Comparing fractions means deciding which fraction represents the greater or smaller number.
Ordering fractions means arranging several fractions from least to greatest or from greatest to least.
In this lesson you will use fraction size, common denominators, equivalent fractions, benchmarks and number lines to compare fractions accurately.
Goal: compare and order fractions by reasoning about their size instead of relying on unreliable shortcuts.
Before you start
Review Fractions Explained with Everyday Examples.
You should already understand numerators, denominators, equivalent fractions and the idea that fractions are numbers.
Paper or a private note may help with the practice activities.
Fractions must refer to the same whole
Before comparing fractional parts, make sure the fractions describe the same whole or equal sized wholes.
For example, one half of a small pizza might be smaller than one quarter of a much larger pizza.
But as numbers:
1/2 is greater than 1/4
That numerical comparison assumes the same unit whole.
Think about fraction size
A fraction represents a single number.
Consider:
3/5
This means three parts, each of size one fifth.
To compare it with another fraction, think about the total amount represented rather than treating the numerator and denominator as unrelated whole numbers.
Strategy 1: Compare fractions with the same denominator
Consider:
3/8 and 5/8
Both fractions are made from eighths.
The parts are therefore the same size.
Five eighths contains more eighth sized parts than three eighths.
So:
5/8 is greater than 3/8
Same denominator rule
When positive fractions have the same denominator, compare their numerators.
The greater numerator represents more parts of the same size.
For example:
7/10 is greater than 4/10
Practice 1
Which is greater?
2/9 or 7/9
Answer
7/9.
Both fractions use ninths, and seven ninth sized parts are greater than two.
Strategy 2: Compare unit fractions
A unit fraction has numerator 1.
Consider:
1/3 and 1/7
Imagine dividing the same whole into three equal pieces and then into seven equal pieces.
Thirds are larger pieces than sevenths.
Therefore:
1/3 is greater than 1/7
Why a larger denominator can mean a smaller unit fraction
The denominator tells how many equal parts make the whole.
If the same whole is divided into more parts, each individual part becomes smaller.
So:
1/4 is greater than 1/8
This does not mean that a larger denominator always makes every fraction smaller.
That reasoning works directly for unit fractions because each numerator is 1.
Practice 2
Which is smaller?
1/5 or 1/12
Answer
1/12.
Twelfths are smaller parts than fifths when the whole is the same.
Strategy 3: Compare fractions with the same numerator
Consider:
3/4 and 3/7
Both fractions contain three parts.
But fourths are larger pieces than sevenths.
Three fourths therefore represents a greater quantity than three sevenths.
So:
3/4 is greater than 3/7
Think about the size of each part
When the numerators are equal, the fraction with the smaller positive denominator has larger individual parts.
For example:
5/6 is greater than 5/9
because sixths are larger than ninths.
Practice 3
Which fraction is greater?
4/5 or 4/11
Answer
4/5.
Both fractions contain four parts, but fifths are larger than elevenths.
Strategy 4: Use one half as a benchmark
One half is a useful reference point.
Consider:
3/8 and 5/9
For 3/8, half of eight eighths is four eighths.
So:
3/8 is less than 1/2
For 5/9, half of nine is four and a half.
Five ninths is therefore greater than one half.
So we can conclude:
5/9 is greater than 3/8
We did not need to create common denominators.
Another benchmark example
Compare:
7/12 and 4/9
Six twelfths equals one half.
So:
7/12 is greater than 1/2
Half of nine is four and a half.
So:
4/9 is less than 1/2
Therefore:
7/12 is greater than 4/9
Practice 4
Which is greater?
4/10 or 6/11
Answer
6/11.
Four tenths is less than one half.
Six elevenths is greater than one half.
Strategy 5: Use equivalent fractions
Sometimes both fractions lie on the same side of a useful benchmark.
Then equivalent fractions can make the comparison clearer.
Compare:
2/3 and 3/4
Express both using twelfths.
2/3 = 8/12
3/4 = 9/12
Now the denominators are equal.
