2}{3} > \dfrac{2}{5}\).
Three quarters and four sixths in twelfths Four bars the same length: 3 of 4 parts shaded, 9 of 12 parts shaded, 4 of 6 parts shaded and 8 of 12 parts shaded. The first two match and the last two match. 34 912 46 812
\(\dfrac{3}{4} = \dfrac{9}{12}\) and \(\dfrac{4}{6} = \dfrac{8}{12}\), so \(\dfrac{3}{4}\) is larger.
Fractions on a number line in tenths A number line from 0 to 1 marked in tenths. Point A is at 2 tenths (one fifth), B at 5 tenths (one half), C at 6 tenths (three fifths) and D at 7 tenths. 01102103104105106107108109101 A B C D
A \(= \dfrac{1}{5}\), B \(= \dfrac{1}{2}\), C \(= \dfrac{3}{5}\), D \(= \dfrac{7}{10}\): further right is larger.

Ways to compare:

FractionsMethod
same denominatorlarger numerator is larger
same numeratorsmaller denominator is larger
one each side of a halfcompare with one half
anything elserename with a common denominator
\[ \dfrac{a}{b} > \dfrac{1}{2} \text{ when } a > \dfrac{b}{2} \]
a / b >1/2 when a > b /2
Bigger denominator, smaller parts. \(\dfrac{1}{10}\) is smaller than \(\dfrac{1}{3}\), even though \(10 > 3\).
Common denominator. Use a number both denominators divide into, such as 12 for quarters and sixths.

How to compare two fractions

  1. Check for the same denominator or the same numerator.
  2. Try the benchmark one half.
  3. Otherwise rename both with a common denominator and compare the numerators.
  4. Write the answer with \(>\), \(<\) or \(=\).

How to order several fractions

  1. Rename them all with one common denominator.
  2. Order the numerators, then write the original fractions in that order.
Example 1 — Same denominator
Write \(>\) or \(<\): \(\dfrac{2}{5}\) ___ \(\dfrac{4}{5}\).
Solution
\(\dfrac{2}{5}\)\(<\)\(\dfrac{4}{5}\)
2/5<4/5

Same-sized parts: \(2 < 4\).

Example 2 — Same numerator
Which is larger, \(\dfrac{2}{3}\) or \(\dfrac{2}{5}\)?
Solution
\(\dfrac{2}{3}\)\(>\)\(\dfrac{2}{5}\)
2/3>2/5

Thirds are bigger than fifths, so \(\dfrac{2}{3}\) (first diagram).

Example 3 — Use one half
Which is larger, \(\dfrac{3}{10}\) or \(\dfrac{4}{6}\)?
Solution

\(3\) is less than half of \(10\); \(4\) is more than half of \(6\).

\(\dfrac{3}{10}\)\(<\)\(\dfrac{1}{2} < \dfrac{4}{6}\)
3/10<1/2<4/6

\(\dfrac{4}{6}\).

Example 4 — Unit fractions
Order from smallest to largest: \(\dfrac{1}{5}\), \(\dfrac{1}{10}\), \(\dfrac{1}{3}\).
Solution

More parts means smaller parts: \(\dfrac{1}{10}, \dfrac{1}{5}, \dfrac{1}{3}\).

Example 5 — Twelfths
Compare \(\dfrac{3}{4}\) and \(\dfrac{4}{6}\).
Solution
\(\dfrac{3}{4}\)\(=\)\(\dfrac{9}{12}\)
\(\dfrac{4}{6}\)\(=\)\(\dfrac{8}{12}\)
3/4=9/12,4/6=8/12

\(9 > 8\), so \(\dfrac{3}{4} > \dfrac{4}{6}\).

Example 6 — Thirtieths
Which is larger, \(\dfrac{2}{3}\) or \(\dfrac{7}{10}\)?
Solution
\(\dfrac{2}{3}\)\(=\)\(\dfrac{20}{30}\)
\(\dfrac{7}{10}\)\(=\)\(\dfrac{21}{30}\)
2/3=20/30,7/10=21/30

\(\dfrac{7}{10}\), by just \(\dfrac{1}{30}\).

Example 7 — Order four fractions
Order from smallest to largest: \(\dfrac{3}{5}\), \(\dfrac{1}{2}\), \(\dfrac{7}{10}\), \(\dfrac{1}{5}\).
Solution

In tenths: \(\dfrac{6}{10}\), \(\dfrac{5}{10}\), \(\dfrac{7}{10}\), \(\dfrac{2}{10}\).

\(\dfrac{1}{5}, \dfrac{1}{2}, \dfrac{3}{5}, \dfrac{7}{10}\) (see the number line).

Example 8 — Pizza problem
Jo ate \(\dfrac{3}{8}\) of a pizza and Sam ate \(\dfrac{1}{3}\) of a pizza the same size. Who ate more?
Solution
\(\dfrac{3}{8}\)\(=\)\(\dfrac{9}{24}\)
\(\dfrac{1}{3}\)\(=\)\(\dfrac{8}{24}\)
3/8=9/24,1/3=8/24

Jo.

Example 9 — Find the mistake
Tia says \(\dfrac{3}{10}\) is larger than \(\dfrac{3}{6}\) because \(10\) is larger than \(6\). What is right?
Solution
\(\dfrac{3}{10}\)\(<\)\(\dfrac{3}{6}\)
3/10<3/6

Both have \(3\) parts, but tenths are smaller than sixths, so \(\dfrac{3}{6}\) is larger.

Common pitfalls

Bigger denominator means bigger fraction. Not true: \(\dfrac{1}{10} < \dfrac{1}{3}\).
Comparing numerators with different denominators. \(\dfrac{4}{6}\) and \(\dfrac{3}{4}\) need a common denominator first.
Changing only the denominator. When renaming, multiply the numerator by the same number: \(\dfrac{3}{4} = \dfrac{9}{12}\), not \(\dfrac{3}{12}\).
Forgetting the original fractions. After ordering the renamed fractions, write the answer with the fractions you were given.

Frequently asked questions

How do I compare fractions with the same denominator?

Compare the numerators: more parts of the same size is bigger.

How do I compare fractions with the same numerator?

The one with the smaller denominator is bigger, because its parts are bigger.

What is a common denominator?

A number that both denominators divide into, such as 12 for quarters and sixths.

How can one half help?

If one fraction is less than a half and the other is more, you can tell which is bigger straight away.

How do I order fractions?

Rename them all with one common denominator and order the numerators.

More in Fractions
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