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.

Algebraic fractions are the natural meeting point of everything covered earlier in this section — factorising, expanding, and simplifying all show up together the moment a fraction has algebra in its numerator or denominator. This makes them a genuine capstone skill: rational functions built from algebraic fractions are used throughout physics and engineering to model quantities like resistance in parallel electrical circuits, rates of chemical reactions, and lens equations in optics, wherever one quantity depends on the ratio of two changing values.

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.

Simplify (x squared - 9) / (x + 3).

Numerator: x squared - 9 = (x+3)(x-3). Cancel (x+3): Answer = x - 3. (Valid for x not equal to -3.)

Simplify (6x squared + 3x) / (9x).

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.

Multiply (2x / 5) times (3 / (4x)).

(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.

Divide (x / 6) by (2x / 3).

(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.

Add (1/x) + (2/(x+1)).

Common denominator: x(x+1). Numerators: (x+1) + 2x = 3x + 1. Answer: (3x + 1) / x(x+1).

Simplify (3/(x-2)) - (1/(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: Algebraic Fractions

Simplify the Algebraic Fraction

Congratulations!

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