Sec 2 · G3 (Express)

Algebraic fractions and formulae

Practise algebraic fractions and formulae with step-by-step help. Free account.

1. Simplifying algebraic fractions

Cancel common factors, never single terms

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

e.g.

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

\displaystyle = \dfrac{2x}{3y^{2}}

Factorise first, then cancel the bracket

\displaystyle \dfrac{x^{2} - 9}{x + 3}

e.g.

\displaystyle \dfrac{(x + 3)(x - 3)}{x + 3}

\displaystyle = x - 3

📖 Worked example

Simplify $\dfrac{x^{2} - 11x + 30}{x - 5}$ .

Step 1 of 2

Factorise every quadratic

\dfrac{x^{2} - 11x + 30}{x - 5} = \dfrac{(x - 5)(x - 6)}{(x - 5)}

Practise this →

2. Multiplying, dividing, adding and subtracting

Divide: multiply by the reciprocal

\displaystyle \dfrac{4a}{5b} \div \dfrac{2a^{2}}{15}

e.g.

\displaystyle \dfrac{4a}{5b} \times \dfrac{15}{2a^{2}}

\displaystyle = \dfrac{6}{ab}

Add: a common denominator

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

e.g.

\displaystyle \dfrac{2(x + 1) + 3x}{x(x + 1)}

\displaystyle = \dfrac{5x + 2}{x(x + 1)}

📖 Worked example

Express $\dfrac{1}{x - 3} + \dfrac{3}{x - 2}$ as a single fraction in its simplest form.

Step 1 of 3

Use a common denominator

\dfrac{1}{x - 3} + \dfrac{3}{x - 2} = \dfrac{1(x - 2) + 3(x - 3)}{(x - 3)(x - 2)}

Practise this →

3. A bracket the other way round, and quadratic denominators

Look for	Do	Example
$(k - x)$ and $(x - k)$	$k - x = -(x - k)$	$\dfrac{3}{x - 2} + \dfrac{1}{2 - x} = \dfrac{3}{x - 2} - \dfrac{1}{x - 2}$
$x^{2} - k^{2}$ underneath	factorise it first	$x^{2} - 9 = (x + 3)(x - 3)$
$\dfrac{k - x}{x - k}$	it is $-1$	$\dfrac{(3 - x)(3 + x)}{(x - 3)(x + 2)} = -\dfrac{x + 3}{x + 2}$

📖 Worked example

Express $\dfrac{3}{x - 2} + \dfrac{1}{2 - x}$ as a single fraction in its simplest form.

Step 1 of 2

Use 2 − x = −(x − 2): the sign in front of the second fraction changes

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

Practise this →

4. Formulae

Substitute: v² = u² + 2as with u = 3, a = 2, s = 4

\displaystyle v^{2} = u^{2} + 2as

e.g.

\displaystyle 3^{2} + 2(2)(4)

\displaystyle = 25

$v^{2} = 25$ , so $v = 5$ or $v = -5$

5. Changing the subject

Undo in reverse order; a square becomes ± a square root

\displaystyle v^{2} = u^{2} + 2as

e.g.

\displaystyle u^{2} = v^{2} - 2as

\displaystyle u = \pm\sqrt{v^{2} - 2as}

The subject appears twice: collect, then factorise

\displaystyle ax + b = cx

e.g.

\displaystyle ax - cx = -b

\displaystyle x(a - c) = -b

\displaystyle x = \dfrac{b}{c - a}

📖 Worked example

Make $x$ the subject of $y = \dfrac{3x - 6}{4}$ .

Step 1 of 3

Clear the fraction

4y = 3x - 6

Practise this →

6. Multiplying and dividing algebraic fractions

Cancel	How	Example
numbers	divide the top and the bottom by their HCF	$\dfrac{6}{36} = \dfrac{1}{6}$
letters	cross out a letter on top with the same letter underneath	$\dfrac{a \times a}{a} = a$

Write the letters out, then cross out pairs

\displaystyle \dfrac{a^{2}b}{ab^{3}}

a × a × b over a × b × b × b: one a is crossed out on top and underneath, and one b on top and underneath, leaving a over b squared.

e.g.

