Algebraic Fractions

In short: An algebraic fraction is a fraction with letters in it, like $\frac{x + 2}{x ^{2} - 4}$ . Factorise first and cancel common factors (never single terms), add and subtract over a common denominator, and remember that a value making a denominator zero is never allowed.

Starting point: Years 9 and 10

Simplifying

Cancel common factors, never single terms. Factorise first, then cancel.

$\frac{6 x ^{2}}{9 x} = \frac{2 x}{3} \frac{4 x + 8}{4} = \frac{4 ( x + 2 )}{4} = x + 2$

Multiplying and dividing

Multiply: factorise, cancel across, then multiply tops and bottoms.
Divide: flip the second fraction and multiply.

$\frac{2 x}{5} \times \frac{10}{x ^{2}} = \frac{20 x}{5 x ^{2}} = \frac{4}{x}$

$\frac{x + 1}{6} \div \frac{x + 1}{3} = \frac{x + 1}{6} \times \frac{3}{x + 1} = \frac{1}{2}$

Adding and subtracting

Find a common denominator, rewrite each fraction, then add the tops.

$\frac{c}{3} - \frac{c}{5} = \frac{5 c}{15} - \frac{3 c}{15} = \frac{2 c}{15}$

So each of Josh's first three friends saved $\frac{2 c}{15}$ .

Solving equations with fractions

Multiply every term by the lowest common denominator to clear the fractions.

Algebraic fractions

Simplify

Factorise first
Cancel factors only

Multiply

Cancel across

Divide

Flip and multiply

Add and subtract

Common denominator
Brackets on tops

Equations

Multiply every term by LCD
Check the answer

Year 11Letters in the denominator

With algebraic denominators, the common denominator is usually their product.

$\frac{2}{x} + \frac{3}{x + 1} = \frac{2 ( x + 1 ) + 3 x}{x ( x + 1 )} = \frac{5 x + 2}{x ( x + 1 )}$

Equations with $x$ on the bottom. Solve $\frac{4}{x} = \frac{3}{x - 1}$ . Cross-multiply: $4 (x - 1) = 3 x$ , so $x = 4$ .

Solve $x + \frac{6}{x} = 5$ . Multiply by $x$ : $x^{2} - 5 x + 6 = 0$ , so $x = 2$ or $x = 3$ .

Always check the answers do not make a denominator zero. Those values are not in the domain.

Finding unknown numerators. To find $a$ and $b$ in $\frac{a}{x - 1} + \frac{b}{x + 3} = \frac{4 x}{( x - 1 ) ( x + 3 )}$ , join the left side: $a (x + 3) + b (x - 1) = 4 x$ . Put in $x = 1$ : $4 a = 4$ , so $a = 1$ . Put in $x = - 3$ : $- 4 b = - 12$ , so $b = 3$ .

Splitting off a whole number. Rewrite the top so the bottom appears in it: $\frac{x + 5}{x + 1} = \frac{( x + 1 ) + 4}{x + 1} = 1 + \frac{4}{x + 1}$

Inequalities with $x$ on the bottom. Do not multiply by $x - 4$ , because it might be negative. Multiply by its square, which is always positive. For $\frac{x - 2}{x - 4} \leq 3$ with $x \neq = 4$ : $(x - 2) (x - 4) \leq 3 (x - 4)^{2}$ , so $(x - 4) [(x - 2) - 3 (x - 4)] \leq 0$ , which is $(x - 4) (10 - 2 x) \leq 0$ . Sketch this parabola: it is at or below zero for $x < 4$ or $x \geq 5$ .

Quick summary

Factorise first, then cancel common factors, never single terms.
Multiply: cancel across, then multiply. Divide: flip the second fraction and multiply.
Add and subtract with a common denominator. Put brackets around each top.
Clear fractions in an equation by multiplying every term by the lowest common denominator.
Values that make a denominator zero are never allowed.

Quiz

13 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.

Try 3 questions now

Simplify $\frac{12 x ^{3}}{4 x}$ .
1. A $3 x^{3}$
2. B $3 x^{2}$
3. C $8 x^{2}$
4. D $\frac{3}{x ^{2}}$
Show answer
B: $3 x^{2}$
$12 \div 4 = 3$ and $x^{3} \div x = x^{2}$ .
Simplify $\frac{5 x - 10}{5}$ .
1. A $x - 10$
2. B $x - 2$
3. C $5 x - 2$
4. D $x$
Show answer
B: $x - 2$
$\frac{5 ( x - 2 )}{5}$ .
Simplify $\frac{x ^{2} - 4}{x - 2}$ .
1. A $x - 2$
2. B $x + 2$
3. C $x^{2} - 2$
4. D $x$
Show answer
B: $x + 2$
$\frac{( x - 2 ) ( x + 2 )}{x - 2}$ .