Algebraic Fractions: Adding, Subtracting and Solving Equations
Adding or subtracting algebraic fractions, such as 3/(x + 1) + 2/(x - 2), means rewriting each fraction over a common denominator before combining the numerators into one fraction. Equations containing algebraic fractions are solved by multiplying every term by the common denominator to clear the fractions, often producing a linear or quadratic equation. This builds directly on simplifying algebraic fractions.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify a common denominator for the fractions, usually the product of the individual denominators (factorised first if needed).
- Rewrite each fraction over the common denominator, adjusting each numerator to match.
- Combine the numerators over the single common denominator, expanding brackets carefully and collecting like terms.
- To solve an equation containing algebraic fractions, multiply every term on both sides by the common denominator to clear all the fractions at once.
- Expand and rearrange the resulting equation, which may be linear or a quadratic that needs factorising or the quadratic formula.
- State any value(s) of x for which the original expression or equation is undefined, where a denominator would equal zero.
Worked example
Write 5/(x + 3) - 2/(x - 1) as a single fraction in its simplest form.
- Identify the common denominator (x + 3)(x - 1).
- Rewrite the expression as [5(x - 1) - 2(x + 3)] / [(x + 3)(x - 1)].
- Expand the numerator: 5(x - 1) = 5x - 5 and 2(x + 3) = 2x + 6.
- Combine like terms in the numerator: 5x - 5 - 2x - 6 = 3x - 11.
- Final answer: (3x - 11)/(x^2 + 2x - 3).
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Write x/4 + x/6 as a single fraction in its simplest form.Show answer
Answer: 5x/12 (common denominator 12: 3x/12 + 2x/12)
Q2Solve x/2 + x/3 = 10.Show answer
Answer: x = 12 (multiply through by 6 to get 5x = 60)
Q3Write 3/(x + 2) + 2/(x - 3) as a single fraction in its simplest form.Show answer
Answer: (5x - 5)/(x^2 - x - 6) (numerator 3(x-3) + 2(x+2) = 5x - 5)
Q4Solve 6/(x + 1) = 3.Show answer
Answer: x = 1 (multiply through by (x+1): 6 = 3x + 3)
Q5Simplify (x^2 + 3x + 2)/(x^2 - 1) fully, stating any value(s) of x for which it is undefined.Show answer
Answer: (x + 2)/(x - 1), undefined at x = 1 or x = -1
Q6Solve 3/(x - 1) + 5/(x + 1) = 2, giving all solutions.Show answer
Answer: x = 0 or x = 4 (clears fractions to 2x^2 - 8x = 0)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Write 4/(x + 3) - 1/(x - 2) as a single fraction in its simplest form.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 3 available
Solve 5/(x + 2) = 3/(x - 4), stating any value(s) of x for which the equation is not defined.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Maya cycles 18 km from her home to the lake at an average speed of x km/h. Due to a flat tyre, her average speed on the return journey falls to (x - 3) km/h, and the return journey takes 1 hour longer than the outward journey. (a) Show that x^2 - 3x - 54 = 0. (b) Solve the equation to find the value of x, given that x is greater than 3.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 6 available
See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the algebraic fractions: adding, subtracting and solving equations worksheet pack - 9 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Other cuts of this worksheet:
Next topics
Not quite what you needed?
Tell us what is missing on algebraic fractions: adding, subtracting and solving equations, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.