Cancelling common factors

1 Cancelling common factors

Consider the fraction $\frac{10}{35}$ . To simplify it we can factorise the numerator and the denominator and then cancel any common factors. Common factors are those factors which occur in both the numerator and the denominator. Thus

$\frac{10}{35} = \frac{5̸ \times 2}{7 \times 5̸} = \frac{2}{7}$

Note that the common factor 5 has been cancelled. It is important to remember that only common factors can be cancelled. The fractions $\frac{10}{35}$ and $\frac{2}{7}$ have identical values - they are equivalent fractions - but $\frac{2}{7}$ is in a simpler form than $\frac{10}{35}$ .

We apply the same process when simplifying algebraic fractions.

Example 49

Simplify, if possible,

$\frac{y x}{2 x}$ ,
$\frac{x}{x y}$ ,
$\frac{x}{x + y}$

Solution

In the expression $\frac{y x}{2 x}$ , $x$ is a factor common to both numerator and denominator. This common factor can be cancelled to give
$\frac{y x̸}{2 x̸} = \frac{y}{2}$
Note that $\frac{x}{x y}$ can be written $\frac{1 x}{x y}$ . The common factor of $x$ can be cancelled to give
$\frac{1 x̸}{x̸ y} = \frac{1}{y}$
In the expression $\frac{x}{x + y}$ notice that an $x$ appears in both numerator and denominator. However $x$ is not a common factor. Recall that factors of an expression are multiplied together whereas in the denominator $x$ is added to $y$ . This expression cannot be simplified.

Task!

Simplify, if possible,

$\frac{a b c}{3 a c}$ ,
$\frac{3 a b}{b + a}$
When simplifying remember only common factors can be cancelled.

$\frac{b}{3}$
This cannot be simplified.

Task!

Simplify $\frac{21 x^{3}}{14 x}$ ,

Factorising and cancelling common factors gives:

$\frac{21 x^{3}}{14 x} = \frac{7̸ \times 3 \times x̸ \times x^{2}}{7̸ \times 2 \times x̸} = \frac{3 x^{2}}{2}$

Task!

Simplify $\frac{36 x}{12 x^{3}}$

Factorising and cancelling common factors gives:

$\frac{36 x}{12 x^{3}} = \frac{12 \times 3 \times x}{12 \times x \times x^{2}} = \frac{3}{x^{2}}$

Example 50

Simplify $\frac{3 x + 6}{6 x + 12}$ .

Solution

First we factorise the numerator and the denominator to see if there are any common factors.

$\frac{3 x + 6}{6 x + 12} = \frac{3 (x + 2)}{6 (x + 2)} = \frac{3}{6} = \frac{1}{2}$

The factors $x + 2$ and 3 have been cancelled.

Task!

Simplify $\frac{12}{2 x + 8}$ .

Factorise the numerator and denominator, and cancel any common factors. $\frac{6 \times 2}{2 (x + 4)} = \frac{6}{x + 4}$

Example 51

Show that the algebraic fraction $\frac{3}{x + 1}$ and $\frac{3 (x + 4)}{x^{2} + 5 x + 4}$ are equivalent.

Solution

The denominator, $x^{2} + 5 x + 4$ , can be factorised as $(x + 1) (x + 4)$ so that

$\frac{3 (x + 4)}{x^{2} + 5 x + 4} = \frac{3 (x + 4)}{(x + 1) (x + 4)}$

Note that $(x + 4)$ is a factor common to both the numerator and the denominator and can be cancelled to leave $\frac{3}{x + 1}$ . Thus $\frac{3}{x + 1}$ and $\frac{3 (x + 4)}{x^{2} + 5 x + 4}$ are equivalent fractions.

Task!

Show that $\frac{x - 1}{x^{2} - 3 x + 2}$ is equivalent to $\frac{1}{x - 2}$ .

First factorise the denominator:

$(x - 1) (x - 2)$

Now identify the factor common to both numerator and denominator and cancel this common factor:

$\frac{1}{x - 2}$ . Hence the two given fractions are equivalent.

