Algebraic
Fractions
Unit 1
Simplifying
Fractions
WHAT THIS UNIT IS
ABOUT
In this unit you will be learning about
simplifying algebraic fractions. You
will be using a fraction board and
fraction strips to compare and to
identify equivalent fractions.
You will then learn how to simplify
fractions that use algebraic
expressions instead of numbers.
The amount you pay every month for
a home loan is calculated using an
algebraic fraction where the main
variables are the Interest rate, the number of instalments and the Deposit.
In this unit you will

Use a Fraction Board and Fraction
Strips to compare fractions and
identify equivalent fractions.

Simplify fractions that contain
binomial expressions.

Recognise the pattern that links
equivalent fractions and calculate new
fractions from that pattern.

Explain why there is a problem if we
try to divide a fraction by zero.

Simplify algebraic fractions using the
pattern used for equivalent fractions.

Identify the values of variables that
make a denominator zero and cause a
fraction to be undefined.
©PROTEC 2001
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You will need the
following fraction
board, some fraction
strips and scissors,
A fraction strip
is one row of the
fraction board
below.
1 Whole
1
2
1
2
1
3
1
3
1
4
1
4
1
5
1
6
1
9
1
12
1
10
1
11
1
12
1
10
1
11
1
12
1
11
1
12
1
9
1
10
1
10
1
11
1
12
1
9
1
10
1
11
1
12
1
8
1
9
1
10
1
11
1
12
1
7
1
8
1
9
1
11
1
12
1
7
1
8
1
9
1
10
1
11
1
12
1
8
1
9
1
6
1
7
1
8
1
9
1
10
1
6
1
7
1
8
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
11
1
4
1
5
1
10
1
3
1
10
1
11
1
12
1
12
1
11
1
12
How to make a fraction
strip
1.
Take a strip of paper and place it on top of one of the rows of fractions in
the table above.
2. Mark off the divisions showing the length of each fraction on your strip.
3. Fold the strip along the divisions so that the folded length is the same as
one division.
4 Unfold the strip. You can now use it to represent any number of divisions for
4
that fraction (e.g.
)
12
©PROTEC 2001
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Activity 1
Equivalent Fractions
In this activity you will identify fractions that are the same or
equivalent to other fractions. You will be using scissors, a Fraction
Board and Fraction strips that you have made to do this.
1.1 Use the supplied fraction board and
fraction strips to find all the fractions
1
equivalent to .
3
1.2 Use your results to make a table like
the one below and to make
predictions about other fractions
1
that might also be equivalent to
3
Fraction Equivalent to
1
2
6
1
3
Fractions equivalent
1
to
will be the same
3
1
length as the
strip.
3
Relationship between 13 and the
fraction equivalent to it
To obtain 26 , multiply both the numerator and
denominator of 13 by 2.
You can also divide both the denominator and
numerator of 26 by 2, you get 13 .
2.
3.
4.
5.
6.
©PROTEC 2001
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If we multiply or divide both the
numerator and the denominator of any
fraction by the same number, we get a
fraction equivalent to the given
fraction.
i.e.
1 2 3 10
  
3 6 9 30
and
8
4 2 1

 
24 12 6 3
Activity 2
Fractions and Algebra
On a fraction board, the ‘whole’ could mean the whole apple (or simply the whole ‘a’). In
this activity you will be learning how to simplify fractions that are contained in algebraic
1
1
x
expressions, e.g. , (that means of x) or .
3
3
3
2.1 Use the supplied fraction board and
fraction strips to find all the
8x
fractions equivalent to
.
12
Fraction Equivalent to
4x
12
2.2 Use your results to
make a table like the
one below.
Relationship between the
fraction and equivalent fractions
2x
, divide both the numerator
6
(top) and the denominator (bottom) of
4x
by 2.
12
To obtain
1
4x
?
12
2.
In summary, to simplify a fraction, divide both the
numerator and the denominator by the common factor.
The common factor may be a number or a variable.
©PROTEC 2001
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2.2 Simplify the
algebraic fractions
below by dividing the
numerator (top) and
the denominator
(bottom) by the
common factors.
1.
4.
Example
2
8ax
12 x
Simplify
1
Divide the top and the bottom by x
2
Divide the top and the bottom by 2
3
Divide the top and the bottom by 2 again
3a  6
9
a ( a  1)
5.
a 2 (a  1)
6x
15
6a 3
15a 2 b 3
8ax
12
4ax
6
2ax
3
15a 2
21a 3
3x 2 ( x  3)
6.
18x(3  x)
2.
3.
Activity 3
Complex Fractions
This activity involves simplifying
fractions that contain at least one binomial
factor in the numerator or the
denominator. These are known as
Complex Fractions. You have already
come across two examples of these
fractions in activity 2.2 above, can you say
which two?
Example
x 1
Simplify 2
x 1
1
Factorise the bottom (difference of two squares)
2
Divide the top and the bottom by (x-1)
3a  6
a2  4
p2  1
4.
( p  1) 2
1.
3.1 Simplify the complex algebraic
fractions below by factorising the
numerator (top) and the
denominator (bottom) and then
dividing both by the common
algebraic factors.
x2  x  6
3x  9
5x2  5x
5.
5( x  1)
2.
x 1
( x  1)( x  1)
1
x 1
2 x 2  2 xy  3x  3 y
4x2  9
2 y 2  7 xy  3x 2
6.
2 x 2  4 xy
3.
©PROTEC 2001
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You cannon split up the denominator a
fraction.
1
1 1
 
(2  3) 2 3
Check this for yourself.
x
1
3
2
1
3
2
4
6
1
x3
3.2
1 1

x 3
Calculate and compare
1
1 1
 for the
and
x3
x 3
values of x shown in the
table.
Are they equal? Yes or No
1
x 1
x
and 2 for the
2
1
x 1
x
values of x shown in the table.
3.2 Calculate and compare
x
1
10
2
5
3
2
4
1
x 1
x2  1
x
1

2
x
1
Is
x 1
x
1
 2
? Yes or No
2
x 1 x
1
Remember
You cannot split up a denominator of a fraction, i.e.
©PROTEC 2001
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1 1
 
x y x y
6
Activity 4
The problem with dividing by zero
This activity explores why dividing by zero is a
problem in algebraic fractions.
1
2
3
4
2
1
2
0
1 0  0   1
0
0
20  0   2
0
0
40  0   4
0
1 2  2 
4,1 Consider the arguments
below and discuss why you
think they are true or false.
(true or false?)
(true or false?)
(true or false?)
(true or false?)
Non of these statements are
false.
n
is not
0
defined because no number of zeros can
be added to give n.
For any non zero number n,
4.2 Simplify the
algebraic fractions
below and identify
the values that will
make the
denominator (bottom)
equal to zero (not
defined).
1
4.
x
x( x  2)  1
x4
(2 x  1)(3x  2)
The problem with
dividing by zero is that
the answer is not
definite, i.e. it cannot
be defined.
Example
x3
( x  2)( x  3)
If x = 2, x-2 = 0 and the fraction cannot be defined.
For the fraction
1
2
Or if x = -3, x+3 = 0 and the fraction cannot be
defined
i.e. the fraction cannot be defined if x = 2 or –3.
2.
2 x( x  3)
(2 x  1)  ( x  5)
3.
1
( x  1) 2
5
x2  7 x 1
8x3  7 x 2  x
6
3a 3  24
2 x 2  6 x  20
©PROTEC 2001
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