fractions and decimals - overton grange maths ks44... · chapter fractions and decimals 4.1 what is...
TRANSCRIPT
64
4C H A P T E R
Fractions and decimals
4.1 What is a fraction?
�27� of this circle is shaded.
�27� is a fraction.
Write down the fraction of the shape that is shaded.
Solution 13 parts are shaded.So the top number (numerator) of the fraction is 3
The circle is divided into 5 equal parts.So the bottom number (denominator) of the fraction is 5
�35� of the shape is shaded.
There are 30 students in a class.17 of the students walk to school.Write down the fraction of the students that
a walk to school b do not walk to school.
Solution 2a There are 17 students out of 30 that walk to school.
The fraction of students that walk to school is �1370�
b 30 � 17 � 1313 students out of 30 do not walk to school.
The fraction of students that do not walk to school is �1330�
Example 2
Example 1
27The bottom number shows that the
circle is divided into 7 equal parts.
The top number shows that 2 parts of the circle are shaded.
The bottom number of the fractionis called the denominator.
The top number of the fractionis called the numerator.
65
4.1 What is a fraction? CHAPTER 4
Exercise 4A
In questions 1 to 8, write down the fraction of the shape that is shaded.
1 2 3 4
5 6 7 8
9 Write down the fraction of the shape that is
a shaded b unshaded.
10 Write down the fraction of the shape that is
a shaded b unshaded.
11 On the diagrams on the resource sheet shade in the fraction given next to each diagram.
a b c
12 There are 29 students in a class, 13 of the students are girls.What fraction of the class are girls?
13 In a family of five people, two people are left handed. What fraction of the family are
a left handed b not left handed?
14 There are two red, three blue and six black beads in a box.What fraction of the beads are
a blue b black?
15 There are 75 cars in a car park. 32 of the cars are white.Write down the fraction of the cars that are
a white b not white.
16 Lesley has one geography, three history and five science books.What fraction of her books are
a geography books b geography or history books?
17 Amy says that �13� of this flag is white.
Is Amy correct? Give a reason for your answer.
34
38
15
66
CHAPTER 4 Fractions and decimals
18 Lee chooses some tiles. He wants �35� of each tile he chooses to be blue.
Which of these tiles could Lee choose?A B C D E
4.2 Equivalent fractionsEquivalent fractions are fractions that are equal.
These rectangles areall the same size.
One half of each rectangle is shaded.
The diagrams show that
�12� is equal to �24� and �48�
�12�, �24� and �48� are equivalent fractions.
To find an equivalent fraction, multiply the numerator and the denominator by the samenumber.
Use the diagrams to write down
a fraction that is equivalent to �34�.
Solution 3Shade the same area of the second circle as is shaded in the first. 6 parts will be shaded.
This shows that
�68� is equivalent to �34�
34
68�
�2
�2
Example 3
12
24
48
12
24�
�2
�2
12
48�
�4
�4
67
4.2 Equivalent fractions CHAPTER 4
Complete �23� = �12�
Solution 4
Shade �13� of the circle.
Solution 5The circle has six equal parts so change �13� to sixths.
�13� is the same as �26� so shade two of the six parts of the circle.
Exercise 4B
In questions 1 to 15, copy the fractions and fill in the missing number to make thefractions equivalent.
1 �13� � �9� 2 �15� � �10� 3 �17� � �2� 4 �23� � �9�
5 �34� � �8� 6 �47� � �14� 7 �56� � �15� 8 �35� � �12�
9 �38� � �32� 10 �49� � �45� 11 �78� � �28� 12 �170� � �100�
13 �152� � �15� 14 �2
90� � �60� 15 �1
85� � �40�
In questions 16 to 21 shade the given fraction on the diagrams on the resource sheet.
16 17 18
19 20 21
45
710
58
35
34
16
13
26�
�2
�2
Example 5
23
812�
�4
�4
Example 4
�
4.3 Simplifying fractionsA fraction can be simplified if the numerator and denominator can be divided by the samenumber.This process is called cancelling.When a fraction cannot be simplified, it is in its simplest form or in its lowest terms.
Find the simplest form of the fractions a �150� b �13
80�
Solution 6a Divide both 5 and 10 by 5
�12� cannot be simplified.
