Operations with algebraic fractions | Year 10

01 / LEARN

The idea, step by step

Products and quotients

Multiply numerators together and denominators together, cancelling common factors where allowed. For division, multiply by the reciprocal of the entire second fraction. The divisor must be defined and nonzero, so its numerator can create an extra restriction.

Sums need a shared denominator

For addition or subtraction, first express the fractions with a common denominator. Then combine the numerators and keep that denominator. Put brackets around a subtracted numerator so each sign is handled correctly. State restrictions from the original fractions, not only the simplified answer.

A common trap

$\frac{1}{x} + \frac{1}{y} = \frac{2}{x + y}$ .

Instead: Add using denominator $x y$ : the correct result is $(x + y) / (x y)$ , with $x, y \neq 0$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Add unlike denominators

Simplify $\frac{2}{x} + \frac{3}{x + 1}$ .

Restrictions: $x \neq 0, - 1$ .
The common denominator is $x (x + 1)$ .
The numerator is $2 (x + 1) + 3 x = 5 x + 2$ .

\frac{5 x + 2}{x (x + 1)}

x \neq 0, - 1

Example 2

Division adds a restriction

Simplify $\frac{x}{x - 1} \div \frac{2 x}{x + 3}$ .

The fractions require $x \neq 1, - 3$ .
The divisor cannot be zero, so also $x \neq 0$ .
Multiply by the reciprocal: $\frac{x (x + 3)}{(x - 1) 2 x}$ , then cancel $x$ .

\frac{x + 3}{2 (x - 1)}

x \neq 1, - 3, 0

Check your understanding

Simplify $\frac{5}{a} - \frac{2}{a}$ .

Show answer

$\frac{3}{a}$ , $a \neq 0$

The denominator is already shared, so subtract only the numerators.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $\frac{3 x}{4} \times \frac{8}{9 x}$ and retain restrictions.
2 marks
Simplify $\frac{2}{a} + \frac{5}{a}$ .
2 marks
Simplify $\frac{3}{x} - \frac{1}{x + 2}$ .
2 marks
Simplify $\frac{a + 1}{3} \div \frac{2 (a + 1)}{5}$ .
2 marks
Simplify $\frac{4}{p - 1} - \frac{p + 2}{p - 1}$ .
2 marks
A printer completes a job in $x$ minutes and another in $x + 5$ minutes. Write their combined job rate as one fraction.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Simplify $\frac{3 x}{4} \times \frac{8}{9 x}$ and retain restrictions.
2 marks
The original second denominator excludes zero; reduce $24 x / (36 x)$ .
$\frac{2}{3}$ , $x \neq 0$
Simplify $\frac{2}{a} + \frac{5}{a}$ .
2 marks
Add the numerators over the common denominator.
$\frac{7}{a}$ , $a \neq 0$
Simplify $\frac{3}{x} - \frac{1}{x + 2}$ .
2 marks
The combined numerator is $3 (x + 2) - x = 2 x + 6$ .
$\frac{2 x + 6}{x (x + 2)}$ , $x \neq 0, - 2$
Simplify $\frac{a + 1}{3} \div \frac{2 (a + 1)}{5}$ .
2 marks
The divisor cannot be zero; multiply by $5 / [2 (a + 1)]$ .
$\frac{5}{6}$ , $a \neq - 1$
Simplify $\frac{4}{p - 1} - \frac{p + 2}{p - 1}$ .
2 marks
The numerator is $4 - (p + 2) = 2 - p$ .
$\frac{2 - p}{p - 1}$ , $p \neq 1$
A printer completes a job in $x$ minutes and another in $x + 5$ minutes. Write their combined job rate as one fraction.
3 marks
The separate rates are $1 / x$ and $1 / (x + 5)$ .
Add using the common denominator and apply the practical positive-time restriction.
$\frac{2 x + 5}{x (x + 5)}$ jobs/min, $x > 0$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 algebra, including the approved fuller school-course treatment of rational expressions and formula restrictions; quadratic methods are in their own chapter.

Official curriculum source ↗

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