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Unit 1:
Negative Numbers
UNIT 2
FRACTIONS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Addition and Subtraction of Fractions 2
1.0 Addition and Subtraction of Fractions with the Same Denominator 5
1.1 Addition of Fractions with the Same Denominators 5
1.2 Subtraction of Fractions with The Same Denominators 6
1.3 Addition and Subtraction Involving Whole Numbers and Fractions 7
1.4 Addition or Subtraction Involving Mixed Numbers and Fractions 9
2.0 Addition and Subtraction of Fractions with Different Denominator 10
2.1 Addition and Subtraction of Fractions When the Denominator
of One Fraction is A Multiple of That of the Other Fraction 11
2.2 Addition and Subtraction of Fractions When the Denominators
Are Not Multiple of One Another 13
2.3 Addition or Subtraction of Mixed Numbers with Different
Denominators 16
2.4 Addition or Subtraction of Algebraic Expression with Different
Denominators 17
Part B: Multiplication and Division of Fractions 22
1.0 Multiplication of Fractions 24
1.1 Multiplication of Simple Fractions 28
1.2 Multiplication of Fractions with Common Factors 29
1.3 Multiplication of a Whole Number and a Fraction 29
1.4 Multiplication of Algebraic Fractions 31
2.0 Division of Fractions 33
2.1 Division of Simple Fractions 36
2.2 Division of Fractions with Common Factors 37
Answers 42
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
1
Curriculum Development Division
Ministry of Education Malaysia
PART 1
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding of the concept
of fractions.
2. It serves as a guide for teachers in helping pupils to master the basic
computation skills (addition, subtraction, multiplication and division)
involving integers and fractions.
3. This module consists of two parts, and each part consists of learning
objectives which can be taught separately. Teachers may use any parts of the
module as and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
2
Curriculum Development Division
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PART A:
ADDITION AND SUBTRACTION
OF FRACTIONS
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. perform computations involving combination of two or more operations
on integers and fractions;
2. pose and solve problems involving integers and fractions;
3. add or subtract two algebraic fractions with the same denominators;
4. add or subtract two algebraic fractions with one denominator as a
multiple of the other denominator; and
5. add or subtract two algebraic fractions with denominators:
(i) not having any common factor;
(ii) having a common factor.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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TEACHING AND LEARNING STRATEGIES
Pupils have difficulties in adding and subtracting fractions with different
denominators.
Strategy:
Teachers should emphasise that pupils have to find the equivalent form of
the fractions with common denominators by finding the lowest common
multiple (LCM) of the denominators.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
4
Curriculum Development Division
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numerator
denominator
Fraction is written in the form of:
b
a
Examples:
3
4 ,
3
2
Proper Fraction Improper Fraction Mixed Numbers
The numerator is smaller
than the denominator.
Examples:
20
9 ,
3
2
The numerator is larger
than or equal to the denominator.
Examples:
12
108 ,
4
15
A whole number and
a fraction combined.
Examples:
65
71 8 ,2
Rules for Adding or Subtracting Fractions
1. When the denominators are the same, add or subtract only the numerators and
keep the denominator the same in the answer.
2. When the denominators are different, find the equivalent fractions that have the
same denominator.
Note: Emphasise that mixed numbers and whole numbers must be converted to improper
fractions before adding or subtracting fractions.
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
5
Curriculum Development Division
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1.0 Addition And Subtraction of Fractions with the Same Denominator
1.1 Addition of Fractions with the Same Denominators
8
5
8
4
8
1 i)
2
1
8
4
8
3
8
1 ii)
fff
651 iii)
EXAMPLES
Add only the numerators and keep the
denominator same.
Write the fraction in its simplest form.
Add only the numerators and keep the
denominator the same.
Add only the numerators and keep the
denominator the same.
8
1
8
4
8
5
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
6
Curriculum Development Division
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1.2 Subtraction of Fractions with The Same Denominators
2
1
8
4
8
1
8
5 i)
7
4
7
5
7
1 ii)
nnn
213 iii)
Write the fraction in its simplest form.
Subtract only the numerators and keep
the denominator the same.
Subtract only the numerators and keep
the denominator the same.
Subtract only the numerators and keep
the denominator the same.
8
5
8
1
2
1
8
4
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UNIT 2: Fractions
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1.3 Addition and Subtraction Involving Whole Numbers and Fractions
.8
11 Calculate i)
7
29
7
1
7
28
7
14
7
14
5
18
5
2
5
20
5
24
5
33
3
12
3
1
3
12
3
14
y
yy
First, convert the whole number to an improper fraction with the
same denominator as that of the other fraction.