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

Then cancel any common factors:

$\frac{x + 5}{2 x + 1}$

Exercises

Simplify, if possible,
1. $\frac{19}{38}$ ,
2. $\frac{14}{28}$ ,
3. $\frac{35}{40}$ ,
4. $\frac{7}{11}$ ,
5. $\frac{14}{56}$
Simplify, if possible,
1. $\frac{14}{21}$ ,
2. $\frac{36}{96}$ ,
3. $\frac{13}{52}$ ,
4. $\frac{52}{13}$
Simplify
1. $\frac{5 z}{z}$ ,
2. $\frac{25 z}{5 z}$ ,
3. $\frac{5}{25 z^{2}}$ ,
4. $\frac{5 z}{25 z^{2}}$
Simplify
1. $\frac{4 x}{3 x}$ ,
2. $\frac{15 x}{x^{2}}$ ,
3. $\frac{4 s}{s^{3}}$ ,
4. $\frac{21 x^{4}}{7 x^{3}}$
Simplify, if possible,
1. $\frac{x + 1}{2 (x + 1)}$ ,
2. $\frac{x + 1}{2 x + 2}$ ,
3. $\frac{2 (x + 1)}{x + 1}$ ,
4. $\frac{3 x + 3}{x + 1}$ ,
5. $\frac{5 x - 15}{5}$ ,
6. $\frac{5 x - 15}{x - 3}$ .
Simplify, if possible,
1. $\frac{5 x + 15}{25 x + 5}$ ,
2. $\frac{5 x + 15}{25 x}$ ,
3. $\frac{5 x + 15}{25}$ ,
4. $\frac{5 x + 15}{25 x + 1}$
Simplify
1. $\frac{x^{2} + 10 x + 9}{x^{2} + 8 x - 9}$ ,
2. $\frac{x^{2} - 9}{x^{2} + 4 x - 21}$ ,
3. $\frac{2 x^{2} - x - 1}{2 x^{2} + 5 x + 2}$ ,
4. $\frac{3 x^{2} - 4 x + 1}{x^{2} - x}$ ,
5. $\frac{5 z^{2} - 20 z}{2 z - 8}$
Simplify
1. $\frac{6}{3 x + 9}$ ,
2. $\frac{2 x}{4 x^{2} + 2 x}$ ,
3. $\frac{3 x^{2}}{15 x^{3} + 10 x^{2}}$
Simplify
1. $\frac{x^{2} - 1}{x^{2} + 5 x + 4}$ ,
2. $\frac{x^{2} + 5 x + 6}{x^{2} + x - 6}$ .

1. $\frac{1}{2}$ ,
2. $\frac{1}{2}$ ,
3. $\frac{7}{8}$ ,
4. $\frac{7}{11}$ ,
5. $\frac{1}{4}$ .
1. $\frac{2}{3}$ ,
2. $\frac{3}{8}$ ,
3. $\frac{1}{4}$ ,
4. $4$
1. 5,
2. 5,
3. $\frac{1}{5 z^{2}}$ ,
4. $\frac{1}{5 z}$ .
1. $\frac{4}{3}$ ,
2. $\frac{15}{x}$ ,
3. $\frac{4}{s^{2}}$ ,
4. $3 x$
1. $\frac{1}{2}$ ,
2. $\frac{1}{2}$ ,
3. 2,
4. 3,
5. $x - 3$ ,
6. 5
1. $\frac{x + 3}{5 x + 1}$ ,
2. $\frac{x + 3}{5 x}$ ,
3. $\frac{x + 3}{5}$ ,
4. $\frac{5 (x + 3)}{25 x + 1}$
1. $\frac{x + 1}{x - 1}$ ,
2. $\frac{x + 3}{x + 7}$ ,
3. $\frac{x - 1}{x + 2}$ ,
4. $\frac{3 x - 1}{x}$ ,
5. $\frac{5 z}{2}$
1. $\frac{2}{x + 3}$ ,
2. $\frac{1}{2 x + 1}$ ,
3. $\frac{3}{5 (3 x + 2)}$ .
1. $\frac{x - 1}{x + 4}$ ,
2. $\frac{x + 2}{x - 2}$ .

1 Cancelling common factors

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