Algebraic Fractions – Fractions with Variables
Algebraic fractions contain variables in the numerator, denominator, or both. They follow all the same rules as numerical fractions — but you also need the factorising and simplifying skills you have built throughout this section.
What Is an Algebraic Fraction?
An algebraic fraction has an algebraic expression in the numerator, denominator, or both. Examples: x/3, (2x+1)/(x-4), and (x squared - 9)/(x + 3). The denominator can never equal zero.
Simplifying Algebraic Fractions
Factorise the numerator and denominator, then cancel common factors.
Numerator: x squared - 9 = (x+3)(x-3). Cancel (x+3): Answer = x - 3. (Valid for x not equal to -3.)
Numerator: 3x(2x + 1). Cancel 3x: Answer = (2x + 1) / 3. (Valid for x not equal to 0.)
Multiplying Algebraic Fractions
Multiply numerators together and denominators together, then simplify.
(2x times 3) / (5 times 4x) = 6x / 20x = 3/10.
Dividing Algebraic Fractions
Dividing by a fraction means multiplying by its reciprocal. Flip the second fraction and multiply.
(x / 6) times (3 / 2x) = 3x / 12x = 1/4.
Adding and Subtracting Algebraic Fractions
Find a common denominator, write both fractions over it, then add or subtract the numerators.
Common denominator: x(x+1). Numerators: (x+1) + 2x = 3x + 1. Answer: (3x + 1) / x(x+1).
Common denominator: (x-2)(x+1). Numerators: 3(x+1) - 1(x-2) = 3x+3 - x+2 = 2x+5. Answer: (2x + 5) / ((x-2)(x+1)).
Common Mistakes
- Cancelling terms instead of factors: (x squared + 4) / (x + 4) cannot be simplified — they share no common factor.
- Forgetting to state the values of x for which a simplification is invalid (where the original denominator is zero).
- Adding fractions by adding numerators and denominators separately.
Key Takeaways
- Simplify by factorising top and bottom, then cancelling common factors.
- Multiply: multiply tops, multiply bottoms, then simplify.
- Divide: flip the second fraction and multiply.
- Add/subtract: find a common denominator first, then combine numerators.
Practice Questions
- Simplify (x squared - 4) / (x - 2).
- Simplify (3x squared + 6x) / (9x).
- Multiply (x/4) times (8/(2x)).
- Add (2/x) + (3/(x+2)).
- Simplify (2/(x-1)) - (1/(x+3)) and state any values of x for which it is undefined.
Congratulations!
You have completed the full Algebra section on MathsFamily. Head back to the Resources page to continue your mathematics journey.