Then, add or subtract only the numerators and keep the denominator
the same.
1 8
1
8
11
8
9
+
8
8
+
8
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
8
Curriculum Development Division
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n
n
nn
n
n
52
5252
k
k
k
k
kk
32
323
2
First, convert the whole number to an improper fraction with
the same denominator as that of the other fraction.
Then, add or subtract only the numerators and keep the
denominator the same.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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1.4 Addition or Subtraction Involving Mixed Numbers and Fractions
.8
4
8
11 Calculate i)
7
5
7
15
7
5
7
12
= 7
20 =
7
62
9
4
9
29
9
4
9
23
= 9
25 =
9
72
88
11
88
31
xx
= 8
11 x
First, convert the mixed number to improper fraction.
Then, add or subtract only the numerators and keep the denominator the same.
8
11
8
4
8
51
8
13
+
8
9
+
8
4
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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2.0 Addition and Subtraction of Fractions with Different Denominators
.2
1
8
1 Calculate i)
To make the denominators the same, multiply both the numerator and the denominator of
the second fraction by 4:
Now, the question can be visualized like this:
?
The denominators are not the same.
See how the slices are different in
sizes? Before we can add the
fractions, we need to make them the
same, because we can't add them
together like this!
8
1
8
4
+
8
5
8
4
2
1
4
4
Now, the denominators
are the same. Therefore,
we can add the fractions
together!
8
1
2
1
+
?
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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Hint: Before adding or subtracting fractions with different denominators, we must
convert each fraction to an equivalent fraction with the same denominator.
2.1 Addition and Subtraction of Fractions When the Denominator of One Fraction is
A Multiple of That of the Other Fraction
Multiply both the numerator and the denominator with an integer that makes the
denominators the same.
(i) 6
5
3
1
6
5
6
2
6
7
= 6
11
(ii) 4
3
12
7
12
9
12
7
12
2
6
1
Change the first fraction to an equivalent
fraction with denominator 6.
(Multiply both the numerator and the denominator of the first fraction by 2):
6
2
3
1
2
2
Add only the numerators and keep the
denominator the same.
Change the second fraction to an equivalent fraction with denominator 12.
(Multiply both the numerator and the
denominator of the second fraction by 3):
12
9
4
3
3
3
Subtract only the numerators and keep the
denominator the same.
Write the fraction in its simplest form.
Convert the fraction to a mixed number.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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(iii) vv 5
91
vv 5
9
5
5
v5
14
Change the first fraction to an equivalent
fraction with denominator 5v.
(Multiply both the numerator and the denominator of the first fraction by 5):
vv 5
51
5
5
Add only the numerators and keep the
denominator the same.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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2.2 Addition and Subtraction of Fractions When the Denominators Are Not Multiple of
One Another
Method I
4
3
6
1
(i) Find the Least Common Multiple (LCM)
of the denominators.
2) 4 , 6
2) 2 , 3
3) 1 , 3
- , 1
LCM = 2 2 3 = 12
The LCM of 4 and 6 is 12.
(ii) Change each fraction to an equivalent
fraction using the LCM as the
denominator.
(Multiply both the numerator and the
denominator of each fraction by a whole
number that will make their
denominators the same as the LCM
value).
= 4
3
6
1
= 12
9
12
2
= 12
11
Method II
4
3
6
1
(i) Multiply the numerator and the
denominator of the first fraction with
the denominator of the second fraction
and vice versa.
= 4
3
6
1
= 24
18
24
4
= 24
22
= 12
11
Write the fraction in its
simplest form.
This method is preferred but you
must remember to give the
answer in its simplest form. 3
3 2
2
4
4 6
6
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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Multiply the first fraction with the second denominator and
multiply the second fraction with the first denominator.
1. 5
1
3
2
= 5
5
3
2
+
3
3
5
1
15
3
15
10
= 15
13
2. 8
3
6
5
=
8
8
6
5
–
6
6
8
3
= 48
18
48
40
= 48
22
= 24
11
Write the fraction in its simplest form.
EXAMPLES
Multiply the first fraction by the
denominator of the second fraction and multiply the second fraction by the
denominator of the first fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
Add only the numerators and keep the
denominator the same.
Subtract only the numerators and keep
the denominator the same.
