New Mathematics Connection Textbook P.1-P.6 sample chapter

The Role of Zero What is 10,000 more
than 3,592,643?

Millions Hundred Ten Thousands Hundreds Tens Ones

Thousands Thousands

We add 1 ten thousand to the ten thousands place.

9 ten thousands + 1 ten thousand = 10 ten thousands

Millions Hundred Ten Thousands Hundreds Tens Ones
Thousands Thousands

The digit in the ten thousands place is now zero, because ten ten thousands is
one hundred thousand and it is added to the hundred thousands place.

5 hundred thousands + 1 hundred thousand = 6 hundred thousands
So, 10,000 more than 3,592,643 is 3,602,643.

6 Chapter 1 Numbers

Example 1

Find the place value of the underlined digits.

(a) 547,620 (b) 617,309

Solution: Solution:

The place value of 4 is 40,000. The place value of 6 is 600,000.

Example 2

Write the number names in Arabic and Thai numerals.

(a) Five hundred and seventeen (b) One hundred and sixty-five thousand,
thousand and forty-nine two hundred and thirty-eight

Solution: Solution:

Arabic: 517,049 Arabic: 165,238
Thai : ,
Thai : ,

Fun Facts

Bangkok has a population of about 6,500,000.

Try It

1. Write the place value of the underlined digits.

(a) 346,203 (b) 982,081

(c) 210,000 (d) 7,870,624

(e) 9,103,759 (f) 4,398,542

New Mathematics Connection 7

2. Identify the number represented by the bead counter.
(a)

Millions Hundred Ten Thousands Hundreds Tens Ones

Thousands Thousands

In Arabic numerals: In Thai numerals:
In words:
(b)

Hundred Ten Thousands Hundreds Tens Ones

Thousands Thousands

In Arabic numerals: In Thai numerals:
In words:

(c)

Hundred Ten Thousands Hundreds Tens Ones

Thousands Thousands

In Arabic numerals: In Thai numerals:
In words:

8 Chapter 1 Numbers

3. Fill in the blanks with numbers.

(a) 10 tens = ones

(b) 10 ten thousands = thousands

(c) 1 million = hundred thousands

4. Write the number names in Arabic numerals.
(a) Eight hundred and fifty-five thousand, six hundred and ninety-four.

(b) Two hundred and seventy thousand, eight hundred and three.

(c) Four million, seven hundred and sixteen thousand and thirty.

(d) Nine million, six hundred and forty-nine thousand, five hundred and
twelve.

5. Write the numbers in words.
(a) 1,701,940
(b) 6,920,027
(c) 8,283,005
(d) 7,473,102

6. Write the following number names in numerals:

(a) Seven million, six hundred and five thousand, seven hundred and two.

In Arabic: In Thai:

(b) Eight hundred and sixty-seven thousand and twenty.

In Arabic: In Thai:

(c) One million, one hundred and forty-three thousand, eight hundred and three.

In Arabic: In Thai:

(d) Two hundred and ninety-eight thousand, four hundred and fifteen.

In Arabic: In Thai:

Go to Exercise 1

New Mathematics Connection 9

B Expanded Form

Now, let us learn to write numbers in the expanded form.
Example 3

Expand 1,653,264.
Solution:

Place the digits in the place value chart as shown below.

Millions Hundred Ten Thousands Hundreds Tens Ones
1
thousands thousands

6 5 3 2 64

1,653,264 = 1 million + 6 hundred thousands + 5 ten thousands + 3 thousands
+ 2 hundreds + 6 tens + 4 ones.