The simplest form of �150� is �12�.
b Method 1Divide both 18 and 30 by 2
Then divide both 9 and 15 by 3
There is no number that will divide exactly into both 3 and 5 so the
simplest form of �1380� is �35�
Method 26 is the largest number that goes exactly into both 18 and 30 (in other words, 6 is the HCF of 18 and 30)If the HCF is used, then only one step is needed to simplify the fraction.
The simplest form of �1380� is �35�
Prateek has 24 toy cars. 10 of these cars are blue.What fraction of Prateek’s toy cars are blue? Give your fraction in its simplest form.
Solution 7Prateek has 10 blue cars out of 24 toy cars.
�1204� of the toy cars are blue.
Both 10 and 24 are even numbers so divide both numbers by 2
�1204� � �1
52�
�152� cannot be simplified further so �1
52� of the toy cars are blue.
Example 7
Example 6
68
CHAPTER 4 Fractions and decimals
510
12�
�5
�5
915
35�
�3
�3
1830
915�
�2
�2
1830
35�
�6
�6
1024
512�
�2
�2
69
4.4 Ordering fractions CHAPTER 4
Exercise 4C
In questions 1 to 15 write each fraction in its simplest form.
1 �36� 2 �155� 3 �68� 4 �2
71� 5 �14
00�
6 �2205� 7 �11
26� 8 �12
60� 9 �1
2050� 10 �39
00�
11 �2540� 12 �44
28� 13 �11
05
00� 14 �1
8200� 15 �2
7050�
In questions 16 to 20 write down the fraction of the shape that is shaded.Give each fraction in its simplest form.
16 17 18
19 20
In questions 21 to 25 give each fraction in its simplest form.
21 A class contains 30 students, 12 of these students are girls.Write down the fraction of the class that are girls.
22 Jack has 40 model farm animals. Eight of the animals are horses.Write down the fraction of the model farm animals that are horses.
23 There are four red, three blue and five yellow counters in a bag.Write down the fraction of the counters that are blue.
24 In a car park there are 20 silver, 16 blue, nine red and three green cars.Write down the fraction of the cars that are
a silver b blue c red or green
25 There are 30 cakes in a shop. Five of the cakes are chocolate.Write down the fraction of the cakes that are not chocolate.
4.4 Ordering fractionsHere are two rectangles which are the same size.
The second rectangle has more parts shaded than the first rectangle.
This shows that �170� is bigger than �1
30�
When fractions have the same denominator,you can compare the numerators to put the fractions in order.
�130� of this rectangle is shaded
�170� of this rectangle is shaded
70
CHAPTER 4 Fractions and decimals
Put the fractions �79�, �49�, �89�, and �29� in order of size.
Start with the smallest fraction.
Solution 8All the fractions have the same denominator so compare the numerators to put thefractions in order of size.
�29�, �49�, �79�, �89�
Ben shades �34� of a rectangle. Lucy shades �45� of an identical rectangle.
Who has shaded in more of their rectangle? Give a reason for your answer.
Solution 9Compare the fractions by writing them with a common denominator.
The denominators 4 and 5 both divide exactly into 20
Find a fraction equivalent to �34� that has a denominator of 20
Find a fraction equivalent to �45� that has a denominator of 20
�1260� is bigger than �12
50� so �45� is bigger than �34�
As �45� is bigger than �34�, Lucy shaded in more than Ben.
Which fraction is bigger �13� or �25�?
Solution 10The smallest number that the denominators 3 and 5 both divide exactly into is 15
�165� is bigger than �1
55�
So �25� is bigger than �13�
13
515�
�5
�5
25
615�
�3
�3
Example 10
Example 9
Example 8
34
1520�
�5
�5
45
1620�
�4
�4
Write the fractions �14�, �120� and �35� in order of size. Start with the smallest fraction.
Solution 11The smallest number that the denominators 4, 10 and 5 all divide exactly into is 20
Find an equivalent fraction for each of �14�, �120� and �35� with a denominator of 20
Starting with the smallest fraction, the order is �240�, �2
50�, �12
20�
that is �120�, �14�, �35�
Exercise 4D
In questions 1 to 10 write the fractions in order of size. Start with the smallest fraction.
