MATHEMATICS - FORM 1

DECIMALS 

                                
                                       
              
       
DECIMAL AND FRACTION


1. A decimal is a fraction in which the denominator

   
is 10 or a power of 10.

   For example :-

   
2. The decimal point ( the dot ) separates 

    the whole number from its fractional part.

    A number written with a decimal point is

    known as a decimal.

    For example:-

    
3. The number of digits after the decimal point

    is the same as the number of zeros in the

    denominator.

    For example:-

   

A) Representing Decimals with Diagrams

Decimals can be represented with diagrams.

For example :-

      1                      3                     9
     10                   100                1 000

(1 out of 10)  ( 3 out of 100)  ( 7 out of 1 000)

B) Writing Decimals based on given Diagrams

Worked Example

     
Write a decimal to represent the shaded

area in the diagram above.

Solution

The shaded area is 3 .
                               10

Therefore, the decimal is 0. 3.

C) Conversion of Fractions into Decimals

 and vice versa


1. A fraction in which the denominator is

   10 or a power if 10 can be converted to a

   decimal mentally.


Worked Example

Express the following as decimals.

(a) 7          (c) 138
    10              1 000
(b) 52
    100

Solution



Worked Example

Change the following into fractions with

denominators of 10 or powers of 10.

(a) 0.6      (b) 0.55      (c) 0.374

Solution



2. A fraction can also be converted to a

    decimal by dividing the numerator by

    the denominator.

3. Mixed numbers can also be expressed

    as decimals.

Worked Example

Express the following fractions as decimals.

(a) 3        (b) 7 2
     4                  5

Solution


Worked Example

Express the following as decimals.

(a) 1 14           (b) 4 5
         100                  20

Solution


Worked Example

Express the following decimals as mixed

numbers.

(a) 3. 6          (b) 11. 025

Solution

(a) 3. 6 = 3 + 0. 6

            = 3 + 6
                     10
            = 3 6
                  10

PLACE VALUE AND DIGIT VALUE
 IN DECIMALS

A) Place Value and Digit Value

1. Each digit in a decimal has its own

    place value and digit value.

    Look at the decimal 8.235 below.

   
Worked Example

State the place value and digit value

of the underlined digit in each of the

follow decimals.

(a) 13.46        (b) 7.095

Solution

(a) Tenths ; 0.4

(b) Thousandths ; 0.005

2. Decimal places are the number of

    places occupied by the digits after

    the decimal point.

    For example :-

    (a) 0.3

    1 decimal place

    (b) 0.06

    2 decimal places

    (c) 3.152

    3 decimal places

    (d) 9.7420

    4 decimal places

B) Representing Decimals on Number Lines

    A decimal can be represented on a number line.

    For example :-
 
   

Worked Example



State the decimal represented by each

of the letters on the number line above.

Solution



One part = 4.95 - 4.80

               = 0.15

F = 4.95 + 0.15 + 0.15

   = 5.25

G = 5.25 + 0.15

   = 5.40

Therefore, F = 5.25 and G = 5.40.

C) Comparing Two Decimals


When comparing two decimals, arrange the

digits of the decimals according to their place

values and then compare their digit values

from left to right.

Worked Example

Which is greater, 24.853 or 24.832

Solution

Arrange the decimals as shown below.

 
At the place value of hundredths, 5 is larger  than 3.

Therefore, 24.853 is greater than 24.832.

Worked Example 

Fill in each box with ' >' or ' <'.


Solution



D) Order of Decimals

We can arrange decimals in increasing or

decreasing order according to their values.

Worked Example

Arrange 7.1, 6.4, 6.83, 6.81 in decreasing order.

Solution



Worked Example

Arrange 11 , 2.73 and 2.732 in increasing order.
                4
Solution


E) Rounding Off Decimals. 
 

A Decimal can be rounded off to a whole number

or certain decimal places. The rules for rounding

off decimals are as follows.    

(a) Look at the digit to the right of the digit that

     is to be rounded off.        

(b) If the digit is more than or equal to 5, add 1

     to the digit that is to be rounded off and ignore

     the rest of the digits after it.

(c) If the digit is less than 5, maintain the digit

     that is to be rounded off and ignore the rest of

     the digits after it.