= (1 × 1,000,000) + (6 × 100,000) + (5 × 10,000) + (3 × 1,000)
+ (2 × 100) + (6 × 10) + (4 × 1)

= 1,000,000 + 600,000 + 50,000 + 3,000 + 200 + 60 + 4

Look at the expanded form of the following numbers:
(a) 256,432 = 200,000 + 50,000 + 6,000 + 400 + 30 + 2
(b) 6,836,452 = 6,000,000 + 800,000 + 30,000 + 6,000 + 400 + 50 + 2

Try It

1. Write the expanded form of the following number:
139,645 = 1 hundred thousand + 3 ten thousands
+ 9 thousands + 6 hundreds + 4 tens + 5 ones
= (1 × 100,000) + (3 × 10,000) + (9 × 1,000)
+ (6 × 100) + (4 × 10) + (5 × 1)
=++
+++

10 Chapter 1 Numbers

2. Write the numerals for the numbers in the expanded form.
(a) 800,000 + 50,000 + 0 + 400 + 70 + 5 =
(b) 8,000,000 + 700,000 + 60,000 + 9,000 + 300 + 20 + 4 =
(c) 9,000,000 + 900,000 + 40,000 + 5,000 + 0 + 80 + 6 =

3. Write the missing numbers. + 800 + 30 +
(a) 540,830 = 500,000 + 40,000 +

(b) 851,099 = 800,000 + + 1,000 + 0 + +9
+ 10 + 3
(c) 9,708,613 = + 700,000 + 0 + 8,000 +

4. Expand the following:
(a) 230,945
(b) 4,694,321

C Comparing and Ordering Go to Exercise 2

Steps for Comparing Two Numbers Recall what we
learnt earlier.

Step 1
• Check the number of digits.
• The number with more digits is the greater number.

Step 2
• If the number of digits is equal, start comparing the digits on the left-most.
• The number with the bigger digit is the greater number.

Step 3

• If the digits on the left-most are the same, we compare the next digits to the
right and keep doing this until the digits are different.

New Mathematics Connection 11

Comparing
The examples 4 and 5 teach us how to compare numbers.

Example 4
Compare 183,715 and 1,874,325 using the > or < sign.

Solution:

Millions Hundred Ten Thousands Hundreds Tens Ones

thousands thousands

1 8 3 7 15

1 8 7 4 3 25

The first number is a 6-digit number, while the second number is a 7-digit number.
So, 183,715 is less than 1,874,325.
We write: 183,715 < 1,874,325

Example 5
Compare the numbers 756,683 and 756,983 using the > or < sign.

Solution:

Hundred Ten Thousands Hundreds Tens Ones

thousands thousands

7 5 6 6 83

7 5 6 9 83

The first three digits are the same. So we compare the digits in the hundreds place.
In the hundreds place, 6 is less than 9.

So, 756,683 is less than 756,983.

We write: 756,683 < 756,983

12 Chapter 1 Numbers

Ordering ‘Increasing’ means
‘small to big’.
There are two types of ordering:
1. Increasing order ‘Decreasing’ means
2. Decreasing order ‘big to small’.

Let us look at the pattern below.

1,000,000 10,000Decreasing

Increasi1ng,0o0r0de1r0,000100,000 100,000 1,000

order

100 100

10 10

Example 6

Arrange the numbers in increasing order.
750,632; 44,526; 774,982

Solution:

Hundred Ten Thousands Hundreds Tens Ones

thousands thousands

7 5 0 6 32

4 4 5 26

7 7 4 9 82

Look at the hundred thousands column.There are zero hundred thousands in the
number 44,526.Therefore, 44,526 is the smallest number.

Now, consider the other two numbers.

Hundred Ten Thousands Hundreds Tens Ones

thousands thousands

7 5 0 6 32

7 7 4 9 82

The two numbers have the same digits in the hundred thousands column.

New Mathematics Connection 13

So, we compare the digits in the ten thousands column.
7 is greater than 5.
So, 774,982 is greater than 750,632.
Arranging the numbers in decreasing order, we have
774,982; 750,632; 44,526
Using the sign, we write: 774,982 > 750,632 > 44,526
In increasing order, we write: 44,526; 750,632; 774,982

Try It

1. Compare the numbers using the > or < sign.

(a) 406,250 405,260 (b) 6,918,790 6,719,890
4,267,201
(c) 8,253,420 8,253,425 (d) 4,276,672
5,467,226
(e) 99,998 100,000 100,000

