Simplify x^2 + 5xx + 5. [2 marks]

Express 3x + 12 as a single fraction. [2 marks]

SQA SQA National 5 2020-style practice: Simplify x^2 - 9x^2 + 7x + 12.

Factorise the top as a difference of two squares: x^2 - 9 = (x + 3)(x - 3) (1 mark). Factorise the bottom trinomial: x^2 + 7x + 12 = (x + 3)(x + 4). Cancel the common factor (x + 3), leaving x - 3x + 4 (1 mark). Markers reward correct factorisation of both top and bottom and cancelling the common bracket.

SQA SQA National 5 2023-style practice: Express 2x + 3x + 1 as a single fraction in its simplest form.

The common denominator is x(x + 1) (1 mark). Rewrite each fraction: 2(x + 1)x(x + 1) + 3xx(x + 1) (1 mark). Add the numerators: 2(x + 1) + 3xx(x + 1) = 2x + 2 + 3xx(x + 1) = 5x + 2x(x + 1) (1 mark). Markers reward the common denominator, correct rewriting, and the simplified single fraction.

ScotlandMathsSyllabus dot point

How do you simplify, add, subtract, multiply and divide algebraic fractions?

Working with algebraic fractions: simplifying by factorising and cancelling, and adding, subtracting, multiplying and dividing algebraic fractions.

A focused answer to the SQA National 5 Mathematics algebraic fractions content, covering simplifying by factorising and cancelling, multiplying and dividing fractions, and adding and subtracting with a common denominator.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-14

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
Simplifying by factorising and cancelling
Multiplying and dividing
Adding and subtracting
Putting the skills together
Examples in context
Try this

What this dot point is asking

The SQA wants you to simplify an algebraic fraction by factorising and cancelling, and to add, subtract, multiply and divide algebraic fractions, giving the answer in its simplest form. The skills mirror ordinary fraction work; the only new step is factorising before you cancel.

Simplifying by factorising and cancelling

You can only cancel a factor that divides the whole of the top and the whole of the bottom. So the first step is always to factorise both fully; only then can complete factors be cancelled.

Simplify

\dfrac{4x + 8}{x^2 - 4}

You can only cancel whole factors, never individual terms. The correct approach is to factorise both the numerator and denominator fully, then cancel any bracket that appears in both.

Step 1: Factorise the numerator and denominator

The numerator $4x + 8$ has a common factor of $4$ : $4x + 8 = 4(x + 2)$ .

The denominator $x^2 - 4$ is a difference of two squares (since $4 = 2^2$ ): $x^2 - 4 = (x + 2)(x - 2)$ .

Step 2: Cancel the common bracket

The bracket $(x + 2)$ appears in both numerator and denominator and can be cancelled, because it divides the whole of each:

\dfrac{4(x + 2)}{(x + 2)(x - 2)} = \dfrac{4}{x - 2}

Final answer: $\dfrac{4x + 8}{x^2 - 4} = \dfrac{4}{x - 2}$ .

Multiplying and dividing

Multiplying is the same as for numbers: multiply the numerators together and the denominators together, cancelling any common factors first. Dividing means multiplying by the reciprocal of the second fraction (turn it upside down).

Simplify

\dfrac{3}{x} \div \dfrac{6}{x^2}

Dividing by a fraction is the same as multiplying by its reciprocal. We flip the second fraction upside down, then multiply and cancel.

Step 1: Convert the division to multiplication by flipping the second fraction

The reciprocal of $\dfrac{6}{x^2}$ is $\dfrac{x^2}{6}$ :

\dfrac{3}{x} \div \dfrac{6}{x^2} = \dfrac{3}{x} \times \dfrac{x^2}{6}

Step 2: Cancel common factors and multiply

Cancel the factor of $3$ against the $6$ (leaving $2$ on the bottom) and cancel one $x$ from the $x^2$ against the $x$ in the first denominator:

\dfrac{3 \times x^2}{x \times 6} = \dfrac{x}{2}

Final answer: $\dfrac{3}{x} \div \dfrac{6}{x^2} = \dfrac{x}{2}$ .