Worked Example

Round off  8.3752 correct to

(a) 1 decimal place,

(b) 2 decimal places,

(c) 3 decimal places.

Solution



Worked Example

Round off the following decimals to the

nearest whole numbers.

(a) 8.6168        (c) 352.84

(b) 30.324

Solution



ADDITION AND SUBTRACTION
OF DECIMALS


1. Decimals are added and subtracted in

    the same way as whole numbers.

2. When two decimals are added together

    or subtracted from each other, the deci-

    mal points must be placed directly one

    below the other and the digits written in

    the correct place value columns.

A) Addition of Decimals

Worked Example

Calculate

(a) 0.382 + 4.264   (b) 5 + 8.8

Solution


Worked Example

Calculate

(a) 34.2 + 26.73 + 9.985

(b) 9.25 + 54 + 17.631

Solution



B) Problem Solving involving
Addition off Decimal


Worked Example

Salman is 1.75 m tall and Arjun is 0.19 m

taller than Salman. How tall is Arjun.

Solution

1. Understand the problem

    Given information :

    Salman's height = 1.75 m

    Arjun is taller than Salman by 0.19 m

    Find : Height of Arjun

2. Devise a plan

    Use addition.

3. Carry out the plan

    Arjun's height
   
    = Salman's height  + 0.19 m

    1.75 m + 0.19 m = 1.94 m

          1 . 7 5
       + 0 . 1 9
          1 . 9 4

    Therefore, Arjun is 1.94 m tall.

4. Check

          1 . 9 4
       -  1 . 7 5
          0 . 1 9  ( The difference )

Worked Example

Dewi collected 21.10 litres of latex on

Monday, 26 litres on Tuesday and

23.906 litres on Wednesday. Find the

total amount of latex collected.

Solution


The total amount of latex collected was

69.906 litres.

C) Subtraction of Decimal

Worked Example

Calculate

(a) 0.85 - 0.35

(b) 83 - 26.421

Solution


Worked Example


Calculate

(a) 57.3 - 35.82 - 2.346

(b) 81 - 24.907 - 7.5

Solution



D) Problem Solving involving
Subtraction of Decimals


Worked Example

Suziela weighs 52.03 kg and Azman weighs 60.4

kg. Find the difference in their body mass.

Solution

1. Understand the problem

    Given information :

    Suziela's mass = 52.03 kg

    Azman's mass = 60.4 kg

    Find : Difference in their body mass

2. Devise a plan

    Use subtraction.

3. Carry out the plan
 
    60.4 kg - 52.03 kg

            
    Therefore, the difference in their body

    mass is 8.37 kg.

4. Check

            5 2 . 0 3
         +    8 . 3 7
            6 0 . 4 0

Worked Example

Zaiton bought 6.3 m of cloth. She used 2.15 m

to make a shirt and 3.28 m to make a pair of

pants. How much of the cloth remained ?

Solution

6.3 m - 2.15 m - 3.28 m = 0.87 m
 
           
Therefore, 0.87 m of the cloth remained.

MULTIPLICATION AND DIVISION OF DECIMALS

A) Multiplication of Decimals

i - Generel multiplication of decimals

1. To find the product of decimals, multiply the

    numbers in the same way as for whole num-

    bers first. Then, put in the decimal point.

2.The number of decimal places in the answer

    must correspond to the total number of  decimal

    places in the decimals being multiplied.

Worked Example

Calculate

(a) 0.73 x 0.7

Solution



ii) Multiplying a decimal by a power of 10

When multiplying a decimal by 10, 100,

1 000, ect., we move the decimal point 1,

2, 3, ect. places respectively to the right.

Worked Example

Calculate

(a) 1.42            (c) 0.8

(b) 6.973          ( d) 0.0032

Solution



2. Similarly, when we multiply a decimal by

    0.1, 0.01, 0.001, ect., we move the decimal

    point 1, 2, 3, ect. Place respectively to the

    left.

Worked Example

Calculate

(a) 24.8 x 0.1,

(b) 53.62 x 0.01,

(c) 21.73 x 0.001

Solution



Worked Example

Calculate 0.38 x 0.7 x 0.5.

Solution



B) Problem Solving involving Multiplication

of Decimals


Worked Example

Dalina needs 0.95 kg of flour to make a cake.