2. Compare the numbers using the = or ≠ sign.

(a) 532,645 52,645 (b) 5,467,226

(c) 9,151,515 9,155,151 (d) 10,009

(e) 7,654,203 7,654,203

3. Arrange the numbers in increasing order.
(a) 531,260; 531,461; 421,361
(b) 465,216; 465,562; 465,262
(c) 646, 216; 670,262; 646,566
(d) 238,519; 238,520; 238,511
(e) 100,001; 100,010; 100,100

14 Chapter 1 Numbers

4. Arrange the numbers in decreasing order.
(a) 538,103; 853,130; 358,310
(b) 233,124; 235,421; 234,641
(c) 210,359; 310,358; 210,360
(d) 271,012; 217,210; 217,970
(e) 609,312; 906,312; 706,412

Go to Exercise 3

D Counting Forward and Counting Backward

Counting Forward
Look at these numbers.
(a) 605,125; 605,150; 605,175; 605,200;…

The numbers increase by 25. Hence, we count forward by 25.
(b) 140,650; 140,700; 140,750; 140,800;…

The numbers increase by 50. Hence, we count forward by 50.

Example 7
Find the next number.
72,000; 73,000; 74,000;

Solution:
We see that the numbers are in increasing order.
Let us find the difference between numbers.

New Mathematics Connection 15

The difference between the first and 73,000
the second number is 1,000. 72,000 –
1,000

The difference between the second 74,000
and the third number is 1,000. 73,000 –
1,000

If the difference is the same, the numbers increase by equal numbers.

The next number is found by adding 1,000 to the preceding number.
74,000 + 1,000 = 75,000
The next number is 75,000.

Counting Backward

Look at these numbers.

(a) 91,350; 91,300; 91,250; 91,200;…
Here, we count backward by 50.

(b) 130,275; 130,250; 130,225; 130,200;…
Here, we count backward by 25.
Now, let us learn to find the missing numbers by counting backward.

16 Chapter 1 Numbers

Example 8 66,288
Find the next number. 66,188 –
66,288; 66,188; 66,088;
100
Solution:
The numbers are decreasing. 66,188
66,088 –
The difference between the first and
the second number is 100. 100

The difference between the second
and the third number is 100.

If the difference is the same, the numbers decrease by equal numbers.

The next number is 66,088 – 100 = 65,988.

Try It

1. Find the next number by counting forward.
(a) 578,790; 578,890; 578,990;
(b) 173,700; 183,700; 193,700;
(c) 209,805; 214,805; 219,805;
(d) 1,650,205; 1,652,205; 1,654,205;

New Mathematics Connection 17

2. Find the next number by counting backward.

(a) 108,888; 107,888; 106,888;

(b) 835,900; 635,900; 435,900;

(c) 855,987; 845,987; 835,987;

(d) 1,230,075; 1,230,050; 1,230,025;

3. Given the number patterns, complete the sentences.

(a) 365,012; 385,012; 405,012;…

The numbers by .

(b) 68,451; 63,451; 58,451;…

The numbers by .

(c) 90,258; 100,258; 110,258;…

The numbers by .

(d) 23,159; 23,059; 22,959;…

The numbers by .

Go to Exercise 4

18 Chapter 1 Numbers

Contents

1 Fractions 1
3
CHAPTER A Types of Fractions 8
B Fractions and Counting Numbers 9
C Improper Fractions and Mixed Numbers
D Equivalent Fractions and Simplest 12
16
Form of Fractions
E Comparing and Ordering Fractions 18

2 Fraction Operations 22
26
CHAPTER A Adding and Subtracting Multiple 28
Denominator Fractions 33

B Multiplying Fractions 35
C Dividing Fractions 37
D Solving Word Problems on Fractions 39
41
3 Decimals

CHAPTER A Knowing up to Two-Position Decimals
B Comparing and Ordering Decimals
C Converting Decimals

4 Decimal Operations 43
45
CHAPTER A Addition and Subtraction of Decimals 47
B Multiplication of Decimals 49
C Mixed Problems 51
D Solving Word Problems on Decimals
54
5 Data Analysis and Probability 60

