A) Reading Fractions
A fraction is a number that represents a part
of a whole. It is written as p where p and q are
q
whole numbers and q # 0.
p is read as 'p over q'.
q
For example:-
(a) 3 is read as 'three over four' or three quarter.
4
(b) 2 is read as 'two over three' or 'two third'.
3
B) Representing Fractions with Diagrams.
1. Fractions can be represented with diagrams and number lines.
For example:-
In each of the diagrams above, the shaded parts are 2
out of 3 equal parts, that is 2
3
and the bottom number is called the numerator.
For example:-
2 ( Numerator )
3 ( Denominator )
In the fraction 2, 2 is the numerator and 3 is denominator.
3
3. The numerator represents the number of equal parts
that are shaded and the denominator represents the
total number of equal parts in one whole.
4. When the numerator is the same as the denominator,
the value of the fraction is equal to 1. The figure 1
represents all the parts of a fraction.
For example:-
(a)
(b)
Worked example
(a) Shade the diagram to represent 3
5
(b) Write the fraction represented by the
shaded parts in the diagram below.
Solution
(a) Shade any 4 parts.
(b) 4 parts out of 7 equal parts are shaded.
The fraction represented by the shaded
parts is 4 .
7
EQUIVALENT FRACTIONS
A)Finding Equivalent Fractions
1. Equivalent fractions are fractions having
the same value.
For example:-
(a)
The shaded parts in both the triangle are
equal.
Therefore, 1 and 2 have the same value.
2 4
1 and 2 are equivalent fractions.
2 4
Therefore, 1 = 2 .
2 4
( when you fold the two triangle along their
vertical lines, you will find that the shaded
portions are equal.)
(b)
The shaded parts of all the three rectangles
are equal.
Therefore, 1 , 2 , and 4 have the same value.
2 4 16
1 = 2 = 4 are equivalent fractions.
2 4 16
Therefore, 1 = 2 = 4.
4 8 16
2. Equivalent fractions can be obtained by multiplying
the numerator and denominator by the same whole
numbers ( greater than 1 ).
For example:-
2 = 2 x 2 = 2 x 3 = 2 x 4 = 2 x 5
3 3 x 2 3 x 3 3 x 4 3 x 5
= 4 = 6 = 8 = 10
6 9 12 15
Worked example
List the first three equivalent fractions of
Solution
Worked example
List all the equivalent fractions of 3 with
denominators between 30 and 50.
Solution
B) Determining whether Two given
Fractions are Equivalent
To determine whether two given fractions
are equivalent or not, we can use diagrams calculation.
Worked example
Are 2 and 6 equivalent fractions ?
4 8
Solution
2 6
4 8
The shaded parts for 2 and 6 are the same.
4 8
Therefore, 2 and 6 are equivalent fractions.
4 8
Worked example
Are 1 and 3 are not equivalent fractions ?
2 4
Solution
1 and 3 are not equivalent fraction because
2 4
their shaded parts are not the same.
Worked example
Determine whether the following pairs of
fractions are equivalent.
(a) 4 and 1
8 2
(b) 4 and 1
6 3
Solution
(a)
4 and 1 are at the same position on the
8 2
number line.
Therefore, 4 and 1 are equivalent.
8 2
(b)
4 and 1 are at different positions on the
6 3
number line.
Therefore, 4 and 6 are not equivalent.
6 3
Worked example
Determine whether 2 and 4 are equivalent.
6 12
Solution
Worked example
Determine whether 3 and 5 are equivalent.
6 12
Solution
3 and 5 are not equivalent.
6 12
C) Comparing the values of Two Fractions
1. When comparing two fractions having the same
denominator, the fraction with the bigger numerator
is greater in value.
For example:-
2. When comparing two fractions having the same
numerator, the fraction with the smaller denominator
is greater in value.
For example:-
3. To compare two fractions having different numerators,
and denominators, carry out the following steps.
Step 1 : Find the LCM of the two denominators.
Step 2 : Convert each of the given fraction with the
LCM as its denominator.
Step 3 : Compare the numerators of the fractions.
Worked example
Which is greater, 7 or 4 ?
