"Ordering fractions" is searched by children in grades 4 to 6 (CBSE Class 5 and 6, UK Years 5 and 6) with a worksheet in front of them, and by parents who have forgotten which method the school uses. There are five methods, each is fastest in a different situation, and the children who find ordering fractions hard are usually using the one method they were taught for every question. This page gives the five with a worked example each and the situation it suits, then the harder cases (mixed numbers, negatives, fractions with decimals and percentages in one list), where the topic sits in each curriculum, and the four mistakes. The printable worksheet has twenty questions in rising difficulty with answers.
Download the ordering fractions worksheet (PDF) (the five methods on one side with an example each, twenty questions from easy to hard on the other, with answers; two pages, free).
First, what "bigger" means
A fraction is bigger when it is further to the right on the number line. 3/4 is bigger than 2/3 because 3/4 = 0.75 and 2/3 = 0.667. The denominator says how many equal pieces the whole is cut into (bigger denominator, smaller pieces); the numerator says how many pieces. Children who think 1/8 is bigger than 1/4 "because 8 is bigger" have not yet seen that eighths are smaller pieces, and every method below starts from that fact.
The five methods, and when each is fastest
1. Common denominator (always works)
Rewrite every fraction over the least common multiple of the denominators, then order by the numerators.
Order 2/3, 3/4, 5/8 from least to greatest. The LCM of 3, 4 and 8 is 24: 2/3 = 16/24, 3/4 = 18/24, 5/8 = 15/24. Order: 5/8, 2/3, 3/4.
Fastest when the denominators share factors (halves, quarters, eighths; thirds, sixths, twelfths). Slow when the LCM is large (7ths and 11ths).
2. Same numerator (when the tops match, or can be made to)
With the same numerator, the fraction with the smaller denominator is bigger, because its pieces are bigger.
Order 3/5, 3/8, 3/4. Same numerator 3; denominators 5, 8, 4; smaller denominator is bigger. Order: 3/8, 3/5, 3/4.
Fastest for unit fractions (1/3, 1/5, 1/4) and any list where the numerators match. It also works for 2/5 and 4/9 by doubling the first: 4/10 against 4/9, so 4/9 is bigger.
3. Benchmarks (compare each to 0, ½ and 1)
Decide whether each fraction is less than a half, about a half, or more than a half, and how close to 0 or 1 it is. Half means the numerator is half the denominator.
Order 1/6, 5/8, 4/9, 7/8. 1/6 is close to 0; 4/9 is a little under half (4.5/9 would be half); 5/8 is a little over half; 7/8 is close to 1. Order: 1/6, 4/9, 5/8, 7/8, with no calculation.
Fastest for a rough order and for mental maths; it fails when two fractions are both just over a half (5/8 and 7/12), and then method 1 or 4 settles it.
4. Convert to decimals
Divide each numerator by its denominator, then order the decimals.
Order 5/7, 2/3, 7/10, 11/16. 5/7 = 0.714, 2/3 = 0.667, 7/10 = 0.7, 11/16 = 0.6875. Order: 2/3, 11/16, 7/10, 5/7.
Fastest when a calculator is allowed, when the denominators have no common factors, or when decimals and percentages are in the same list. The conversions are in fractions to decimals to percentages.
5. Cross-multiplying (two fractions at a time)
To compare a/b with c/d, compare a × d with c × b. The side with the bigger product is the bigger fraction.
Compare 5/7 and 7/10: 5 × 10 = 50 and 7 × 7 = 49, so 5/7 is bigger. Compare 4/9 and 3/7: 4 × 7 = 28 and 3 × 9 = 27, so 4/9 is bigger.
Fastest for a single pair, and for checking an order found another way. For a list of five fractions it means many comparisons, so use method 1 or 4.
The harder cases
Mixed numbers and improper fractions. Order by the whole number first; compare the fraction parts only when the whole numbers tie. Convert improper fractions to mixed first so the whole numbers show. Order 2 1/2, 11/4, 2 2/3, 3 1/8: 11/4 = 2 3/4, so the three "2 and a bit" fractions are compared by 1/2, 3/4, 2/3 (6/12, 9/12, 8/12). Order: 2 1/2, 2 2/3, 11/4, 3 1/8.
Negative fractions. Order the sizes, then reverse, because a bigger negative is further below zero. Order −1/2, −3/4, −1/4, 1/3: the negatives from least are −3/4, −1/2, −1/4, then the positive. Order: −3/4, −1/2, −1/4, 1/3.
Fractions, decimals and percentages together. Convert everything to one form, usually decimals. Order 3/5, 0.55, 62%, 5/8: 0.6, 0.55, 0.62, 0.625. Order: 0.55, 3/5, 62%, 5/8.
Greatest to least. Every method gives least to greatest; read the list backwards. Exam questions say which order they want, and about one child in five answers the other way.
Where it sits in the curriculum
| System | Where |
|---|---|
| US Common Core | Grade 4 (comparing two fractions with different numerators and denominators, by benchmarks and common denominators); grade 5 (ordering, with decimals); grade 6 (with negatives on the number line) |
| CBSE | Class 5 (comparing fractions) and Class 6 (ordering like and unlike fractions, the number line) |
| England | Year 5 (compare and order fractions whose denominators are multiples of the same number) and Year 6 (any denominators, using equivalents) |
| Australia | Years 5 and 6 |
The grade 4 and grade 6 content in full is in the grade 4 math syllabus and grade 6 math syllabus guides.
The four mistakes
- "The bigger numbers make the bigger fraction." 1/8 is not bigger than 1/4. Draw the pieces once and the mistake ends.
- Changing the denominator without changing the numerator. 2/3 over 24 is 16/24, not 2/24; multiply top and bottom by the same number.
- Comparing only the numerators or only the denominators of unlike fractions. 3/5 and 4/9: 4 is more than 3 and 9 is more than 5, and neither fact decides it (3/5 is bigger).
- Reversing the order. "Least to greatest" and "greatest to least" are different questions; underline which one before starting.
Download the ordering fractions worksheet (PDF)
Where to get help
A 1-on-1 online maths class with a tutor who holds a master's in mathematics, for grades 4 to 6, CBSE Class 5 and 6 and Years 5 and 6: fractions taught with pictures and the number line before any rule, the five methods matched to the question, and the school's own worksheet worked on the shared whiteboard. The recording and notes after every class, the same tutor all year, US$10 to $13 an hour. The grade 5 math page has the units; the free 30-minute first class is a real lesson on this week's homework.
Questions parents ask
1How do you order fractions from least to greatest?
2What is the easiest way to compare fractions?
3How do you order fractions with different denominators?
4How do you order mixed numbers and improper fractions?
5How do you order negative fractions?
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.