CHAPTER A Collection and Classification of Data 63
B Drawing Bar Graphs by Reducing 66
67
the Distance on the Number Line 70
C Comparison of Bar Graphs
D Probability of Events 71
E Reading and Writing Line Graph
72
6 Rule of Three 74
78
CHAPTER A Understanding Mathematical Rule:
Rule of Three 79

B Solution of an Equation with One Unknown
C Properties of Equality
D Solving Word Problems Using an Equation

in One Variable

7 Percentage 80
81
CHAPTER A Knowing Percents 82
B Converting Percents 84
C Solving Word Problems on Percentage 89
D Discount as Percent 90
E Profit and Loss
92
8 Parallel Lines and Angles 94

CHAPTER A Parallel Lines 97
B Drawing a Parallel Line passing through 101
103
a Point not on the Line
C Measuring Angles 105
D Drawing Angles 107
108
9 Quadrilaterals and Circles 110
115
CHAPTER A Drawing Squares and Rectangles
B Perimeter and Area of Quadrilaterals
C Solving Word Problems
D Components of a Circle

10 Volume of Solids 117
118
CHAPTER A Three-Dimensional Solids 119
B Measuring Volume and Capacity 121
C Volume Conversion
D Finding Volume or Capacity of Cuboids 124

Using Formula

1 Fractions

Objectives

◆ To identify the types of fractions;
◆ To describe the meaning of mixed numbers;
◆ To tell the meaning of the equivalence of fractions and counting numbers;
◆ To express improper fractions as mixed numbers and vice versa;
◆ To write equivalent fractions and simplest form of fractions;
◆ To compare and order fractions;
◆ To apply knowledge to daily life.

Getting Started Fraction is a part
of the whole.
What is the meaning of fraction?

Can you give me
an example?

Good. Now, let us discuss the types of fractions, I ate half of the cake.
mixed numbers and equivalent fractions.

New Mathematics Connection 1

Review

Let us take a quick review on fractions.
Fraction is a part of the whole.

Example 1

There are 4 parts with 1 part shaded. So, the shaded
part of the fraction is 1 .

4

Read and Write a Fraction

There are 6 parts with 3 shaded parts. So, the shaded
part of the fraction is 3 .

6
We write it as 3 and read it as three sixths or three out of six.

6

Example 2

Read and write the proper fractions given below.

Read it as five eighths Remember
5 numerator
5 8 denominator

Write it as 8

(a)
Read it as
Write it as

(b)
Read it as

Write it as

2 Chapter 1 Fractions

(c)
Read it as
Write it as

(d)
Read it as
Write it as

A Types of Fractions

Understanding the Meaning of Proper FractionsMethod
A fraction whose numerator is less than the denominator is called a proper fraction.

Example 3

Read it as one fourth three eighths two sixths one half

numerator 1 3 2 1
Write it as 8 6 2

denominator 4

New Mathematics Connection 3

Understanding the Meaning of Improper Fractions

A fraction whose numerator is greater than the denominator is called an improper
fraction.
The fractions 5 , 6 , 7 are improper fractions.

444

Picture

Read five fourths six fourths seven fourths
and
write 5 6 7
4 4 4
Read it as

Write it as

Understanding the Meaning of Mixed Numbers
A mixed number is made up of a whole number and a proper fraction.

Example 4

1+1+1<
4

2 + 1 =21
1 44
2 can be read as two and one fourth.
4

4 Chapter 1 Fractions

Example 5

4 oranges 1
orange
1
So, we read it as four and a half and write it as 4 . 2

12
Therefore, 4 is a mixed number.

2

Try It

1. Circle all the proper fractions. (There can be more than one answer.)

(a) 1 398 (b) 30 6 78
2 2 7 15 10 5 9 11
574
(c) 3 459 (d) 20 2 17 9
4 14 12 16 17 7 8 13
12 20 11
(e) 7 (f) 4 17 13 18
3 9 12 10 17

2. Circle all the improper fractions. (There can be more than one answer.)

(a) 3 15 9 3 (b) 13 14 15 21
5 18 25 24
4 7 5 17
(d) 19 10 23 12
(c) 11 45 42 13 4 5 8 17
7 41 49 27
(f) 25 14 7 16
(e) 19 9 5 1 30 12 3 20
30 7 2 8