How much flour is needed to make 8 cakes ?

Solution

1. Understand the problem

    Given information :

    Amount of flour to make 1 cake = 0.95

    Find : Amount of flour to make 8 cakes

2. Devise a plan

    Use multiplication.

3. Carry out the plan

    8 x 0.95 kg = 7.60

               0 . 9 5
            x        8
               7 . 6 0

    Therefore, 7. 60 kg of flour is needed to make

    8 cakes.

4. Check

    Multiply again to verify the answer.

Worked Example

A apple costs RM0.70. The price of a jackfruit

is 4.8 times the price of the apple. What is the

cost of 5 jackfruits ?

Solution

Cost of a apple = RM0.70

Cost of a jackfruit = 4.8 x RM0.70

Cost of 6 jackfruits = 5 x 4.8 x 0.70

                              = RM16.80

               0 . 7 0                 3 . 3 6
            x    4 . 8            x          5
                 5 6 0              1 6 . 8 0
              2 8 0__
            3 . 3 6 0   

    Therefore, the cost of 6 jackfruits is RM16.80.

C) Division of Decimals

i - Dividing a whole number or a decimal by

   10 or a power of 10

When a whole number or a decimal is divided

by 10, 100, 1 000,... the decimal point is moved

1, 2, 3,,... places respectively to the left.

Worked Example

Calculate

(a) 5 ÷ 10,          (b) 270 ÷ 1 000.

Solution



Worked Example

Calculate

(a) 0.7 ÷ 10,            (c) 38.46 ÷ 1 000.    

(b) 231.4 ÷ 100,

Solution



ii - Dividing a whole number by a whole number

Worked Example

Calculate 35 ÷ 4

Solution


Therefore, 35 ÷ 4 = 8.75

Worked Example

Calculate 8 ÷ 3 and give your answer correct

to 4 decimal places.

Solution



Therefore, 8 ÷ 3 = 2. 66666...

                         = 2. 6667 ( 4 d.p )

iii - Dividing a decimal by a whole number

     and vice versa

1. Dividing a decimal by a whole number

    is equal sharing.

Worked Example

Find the value of each of the following.

(a) 38.4 ÷ 4

(b) 0.22 ÷ 22

Solution



2. When dividing a whole number by a decimal,

    convert the divisor to a whole number first by

    moving its decimal point to the right.

Worked Example

Find the value of each of the following.

(a) 6 ÷ 0.2

(b)  33
     0.11

Solution



iv - Dividing a decimal by a decimal

When dividing a decimal by a decimal, use the

idea of equivalent fractions to convert the divisor

to a whole number. Shift the decimal points the

same number of places in the dividend and divi-

sor to make the divisor a whole number.

Worked Example

Find the value of each of the following.

(a) 5.52 ÷ 0.6

(b) 0.072
       0.8

Solution



v - Dividing a decimal by a fraction and vice versa

When dividing a decximal by a fraction,

change the operation from division to

multiplication and invert the fraction at

the same time.

Worked Example

Find the value of each of the following.

(a) 3.72 ÷ 3
              4

Solution


D) Problem Solving involving
Division of Decimals


Worked Example

Akmal buys 6 tickets for a circus show

costing RM85.80. How much does each

ticket cost ?

Solution

1. Understand the problem

    Given information :

    Number of tickets bought = 6

    Cost of 6 tickets = RM 85.80

    Find : Cost of 1 tickets

2. Devise a plan

    Use division

3. Carry out the plan

    RM85.80 ÷ 6 = RM14.30

               

    Therefore, the cost of 1 ticket is RM14.30.

4. Check

          1 4 . 3 0
       x            6
          8 5 . 8 0

COMBINED OPERATIONS
OF +, -, x, ÷ OF DECIMALS


A) Combined Operations of Addition
and Subtraction


Worked Example

Find the value of each of the following.

(a) 7.41 + 6.35 - 10.02

(b) 8.41 - 31 + 2.07
                 4
Solution



B) Problem Solving involving Addition
 and Subtraction


Worked Example

A vessel weighing 0.98 kg contains 25.8

kg of rice. If 10.5 kg of the rice is used,

find the total mass of the vessel with the

remaining rice.