9 5
Solution
7 = 35 ( LCM of 9 and 5 is 45 )
9 45
4 = 36
5 45
35 < 36
45 45
Therefore, 4 is greater.
5
D) Arranging Fractions in Order
(a) Arrange 2 , 6 , 4 , 8 in increasing order.
9 9 9 9
(b) Arrange 1 , 1 , 3 and 1 in decreasing order.
2 3 4 6
Solution
(a) 2 , 4 , 6 , 8
9 9 9 9
(b) 1 = 1 x 6 = 6 ; 1 = 1 x 4 = 4 ;
2 1 x 6 12 3 3 x 4 = 12
3 = 3 x 3 = 9 ; 1 = 1 x 2 = 2 ;
4 4 x 3 12 6 6 x 2 12
Therefore, the fractions arranged decreasing order
are 3 , 1 , 1 , 1 .
4 2 3 6 ( from the biggest to the smallest )
E) Simplifying Fractions
1. A fraction is in its lowers terms if the numerator and
denominator have no common factor except 1.
2. To simplify a fraction, divide the numerator and deno-
minator repeatedly by their common factors.
For example:-
3. To simplify a fraction to its lowest terms, divide the nume-
rator and the denominator by their HCF.
4. All answer must be given in their lowest terms.
Worked example
Simplify 12 to its lowest terms.
18
Solution
MIXED NUMBERS
A) Recognising Mixed Numbers
1. A mixed number is a number consisting of a
whole number and a fraction.
For example:-
2. All mixed numbers are greater than 1.
B) Representing Mixed Numbers with Diagrams
Mixed numbers can be represented by diagrams.
For example:-
C) Writing Mixed Numbers based on the given Diagrams
For example:-
D) Comparing andd Arranging Mixed Numbers
1. Like fractions, mixed numbers can be compared and
arranged by using a number line. Any number on the
number line is greater than the numbers to its left.
2. To compare and arrange mixed numbers having the
same whole number part but different fractional part,
carry out following steps.
Step 1 : Find the LCM of the dinominators.
Step 2 : Convert each of the given fractions to an equivalent
fraction with the LCM as its denominator.
Step 3 : Compare and arrange the mixed numbers.
Worked Example
Which is greater,
Arrange both mixed numbers on a number line.
Solution
The LCM of 9 and 3 is 9.
Since 3 6 is greater than 3 5 , therefore 3 2 is
9 9 3
Worked Example
(a) Arrange 2 1 , 2 2 , 2 1 in increasing order.
2 3 6
(b) Arrange 2 3 , 1 1 , 2 1 in decreasing order.
4 3 2
Solution
(a) The LCM of 2, 3 and 6 is 6.
2 1 = 2 3 ; 2 2 = 2 4
2 6 3 6
Therefore, the mixed numbers arranged increasing
order are 2 1 , 2 1 , 2 2 .
6 2 3
(b) The LCM of 4, 3 and 2 is 2.
1 1 is smaller than 2 1 and 2 3 .
3 2 4
2 1 = 2 2
2 4
Therefore, the mixed numbers arranged in decreasing
order are 2 3, 2 1 , 1 1 .
4 2 3
3.4 PROPER FRACTIONS AND IMPROPER FRACTIONS
A) Determining Proper Fractions and
Improper Fractions
1. A proper fraction has a numerator which is
smaller than the denominator.
For example:-
1 , 3 , 5 , 7 , 13
2 4 7 12 30
2. An improper fraction has a numerator which
is the same as or greater than the denominator.
For example:-
3 , 4 , 7 , 10 , 12 , 15
2 3 3 10 5 9
Worked Example
Determine whether each fraction below is a proper
fraction or an improper fraction.
(a) 6 (b) 7 (c) 13
4 7 16
Solution
(a) 6 is an improper fraction.
4
(b) 7 is an improper fraction.
7
(c) 13 is a proper fraction.
16
B) Converting Whole Numbers to Improper Fraction.
1. All whole numbers are improper fractions with
1 as their denominators.
For example:-
5 = 5 , 13 = 13 , 32 = 32
1 1 1
2. Whole numbers can be converted to improper
fractions with other denominators.
