Year 10 - Algebraic fractions - Free practice

Year 10 Algebraic Fractions Worksheets

Add, subtract, multiply, divide - and simplify cleanly.

Quick answer

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

Treat algebraic fractions like numerical fractions. To multiply, multiply numerators and denominators then simplify. To divide, multiply by the reciprocal. To add or subtract, find a common denominator first, then combine numerators. For example, x/3 + x/6 = 2x/6 + x/6 = 3x/6 = x/2.

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 10 algebraic fractions: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Adding algebraic fractions with a common denominator
Simplify $\frac{3x}{8} + \frac{x}{8}.$
Show worked solution
Key idea: When denominators are the same, add the numerators and keep the denominator.
1. Add the numerators: $3x + x = 4x$
2. Write over the common denominator: $\frac{4x}{8}$
3. Simplify by dividing numerator and denominator by 4: $\frac{x}{2}$
Answer: $\dfrac{x}{2}$
Example 2 · Multiplying and simplifying algebraic fractions
Simplify $\frac{4x}{9} \times \frac{3}{2x}.$
Show worked solution
Key idea: Multiply numerators together and denominators together, then cancel common factors before or after multiplying.
1. Write as a single fraction: $\frac{4x \times 3}{9 \times 2x}$
2. Simplify numerator and denominator: $\frac{12x}{18x}$
3. Cancel $x$ (assuming $x \neq 0$ ): $\frac{12}{18}$
4. Simplify the numerical fraction by dividing by 6: $\frac{2}{3}$
Answer: $\dfrac{2}{3}$
Example 3 · Algebraic fractions in a real-life problem
Mia and Theo are painting a fence together. Mia can paint $\dfrac{1}{x}$ of the fence per hour and Theo can paint $\dfrac{1}{3x}$ of the fence per hour, where $x$ is the number of hours it would take Mia to paint the fence alone. Write a simplified expression for the fraction of the fence they complete together in one hour.
Show worked solution
Key idea: Add the two rates. Since the denominators are different, find a common denominator first.
1. Identify the two rates: $\dfrac{1}{x}$ and $\dfrac{1}{3x}$
2. The lowest common denominator of $x$ and $3x$ is $3x$ .
3. Rewrite the first fraction: $\frac{1}{x} = \frac{3}{3x}$
4. Add the fractions: $\frac{3}{3x} + \frac{1}{3x} = \frac{4}{3x}$
Answer: $\dfrac{4}{3x}$ of the fence per hour

Common mistakes

Where students usually lose marks

Adding denominators instead of finding a common denominator

$\dfrac{1}{x} + \dfrac{1}{3x} = \dfrac{2}{4x}$

Never add the denominators. Find the lowest common denominator, rewrite each fraction with that denominator, then add only the numerators.

Cancelling terms instead of factors

$\dfrac{x + 6}{6} = x + 1$ (cancelling the 6s)

You can only cancel a factor that multiplies the entire numerator and the entire denominator. Here, 6 is added to $x$ , not multiplied, so nothing cancels.

Forgetting to flip the second fraction when dividing

$\dfrac{2x}{5} \div \dfrac{x}{3} = \dfrac{2x}{5} \times \dfrac{x}{3}$

When dividing by a fraction, multiply by its reciprocal. Flip the second fraction first, then multiply: $\dfrac{2x}{5} \times \dfrac{3}{x}$ .

Cancelling x when it could equal zero

$\dfrac{3x}{5x} = \dfrac{3}{5}$ with no condition stated, then substituting $x = 0$ later

When you cancel a variable factor such as $x$ , note that $x \neq 0$ . In exam working, state this restriction so your simplification is valid.

Try one yourself

Your turn

Simplify the following expression. $\frac{x}{4} + \frac{3x}{8}$

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About this worksheet

Why we built it

Algebraic fractions are a foundational Year 10 skill that senior Methods maths leans on heavily. These worksheets isolate the four operations on simple algebraic fractions so your student gets the repetition needed to make finding common denominators automatic.

What's covered

Sub-skills your student will practise

✓Multiplying and dividing algebraic fractions
✓Adding and subtracting with a common denominator
✓Simplifying before multiplying / dividing
✓Single-variable denominators
✓Mixed-operation problems

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Frequently asked

Questions parents ask about algebraic fractions

What's an algebraic fraction?

A fraction with at least one variable in it - 3/x, (x + 1)/2, (x^2 - 1)/(x + 1). Year 10 covers the four operations on simple algebraic fractions plus solving equations that involve them.

How do students add algebraic fractions?

Find a common denominator, just like with numeric fractions. For 2/x + 3/y, the common denominator is xy, giving (2y + 3x) / xy. The common denominator step is what most students forget.

How do students multiply algebraic fractions?

Multiply numerators, multiply denominators, then simplify by cancelling common factors. The simplification step often involves factorising first - which is why fluent factorising matters.

Why are algebraic fractions a Year 10 priority?

Because senior Methods maths uses them constantly - derivative quotients, rational functions, partial fractions in Specialist. Students who can't simplify algebraic fractions struggle from Methods Unit 1 onwards.

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