GCSE Maths • Higher Tier
GCSE Algebraic Fractions Revision Guide
Build confidence with simplifying algebraic fractions, factorising expressions, using index laws, clearing denominators and solving algebraic fraction equations with this Higher Tier GCSE guide.
Foundation
Exam practice
Worked examples
What You'll Learn
Recognise Algebraic Fractions
Understand fractions that contain algebraic expressions in the numerator, denominator or both.
Simplify Using Common Factors
Simplify algebraic fractions by identifying and cancelling common factors.
Factorise Before Cancelling
Factorise expressions first so that common factors become visible and can be cancelled correctly.
Use Index Laws
Apply index laws when dividing powers of the same variable, such as x³ ÷ x² = x.
Solve Algebraic Fraction Equations
Remove fractions, isolate the variable and check solutions by substitution.
Key Algebraic Fraction Rules
1
Cancel factors, not terms
Only common factors can be cancelled. Terms that are being added or subtracted cannot be cancelled.
2
Factorise first
Factorise the numerator and denominator where possible before looking for common factors.
3
Use index laws
When dividing powers with the same base, subtract the powers; for example, x³ ÷ x² = x.
4
Clear the denominator
To solve an algebraic fraction equation, multiply both sides by the denominator. If there are different denominators, use their lowest common multiple and multiply every term.
5
Check the result
Check that a simplified fraction cannot be reduced further, and check an equation solution by substituting it into the original equation.
Algebraic Fractions Lessons
Understanding Algebraic Fractions
Learn what an algebraic fraction is and how it behaves like a numerical fraction while also requiring algebraic rules.
Simplifying Basic Algebraic Fractions
Simplify the numerical coefficients and variables, then write the answer in its simplest form.
Factorising Before Cancelling
Use common factors, quadratics and the difference of two squares to reveal factors that can be cancelled.
Solving Algebraic Fraction Equations
Multiply both sides by the denominator, simplify the equation and solve using standard equation methods.
Equations with Different Denominators
Multiply every term by the lowest common multiple of the denominators to remove all fractions simultaneously.
Worked Examples
Example 1: Add Fractions with Different Denominators
3/4 + 5/6 = 1 7/12
Step 1: Find the lowest common multiple of 4 and 6.
LCM = 12
Write both fractions with denominator 12.
3/4 = 9/12
5/6 = 10/12
Step 3: Add the numerators.
9/12 + 10/12 = 19/12
Step 4: Convert the improper fraction. 19/12 = 1 7/12
LCM = 12
Write both fractions with denominator 12.
3/4 = 9/12
5/6 = 10/12
Step 3: Add the numerators.
9/12 + 10/12 = 19/12
Step 4: Convert the improper fraction. 19/12 = 1 7/12
Example 2: Multiply & Simplify Fractions
3/8 × 4/9 = 1/6
Step 1: Cancel common factors.
3 and 9 divide by 3.
4 and 8 divide by 4.
Step 2: Step 2: Multiply the simplified fractions.
1/2 × 1/3 = 1/6
3 and 9 divide by 3.
4 and 8 divide by 4.
Step 2: Step 2: Multiply the simplified fractions.
1/2 × 1/3 = 1/6
Example 3: Divide a Mixed Number by a Fraction
1 1/2 ÷ 3/4 = 2
Step 1: Convert the mixed number to an improper fraction.
1 1/2 = 3/2
Step 2: Keep, Change, Flip.
3/2 ÷ 3/4 = 3/2 × 4/3
Step 3: Multiply and simplify.
3/2 × 4/3 = 12/6 = 2
1 1/2 = 3/2
Step 2: Keep, Change, Flip.
3/2 ÷ 3/4 = 3/2 × 4/3
Step 3: Multiply and simplify.
3/2 × 4/3 = 12/6 = 2
- Common GCSE Mistakes
- Adding or subtracting the denominators instead of keeping or matching them.
- Forgetting to find a common denominator before adding or subtracting unlike fractions.
- Multiplying or dividing only the numerator or only the denominator when creating equivalent fractions.
- Looking only at numerators or only at denominators when comparing fractions.
- Forgetting to flip the second fraction when dividing, or flipping the wrong fraction.
- Failing to convert mixed numbers before multiplying or dividing.
- Not simplifying the final answer fully.
- Forgetting units in fractions-of-an-amount questions.
- Exam Tips
- Check whether a fraction can be simplified before giving your final answer.
- Use the highest common factor to simplify efficiently.
- Find a common denominator before adding, subtracting or comparing fractions with different denominators.
- Cancel common factors before multiplying to keep the numbers smaller.
- Convert mixed numbers to improper fractions before multiplying or dividing.
- For a fraction of an amount, divide by the denominator and multiply by the numerator.
- Include units when the question uses money, length or a measured quantity.
- Practise equivalent fractions and common denominators so that the methods become quicker and more accurate.
Ready to Practice?
Printable Worksheet
Practise simplifying, comparing, converting and calculating with fractions, then check your answers.
Interactive Quiz
Test the key fraction rules using short questions on equivalent fractions, operations, mixed numbers and fractions of amounts.
Exam-Style Questions
Apply fraction methods to Foundation and Higher Tier calculations and problem-solving questions.
Watch the Fractions Tutorial
Follow a clear, step-by-step explanation of equivalent fractions, simplifying, comparing, the four operations, mixed numbers and fractions of amounts.
- Equivalent fractions and simplifying
- Adding, subtracting, multiplying and dividing
- Mixed numbers, improper fractions and fractions of amounts
Frequently Asked Questions
Equivalent fractions have the same value even though they look different. To create an equivalent fraction, multiply the numerator and denominator by the same number. For example, 1/2 = 2/4 = 3/6 = 4/8.
Find the highest common factor of the numerator and denominator, then divide both by that number. For example, the HCF of 18 and 24 is 6, so 18/24 simplifies to 3/4.
Find a common denominator and compare the equivalent fractions. You can also compare fractions by converting them to decimals or percentages. On a number line, the fraction further to the right is larger.
Find a common denominator, rewrite both fractions as equivalent fractions, then add or subtract the numerators. Keep the common denominator and simplify the final answer.
Multiply the numerators, multiply the denominators and simplify. A common denominator is not needed. You can cancel common factors before multiplying to make the calculation easier.
Use Keep, Change, Flip: keep the first fraction, change division to multiplication and flip the second fraction. Then multiply and simplify.
To convert a mixed number, multiply the whole number by the denominator, add the numerator and keep the same denominator. To convert an improper fraction, divide the numerator by the denominator; the quotient is the whole number and the remainder becomes the new numerator.
To convert a mixed number, multiply the whole number by the denominator, add the numerator and keep the same denominator. To convert an improper fraction, divide the numerator by the denominator; the quotient is the whole number and the remainder becomes the new numerator.
Divide the amount by the denominator, then multiply by the numerator. For example, 3/4 of 20 is 20 ÷ 4 × 3 = 15.
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