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UNIT 2: Fractions
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3. 7
1
3
2g
= 3
3
7
7
7
1
3
2
g
= 21
3
21
14
g
= 21
314 g
4. 53
2 hg
3
3
55
5
3
2
hg
15
3
15
10 hg
15
310 hg
5. dc
46
= c
c
d
d
dc
46
cd
c
cd
d 46
= cd
cd 46
Multiply the first fraction by the denominator of the second fraction and
multiply the second fraction by the
denominator of the first fraction.
Write as a single fraction.
Write as a single fraction.
Write as a single fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Convert the mixed numbers to improper fractions.
Convert the mixed numbers to improper fractions.
2.3 Addition or Subtraction of Mixed Numbers with Different Denominators
1. 4
32
2
12
= 4
11
2
5
= 4
11
2
5
2
2
= 4
11
4
10
= 4
21
4
15
2. 4
31
6
53
= 4
7
6
23
= 6
6
4
4
4
7
6
23
= 24
42
24
92
= 24
50
= 12
25
= 12
12
Change the first fraction to an equivalent fraction
with denominator 4. (Multiply both the numerator and the denominator
of the first fraction by 2)
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Convert the mixed numbers to improper fractions.
Convert the mixed numbers to improper fractions.
Add only the numerators and keep the
denominator the same.
Change the fraction back to a mixed number.
Add only the numerators and keep the
denominator the same.
Change the fraction back to a mixed number.
Write the fraction in its simplest form.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
2.4 Addition or Subtraction of Algebraic Expression with Different Denominators
1. 22
m
m
m
= )2(
)2(
2
2
22
m
mm
m
m
=
22
2
22
2
m
mm
m
m
= )2(2
)2(2
m
mmm
= )2(2
22 2
m
mmm
= )2(2
2
m
m
2. y
y
y
y 1
1
= )1(
)1(1
1
y
y
y
y
y
y
y
y
= )1(
)1)(1(2
yy
yyy
= )1(
)1( 22
yy
yy
= )1(
122
yy
yy
= )1(
1
yy
Remember to use brackets
Write the above fractions as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Expand:
m (m – 2) = m2 – 2m
Expand:
(y – 1) (y + 1) = y2 + y – y – 1
2
= y2 – 1
Expand:
– (y2 – 1) = –y
2 + 1
Write the fractions as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
3. 24
5
8
3
n
n
n
= n
n
n
n
n
n
n 8
8
24
4
4
5
8
3
2
2
= )4(8
)5(8
)4(8
1222
2
nn
nn
nn
n
= )4(8
)5(812
2
2
nn
nnn
= )4(8
84012
2
22
nn
nnn
= )4(8
404
2
2
nn
nn
= )8(4
)10(42nn
nn
= 28
10
n
n
Factorise and simplify the fraction by canceling
out the common factors.
Expand:
– 8n (5 + n) = –40n – 8n2
Subtract the like terms.
Write as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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Calculate each of the following.
1. 7
1
7
2
2. 12
5
12
11
3. 14
1
7
2
4. 12
5
3
2
5. 5
4
7
2
6. 7
5
2
1
7. 313
22
8. 9
72
5
24
9. ss
12
10. ww
511
TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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11. aa 2
12
12. ff 3
52
13. ba
42
14. qp
51
15. nmnm5
3
7
2
5
2
7
5
16.
)2(2
1p
p
17.
5
3
2
32 yxyx
18.
xx
x 5
2
412
19.
x
x
x
x 1
1
20.
2
4
2 x
x
x
x
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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21.
4
84
2
36 yxyx
22.
29
4
3
2
n
n
n
23.
r
rr
15
25
5
2
24.
p
p
p
p
2
232
25.
n
n
n
n
10
34
5
322
26.
n
n
mn
nm 33
27.
mn
nm
m
m
5
5
28.
mn
mn
m
m
3
3
29.
24
5
8
3
n
n
n
30.
m
p
m
p 1
3
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UNIT 2: Fractions
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PART B:
MULTIPLICATION AND DIVISION
OF FRACTIONS
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to:
1. multiply:
(i) a whole number by a fraction or mixed number;
(ii) a fraction by a whole number (include mixed numbers); and
(iii) a fraction by a fraction.
2. divide:
(i) a fraction by a whole number;
(ii) a fraction by a fraction;
(iii) a whole number by a fraction; and
(iv) a mixed number by a mixed number.
