Maths Form 1 Chapter 4 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 shadedarea 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 mixednumbers.
(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
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