1 �35�, �170� 2 �58�, �34� 3 �34�, �23� 4 �56�, �34� 5 �23�, �56�, �1
72� 6 �2
90�, �45�, �34�
7 �145�, �13�, �1
30� 8 �34�, �1
96�, �58� 9 �24
30�, �1
70�, �35�, �12
30� 10 �12�, �35�, �1
52�, �13
10�, �1
75�
11 Julie and Susan have identical chocolate bars.
Julie eats �34� of her chocolate bar. Susan eats �78� of her chocolate bar.
Who eats more chocolate? You must give a reason for your answer.
12 Ahmid says that �172� is bigger than �56� because 7 is bigger than 5
Is Ahmid correct? You must give a reason for your answer.
4.5 Reading and writing decimalsThe lengths of two pencils are measured.
The length of the red pencil is exactly 8 cm. This can also be written as 8.0 cm.
The blue pencil does not measure a whole number of centimetres.
Look at the diagram. Each centimetre is divided into ten equal parts called tenths of acentimetre, also known as millimetres.
A decimal point is used to separate the whole number of centimetres from the number oftenths of a centimetre.The length of the blue pencil is 9.3 cm.
14
520�
�5
�5
210
420�
�2
�2
35
1220�
�4
�4
Example 11
71
4.5 Reading and writing decimals CHAPTER 4
0 1 2 3 4 5 6 7 8 9 10 cm
0 1 2 3 4 5 6 7 8 9 10 cm
72
CHAPTER 4 Fractions and decimals
Write down the length of the key.
Solution 12The length of the key is 6 whole centimetres and 8 tenths of a centimetre.The length of the key is 6.8 cm.
Write down the weight of the parcel.
Solution 13The scale measures weight in kilograms.Each kilogram is divided into tenths.The parcel weighs 1.7 kg.
Exercise 4E
In questions 1 to 5 write down the length of each pencil.
1
2
3 4
5
In questions 6 to 8 write down the weight of each parcel.
6 7 8
1
2 3 4
6
5
0kg1
2 3 4
6
5
0kg
1 2
30kg
0 1 2 3 4 5 6cm0 1 2 3 4 5cm
0 1 2 3 4 5 6 7 8 9 10cm
0 1 2 3 4 5 6 7 8 9 10cm
1 2
30kg
Example 13
0 1 2 3 4 5 6 7 8 9 10 cm
Example 12
0 1 2 3 4 5 6 7 8 9 10cm 11 12 13 14 15
9 Write down the number that each arrow is pointing to on the scales.
a
b
In questions 10 to 12 write down the weight shown on each scale.
10 11 12
4.6 Understanding place valueThe decimal point separates the whole number part from the part that is less than 1
Look at the table. The first number on the right of the decimal point tells us how manytenths there are.
There can be more numbers after the decimal point.
The column headings tell us the place value of each figure.
This number in the table is read as ‘four hundred and thirty two point six nine five’.
The column headings tell us that
● the 4 has a value of four hundreds● the 3 has a value of three tens● the 2 has a value of two units● the 6 has a value of six tenths● the 9 has a value of nine hundredths● the 5 has a value of five thousandths
0
12
3
kg
0
1
2
3
4
5kg
0
kg1
2
3
45
6
7
8
29 30 31 32 33
F G H I
0 1 2 3
A B C D
73
4.6 Understanding place value CHAPTER 4
4 3 2 . 6 9 5
thou
sand
shu
ndre
dste
ns
unit
s
tent
hshu
ndre
dths
thou
sand
ths
74
CHAPTER 4 Fractions and decimals
Write down the value of the 2 in the number 34.72
Solution 14The 2 has a value of two hundredths.
Write down the number that each arrow is pointing to on the scale.
Solution 15
Exercise 4F
1 Write down the value of the 6 in each number.
2 Write down the value of the 4 in each number.a 56.43 b 4521.8 c 98.243 d 0.814 e 342.1
3 Write down the value of the 9 in each number.a 3.19 b 792.3 c 0.039 d 79.3 e 1.9
4 Write down the number that each arrow is pointing to on the scales.a b
c d
0.4 0.5 0.6
M N P Q
24.5 24.6 24.7 24.8
I J K L
10.1 10.2 10.3
E F G H
6.3 6.4 6.5 6.6
A B C D
2.6
2.64 2.75 2.79 2.81
2.7 2.8
2.6 2.7 2.8
Example 15
Example 14
a 6 5 2 . 8 1
b 8 7 . 6 3 4
c 1 3 5 . 4 2 6
d 7 5 8 9 . 0 6
e 6 . 5 2
thou
sand
shu
ndre
dste
ns
unit
s
tent
hshu
ndre
dths
thou
sand
ths
3 4 . 7 2
tens
unit
s
tent
hshu
ndre
dths
75
4.7 Ordering decimals CHAPTER 4
4.7 Ordering decimals
Five boys took part in a long jump competition.The table shows the distance each boy jumped.