New Mathematics Connection 5

3. Circle all the mixed numbers.

(a) 7 2 1 1 9 (b) 3 1 2 11 1 2
9 73 12
15 3 12 12 13 1 2
(d) 12 1 6
(c) 17 7 4 3 51 6 6 12 7
2 16 12 7 6 53
(f) 3 1 77 11
(e) 1 1 7 2 1 17 7 34 4 28
12 12 49
(h) 2 3 57 8
(g) 4 2 5 6 8 1 4 9
56 7 9

4. Write the mixed fractions.

1 3 1
(a) 3 + (b) 4 + (c) 5 +

2 4 7
4
3 6 (f) 6 +
(d) 9 + (e) 13 + 5
9
8 17 (i) 8 +
10
7 8
(g) 6 + (h) 7 + 7
(c) 4
8 9
9
5. Write the mixed fractions in words. 1
(f) 3
1 1 4
(a) 7 (b) 6 2
(i) 5
8 3 3

1 4
(d) 40 (e) 3

4 5

3 5
(g) 7 (h) 4

5 6

6 Chapter 1 Fractions

6. Group the fractions in the correct boxes.

1 89 6 5 7
3 10 16 27 24

7 1 6 7 1 7
3 3 17
4 4 27
1 30
10 6 17 1
26 21 6 9
1 1 1 11
1 3 4 5 12 2 15
7 5
7 51
20 2
16 9 1 17 4
24 7 46 9 40

22 23 15 8
16
107 4 1 21 21
11 9
31

Mixed fractions Proper fractions Improper fractions

Go to Exercise 1

New Mathematics Connection 7

B Fractions and Counting Numbers

Representing Counting Numbers as Fractions

We can represent the counting numbers as an improper fraction.

Example 6 Example 7

2 8 can be written as 8 . (8 ÷ 1 = 8)
2 can be written as . (2 ÷ 1 = 2) 1

1

Representing Fractions as Counting Numbers

If the numerator is divisible by the denominator then the fraction can be represented
as a whole number or a counting number.

Example 8 Example 9

42 14 21
= = 2 (4 ÷ 2 = 2 and 2 ÷ 1 = 2) = 7 or = 7

21 23

Try It

1. Express counting numbers as fractions with denominators not equal to 1.

(a) 13 (b) 22 (c) 17 (d) 29

(e) 15 (f) 12 (g) 16 (h) 20

2. Express fractions as counting numbers.

(a) 21 (b) 12 40 24
7 4 (c) (d)

5 8
81
55 56 64 (h)
(e) (f) (g) 9

11 7 16

Go to Exercise 2

8 Chapter 1 Fractions

C Improper Fractions and Mixed Numbers

Expressing Improper Fractions as Mixed Numbers

To express an improper fraction as a mixed number, the following steps are followed:

Divide the numerator by the denominator.

Write the quotient as the whole number.

Then, write down the remainder as the numerator.

For example:

3
is an improper fraction. Express this as a mixed number.

2

Divide the numerator 3 by the denominator 2. Write the remainder as
1 the numerator, that is 21.

2 3 Quotient = 1
2 Remainder = 1
31
1 Remainder Answer: = 1
22

Example 10
5

Suppose the fraction is .
4

5 11 (Mixed number)
=1+ =1

4 44
51
So, = 1
44

51
Answer: = 1

44

New Mathematics Connection 9

Expressing Mixed Numbers as Improper Fractions
Consider the following figure:

1
The above figure represents the fraction 3 .

3
1
There are 10 shaded parts in 3 .
3

11 or 3 1 = (3 × 3) + 1
3 =3+ 33
10
33 =
9 1 10 3

=+= 10
33 3 Answer:
10
3
Answer:
3

Example 11

1
The figure shown above represents the fraction 2 .

4
1

The total number of parts that are shaded is 9 and represents 2 .
4

10 Chapter 1 Fractions

21 =2+ 1 or 2 1 = (2 × 4) + 1
44 44
=8+1=9
444 =9
9 4
9
Answer:
4 Answer:
4
Try It