For example:-
(a) 4 = 4 4 = 4 x 12
1 1 x 12
= 4 x 8 = 48
1 x 8 ( 8 as denominator ) 12 ( 12 as denominator )
= 32
8
C) Converting Mixed Numbers to Improper Fractions.
To change a mixed number to an improper fraction,
multiply the whole number by the denominator and
then add the product to the numerator. The denomi-
nator remains the same.
Worked Example
Convert these mixed numbers to improper fractions.
(a) 5 5 (b) 10 6
9 7
Solution
(b) 10 6 = 10 x 7 + 6
7 7
= 76
7
D) Converting Improper Fractions to Mixed Numbers.
1. To change an improper fraction to a mixed number,
divide the numerator by the denominator.
2. The quotient obtained is the whole number part and
the remainder is the numerator of the fractional part.
Worked Example
Convert these improper fractions to mixed numbers.
(a) 57 (b) 92
4 8
Solution
(a) 57 = 14 1
4 4
(b) 92 = 11 4
8 8
3. Where possible, simplify the improper fraction to
its lowest terms before converting it to the mixed
number.
Worked Example
Change these improper fractions to mixed numbers
or whole numbers.
(a) 129 (b) 96
9 8
Solution
ADDITION AND SUBTRACTION OF FRACTIONS
A) Addition of Fractions
i) Adding two fractions with the same or common
denominator
1. To add two fractions with the same denominator,
keep the denominator and add the numerators.
2. Answer are always written in their lowest terms.
Worked Example
Find the value of 5 + 7 .
9 9
Solution
5 + 7 = 12
9 9 9 ( No change in the denominator. )
= 1 3 ( Change improper fraction
9 to mixed number )
= 1 1
3 ( Simplify to lowest terms )
ii) Adding two fractions with different denominators
To add two fractions with different denominators, first
find the LCM of the denominators and then convert
both the fractions with the same denominator.
Worked Example
Solve the following.
(a) 5 + 2
12 3
Solution
(a) 5 + 2 = 5 + 2 x 4
12 3 12 3 x 4 ( LCM of 3 and 12 is 12 )
= 5 + 8
12 12
= 13
12
= 1 1
12 ( Change to mixed number. )
iii) Adding whole number and fraction, a mixed
number is produced
Worked Example
Solve the following.
(a) 12 + 3 (b) 7 + 6
5 9
Solution
(a) 12 + 3 = 12 3
5 5
( Answer are mixed numbers. )
(b) 7 + 6 = 6 7
9 9
iv) Adding fractions and mixed numbers
1. To add a fraction and a mixed number, keep the
whole number and add the fractional parts like
adding the fractions.
2. Answer are always written in their lowest terms.
Worked Example
Simplify
(a) 1 + 4 2 = 1 + 4 + 2
3 3 3 3
Solution
(a) 1 + 4 2 = 1 + 4 + 2
3 3 3 3
= 4 + 3
3
= 4 + 1
= 5
v) Adding two mixed numbers
Worked Example
Solve 3 5 + 4 5
9 6
Solution
vi) Adding three fractions
Before performing the addition, convert the fractions
to their respective equivalent fractions with the same
denominator.
Worked Example
Find the value of
(a) 5 + 4 + 3 1
5 2
Solution
(a) 5 + 4 + 3 1 = 5 + 4 + 7
5 2 1 5 2
= 50 + 8 + 35
10
= 93
10
= 9 3
10
B) Problem Solving involving Addition of Fractions
Worked Example
Bag P weighs 1 4 kg. Bag R is 2 3 kg heavier than
5 10
bag P. Calculate the total mass of the two bags.
Solution
1. Understand the problem
Given information :
Bag P weighs 1 4 kg.
5
Bag R is 2 3 kg heavier than bag P.
10
Find : Total mass of bags P and R.
2. Devise a plan
Use addition.
3. Carry out the plan
1 4 + 2 3 + 1 4 = 9 + 23 + 9
5 10 5 5 10 5
= 18 + 23 + 18
10
= 59
10
= 5 9
10
Therefore, the total mass of bags P and R is 5 9 kg.