To decide who jumped the furthest,put the distances in order.To put decimals in order,look at the place values.
Use column headings to show the valueof each figure.
Write a 0 in each empty box.
In order of size, the five numbers are 4.2, 4.3, 4.33, 4.39, 4.4.
Putting the boys’ distances in order gives:
So Daneep jumped the furthest.
Write the numbers 7.53, 7.5, 7.6, 7.65, 7.56 in order of size starting with the biggest.
Solution 16Use column headings to show the value of each figure.Write a 0 in each empty box.As all the numbers have a 7 in the units column, look at the tenths.7.6 and 7.65 both have a 6 in the tenths column but 7.65 has a 5 in the hundredths column so 7.65 is bigger than 7.607.53, 7.50 and 7.56 all have a 5 in the tenths column.To order these numbers use the hundredths column.Starting with the biggest, the order of the numbers is
7.65, 7.6, 7.56, 7.53, 7.5
Example 16
Adam 4.3 m
Brian 4.2 m
Colin 4.39 m
Daneep 4.4 m
Elliot 4.33 m
4 . 3 0
4 . 3 9
4 . 3 3
unit
s
tent
hshu
ndre
dths
4 . 3 0
4 . 2 0
4 . 3 9
4 . 4 0
4 . 3 3
unit
s
tent
hshu
ndre
dths
Three of the numbers have a 3 in the tenths column.To order these numbers use the hundredths column.
The numbers in the hundredths column are 0, 9 and 3In order, these are 0, 3, 9
So the three numbers in order are 4.30, 4.33, 4.39.That is 4.3, 4.33, 4.39
7 . 5 3
7 . 5 0
7 . 6 0
7 . 6 5
7 . 5 6
unit
s
tent
hshu
ndre
dths
All the numbers have a 4 in the units column.
In the tenths column, the smallest number is2 and the largest number is 4
So, 4.2 is the smallest number and 4.4 is thelargest number.
Brian 4.2 m
Adam 4.3 m
Elliot 4.33 m
Colin 4.39 m
Daneep 4.4 m
76
CHAPTER 4 Fractions and decimals
Exercise 4G
In questions 1 to 12 write the numbers in order of size.Start with the smallest number each time.
1 2
3 6.76, 6.66, 6.67 4 8.11, 8, 8.1, 8.01
5 0.09, 0.9, 0.92, 0.2 6 73.24, 73.2, 73.42, 73.4
7 2.314, 2.413, 2.134, 2.341, 2.431 8 0.373, 0.37, 0.73, 0.333, 0.733
9 15.8, 15.38, 15.3, 15.833, 15.803 10 0.045, 0.05, 0.0545, 0.055, 0.0454
11 6.067, 6.006, 6.07, 6.06, 6.077, 6.076 12 8.092, 8.9, 8.02, 8.09, 8.2, 8.29, 8.92
4.8 Converting decimals to fractionsUsing place values, some decimals can be converted to fractions
0.7 � �170� 0.06 � �1
600�
0.76 � �170� � �1
600�
� �17000� � �1
600�
� �17060�
To convert decimals to fractions, use the place values of the figures.
14 The table shows the time, in seconds,in which five runners ran 100 mWrite down the order in which therunners finished the race.
13 The table shows the heights, in metres,of five children.Write down the children in order of height.Start with the tallest child.
Linford 10.2
Dwain 10.02
Roger 10.23
Steve 10.12
Maurice 10.21
Linda 1.34m
Anthony 1.4m
Chris 1.43m
Ian 1.33m
Julie 1.3m
6 8 . 3 8 3
6 8 . 3 8 7
6 8 . 3 7
tens
unit
s
tent
hshu
ndre
dths
thou
sand
ths
5 . 7 7
5 . 0 7
5 . 7
unit
s
tent
hshu
ndre
dths
77
4.8 Converting decimals to fractions CHAPTER 4
Write 0.13 as a fraction.