3. solve problems involving combined operations of addition, subtraction,
multiplication and division of fractions, including the use of brackets.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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TEACHING AND LEARNING STRATEGIES
Pupils face problems in multiplication and division of fractions.
Strategy:
Teacher should emphasise on how to divide fractions correctly. Teacher should
also highlight the changes in the positive (+) and negative (–) signs as follows:
Multiplication Division
(+) (+) = + (+) (+) = +
(+) (–) = – (+) (–) = –
(–) (+) = – (–) (+) = –
(–) (–) = + (–) (–) = +
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UNIT 2: Fractions
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1.0 Multiplication of Fractions
Recall that multiplication is just repeated addition.
Consider the following:
32
First, let’s assume this box as 1 whole unit.
Therefore, the above multiplication 32 can be represented visually as follows:
This means that 3 units are being repeated twice, or mathematically can be written as:
6
33 32
Now, let’s calculate 2 x 2. This multiplication can be represented visually as:
This means that 2 units are being repeated twice, or mathematically can be written as:
4
22 22
LESSON NOTES
3 + 3 = 6
2 + 2 = 4
2 groups of 3 units
2 groups of 2 units
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Curriculum Development Division
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Now, let’s calculate 2 x 1. This multiplication can be represented visually as:
This means that 1 unit is being repeated twice, or mathematically can be written as:
211 12
It looks simple when we multiply a whole number by a whole number. What if we
have a multiplication of a fraction by a whole number? Can we represent it visually?
Let’s consider .2
12
Since represents 1 whole unit, therefore 2
1unit can be represented by the
following shaded area:
Then, we can represent visually the multiplication of 2
12 as follows:
This means that 2
1unit is being repeated twice, or mathematically can be written as:
1
2
2
2
1
2
1
2
12
1 + 1 = 2
2
1 +
2
1 = 1
2
2
2 groups of 1 unit
2 groups of 2
1 unit
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UNIT 2: Fractions
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Let’s consider again .22
1 What does it mean? It means ‘
2
1 out of 2 units’ and the
visualization will be like this:
Notice that the multiplications2
12 and 2
2
1 will give the same answer, that is, 1.
How about ?23
1
Since represents 1 whole unit, therefore 3
1unit can be represented by the
following shaded area:
Then, we can represent visually the multiplication 23
1 as follows:
This means that 3
1unit is being repeated twice, or mathematically can be written as:
3
2
3
1
3
1 2
3
1
3
1 +
3
1 =
3
2
The shaded area is 3
1unit.
2
1 out of 2 units 12
2
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
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Let’s consider 23
1 . What does it mean? It means ‘
3
1out of 2 units’ and the visualization
will be like this:
Notice that the multiplications3
12 and 2
3
1 will give the same answer, that is,
3
2.
Consider now the multiplication of a fraction by a fraction, like this:
2
1
3
1
This means ‘3
1 out of
2
1 units’ and the visualization will be like this:
Consider now this multiplication:
2
1
3
2
This means ‘3
2 out of
2
1 units’ and the visualization will be like this:
2
1unit
3
1 out of 2 units
3
22
3
1
3
1 out of
2
1 units
6
1
2
1
3
1
2
1unit
3
2 out of
2
1 units
6
2
2
1
3
2
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UNIT 2: Fractions
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What do you notice so far?
The answer to the above multiplication of a fraction by a fraction can be obtained by
just multiplying both the numerator together and the denominator together:
6
1
2
1
3
1
9
2
3
1
3
2
So, what do you think the answer for 3
1
4
1 ? Do you get
12
1 as the answer?
The steps to multiply a fraction by a fraction can therefore be summarized as follows:
1.1 Multiplication of Simple Fractions
Examples:
a) 35
6
7
3
5
2
b) 35
6
5
3
7
2
c) 35
12
5
2
7
6
d) 35
12
5
2
7
6
Steps to Multiply Fractions:
1) Multiply the numerators together and
multiply the denominators together.
2) Simplify the fraction (if needed).
Remember!!!
(+) (+) = +
(+) (–) = –
(–) (+) = –
(–) (–) = +
Multiply the two numerators together and the two denominators together.
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UNIT 2: Fractions
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1.2 Multiplication of Fractions with Common Factors
6
5
7
12 or
6
5
7
12
1.3 Multiplication of a Whole Number and a Fraction
6
152
=
6
31
1
2
=
6
31
1
2
= 3
31
= 3
110
Second Method:
(i) Simplify the fraction by canceling
out the common factors.