10
4. Check
1 4 + 2 3 = 9 + 23
5 10 5 10
= 18 + 23
10
= 41
10
41 + 9 = 41 + 18
10 5 10
= 59 = 5 9
10 10
c) Subtraction of Fractions
i) Subtracting fraction with the same or common
denominator.
1. To subtract two fractions with the same denominator,
keep the denominator and subtract the numerators.
2. Answers are always written in their lowest terms.
Worked Example
Solve 9 - 3 .
14 14
Solution
9 - 3 = 9 - 3
14 14 14 ( subtract. )
= 6
14
= 3 ( lowest terms )
7
ii) Subtracting fractions with different denominators
To subtract two fractions with different denominators,
first find the LCM of the denominators and then convert
both the fractions to their respective equivalent fractions
with the same denominator.
Worked Example
Solve 5 - 3
6 8
Solution
5 - 3 = 5 x 4 - 3 x 3
6 8 24 24 ( LCM of 6 and 8 is 24. )
= 20 - 9
24
= 11
24
iii) Subtracting a fraction from a whole number
To subtracting two fraction from a whole number,
convert the whole number to an improper fraction
with a common denominator as the fraction.
Worked Example
Solve the following.
Solution
iv) Subtracting a fraction from a mixed number
To subtract a fraction from a mixed number, carry
out the following steps.
Step 1 : Convert the mixed number to an improper
fraction.
Step 2 : Find the LCM of the denominators.
Step 3 : Convert both the fractions to their respective
equivalent fractions with the same denominator.
Worked Example
Simplify the following.
(a) 5 5 - 3
6 4
Solution
(a) 5 5 - 3 = 35 - 3
6 4 6 4
= 70 - 9
12
= 61
12
= 5 1
12
v) Subtracting two mixed numbers
Worked Example
Simplify 5 1 - 3 5
4 6
Solution
vi) Subtracting three fractions
Carry out the subtractions of three fractions
from left to right.
Worked Example
Simplify
(a) 9 - 1 - 1
10 2 5
Solution
D) Problem Solving involving Subtraction of Fractions
Worked Example
Puan Aishah bought 5 kg of cooking oil. She used
up 3 2 kg. Find the amount of oil remained.
5
Solution
1. Understand the problem
Given information :
3 2 kg from the 5 kg of oil was used.
5
Find : Amount of oil remained
2. Devise a plan
Use subtraction.
3. Carry out the plan
5 - 3 2 = 5 17
5 5
= 25 - 17
5
= 8
5
= 1 3
5
Therefore, amount of oil remained was 1 3 kg.
5
4. Check
3 2 + 1 3 = 17 + 8
5 5 5 5
= 25
5
= 5
MULTIPLICATION AND DIVISION OF FRACTIONS
A) Multiplication of Fractions
i) Multiplying a whole number by a fraction
Multiplying of a whole number by a fraction
or a mixed number is the repeated addition
of the fraction or the mixed number.
Worked Example
Solve 6 x 3
4
Solution
Method 1 : By using diagrams
6 x 3
4
Method 2 : By repeated addition
6 x 3 = 3 + 3 + 3 + 3 + 3 + 3
4 4 4 4 4 4 4
= 18
4
= 4 1
2
Method 3 : By multiplying directly
Method 4 : By cancellation
ii) Multiplying a fraction by a whole number
Worked Example
Find the value of 4 x 25.
5
Solution
Method 1 : By multiplying directly
4 x 25 = 100
5 5
= 20
Method 2 : By cancellation
iii) Multiplying two fractions
Evaluate 2 x 3
3 8
Solution
Method 1 : By using diagrams
2 x 3
3 8
Method 2 : By cancellation
2 x 3
3 8
iv) Multiplying two mixed numbers ( including
whole numbers )
In multiplication involving a mixed number, change
the mixed number to an improper fraction first.
Worked Example
Simplify each of the following.
(a) 2 4 x 20
5
Solution
v) Multiplying three fractions ( including whole
numbers and mixed numbers )
Worked Example
Evaluate 2 1 x 9 x 1 1
3 14 12
Solution
B) Problem Solving involving Multiplication of Fractions
Worked Example
In a class of 48 pupils, 5 are girls. How many
12
girls are there in the class ?