Adding and subtracting

To add or subtract, both fractions need the same denominator. Find the common denominator (usually the product of the two denominators), rewrite each fraction over it, then combine the numerators.

Simplify

\dfrac{5}{x} - \dfrac{2}{3}

The two fractions have different denominators ( $x$ and $3$ ), so they cannot be subtracted as they stand. We need to rewrite them over the same denominator before combining the numerators.

Step 1: Find the common denominator

Neither $x$ nor $3$ divides the other, so the common denominator is their product: $3 \times x = 3x$ .

Step 2: Rewrite each fraction over the common denominator

To convert $\dfrac{5}{x}$ to denominator $3x$ , multiply top and bottom by $3$ : $\dfrac{5}{x} = \dfrac{15}{3x}$ .

To convert $\dfrac{2}{3}$ to denominator $3x$ , multiply top and bottom by $x$ : $\dfrac{2}{3} = \dfrac{2x}{3x}$ .

Step 3: Subtract the numerators, keeping the common denominator

\dfrac{15}{3x} - \dfrac{2x}{3x} = \dfrac{15 - 2x}{3x}

Final answer: $\dfrac{5}{x} - \dfrac{2}{3} = \dfrac{15 - 2x}{3x}$ .

When both denominators contain a variable, the common denominator is usually their product, and each numerator is multiplied by the denominator it is missing. The harder questions combine the skills: you may need to multiply out brackets in the numerator and then collect like terms, but you never multiply out the denominator, because leaving it factorised keeps the answer in its simplest form.

Putting the skills together

A full question often asks you to simplify a fraction, then carry out an operation. The order that keeps the working tidy is: factorise everything first, cancel any common factors, then perform the multiplication, division, addition or subtraction. Cancelling early keeps the numbers and expressions small, which reduces the chance of a slip and makes the final simplification obvious. Always finish by checking that nothing in your answer can be factorised and cancelled further.

Examples in context

Algebraic fractions appear whenever rates are combined. If one pump fills a tank in $x$ hours and a second in $x + 1$ hours, in one hour they fill $\dfrac{1}{x} + \dfrac{1}{x + 1}$ of the tank. Combining over a common denominator gives $\dfrac{(x + 1) + x}{x(x + 1)} = \dfrac{2x + 1}{x(x + 1)}$ , the fraction filled per hour, from which the time to fill the tank together can be found.

Try this

Q1. Simplify $\dfrac{x^2 + 5x}{x + 5}$ . [2 marks]

Cue. $\dfrac{x(x + 5)}{x + 5} = x$ .

Q2. Simplify $\dfrac{a}{4} \times \dfrac{8}{a^2}$ . [2 marks]

Cue. $\dfrac{8a}{4a^2} = \dfrac{2}{a}$ .

Q3. Express $\dfrac{3}{x} + \dfrac{1}{2}$ as a single fraction. [2 marks]

Cue. $\dfrac{6 + x}{2x}$ .

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA National 5 20202 marksSimplify

\dfrac{x^2 - 9}{x^2 + 7x + 12}

Show worked answer →

Factorise the top as a difference of two squares: $x^2 - 9 = (x + 3)(x - 3)$ (1 mark). Factorise the bottom trinomial: $x^2 + 7x + 12 = (x + 3)(x + 4)$ . Cancel the common factor $(x + 3)$ , leaving $\dfrac{x - 3}{x + 4}$ (1 mark). Markers reward correct factorisation of both top and bottom and cancelling the common bracket.

SQA National 5 20233 marksExpress

\dfrac{2}{x} + \dfrac{3}{x + 1}

as a single fraction in its simplest form.

Show worked answer →

The common denominator is $x(x + 1)$ (1 mark). Rewrite each fraction: $\dfrac{2(x + 1)}{x(x + 1)} + \dfrac{3x}{x(x + 1)}$ (1 mark). Add the numerators: $\dfrac{2(x + 1) + 3x}{x(x + 1)} = \dfrac{2x + 2 + 3x}{x(x + 1)} = \dfrac{5x + 2}{x(x + 1)}$ (1 mark). Markers reward the common denominator, correct rewriting, and the simplified single fraction.

Related dot points

Sources & how we know this

SQA National 5 Mathematics Course Specification — SQA (2018)