6
5
7
12
(i) Then, multiply the two
numerators together and the two
denominators together, and
convert to a mixed number, if
needed.
6
5
7
12
7
31
7
10
2
1
Convert the mixed number to improper
fraction.
Simplify by canceling out the common
factors.
Remember
2 = 1
2
First Method:
(ii) Multiply the two numerators
together and the two
denominators together:
6
5
7
12 =
42
60
(ii) Then, simplify.
7
31
7
10
42
60
10
7
3 Multiply the two numerators together and
the two denominators together.
Remember: (+) (–) = (–)
Change the fraction back to a mixed number.
1
1
2
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
30
Curriculum Development Division
Ministry of Education Malaysia
1. Find 10
15
12
5
Solution: 10
15
12
5
= 8
5
2. Find 5
2
6
21
Solution : 5
2
6
21
= 5
2
6
21
5
7
= 5
21
Simplify by canceling out the common
factors.
Note that 3
21 can be further simplified.
Simplify further by canceling out the
common factors.
3
1
Simplify by canceling out the common factors.
EXAMPLES
Multiply the two numerators together and the
two denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and
the two denominators together.
Remember: (+) (–) = (–)
3
1
1
7
Change the fraction back to a mixed
number.
2
1
4
5
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
31
Curriculum Development Division
Ministry of Education Malaysia
1.4 Multiplication of Algebraic Fractions
1. Simplify 4
52 x
x
Solution : 4
52 x
x
= 2
5
= 2
12
2. Simplify
m
n
n4
9
2
Solution:
m
n
n4
9
2
=
1
4
2
9
2
mn
n
n
= 1
)2(
2
9 mn
= nm22
9
1 2
1 1 Simplify the fraction by canceling out the x’s.
Multiply the two numerators together and
the two denominators together.
Simplify the fraction by canceling the
common factor and the n.
Multiply the two numerators together
and the two denominators together.
Write the fraction in its simplest form.
Change the fraction back to a mixed
number.
2
1
1
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
32
Curriculum Development Division
Ministry of Education Malaysia
1. Calculate 27
25
5
9
2. Calculate – 20
14
7
3
12
45
3. Calculate
4
112
4. Calculate
5
14
3
1
5. Simplify
k
m3
6. Simplify )5(2
mn
7. Simplify
14
3
6
11
x
8. Simplify )32(2
dan
9. Simplify
yx
10
95
3
2
10. Simplify
x
x 120
4
TEST YOURSELF B1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
33
Curriculum Development Division
Ministry of Education Malaysia
2.0 Division of Fractions
Consider the following:
36
First, let’s assume this circle as 1 whole unit.
Therefore, the above division can be represented visually as follows:
This means that 6 units are being divided into a group of 3 units, or mathematically
can be written as:
2 36
The above division can also be interpreted as ‘how many 3’s can fit into 6’. The answer is
‘2 groups of 3 units can fit into 6 units’.
Consider now a division of a fraction by a fraction like this:
.8
1
2
1
LESSON NOTES
How many 8
1 is in
?2
1
6 units are being divided into a group of 3
units:
2 36
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
34
Curriculum Development Division
Ministry of Education Malaysia
This means ‘How many is in ?
8
1
2
1
The answer is 4:
Consider now this division:
.4
1
4
3
This means ‘How many is in ?
4
1
4
3
The answer is 3:
But, how do you
calculate the answer?
How many 4
1 is in ?
4
3
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
35
Curriculum Development Division
Ministry of Education Malaysia
Consider again .236
Actually, the above division can be written as follows:
3
16
3
636
Notice that we can write the division in the multiplication form. But here, we have to
change the second number to its reciprocal.
Therefore, if we have a division of fraction by a fraction, we can do the same, that is,
we have to change the second fraction to its reciprocal and then multiply the
fractions.
Therefore, in our earlier examples, we can have:
4
2
8
1
8
2
1
8
1
2
1 (i)
The reciprocal of a
fraction is found by
inverting the
fraction.
Change the second fraction to its
reciprocal and change the sign to .
The reciprocal
of 8
1 is .
1
8
These operations are the same!
The reciprocal
of 3 is .3
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
36
Curriculum Development Division
Ministry of Education Malaysia
3
1
4
4
3
4
1
4
3 (ii)
The steps to divide fractions can therefore be summarized as follows:
2.1 Division of Simple Fractions
Example:
7
3
5
2
= 3
7
5
2
= 15
14
Change the second fraction to its reciprocal
and change the sign to .