Solution
1. Understand the problem
Given information :
5 of 48 pupils are girls.
12
Find : Number of girls
2. Devise a plan
Use multipcation.
3. Carry out the plan
Therefore, there are 20 girls in the class.
4. Check
C) Division of Fractions
i) Dividing a quantity into parts.
Diagrams can be used to show the division of
a quantity.
For example:-
ii) Dividing a fraction by a whole number
1. To perform division involving fractions, multiply
the dividend by the reciprocal of the divisor.
2. Answer are always written in their lowest terms.
Worked Example
Divide the shaded portion into 4 equal parts.
Solution
Worked Example
Simplify the following.
(a) 2 ÷ 4
5
Solution
iii) Dividing a fraction by a fraction
Diagrams can be used to show the division
of a fraction by another fraction.
For example:-
Worked Example
Simplify 2 ÷ 4
3 5
Solution
iv) Dividing a whole number by a fraction
Division of a whole number by a fraction
is process of finding the number of times
the fraction is contained in that number.
For example:-
Worked Example
Simplify the following.
(a) 6 ÷ 3
8
Solution
v) Dividing a mixed number by a mixed number
1. To perform a division involving mixed numbers,
always convert the mixed numbers into improper
fraction first.
2. Answer are always written in their lowest terms.
Worked Example
Simplify each of the following.
(a) 1 1 ÷ 1 7
4 8
(b) 1 1 ÷ 8 ÷ 1
3 9 6
Solution
D) Problem Solving involving Division of Fractions
Worked Example
2 1 kg of flour is put equally into 15 packets.
4
Find the mass of each packets.
Solution
1. Understand the problem
Given information :
2 1 kg of flour is put equally into 15 packets.
4
Find : Mass of each packet of flour
2. Devise a plan
Use division
3. Carry out the plan
2 1 ÷ 15 = 9 ÷ 15
4 4
= 9 x 1 = 9 = 3 kg.
4 15 60 20
Therefore, the mass of each packet is 3 kg.
20
4. Check
Mass of 15 packets of flour
= 15 x 3 kg = 2 1 kg
20 4
COMBINED OPERATIONS OF +, -, x, ÷ OF FRACTIONS
A) Combined Operations of any Two Operations
Worked Example
Simplify the following.
(a) 11 + 3 - 1 (b) 6 - 4 1 + 2 3
12 4 1 5 10
Solution
Worked Example
Simplify the following.
(a) 3 + 5 x 1 1 (b) 2 5 ÷ 1 1 - 1 1
5 8 5 8 6 2
Solution
B) Combined Operation involving Brackets
1. To perform a calculation involving any two combined
operations of addition,subtraction,multiplication,divi-
sion or brackets, always work the calculation within
the brackets first.
2. Then, do the multiplication or division before the addition
or subtraction, working from left to right.
Worked Example
Simplify
Solution
C) Problem Solving involving any Two Operations
Worked Example
Mr Lee bought 3 1 litres of orange juice. He bought
2
9 litre of water melon less than orange juice. How
10
much fruit juice did he buy altogether?
Solution
1. Understand the problem
Given information :
Volume of orange juice bought = 3 1
2
Volume of water melon juice bought is 9 litre less
10
than volume of orange juice.
Find : The volume of fruit juice bought.
2. Devise a plan
Perform subtraction followed by addition.
3. Carry out the plan
3 1 - 9 + 3 1 = 7 - 9 + 7
2 10 2 2 10 2
= 35 - 9 + 35
10
= 61
10
= 6 1
10
Therefore, he bought 6 1 litres of fruit juice.
10
altogether.
4. Check
3 1 - 9 = 7 - 9
2 10 2 10
= 35 - 9
10
= 26
10
26 + 7 = 26 + 35
10 2 10
= 61 = 6 1
10 10
Worked Example
Fatimah bought 10m of coth. She cut out 3 pieces,
each 2 3 long. Find the length of the remaining cloth.
4
Solution
10 - 3 x 2 3 = 10 - 3 x 11
4 4
= 10 - 33
4
= 40 - 33
4
= 7 = 1 3 m
4 4
Lenght of remaining cloth is 1 3 m.
4
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