\displaystyle \dfrac{a^{2}b}{ab^{3}}

\displaystyle = \dfrac{a \times a \times b}{a \times b \times b \times b}

\displaystyle = \dfrac{a}{b^{2}}

Multiply: tops together, bottoms together, then cancel

\displaystyle \dfrac{3a}{4b} \times \dfrac{2b}{9}

e.g.

\displaystyle \dfrac{3a}{4b} \times \dfrac{2b}{9}

\displaystyle = \dfrac{6ab}{36b}

\displaystyle = \dfrac{a}{6}

Divide: flip the second fraction and multiply

\displaystyle \dfrac{5x}{6} \div \dfrac{10x^{2}}{3y}

e.g.

\displaystyle \dfrac{5x}{6} \div \dfrac{10x^{2}}{3y}

\displaystyle = \dfrac{5x}{6} \times \dfrac{3y}{10x^{2}}

\displaystyle = \dfrac{15xy}{60x^{2}}

\displaystyle = \dfrac{y}{4x}

📖 Worked example

Simplify $\dfrac{11}{6x} \times \dfrac{4xy}{9}$ , giving your answer in its simplest form.

Step 1 of 3

Multiply the tops together and the bottoms together (write the letters out in full)

\dfrac{11}{6x} \times \dfrac{4xy}{9} = \dfrac{44 \times x \times y}{54 \times x}

📖 Worked example

Simplify $\dfrac{5}{11m} \div \dfrac{12n}{11m^2}$ , giving your answer in its simplest form.

Step 1 of 4

Change the division into a multiplication

\dfrac{5}{11m} \div \dfrac{12n}{11m^2} = \dfrac{5}{11m} \times \dfrac{11m^2}{12n}

Practise this →

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Sec 2 · G3 (Express)

Algebraic fractions and formulae

Practise algebraic fractions and formulae with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Simplifying algebraic fractions

Cancel common factors, never single terms

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

e.g.

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

\displaystyle = \dfrac{2x}{3y^{2}}

Factorise first, then cancel the bracket

\displaystyle \dfrac{x^{2} - 9}{x + 3}

e.g.

\displaystyle \dfrac{(x + 3)(x - 3)}{x + 3}

\displaystyle = x - 3

📖 Worked example

Simplify $\dfrac{x^{2} - 11x + 30}{x - 5}$ .

Step 1 of 2

Factorise every quadratic

\dfrac{x^{2} - 11x + 30}{x - 5} = \dfrac{(x - 5)(x - 6)}{(x - 5)}

Practise this →

2. Multiplying, dividing, adding and subtracting

Divide: multiply by the reciprocal

\displaystyle \dfrac{4a}{5b} \div \dfrac{2a^{2}}{15}

e.g.

\displaystyle \dfrac{4a}{5b} \times \dfrac{15}{2a^{2}}

\displaystyle = \dfrac{6}{ab}

Add: a common denominator

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

e.g.

\displaystyle \dfrac{2(x + 1) + 3x}{x(x + 1)}

\displaystyle = \dfrac{5x + 2}{x(x + 1)}

📖 Worked example

Express $\dfrac{1}{x - 3} + \dfrac{3}{x - 2}$ as a single fraction in its simplest form.

