GCSE Maths
GCSE Maths
Mixed Numbers and Equivalent Fractions: GCSE Maths Guide
Fractions are a guaranteed topic in every GCSE Maths paper. Mastering equivalent fractions GCSE rules and mixed numbers GCSE operations will earn you marks across Foundation and Higher tiers.
This guide covers everything you need — no padding, just the essential rules and one worked example per concept.
Part 1: Equivalent Fractions
What Are Equivalent Fractions?
Equivalent fractions have different numerators and denominators but represent the same value.
The rule is simple:
Multiply or divide both the numerator and denominator by the same number.
Example: Finding an Equivalent Fraction
Find the missing numerator:
3/4 = ?/12
Since:
4 × 3 = 12
Multiply the numerator by the same number:
3 × 3 = 9
Therefore:
3/4 = 9/12
Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their highest common factor (HCF).
Example
Simplify:
18/24
The HCF of 18 and 24 is 6.
18 ÷ 6 = 3
24 ÷ 6 = 4
Therefore:
18/24 = 3/4
The fraction is now fully simplified.
Cross-Multiplication Check
You can check whether two fractions are equivalent using cross-multiplication.
For example:
2/3 = 8/12
Cross-multiply:
2 × 12 = 24
and
3 × 8 = 24
Because both products are equal, the fractions are equivalent.
2/3 = 8/12 ✓
Part 2: Mixed Fractions and Improper Fractions
A mixed fraction, also called a mixed number, contains a whole number and a proper fraction.
For example:
2¾
An improper fraction has a numerator greater than or equal to its denominator.
For example:
11/4
Mixed Number → Improper Fraction
Use the rule:
(Whole number × denominator) + numerator
Keep the same denominator.
Example
Convert:
3½
Multiply the whole number by the denominator:
3 × 2 = 6
Add the numerator:
6 + 1 = 7
Keep the denominator as 2.
Therefore:
3½ = 7/2
Improper Fraction → Mixed Number
Divide the numerator by the denominator.
- The quotient becomes the whole number.
- The remainder becomes the new numerator.
- The denominator stays the same.
Example
Convert:
17/5
Divide:
17 ÷ 5 = 3 remainder 2
Therefore:
17/5 = 3⅖
Part 3: Adding Mixed Numbers
Same Denominator
When the fractional parts have the same denominator:
- Add the whole numbers.
- Add the numerators.
- Keep the denominator the same.
- Simplify if necessary.
Example
Calculate:
1⅖ + 2⅕
Add the whole numbers:
1 + 2 = 3
Add the fractions:
2/5 + 1/5 = 3/5
Therefore:
1⅖ + 2⅕ = 3⅗
Different Denominators
When adding mixed numbers with different denominators:
- Convert them to improper fractions.
- Find a common denominator.
- Add the fractions.
- Simplify if necessary.
- Convert back to a mixed number.
Example
Calculate:
2⅓ + 1⅙
Step 1: Convert to Improper Fractions
2⅓ = 7/3
1⅙ = 7/6
Step 2: Find a Common Denominator
The common denominator of 3 and 6 is 6.
Convert 7/3:
7/3 = 14/6
Step 3: Add
14/6 + 7/6 = 21/6
Step 4: Simplify
21/6 = 7/2
Step 5: Convert Back to a Mixed Number
7/2 = 3½
Therefore:
2⅓ + 1⅙ = 3½
Part 4: Subtracting Mixed Numbers
A reliable method for subtracting mixed numbers is to convert them to improper fractions first.
This helps avoid regrouping errors.
Example
Calculate:
4⅖ − 1³⁄₁₀
Step 1: Convert to Improper Fractions
4⅖ = 22/5
1³⁄₁₀ = 13/10
Step 2: Find a Common Denominator
The common denominator is 10.
Convert 22/5:
22/5 = 44/10
Step 3: Subtract
44/10 − 13/10 = 31/10
Step 4: Convert Back
31/10 = 3¹⁄₁₀
Therefore:
4⅖ − 1³⁄₁₀ = 3¹⁄₁₀
Common Mistakes
Not Simplifying Fractions
Always check whether the numerator and denominator have a common factor after completing a calculation.
For example:
6/8 = 3/4
The fraction 6/8 should be simplified to 3/4.
Adding Denominators
Never add the denominators when adding fractions.
Incorrect:
1/4 + 2/4 = 3/8 ✗
Correct:
1/4 + 2/4 = 3/4 ✓
Once the denominators are the same, only add the numerators.
Forgetting to Convert Back
If the question asks for the answer as a mixed number, make sure you convert an improper fraction at the end.
For example:
7/2 = 3½
If the question asks for a mixed number, the final answer should be:
3½
Quick Reference Table
| Operation | Key Step | Example |
|---|---|---|
| Equivalent fractions | Multiply or divide top and bottom by the same number | 1/2 = 4/8 |
| Simplifying fractions | Divide by the HCF | 6/9 = 2/3 |
| Mixed → improper | (whole × denominator) + numerator | 2¾ = 11/4 |
| Improper → mixed | Divide; remainder becomes new numerator | 11/4 = 2¾ |
| Adding mixed numbers | Find a common denominator and add | 1½ + 1½ = 3 |
| Subtracting mixed numbers | Convert first, find common denominator, subtract | 3½ − 1¼ = 2¼ |
FAQ
Do I Always Need to Convert Mixed Numbers to Improper Fractions?
Not always, but it is often the safest method — especially when subtracting mixed numbers where regrouping may be required.
How Do I Know When a Fraction Is Fully Simplified?
A fraction is fully simplified when the numerator and denominator have no common factor greater than 1.
For example:
12/18
Both numbers are divisible by 6:
12 ÷ 6 = 2
18 ÷ 6 = 3
Therefore:
12/18 = 2/3
The fraction 2/3 is fully simplified.
If you are unsure, check whether both numbers can be divided by common factors such as 2, 3, 5, or 7.
Are Mixed Fractions and Mixed Numbers the Same Thing?
Yes. Both terms describe a whole number combined with a proper fraction.
For example:
3⅖
is both a mixed fraction and a mixed number.