Fractions appear throughout KS3 maths, and being able to compare and order them is a fundamental skill. The most reliable approach is to rewrite fractions with a common denominator, then compare numerators directly. Once you have mastered this technique, you can apply it to mixed numbers and even negative fractions.
How do I compare two fractions with different denominators?
When two fractions have different denominators, you cannot decide which is larger simply by looking at the numerators. The denominator tells you the size of each equal part — a larger denominator means smaller individual pieces. So 1/8 is smaller than 1/3, even though 8 > 3.
The standard method is:
- Find a common denominator — a whole number that both denominators divide into exactly.
- Convert each fraction into an equivalent fraction using that common denominator.
- Compare the numerators. The larger numerator belongs to the larger fraction.
Worked Example 1 — Which is larger: 3/4 or 5/7?
- The lowest common multiple (LCM) of 4 and 7 is 28.
- Convert: 3/4 = 21/28 (multiply numerator and denominator by 7).
- Convert: 5/7 = 20/28 (multiply numerator and denominator by 4).
- Compare: 21 > 20, so 3/4 > 5/7.
Check with decimals: 3 ÷ 4 = 0.75 and 5 ÷ 7 ≈ 0.714, confirming 3/4 is larger.
Alternative: the decimal method
Divide the numerator by the denominator for each fraction, then compare:
- 3/8 = 0.375
- 2/5 = 0.4
Since 0.375 < 0.4, we conclude 3/8 < 2/5. This method is particularly convenient on a calculator paper, but the common-denominator method is more reliable for non-calculator work, especially with recurring decimals.
What is a common denominator?
A common denominator is any shared multiple of two or more denominators. For example, both 4 and 6 divide exactly into 12, so 12 is a common denominator for fractions with denominators 4 and 6.
Equivalent fractions are different representations of the same value. You create them by multiplying both the numerator and denominator by the same non-zero integer — this is equivalent to multiplying by 1, so the value of the fraction is unchanged:
- 3/4 = 6/8 = 9/12 = 21/28 (multiply top and bottom by 2, 3, and 7 respectively)
You can also reduce a fraction to a simpler equivalent by dividing — this is known as simplifying or cancelling down.
How do I find the lowest common denominator?
The lowest common denominator (LCD) is the smallest common denominator available. Working with the LCD keeps the numbers as small as possible and reduces arithmetic errors.
Method 1 — List the multiples
To find LCD(4, 6):
- Multiples of 4: 4, 8, 12, 16 …
- Multiples of 6: 6, 12, 18 …
- LCD = 12
Method 2 — Prime factorisation (useful for larger numbers)
- 4 = 2²; 6 = 2 × 3
- LCM = 2² × 3 = 12
In a timed exam, listing multiples is usually quickest for denominators below 20. Use prime factorisation when the denominators are larger or you need to be certain of your answer.
How do I order a list of fractions from smallest to largest?
Worked Example 2 — Order 2/3, 1/4, 5/6, 3/8 from smallest to largest
Step 1 — Find the LCD of 3, 4, 6, and 8.
LCM(3, 4, 6, 8) = 24. Check: 24 ÷ 3 = 8 ✓, 24 ÷ 4 = 6 ✓, 24 ÷ 6 = 4 ✓, 24 ÷ 8 = 3 ✓.
Step 2 — Convert each fraction to an equivalent fraction with denominator 24.
| Original fraction | Equivalent (denominator 24) | Decimal check |
|---|---|---|
| 1/4 | 6/24 | 0.25 |
| 3/8 | 9/24 | 0.375 |
| 2/3 | 16/24 | ≈ 0.667 |
| 5/6 | 20/24 | ≈ 0.833 |
Step 3 — Order the numerators: 6 < 9 < 16 < 20.
Step 4 — Write the answer using the original fractions:
1/4 < 3/8 < 2/3 < 5/6 ✓
The decimal check column provides an independent confirmation of the ordering.
Further example — ordering 3/4, 2/3, 5/6, 7/12 with LCD = 12
| Original fraction | Equivalent (denominator 12) | Decimal check |
|---|---|---|
| 7/12 | 7/12 | ≈ 0.583 |
| 2/3 | 8/12 | ≈ 0.667 |
| 3/4 | 9/12 | 0.75 |
| 5/6 | 10/12 | ≈ 0.833 |
Order of numerators: 7 < 8 < 9 < 10, so 7/12 < 2/3 < 3/4 < 5/6 ✓
How do I compare mixed numbers?
A mixed number has a whole-number part and a fractional part, for example 2¾ or 5⅓.
Rule: compare the whole-number parts first. If they differ, the larger whole number gives the larger mixed number — no need to examine the fractions at all. If the whole-number parts are equal, compare the fractional parts using a common denominator.
Worked Example 3 — Compare 2 3/5 and 2 7/10
- Both mixed numbers have whole-number part 2, so compare the fractional parts.
- LCD(5, 10) = 10.
- 3/5 = 6/10; compare 6/10 and 7/10.
- Since 6 < 7, the first fraction is smaller.
- Answer: 2 3/5 < 2 7/10 ✓
How do negative fractions work?
On a number line, values increase from left to right, and negative numbers sit to the left of zero. Among negative fractions, the one closest to zero is the largest.
Key rule: −1/4 > −3/4 because −1/4 is closer to zero than −3/4.
Worked Example 4 — Compare −1/3 and −2/5
- LCD(3, 5) = 15.
- −1/3 = −5/15; −2/5 = −6/15.
- On a number line, −5/15 sits to the right of −6/15 (it is closer to zero).
- Answer: −1/3 > −2/5 ✓
A common mistake is to apply the rule "larger denominator = smaller fraction" to negative fractions and then reverse the inequality. If in doubt, always sketch a quick number line — the fraction furthest from zero is the most negative and therefore the smallest.
Frequently Asked Questions
Can I use any common denominator, or does it have to be the lowest?
Any common denominator gives a correct comparison — for example, you could always multiply the two denominators together to obtain a valid one. However, the LCD keeps the numerators smaller and reduces the risk of arithmetic errors, particularly in timed exam conditions. If you do use a larger common denominator, your result will still be correct as long as the arithmetic is right.
What if the numerators are equal after I convert?
If the converted numerators are identical, the two fractions are equal in value — they are equivalent fractions that represent the same point on the number line. For instance, 3/6 and 4/8 both simplify to 1/2, so neither is greater than the other.
Is converting to decimals always a reliable method?
Yes — it is mathematically valid in every case. The difficulty arises with recurring decimals (for example, 1/3 = 0.333… and 2/7 ≈ 0.2857…), which are hard to compare precisely without rounding, and rounding can introduce errors. The common-denominator method avoids this problem entirely and is strongly preferred for non-calculator examination papers.
Do I need to simplify fractions before comparing them?
No — simplification is optional. Working with unsimplified fractions throughout the comparison is perfectly valid and will produce the correct answer. Simplifying first can sometimes make it easier to spot the LCD, but it is never compulsory, and leaving fractions unsimplified costs no marks.
Practise comparing and ordering fractions interactively with Professor Pi at aitutors.me — the AI maths tutor built for UK secondary school students.