Since nine twelfths is greater than eight twelfths:
3/4 is greater than 2/3
Why common denominators help
Equivalent fractions allow us to rename fractions without changing their value.
Once two fractions use the same denominator, they are expressed using parts of the same size.
Then their numerators can be compared directly.
Another equivalent fraction example
Compare:
3/5 and 5/8
A common denominator is 40.
3/5 = 24/40
5/8 = 25/40
Therefore:
5/8 is greater than 3/5
Do not choose a common denominator larger than necessary
Any valid common denominator can work, but a smaller convenient one usually makes the arithmetic easier.
For example, to compare:
1/2 and 3/4
you can use fourths:
1/2 = 2/4
There is no need to convert both fractions into very large equivalent fractions.
Strategy 6: Use a number line
Fractions can be compared by locating them on the same number line.
Numbers farther to the right are greater.
Imagine:
0 1/4 1/2 3/4 1
Since three quarters lies to the right of one half:
3/4 is greater than 1/2
A number line helps show that fractions are numbers
Consider:
2/3
To place it between zero and one, divide that interval into three equal sections.
Two thirds is located at the second division point.
This position represents one number, not two separate whole numbers.
Compare fractions greater than one
Fractions can also be greater than one.
Consider:
5/4 and 6/5
Both are greater than one.
You can rewrite them as mixed numbers:
5/4 = 1 1/4
6/5 = 1 1/5
One quarter is greater than one fifth.
Therefore:
5/4 is greater than 6/5
Use distance from one
Consider:
7/8 and 5/6
Both fractions are less than one.
Seven eighths is missing one eighth from one.
Five sixths is missing one sixth from one.
One eighth is smaller than one sixth.
Therefore seven eighths is closer to one and is greater.
7/8 is greater than 5/6
This can be a useful benchmark strategy
When two positive fractions are both close to one, comparing how much each is missing from one can be easier than finding a common denominator.
Choose the strategy that fits the fractions
There is no single method you must use every time.
Useful strategies include:
- same denominator
- same numerator
- unit fraction reasoning
- comparison with one half
- comparison with one
- equivalent fractions
- a number line
Good fraction sense means recognizing which strategy makes the comparison easiest to understand.
Do not compare only the denominators
Consider:
7/10 and 2/3
You cannot say that 7/10 is smaller simply because 10 is greater than 3.
The fractions represent complete quantities.
Using a common denominator:
7/10 = 21/30
2/3 = 20/30
Therefore:
7/10 is greater than 2/3
Do not compare only the numerators
Consider:
5/12 and 4/5
The numerator 5 is greater than 4, but that does not make 5/12 the greater fraction.
Five twelfths is less than one half.
Four fifths is greater than one half.
Therefore:
4/5 is greater than 5/12
Ordering several fractions
Suppose you need to order:
1/4, 3/4, 1/2
These are familiar benchmark fractions.
From least to greatest:
1/4, 1/2, 3/4
Ordering fractions with a common denominator
Order:
7/12, 2/12, 9/12, 5/12
The denominators are already equal.
Order the numerators:
2, 5, 7, 9
So from least to greatest:
2/12, 5/12, 7/12, 9/12
Ordering fractions with different denominators
Order:
1/2, 2/3, 3/4
Use twelfths:
1/2 = 6/12
2/3 = 8/12
3/4 = 9/12
Therefore:
1/2, 2/3, 3/4
is already ordered from least to greatest.
Everyday example: comparing portions
One container is 3/4 full.
Another equal sized container is 2/3 full.
Convert to twelfths:
3/4 = 9/12
2/3 = 8/12
The first container has the greater proportion filled.
Everyday example: distance completed
Two people travel the same total route.
One has completed 5/8 of the route.
The other has completed 3/5.
Use fortieths:
5/8 = 25/40
3/5 = 24/40
So:
5/8 is greater than 3/5
Everyday example: recipe quantities
A recipe uses 2/3 cup of one ingredient and 3/4 cup of another.