Solution

1. Understand the problem

    Given information :

    Mass of vessel = 0.98 kg

    Mass of rice = 25.8 kg

    Mass of rice used = 10.5 kg

    Find : Mass of the vessel and remaining rice

2. Devise a plan

    Perform addition followed by subtraction.

3. Carry out the plan

    0.98 + 25.8 - 10.5

    = 26.78 - 10.5

    = 16. 28

          0 . 9 6
    + 2 5 . 8__
       2 6 . 7 8
    -  1 0 . 5__
       1 6 . 2 8

    Therefore, the total mass of the vessel

    and remaining rice is 16.28 kg.

C) Combined Operations of
Multiplication and Division


Worked Example

Calculate each of the following.

(a) 4.6 x 0.8 ÷ 1.6  

Solution 
     


Worked Example

Calculate each of the following.

(a) 7 x 0 . 9 ÷ 4
                     5

Solution

(a) 7 x 0 . 9 ÷ 4
                     5

    = 6. 3 ÷ 4  ( Invert the fraction. )
                5

    = 6. 3 x 5
                4

    = 31. 5 ÷ 4

    = 7. 875

D) Problem Solving involving
Multiplication and Division


Worked Example

A shopkeeper buys 5 sacks of sugar

each weighing 64.2 kg. If he packs

the sugar into plastic bags of 11 kg
                                                 2
each, how many bags are required ?

Solution

1. Understand the problem

    Given information :

    Mass of a sack of sugar = 64.2 kg

    5 sacks of sugar are packed into

    plastic bags of 11 kg.

    Find : Number of plastic bags required

2. Devise a plan

    Perform multiplication followed by

    division.

3. Carry out the plan

    5 x 64. 2 ÷ 11
                         2

    = 5 x 64. 2 ÷ 3
                         2

    = 192. 6 ÷ 3     
                     2

    = 192. 6 x 2
                     3

    = 385. 2 ÷ 3

    = 128. 4

4. Check

    3 x 128. 4 = 192.6
    2

    192. 6 ÷ 3

    = 64. 2

Worked Example

Amy bought 20 eggs costing RM3. 80.

Lissa bought 35 of those eggs. How

much did Lissa pay ?

Solution
 
Cost of 20 eggs = RM3.80

Cost of 35 eggs

= 3.80 ÷ 20 x 35

= 0.19 x 35

= 6.65

Therefore, Lissa paid RM6.65.

E) Combined Operations of Addition,
Subtraction, Multiplication and Division


Worked Example

Calculate each of the following.

(a) 12. 5 - 0. 26 x 2. 6

(b) 3. 2 x 4 - 10. 4 ÷ 4

(c) 5. 12 - 2. 8 ÷ 1
                          4
Solution


     


    
F) Problem Solving involving Combined
Operations of Addition, Subtraction,
Multiplication and Division


Worked Example

Amir bought 260 eggs costing RM0.15

each. He sold all the eggs at RM0.18

each. Find his profit.

Solution

1. Understand the problem

    Given information :

    Number of eggs bought = 260

    Cost price of an egg = RM0.15

    Selling price of an egg = RM0.18

    Find : Profit made from 260 eggs

2. Devise a plan

    Use subtraction and multiplication.

3. Carry out the plan

     
    Therefore, Amir's profit was RM7.80.

4.Check

            2 6 0          2 6 0          3 6. 8 0
       x 0 . 1 8     x 0 . 1 5       -  3 9. 0 0
         2 0 8 0        1 3 0 0            7. 8 0
         2 6 0__       2 6 0_
        4 6. 8 0      3 9. 0 0

Worked Example


Haikal bought 7 pencils at RM0.80

each and 6 pens at RM1.30 each.

If he paid with a RM30 note, How  

much change did he receive ?

Solution

Cost of 7 pencils = 7 x 0.80

Cost of 6 pens = 6 x 1.30

Total cost of pencils and pens

= 7 x 0.80 + 6 x 1.30

Change received by Haikal

= 30 - ( 7 x 0.80 + 6 x 1.30 )

= 30 - ( 5.60 + 7.80 )

= 30 - 13.40

= 16.60

          0 . 8 0           1 . 3 0
       x         7         x        6
          5 . 6 0           7 . 8 0

Therefore, the change was RM16.60