Multiply the two numerators together and
the two denominators together.
Steps to Divide Fractions:
1. Change the second fraction to its
reciprocal and change the sign to .
2. Multiply the numerators together and
multiply the denominators together.
3. Simplify the fraction (if needed).
Tips:
(+) (+) = +
(+) (–) = –
(–) (+) = –
(–) (–) = +
Change the second fraction to its
reciprocal and change the sign to .
The reciprocal
of 4
1 is .
1
4
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
37
Curriculum Development Division
Ministry of Education Malaysia
2.2 Division of Fractions With Common Factors
Examples:
9
2
21
10
= 2
9
21
10
= 2
9
21
10
= 7
15
= 7
12
7
6
5
3
6
7
5
3
10
7
7
65
3
1
5 3
7
1
2
Express the fraction in division form.
Change the second fraction to its reciprocal and
change the sign to .
Simplify by canceling out the common factors.
Change the fraction back to a mixed number.
Change the second fraction to its reciprocal
and change the sign to .
Then, simplify by canceling out the common
factors.
Multiply the two numerators together and the
two denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and the
two denominators together.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
38
Curriculum Development Division
Ministry of Education Malaysia
1. Find 6
25
12
35
Solution : 6
25
12
35
= 25
6
12
35
= 10
7
2. Simplify –4
52 x
x
Solution : –xx 5
42
= –25
8
x
3. Simplify 2
x
y
Solution :
2x
y
2
1
x
y
x
y
2
5
7
Change the second fraction to its reciprocal
and change the sign to . Then, simplify by canceling out the common
factors.
Method I
EXAMPLES
Change the second fraction to its reciprocal
and change the sign to .
Multiply the two numerators together and the two
denominators together.
Express the fraction in division form.
Change the second fraction to its reciprocal
and change to .
Multiply the two numerators together and the two
denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and the
two denominators together.
2
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
39
Curriculum Development Division
Ministry of Education Malaysia
Multiply the numerator and the denominator of
the given fraction with x
2
x
y
= 2
x
y
x
x
= x
xx
y
2
= x
y
2
4. Simplify 5
)1( 1r
Solution:
5
)1( 1r
= 5
)1
1(r
r
r
= r
r
5
1
The given fraction.
r is the denominator of r
1.
Multiply the given fraction with r
r.
Note that:
1)1
1( rrr
Method II
The numerator is also
a fraction with
denominator x
Multiply the numerator and the denominator of the
given fraction by x.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
40
Curriculum Development Division
Ministry of Education Malaysia
1. Calculate 2
21
7
3
2. Calculate 16
5
8
7
9
5
3. Simplify 3
48 y
y
4. Simplify
k
2
16
5. Simplify
3
5
2
x
6. Simplify n
m
n
m
3
24 2
7. Simplify 8
1
4
y
8. Simplify
x
x
11
TEST YOURSELF B2
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
41
Curriculum Development Division
Ministry of Education Malaysia
9. Calculate 5
)1(341
10. Simplify y
x15
11. Simplify
32
941 x
12. Simplify
15
1
1
p
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
42
Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF A:
1. 7
3
2. 2
1
3. 14
5
4. 4
1
5. 35
38 or
35
31
6. 14
3
7. 13
67 or
13
25
8. 45
73or
45
281
9. s
3
10. w
6
11. a2
5
12. f3
1
13. ab
ab 42
14. pq
pq 5
15. nm
16. 2
33 p
17. 10
1716 yx
18. x
x 12
19. )1(
1
xx
20. 2
21. 2
8 yx
22. 29
47
n
n
23. r
r
3
12
24. 2
2
2
6
p
p
25. 2
2
10
647
n
nn
26. m
m1
27. n
n
5
5
28. n
n
3
3
29. 28
10
n
n
30. m
p
3
34
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
43
Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF B1:
1. 3
21
3
5or 2.
8
11
8
9 or 3.
2
15
2
11or
4. 5
21
5
7 or 5.
k
m3 6.
2
5mn
7. 4
x 8. ndna
2
3 9. yx
5
3
3
10
10. 4
15 x
TEST YOURSELF B2:
1. 49
2 2.
9
51
9
14 or 3.
2
6
y
4. 8k
5. x5
6 6.
m
6
7. )1(2
1
y 8.
1
2
x
x
9. 20
9
10. xy
x 15 11.
6
13x 12.
p4
5