Solution 17
0.13 � �11030�
The heading of the last column with a figure in it gives the denominator.
Write 0.024 as a fraction.Give your fraction in its simplest form.
Solution 18
0.024 � �120
400�
The heading of the last column with a figure in it is thousandths, so the denominator is 1000
�120
400� � �5
1020� � �2
650� � �1
325�
Write 3.7 as a fraction.
Solution 19
3.7 � 3�170�
The 3 is the whole number part, the .7 is �170�
Exercise 4H
1 Write the decimalsas fractions
Example 19
Example 18
Example 17
0 . 1 3
unit
s
tent
hshu
ndre
dths
thou
sand
ths
0 . 0 2 4
unit
s
tent
hshu
ndre
dths
thou
sand
ths
3 . 7
unit
s
tent
hshu
ndre
dths
thou
sand
ths
a 0 . 3
b 0 . 0 7
c 0 . 1 9
d 0 . 2 5 3
e 0 . 0 8 9
unit
s
tent
hshu
ndre
dths
thou
sand
ths
78
CHAPTER 4 Fractions and decimals
In questions 2 to 15 write each of the decimals as a fraction in its simplest form.
2 0.7 3 0.14 4 0.123 5 0.08 6 0.093 7 0.006 8 0.72
9 0.2 10 0.242 11 2.5 12 25.06 13 12.8 14 6.17 15 2.84
4.9 Converting fractions to decimalsAll fractions can be written as decimals.
The fractions and decimals in the table are ones that are used frequently and should be learnt.
Other fractions can be changed into decimals.
Write the following fractions as decimals
a �190� b �1
2030�
Solution 20
a �190� � 0.9 b �1
2030� � 0.23
Write the following fractions as decimals. a �25� b �1215�
Solution 21Method 1 – using equivalent fractions
a �25� � �140� � 0.4 b �12
15� � �1
4040� � 0.44
Method 2 – using a calculator
a �25� means 2 � 5 b �1215� means 11 � 25
Using a calculator, Using a calculator,
2 5 0.4 11 25 0.44
�25� � 0.4 �1215�
� 0.44
Short division is suitable for changing �25� to a decimal because the denominator issmall.
�25� means 2 � 5
�25� � 0.4
0 . 45 �2 .20
����
Example 21
Example 20
Decimal Fraction
0.01 �1100�
0.1 �110�
0.25 �14�
0.5 �12�
0.75 �34�
2.0 is the same as 2 so divide 2.0 by 5
5 does not divide into 2 so put down a zero and carry 2
5 divides into 20 four times
�
�
79
4.9 Converting fractions to decimals CHAPTER 4
Not all fractions can be written as exact decimals.
�13� � 1 � 3 � 0.33333….
In this decimal, the 3 keeps repeating.
When a decimal has repeating figures, it is called a recurring decimal.
To show that a figure recurs, put a dot above the figure.
So 0.33333… is written as 0.3.
and �13� � 0.3.
Sometimes, more than one figure recurs,
�131� � 3 � 11 � 0.272727….
Put a dot above each recurring figure.
So �131� � 0.2
.7.
Write the following fractions as decimals
a �79� b �1232� c �57�
Solution 22
a �79� means 7 � 9
Using a calculator, 7 9
0.777777… � 0.7.
b �1232� means 13 � 22
Using a calculator, 13 22
0.5909090… � 0.59.0.
c �57� means 5 � 7
Using a calculator, 5 7
0.714285714… � 0.7.14285
.
Example 22
Work out 7 � 9 on a calculator.
The 7 recurs so put a dot above the 7
Work out 13 � 22 on a calculator.
The 90 recurs so put a dot above each of these figures.
Do not put a dot above the 5, as it does not recur.
Work out 5 � 7 on a calculator.
A group of six figures recurs.There isn’t enough room to see all the figures recurring butyou can see that the same pattern of figures is starting again.
When more than two figures recur, just two dots are used,one above the first figure in the recurring group and oneabove the last figure in the group.