Which quantity is greater?
Use twelfths:
2/3 = 8/12
3/4 = 9/12
Therefore 3/4 cup is greater.
Practice 5: Same denominator
Which is greater?
5/11 or 8/11
Answer
8/11.
Practice 6: Same numerator
Which is greater?
3/5 or 3/8
Answer
3/5.
Fifths are larger than eighths.
Practice 7: Use one half
Compare:
4/9 and 5/8
Answer
Four ninths is less than one half.
Five eighths is greater than one half.
Therefore:
5/8 is greater than 4/9
Practice 8: Use equivalent fractions
Compare:
3/4 and 5/6
Answer
Use twelfths:
3/4 = 9/12
5/6 = 10/12
Therefore:
5/6 is greater than 3/4
Practice 9: Order three fractions
Order from least to greatest:
2/3, 1/2, 5/6
Answer
Use sixths:
1/2 = 3/6
2/3 = 4/6
5/6 = 5/6
So:
1/2, 2/3, 5/6
Practice 10: Find the mistake
A learner says:
2/9 is greater than 2/5 because 9 is greater than 5.
What is wrong?
Answer
Both fractions have the same numerator.
Fifths are larger pieces than ninths.
Therefore:
2/5 is greater than 2/9
Activity: Choose the best comparison strategy
For each pair, decide which strategy would make the comparison easiest.
- 4/9 and 7/9
- 3/5 and 3/8
- 3/10 and 5/8
- 2/3 and 3/4
- 11/12 and 7/8
Possible approaches include a common denominator, same numerator reasoning, one half, distance from one or another clear fraction relationship.
After choosing the strategy, complete the comparison.
Possible activity solutions
1. Same denominator:
7/9 is greater than 4/9
2. Same numerator:
3/5 is greater than 3/8
3. Compare with one half:
5/8 is greater than 3/10
4. Equivalent fractions:
3/4 is greater than 2/3
5. Compare distance from one:
Eleven twelfths is missing one twelfth.
Seven eighths is missing one eighth.
One twelfth is smaller than one eighth, so:
11/12 is greater than 7/8
Common mistakes to avoid
- comparing fractional parts that refer to different sized wholes
- looking only at the numerators
- looking only at the denominators
- assuming a larger denominator always means a larger fraction
- changing a numerator or denominator without creating an equivalent fraction
- forgetting that fractions are single numbers
- using a complicated method when a simple benchmark would work
Fraction comparison checklist
When comparing two fractions, ask:
- Do the fractions refer to the same whole?
- Do they already have the same denominator?
- Do they have the same numerator?
- Can I compare each with one half or one?
- Would equivalent fractions make the comparison clearer?
- Would a number line help?
- Does my answer make sense based on the size of each fraction?
Self check
- Why is 1/4 greater than 1/8?
- Why can fractions with the same denominator be compared using their numerators?
- How can one half help compare fractions with different denominators?
- Why are equivalent fractions useful when comparing fractions?
- Why must fractions refer to the same whole when comparing fractional parts?
Suggested answers
- When the same whole is divided into four equal parts, each part is larger than when it is divided into eight equal parts.
- The fractional pieces are the same size, so the fraction with more of those pieces is greater.
- You can determine whether each fraction lies below, at or above one half.
- Equivalent fractions can express both values using parts of the same size, making direct comparison easier.
- Different sized wholes can make the physical parts different sizes even when the written fractions suggest another numerical relationship.
Lesson summary
Fractions should be compared as complete numbers.
When denominators are equal, compare numerators. When numerators are equal, think about the size of each fractional part.
Benchmarks such as one half and one, equivalent fractions and number lines provide reliable ways to compare fractions with different numerators and denominators.
Choose the strategy that makes the size relationship easiest to understand.
Continue learning
Review Fractions Explained with Everyday Examples if you need more practice with equivalent fractions and fraction meaning.
Review Practical Number Sense for Everyday Mathematics for more practice comparing numerical magnitude and using benchmarks.