Topics Grade boundaries Exam dates Yumio GCSE Maths Higher Grade 7 Algebraic fractions
Algebraic fractions – GCSE Maths Higher · Grade 7 · Non-calculator
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On this page Objectives Key points Examples Questions Mistakes Tips Worksheet Simplify, add, subtract, multiply and divide fractions with algebra in them, and solve equations that contain them. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Algebraic fractions follow the same rules as number fractions. Multiply: multiply tops and bottoms, cancelling factors (brackets) first. Divide: flip the second and multiply. Add/subtract: common denominator, which is usually the product of the two denominators (or the LCM of the brackets). Factorise anything that will factorise before you cancel — you may only cancel factors, never terms.
x 2 + 5 x + 6 x 2 − 9 = ( x + 2 ) ( x + 3 ) ( x − 3 ) ( x + 3 ) = x + 2 x − 3 x = − 3 x + 1 2 + x − 1 3 ( x + 1 ) ( x − 1 )
Factorise then cancel common factors. Add/subtract with a common denominator. Multiply tops and bottoms; divide by multiplying by the reciprocal. Cancel factors, never terms.
Objectives Cancel a difference of squares over a linear factor. Cancel a numeric factor in a linear fraction. Add two algebraic fractions. Key points Factorise the numerator and denominator before cancelling. Add or subtract by finding a common denominator — usually the product of the two denominators. Multiply straight across; divide by flipping the second fraction and multiplying. Solve an equation with algebraic fractions by multiplying every term by the common denominator. Worked examples Example 1 Simplify the algebraic fraction (x²-9)/(x-3).
2 marks
Factor the numerator using difference of two squares: x² - 9 = (x - 3)(x + 3). Divide out the common bracket factor (x - 3). State the simplified expression: x+3. Answer: x+3
Common slips Slip: Cancel individual terms: x²/x - 9/(-3). Terms inside sums cannot be cancelled individually. Slip: Cancel x and leave 3. Only common factors of the entire numerator and denominator cancel. Slip: Write x-3. (x-3) divides out leaving (x+3). Example 2 Simplify the algebraic fraction (2x+6)/(x+3).
2 marks
Factorise the numerator: 2x + 6 = 2(x + 3). Divide out the common factor (x + 3). State the simplified integer result: 2. Answer: 2
Common slips Slip: Cancel the x terms: 2x/x = 2. Factor out the highest common factor first. Slip: Leave 2x + 2. The whole bracket (x+3) cancels. Slip: Write 2(x+3). The fraction simplifies to 2. Example 3 Add the algebraic fractions 1/(x+2)+1/(x+3).
3 marks
Find common denominator by multiplying: (x + 2)(x + 3) = x² + 5x + 6. Cross multiply numerators: 1(x + 3) + 1(x + 2) = 2x + 5. Form single algebraic fraction: (2x+5)/(x²+5x+6). Answer: (2x+5)/(x²+5x+6)
Common slips Slip: Add denominators directly: (x + 2) + (x + 3). Fractions must have a common multiple denominator. Slip: Keep numerator as 1 + 1 = 2. Each numerator is multiplied by the missing denominator. Slip: Write 2/(2x+5). The combined fraction is (2x+5)/(x²+5x+6). Questions and answers GCSE Maths algebraic fractions questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy 2 marks Simplify the algebraic fraction (x²-4)/(x-2).
Show answer Answer: x+2
Factor numerator: (x-2)(x+2). Cancel (x-2) to leave x+2. M1 — Factor numerator: (x-2)(x+2). A1 — Cancel (x-2) to leave x+2. Question 2 Easy 2 marks Simplify the algebraic fraction (x²-25)/(x-5).
Show answer Answer: x+5
Factor numerator: (x-5)(x+5). Cancel (x-5) to leave x+5. M1 — Factor numerator: (x-5)(x+5). A1 — Cancel (x-5) to leave x+5. Question 3 Medium 2 marks Simplify the algebraic fraction (4x+8)/(x+2).
Show answer Answer: 4
Factor 4(x+2). Cancel (x+2) to leave 4. M1 — Factor 4(x+2). A1 — Cancel (x+2) to leave 4. Question 4 Medium 2 marks Simplify the algebraic fraction (5x+15)/(x+3).
Show answer Answer: 5
Factor 5(x+3). Cancel (x+3) to leave 5. M1 — Factor 5(x+3). A1 — Cancel (x+3) to leave 5. Question 5 Hard 2 marks Subtract the algebraic fractions 1/(x+2)-1/(x+4).
Show answer Answer: 2/(x²+6x+8)
Numerator is (x+4)-(x+2) = 2. Denominator is (x+2)(x+4) = x²+6x+8. M1 — Numerator is (x+4)-(x+2) = 2. A1 — Denominator is (x+2)(x+4) = x²+6x+8. Question 6 Hard 2 marks Subtract the algebraic fractions 1/(x+1)-1/(x+3).
Show answer Answer: 2/(x²+4x+3)
Numerator is (x+3)-(x+1) = 2. Denominator is (x+1)(x+3) = x²+4x+3. M1 — Numerator is (x+3)-(x+1) = 2. A1 — Denominator is (x+1)(x+3) = x²+4x+3. Question 7 Exam-style 4 marks Write 2/(x+4)+3/(x+5) as a single fraction.
Show answer Answer: (5x+22)/(x²+9x+20)
Identify common denominator as (x + 4)(x + 5). Write each fraction over it: 2(x + 5) and 3(x + 4) on the top. Expand the numerator: 2x + 10 + 3x + 12 = 5x + 22. Expand the denominator: x² + 9x + 20. State fraction: (5x+22)/(x²+9x+20). M1 — Common denominator (x+4)(x+5) M1 — Numerator 2(x+5) + 3(x+4) A1 — Numerator 5x + 22 A1 — (5x+22)/(x²+9x+20) or (5x+22)/((x+4)(x+5)) Common mistakes Most lost marks on this topic are one slip.
Cancelling one letter from a square
Factor first, then cancel a whole bracket
Adding the denominators
Use a common denominator
Leaving a cancelled bracket on the top
Divide it out completely
Spot the slip Sam cancels the x in (x²-9)/(x-3) and writes x-9. What is the simplified fraction?
x+3 x-9 x-3 Show the slip x+3 (x-3)(x+3)/(x-3) leaves x+3.
Exam tips Be ready to do this without a calculator. Paper 1 is the non-calculator paper. If the question says “show that”, the result is given. The marks are for the steps that reach it. The method is the same on AQA-style and Edexcel-style papers. Write the factorised forms on separate lines before cancelling — the factorising is where the method marks are. Keep denominators as products of brackets; it makes cancelling later much easier. Printable worksheet The same questions are in a PDF you can print. Answers start on a later page.
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Questions students ask Is Algebraic fractions Foundation or Higher? Algebraic fractions is Higher only, aimed at grade 7.
What is a common slip on Algebraic fractions? Cancelling one letter from a square. Factor first, then cancel a whole bracket.
Where can I get Algebraic fractions questions and answers? The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.
Related topics More on Algebraic fractions Previous topic Harder factorising (ax² + bx + c) Next topic Harder rearranging formulae AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.
Last updated 9 November 2026.