�
�
�
�
�
�
Exercise 4I
Do not use a calculator in questions 1 to 4
1 Write the following fractions as decimals
a �190� b �1
3070� c �1
300� d �1
506010� e �10
800�
2 Write the following as equivalent fractions and then as decimals
a �45� � �10� b �570� � �100� c �2
85� � �100� d �5
900� � �1000� e �2
30� � �100�
3 Write down the following fractions as decimals
a �12� b �14� c �110� d �1
100� e �34�
4 Use short division to change these fractions to decimals
a �35� b �38�
5 Use a calculator to change these fractions to decimals
a �18� b �490� c �22
35� d �78� e �11
16�
6 Use a calculator to change these fractions to decimals
a �23� b �89� c �151� d �1
72� e �17�
Chapter summary
80
CHAPTER 4 Fractions and decimals
You should know and be able to use these facts
The top number of a fraction is called the numerator.
The bottom number of a fraction is called the denominator.
Equivalent fractions are fractions that are equal.
A fraction can be simplified if the numerator and denominator can both be dividedby the same number. This process is called cancelling.
A fraction that cannot be simplified is in its simplest form.
To compare fractions, first write them with the same denominator.
In a decimal the decimal point separates the whole number part from the part that isless than one.
Decimals can be put in order by looking at the place value of each number. Firstcompare the whole number part, then the tenths, then the hundredths, then thethousandths.
Decimals can be converted to fractions by using their place value.
Fractions can be converted to decimals by using equivalent fractions or division.
Some fractions convert to recurring decimals.
�
�
�
�
�
�
�
�
�
�
�
�
�
81
Chapter 4 review questions CHAPTER 4
Chapter 4 review questions1 Write down the fraction of each shape that is shaded.
Give each fraction in its simplest form.
a b c d
2 Write down the fraction of this shape that is shaded.Give your fraction in its simplest form.
3 a Write down the fraction of this shape that is shaded.Write your fraction in its simplest form.
b Shade �23� of this shape
on the resource sheet.
(1387 June 2003)
4 There are 60 cars in a car park. 35 of the cars are silver.Write down the fraction of cars in the car park that are silver.Give your fraction in its simplest possible form.
5 Copy the fractions and fill in the missing number to make a pair of equivalent fractions.
a �45� � �15� b �38� � �6� c �190� � �50� d �58� � �20�
6 a Shade �14� of this shape.
b Copy the fractions and write a number on the dotted line so that the two fractions are equivalent
�14� � �1…2�
7 Give each fraction in its simplest form
a �48� b �155� c �14
00� d �1
7050�
8 Write down the reading on each of these scales
a b c
1
20 kg
1
20 kg
A B C
3.6 3.7 3.8
�
82
CHAPTER 4 Fractions and decimals
9 Write down the value of the 6 in each of the following numbers.a 56.3 b 9.62 c 0.916 d 45.16
10 Five girls each threw a ball in a competition.The table shows the distance, in metres,each girl threw the ball.Write down the distances in order of size.Start with the longest distance.
11 Write each fraction as a decimal. You may not use your calculator.
a �170� b �1
900� c �1
4030� d �1
60
700� e �25� f �2
70�
12 Write 0.45 as a fraction. Give your fraction in its simplest form.
13 Write 0.028 as a fraction. Give your fraction in its simplest form.
14 Here are two fractions �35� and �23�
Explain which is the larger fraction.You may use the grids to help with your explanation.
(1387 June 2003)
15 Change �78� to a decimal.
16 Amanda and Mary each had the same size of chocolate bar.
Amanda ate �23� of her bar of chocolate. Mary ate �58� of her bar of chocolate. Work out
which girl had eaten the most chocolate. You must give a reason for your answer.
17 Write these five numbers in order of size. Start with the smallest number
2.5, 0.5, 0.52, 2.2, 0.25 (1388 Jan 2003)
18 �172�, �56�, �23�
Write these fractions in order of size. Start with the smallest fraction.(1388 Mar 2002)
19 Write these five fractions in order of size. Start with the smallest fraction.
�25�, �13�, �12�, �38�, �141�
20 a Write 0.35 as a fraction. Give your answer in its simplest form.
b Write �38� as a decimal. (1387 June 2002)
21 Use your calculator to write each fraction as a decimal.
a �2430� b �56� c �11
16� d �1
41� e �9
70�
Anna 20.4
Bianca 19.96
Chaya 19.9
Debbie 20.34
Eloise 20.04
