Algebraic FractionsThis is an essential collections of skills that you need

to succeed at National 5 and progress to Higher

Simplifying Fractions

Fractions of fractions

Multiplying and dividing

Adding and subtracting

Course level

questions

Simplifying algebraic fractions 1Simplifying an algebraic fraction means rewriting it as an equivalent fraction where all of the common factors have been cancelled out.

Tip! Try to

memorise

some easy

number

fractions that

show how each

idea works

With number fractions, you simplify by cancelling numbers that are factors of both the numerator and denominator:

43

33

12

9

4

3

Fractions with algebraic factors simplify in the same way: ba

a

ab

a

7

5

7

5

b7

5

You will usually have to do some factorising. Also, any terms inside brackets must be treated as single objects when simplifying.

)2(3

)2(5

63

105

p

p

p

p

3

5

Beware the classic mistake: k is NOT a factor so DON’T do this:

k

k 12 3

1

12

What can I expect in the

unit test?

Simplifying algebraic fractions 2Your factorisation skills will really be tested out here. Always fully factorise thenumerator and denominator if there are any terms being added or subtracted:

82

162

x

x

Simplify these fractions. Start by factorising as much as you can:

)4(2

)4)(4(

x

xx

2

4x

Tip! You don’t have to have the brackets

here because it is not being multiplied anymore1642

162

2

yy

y

)2)(4(2

)4)(4(

yy

yy

)2(2

4

y

y

66

6332

2

t

tt

)1)(1(6

)1)(2(3

tt

tt)1(2

4

t

t2

Tip! Once

you’ve

cancelled all

you can, don’t

expand

brackets What can I expect in the

unit test?

Unit Assessment of simplifying fractions

At unit assessment level, you will either be given fully factorised fractions or have to do a simple common factor factorisation.

Simplify the following algebraic fractions:

2)2(

)2)(4(

a

aa

2

4

a

a

62

124 2

b

bb

)3(2

)3(4

b

bbb2

2

93

62 23

c

cc

)3(3

)3(2 2

c

cc

3

2 2c

Dealing with fractions of fractionsThis is a very useful algebraic trick for simplifying complicated fractions where the numerator or denominator contains a fraction.

The trick is to multiply the fraction by 1 (which will not change its value).The challenge is to find the best way of writing 1…

Remember, If the numerator and denominator of a fraction are equal then the fraction equals 1:

.1)sin(

)sin(,1,1

5

5etc

x

x

a

a

Example: simplify

x2

3The fraction in the denominator is making this awkward so multiply by 1 in a form that lets you reduce it: x

x

x

2

3 2

332

xx

xx

Example: simplify

xa

x

1

13Simplifying here means finding

an equivalent fraction without fractions in the numerator and denominator. x

x

xa

x

11

1

13

axxaxx 3

1)1(

1)1(3

Simplifying more complicated fractions

x4

32

x

x

1

1

1xx

xx

1

1

x

x

x

4

32

432x

3

3

432

x

612

2 xx

xx

x

x

1

1

1

xxxx

x

1

1

x

xx

xx

xx

11

1

1

xxxx

1)1(

xxx

xx

x

111

)1(

)1(

)1(

xx

xx

Multiplying fractionsThis is much more straightforward process than adding fractions. Remember this pattern:

bd

ac

db

ca

d

c

b

a

The product of the numerators

The product of the denominators

Now try these on paper before pressing the spacebar for the answers.Express these products as fractions in their simplest form

2

22

3

2

3

10

5

6

ab

yx

xy

ba

y

xx

y

x

15

60 2

xyab

ybxa2

22

3

2

y

x2

b

ax

3

2

So how do I divide fractions?

Dividing fractionsConvert divisions to multiplications by replacing the second fraction with itsreciprocal. Remember this pattern:

bc

ad

c

d

b

a

d

c

b

a

Now try these on paper before pressing the spacebar for the answers.Express these products as fractions in their simplest form

p

qpq

xx

105

2

6

4

3

64

3 x

x

q

ppq 10

5

2

8

1

24

3x

x

22

45

20p

q

qp

What can I expect in the

unit test?

Unit Assessment of multiplying and dividing fractions

Simplify the following:

2

14

7 b

b

g

g

3

12

4

6 2

y

x

y

x 22 5

b

b

b

2

14

7 2

2

gg

g

63

12

4

6 2

2

2

5x

y

y

x

2

2

5x

y

y

x

5

1

Calculate the Area and perimeter of this rectangle:

cmx41

cmx34

2343

4 2

cmArea xxx

2619

212316

434

2

2

cm

cm

Perimeter

x

xx

xx

Adding and Subtracting Fractions 1

Fractions can only be added or subtracted if they have the same denominator.To answer this type of questions you will first find equivalent fractions then add or subtract the numerators.

With number fractions, the technique called ‘cross-multiplying’ produces equivalent fractions with the same denominators:

3

2

2

1

32

22

The denominators are 2 and 3 and the smallest number that both divide intois 6. So when each fraction is multiplied (top and bottom) by the oppositedenominator we get equivalent fractions with the same denominator. Now we Can add or subtract as needed.

32

31

6

4

6

3

6

7

Adding and Subtracting Fractions 2

The same idea allows us to add or subtract algebraic fractions.

Cross multiplying works here too:y

x

x 2

1

yx

xx

2

The denominators are x and 2y and the smallest expression that both divide into is 2xy. So when each fraction is multiplied (top and bottom) by the oppositedenominator we get equivalent fractions with the same denominator. Now we Can add or subtract as needed.

yx

y

2

21

xy

x

xy

y

22

2 2

xy

yx

2

22

What can I expect in the

unit test?

Unit Assessment of adding and subtracting fractions

yx

23

Express each of the following as a single fraction in its simplest form.Remember revising means doing questions first then checking the answers.

xy

x

xy

y 23

xy

xy 23

48

7 xx

32

8

32

28 xx

8

5

32

20 xx

432

aaa

12

3

12

4

12

6 aaa

12

5a

Here you can add the fractions by cross-multiplying twice or notice that lcm(2,3,4) = 12

Typical Course level Questions

<insert standard questions>