Operations with algebraic fractions | Year 10

01 / LEARN

Understand the idea.

Products and quotients

Multiply algebraic fractions by multiplying numerators and denominators, then cancel common nonzero factors. For division, invert the whole second fraction and multiply. Before inverting it, check that the divisor is both defined and nonzero.

A denominator excludes any input that makes it zero. A divisor also excludes inputs that make its numerator zero, because a zero divisor cannot be inverted. Keep these restrictions even if their factors disappear from the simplified result.

Sums need a shared denominator

For addition or subtraction, rewrite both fractions over a common denominator. Multiply each numerator by the same factor used on its denominator, then combine numerators. Bracket a subtracted numerator so its negative sign applies to every term.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Add unlike denominators

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

Worked solution

1
State the original restrictions.
$x \neq 0, x \neq - 1$
Neither $x$ nor $x + 1$ may be zero.
2
Rewrite both fractions over a shared denominator.
$\begin{aligned} \frac{2}{x} & = \frac{2 (x + 1)}{x (x + 1)} \\ \frac{3}{x + 1} & = \frac{3 x}{x (x + 1)} \end{aligned}$
Multiplying a numerator and denominator by the same nonzero factor preserves each fraction.
3
Combine and simplify the numerator.
$\frac{2 (x + 1) + 3 x}{x (x + 1)} = \frac{5 x + 2}{x (x + 1)}$
Fractions with the same denominator are added by adding their numerators.
4
Retain both excluded values.
$\frac{5 x + 2}{x (x + 1)}, x \neq 0, - 1$
The denominator factors come from the original fractions.

Answer

$\frac{5 x + 2}{x (x + 1)}$ , $x \neq 0, - 1$

EXAMPLE 02

Division adds a restriction

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

Worked solution

1
Restrict the denominators and the divisor.
$x \neq 1, x \neq - 3, x \neq 0$
The denominators exclude $1$ and $- 3$ ; the divisor is zero at $x = 0$ .
2
Multiply by the reciprocal of the whole divisor.
$\frac{x}{x - 1} \times \frac{x + 3}{2 x}$
Division by a defined nonzero fraction is multiplication by its reciprocal.
3
Cancel the shared factor $x$ .
$\frac{x (x + 3)}{2 x (x - 1)} = \frac{x + 3}{2 (x - 1)}$
The factor $x$ is nonzero under the recorded restrictions.
4
Attach all original restrictions.
$\frac{x + 3}{2 (x - 1)}, x \neq 1, - 3, 0$
The simplified formula does not restore values excluded by the original division.

Answer

$\frac{x + 3}{2 (x - 1)}$ , $x \neq 1, - 3, 0$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

The denominator is already $a$ in both fractions, so subtract the numerators: $\frac{5}{a} - \frac{2}{a} = \frac{5 - 2}{a} = \frac{3}{a}$ . The original denominator requires $a \neq 0$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 $\frac{24 x}{36 x}$ .
$\frac{2}{3}$ , $x \neq 0$
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 $\frac{5}{2 (a + 1)}$ .
$\frac{5}{6}$ , $a \neq - 1$
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 $\frac{1}{x}$ and $\frac{1}{x + 5}$ .
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 ↗

In this lesson

Understand the idea.

Products and quotients

Sums need a shared denominator

See the method in action.

Add unlike denominators

State the original restrictions.

Rewrite both fractions over a shared denominator.

Combine and simplify the numerator.

Retain both excluded values.

Division adds a restriction

Restrict the denominators and the divisor.

Multiply by the reciprocal of the whole divisor.

Cancel the shared factor x.

Attach all original restrictions.

Try it yourself.

Check your understanding

Keep practising

Cancel the shared factor $x$ .