"Numerator and denominator" is searched about 800 times a month in Australia (and the question is asked in every Year 3 to 6 classroom in every country), because the two words are learned at the same time as the idea they describe and are easy to swap for years. This page says which is which, what each one means, gives three ways to remember them, shows what happens to a fraction when each changes, covers proper, improper and mixed fractions and equivalent fractions, explains how to compare fractions that share one number, lists the mistakes, and gives a printable sheet with practice for Years 3 to 6.
Download the numerator and denominator sheet with practice (PDF) (A4, the two words with pictures, the memory tricks and the fraction types on page 1, equivalent fractions, comparing, and the practice set with answers on page 2, free).
Which is which
A fraction is two numbers with a line between them:
- the numerator is the number on top;
- the denominator is the number on the bottom.
In ¾: numerator 3, denominator 4. In ⅝: numerator 5, denominator 8. In 1⁄2: numerator 1, denominator 2.
What each one means
Take a pizza cut into 4 equal slices and eat 3 of them.
- The denominator, 4, says the whole was cut into 4 equal parts. It names the parts: quarters. Denominator 2: halves. 3: thirds. 5: fifths. 10: tenths. 100: hundredths.
- The numerator, 3, says how many of those parts you have: three quarters.
So a fraction is a count (numerator) of a named part (denominator). "Three quarters" reads the numerator, then the denominator, and that is how every fraction is said aloud: five eighths, two thirds, seven tenths.
The line between them is a division sign: ¾ is 3 ÷ 4 = 0.75, which is the link to decimals in Year 5 and 6.
Three ways to remember
- D is for Down. Denominator begins with D, so it is at the bottom.
- N is for North. Numerator begins with N, so it is at the top. Some children prefer "nUmerator Up", since the word has a U in it.
- The denominator names the fraction. De-nomin-ator shares its root with "nominate" and "name": it tells you the name of the parts (quarters, fifths). The numerator numbers them, it counts how many.
Whichever sticks, use the same one every time for a fortnight and the swapping stops.
What happens when each one changes
Change the numerator (keep the denominator): you have more or fewer of the same-sized parts. ¼, 2⁄4, ¾, 4⁄4: the fraction grows as the numerator grows. 4⁄4 is the whole pizza.
Change the denominator (keep the numerator): the parts change size. ½, ⅓, ¼, ⅕: one piece of a pizza cut into 2, then 3, then 4, then 5. A bigger denominator means smaller pieces, so ⅕ is smaller than ½. This is the fact Year 3 and 4 children most often get backwards, because with whole numbers a bigger number is a bigger amount.
| Fraction | Numerator | Denominator | Meaning | Size |
|---|---|---|---|---|
| ½ | 1 | 2 | one of two equal parts | 0.5 |
| ⅓ | 1 | 3 | one of three | 0.33 |
| ¼ | 1 | 4 | one of four | 0.25 |
| ¾ | 3 | 4 | three of four | 0.75 |
| 4⁄4 | 4 | 4 | all four: one whole | 1 |
| 5⁄4 | 5 | 4 | five quarters: more than a whole | 1.25 |
Proper, improper and mixed fractions
- Proper fraction: the numerator is smaller than the denominator: ¾, ⅝, 2⁄9. Worth less than 1.
- Improper fraction: the numerator is equal to or bigger than the denominator: 4⁄4, 7⁄4, 11⁄8. Worth 1 or more.
- Mixed number: a whole number and a proper fraction together: 1¾, 2⅜.
Improper to mixed: divide the numerator by the denominator; the quotient is the whole number, the remainder is the new numerator, the denominator stays. 11⁄8: 11 ÷ 8 = 1 remainder 3, so 1⅜. 7⁄4 = 1¾. 9⁄3 = 3.
Mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator. 2⅜: 2 × 8 + 3 = 19, so 19⁄8. 1¾ = 7⁄4.
Improper fractions are needed for multiplying and dividing fractions in Year 6 and 7; mixed numbers are for saying the answer.
