3.


MATHEMATICS - FORM 1

FRACTIONS


                                      

       
 
FRACTIONS

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)  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 ; 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  ,  , 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 + 1  + 4 2
     3         3      3             3

Solution

(a) 1  + 2  =  1 + 4 + 2
     3          3      3          3
                   
                    = 4 + 3
                             3

                    = 4 + 1
 
                    = 5

v) Adding two mixed numbers

Worked Example

Solve 5 + 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  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