Step 1 of 3

Use a common denominator

\dfrac{1}{x - 3} + \dfrac{3}{x - 2} = \dfrac{1(x - 2) + 3(x - 3)}{(x - 3)(x - 2)}

Practise this →

3. A bracket the other way round, and quadratic denominators

Look for	Do	Example
$(k - x)$ and $(x - k)$	$k - x = -(x - k)$	$\dfrac{3}{x - 2} + \dfrac{1}{2 - x} = \dfrac{3}{x - 2} - \dfrac{1}{x - 2}$
$x^{2} - k^{2}$ underneath	factorise it first	$x^{2} - 9 = (x + 3)(x - 3)$
$\dfrac{k - x}{x - k}$	it is $-1$	$\dfrac{(3 - x)(3 + x)}{(x - 3)(x + 2)} = -\dfrac{x + 3}{x + 2}$

📖 Worked example

Express $\dfrac{3}{x - 2} + \dfrac{1}{2 - x}$ as a single fraction in its simplest form.

Step 1 of 2

Use 2 − x = −(x − 2): the sign in front of the second fraction changes

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

Practise this →

4. Formulae

Substitute: v² = u² + 2as with u = 3, a = 2, s = 4

\displaystyle v^{2} = u^{2} + 2as

e.g.

\displaystyle 3^{2} + 2(2)(4)

\displaystyle = 25

$v^{2} = 25$ , so $v = 5$ or $v = -5$

5. Changing the subject

Undo in reverse order; a square becomes ± a square root

\displaystyle v^{2} = u^{2} + 2as

e.g.

\displaystyle u^{2} = v^{2} - 2as

\displaystyle u = \pm\sqrt{v^{2} - 2as}

The subject appears twice: collect, then factorise

\displaystyle ax + b = cx

e.g.

\displaystyle ax - cx = -b

\displaystyle x(a - c) = -b

\displaystyle x = \dfrac{b}{c - a}

📖 Worked example

Make $x$ the subject of $y = \dfrac{3x - 6}{4}$ .

Step 1 of 3

Clear the fraction

4y = 3x - 6

Practise this →

6. Multiplying and dividing algebraic fractions

Cancel	How	Example
numbers	divide the top and the bottom by their HCF	$\dfrac{6}{36} = \dfrac{1}{6}$
letters	cross out a letter on top with the same letter underneath	$\dfrac{a \times a}{a} = a$

Write the letters out, then cross out pairs

\displaystyle \dfrac{a^{2}b}{ab^{3}}

e.g.

\displaystyle \dfrac{a^{2}b}{ab^{3}}

\displaystyle = \dfrac{a \times a \times b}{a \times b \times b \times b}

\displaystyle = \dfrac{a}{b^{2}}

Multiply: tops together, bottoms together, then cancel

\displaystyle \dfrac{3a}{4b} \times \dfrac{2b}{9}

e.g.

\displaystyle \dfrac{3a}{4b} \times \dfrac{2b}{9}

\displaystyle = \dfrac{6ab}{36b}

\displaystyle = \dfrac{a}{6}

Divide: flip the second fraction and multiply

\displaystyle \dfrac{5x}{6} \div \dfrac{10x^{2}}{3y}

e.g.

\displaystyle \dfrac{5x}{6} \div \dfrac{10x^{2}}{3y}

\displaystyle = \dfrac{5x}{6} \times \dfrac{3y}{10x^{2}}

\displaystyle = \dfrac{15xy}{60x^{2}}

\displaystyle = \dfrac{y}{4x}

📖 Worked example

Simplify $\dfrac{11}{6x} \times \dfrac{4xy}{9}$ , giving your answer in its simplest form.

Step 1 of 3

Multiply the tops together and the bottoms together (write the letters out in full)

\dfrac{11}{6x} \times \dfrac{4xy}{9} = \dfrac{44 \times x \times y}{54 \times x}

📖 Worked example

Simplify $\dfrac{5}{11m} \div \dfrac{12n}{11m^2}$ , giving your answer in its simplest form.

Step 1 of 4

Change the division into a multiplication

\dfrac{5}{11m} \div \dfrac{12n}{11m^2} = \dfrac{5}{11m} \times \dfrac{11m^2}{12n}

Practise this →

Try a question: Algebraic fractions and formulae

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Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise algebraic fractions and formulae with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free