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9.6 Adding and subtracting algebraic fractions

9.6 Adding and subtracting algebraic fractions

An algebraic fraction is a fraction that contains an unknown variable, or letter.
y
For example, x , , z , 2a and 4b are all algebraic fractions.
4 2 8 3 5
1
x
You can write the fraction (say as ‘x over 4’) as x (say as ‘one-quarter of x’).
4 4
2
You can write the fraction 2a (say as ‘2a over 3’) as a (say as ‘two-thirds of a’).
3 3
To add and subtract algebraic fractions, you use the same method as for normal fractions.
t If the denominators are the same, simply add or subtract the numerators.
t If the denominators are di%erent, write the fractions as equivalent fractions with the same
denominator, then add or subtract the numerators.
t Cancel your answer to its simplest form.

Worked example 9.6
Simplify these expressions. a x + x b y − y c 4n + 2n d a + b e 5p − q
6 6 3 9 5 3 8 4 6 4

+
a x + x = xx The denominators are the same, so add the numerators.
6 6 6
= 2x Cancel the fraction to its simplest form.
6
x
= x Write 1x as simply .
3 3 3
y
b y − y = y 3 − y The denominators are different, so change into 3y .
3 9 9 9 3 9
3y − y
= 9 The denominators are now the same, so subtract the numerators.

= 2y
9

c 4n + 2n = 12n + 10n The denominators are different, so change 4n into 12n and 2n into 10n .
5 3 15 15 5 15 3 15
= 12n + 10n The denominators are now the same, so add the numerators.
15
= 22n Leave as an improper fraction in its simplest form.
15
b
d a + b = a + b 2 The denominators are different, so change into 2b .
8 4 8 8 4 8
= a + b2 Now add the numerators. You cannot simplify any further as a and 2b are not
8
like terms.
q
e 5p − q = 10p − 3q The denominators are different, so change 5p into 10p and into 3q .
6 4 12 12 6 12 4 12
−3q
= 10p 12 Now subtract the numerators. You cannot simplify any further as 10p and 3q are
not like terms.

92 9 Expressions and formulae