Equivalent fractions
Two fractions are equivalent when they are the same amount with different numbers: ½ = 2⁄4 = 3⁄6 = 50⁄100. Multiply or divide the numerator and the denominator by the same number:
- ½ × (2⁄2) = 2⁄4; ½ × (5⁄5) = 5⁄10.
- 6⁄8 ÷ (2⁄2) = ¾ (this is simplifying, or writing in lowest terms).
Doing it to only one of the two numbers changes the fraction: 1⁄2 → 2⁄2 is not equivalent. Both, always.
Comparing fractions that share one number
- Same denominator: compare numerators; more parts is bigger. ⅝ > ⅜.
- Same numerator: compare denominators; the smaller denominator is bigger (bigger pieces). ⅓ > ⅕, and 3⁄5 > 3⁄8.
- Neither the same: make the denominators equal with equivalent fractions, or use a benchmark (is it more or less than ½?). ⅗ against ⅔: 9⁄15 against 10⁄15, so ⅔ is bigger. Our ordering fractions page has all five methods.
The mistakes that cost marks
- Swapping the words.
- Thinking a bigger denominator means a bigger fraction.
- Adding fractions by adding tops and bottoms (½ + ½ = 2⁄4). The denominator names the part; you add the counts: 1 half + 1 half = 2 halves = 1.
- Making an "equivalent" fraction by changing one number only.
- Reading ¾ as "three over four" in a context that wants "three quarters". Both are accepted, but the second shows understanding.
- Writing a denominator of 0.
Where it sits in each curriculum
| Curriculum | When | What is expected |
|---|---|---|
| Australia | Year 2 (halves, quarters, eighths), Year 3 (unit fractions, numerator and denominator named), Year 4 (equivalent fractions, counting by fractions), Year 5 (compare and order, add and subtract with the same denominator), Year 6 (all denominators); NAPLAN Years 3 and 5 test fraction pictures and comparisons | the words by Year 3, equivalence by Year 4 |
| England | Year 2 (½, ⅓, ¼, ¾), Year 3 (unit and non-unit fractions, the terms numerator and denominator), Year 4 (equivalent fractions), Year 5 (improper and mixed), Year 6 (simplify, compare, all four operations) | the terms in Year 3 |
| US Common Core | Grade 3 (3.NF.1: a fraction a/b as a parts of size 1/b), Grade 4 (equivalence, comparison, mixed numbers), Grade 5 (operations) | meaning before procedure |
| India (CBSE) | Class 3 and 4 (fractions introduced, numerator and denominator), Class 5 (equivalent, like and unlike), Class 6 (Fractions chapter: proper, improper, mixed) | the vocabulary in Class 4 to 6 |
Practice (answers in the PDF)
- In 5⁄9, which number is the numerator and which is the denominator? What does each tell you?
- Write the fraction for "seven tenths" and for "two fifths".
- Which is bigger, ⅙ or ⅛? Explain using the denominators.
- Change 13⁄5 to a mixed number, and 3⅔ to an improper fraction.
- Write two fractions equivalent to ⅔.
- A cake is cut into 12 equal pieces and 7 are eaten. What fraction was eaten, and what fraction is left? Which number is the denominator in both answers, and why is it the same?
Download the sheet and the practice set (PDF)
Where to get help
A 1-on-1 online maths class in Australian after-school hours with a tutor who holds a master's in mathematics and teaches Years 2 to 6, NAPLAN preparation, KS2 and CBSE Class 3 to 6 every week: fractions taught with pictures first, the two words attached to the picture, and the "bigger denominator, smaller piece" idea rehearsed until it is obvious. The NAPLAN page explains how the classes run; the free 30-minute first class is a real lesson on this week's homework. Related pages: the fractions lesson by grade, ordering fractions and fractions to decimals to percentages.
Questions parents ask
1Which is the numerator and which is the denominator?
2What does the denominator mean?
3What does the numerator mean?
4How do you remember numerator and denominator?
5Can the denominator be zero?
See how we teach this, 1-on-1 online →
Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.