1. Express improper fractions as mixed numbers.

(a) 3 (b) 54 (c) 6
2 12 4
111
75 5
(d) (e) (f)
9
13 2 46

24 246 (i)
(g) (h) 8
61
13 12
(l)
10 46 12
(j) (k)

3 9

2. Express mixed numbers as improper fractions.

(a) 2 1 (b) 4 1 (c) 5 1
6 3 4
1
1 1
(d) 7 (e) 3 (f) 5
3
2 5 1

1 1 (i) 8
(g) 6 (h) 7 3
1
2 3
(l) 6
1 1 4
(j) 4 (k) 5

4 2

Go to Exercise 3

New Mathematics Connection 11

D Equivalent Fractions and Simplest Form of Fractions

Equivalent Fractions
Equivalent fractions have different numerators and denominators but they have the
same value.

Example 12

This figure represents the fraction 1 .
2

When multiplied with 2 in the numerator and denominator, it gives 2 .
4

1×2=2
224

When multiplied with 3 in the numerator and denominator, it gives 3 .
6

1×3=3
236

So, 1 = 2 = 3
246

Such fractions are called equivalent fractions.
There are two possibilities to get the equivalent fractions.
1. We can get the equivalent fractions by multiplying the numerator and the

denominator of a fraction by the same number, except zero.
Example 13
1×6 = 6
2 × 6 12

12 Chapter 1 Fractions

2. We can get the equivalent fraction by dividing the numerator and the
denominator of a fraction by the same number.

Example 14

6÷3 = 2
12 ÷ 3 4

16 2 1
Whether it is , or , they all have the same value and all are equal to .
2 12 4 2

1
2
6
12
2
4

Simplest Form of a Fraction

Any fraction that cannot be divided by any number besides 1 without having a
remainder is called a simple fraction.

9 is divided by 3 .
12 3
9 33

÷=
12 3 4

We can see that there are no other numbers that are greater than 1 that can divide
3

both 3 and 4 without having any remainder.Therefore, is the simplest form of a
4

fraction.

New Mathematics Connection 13

Example 15
Look at the fraction 6 .

12

Step 1: In this case, we can divide the numerator and the denominator exactly by
2, since both the numbers are divisible by 2.

6÷2 = 3
12 ÷ 2 6

Step 2: Change into the simplest form.

If the numerator and the denominator of the fraction is divisible by a common
number, the fraction can be written in a simpler form. We continue this operation with
the common divisor until we get the simplest form of fraction.

6÷2 = 3 3÷3 = 1
12 ÷ 2 6 6÷3 2

1
cannot be divided further.

2

1
So, is the simplest form.

2

Answer: 1 is the simplest from of 6 .
2 12

Example 16

1
Consider the fraction .

31
The numerator and the denominator of the fraction cannot be divided

3
exactly by a common number to get another equivalent fraction.

1
So, is the simplest form.

3
11
Answer: is the simplest from of .
33

14 Chapter 1 Fractions

Try It

1. Write the equivalent fractions. (There can be more than one answer.)

(a) 1 = 2 = 4 = 5 (b) 1 = 2 = =
2 20

(c) 1 = 2 = 3 = 4 (d) 2 = 4 = =
3 5 10

(e) 2 = = = (f) 3 = 12 = =
6 4 80

2. Write the next two equivalent fractions.

12 4 (b) 24 8
(a) , , ,,

36 8 3 6 12
4 8 16
3 6 12 (d)
(c) , , ,,
5 10 20
4 8 16 3 6 12

3 6 12 (f) ,,
(e) , , 7 14 28

5 10 20

3. Reduce the following fractions to simplest form.

(a) 8 (b) 16 (c) 4
10 24 16
36
(d) 9 (e) 15 (f) 18
12 20
9=3
4. Fill in the boxes. (b) 4 = (c) 15
(a) 6 = 1 10 5
12

(d) 20 = (e) 27 = (f) 5 =
30 30 10 20

Go to Exercise 4 15
New Mathematics Connection

E Comparing and Ordering Fractions

Comparing Fractions with Unequal Denominators
First, we make the denominators the same.
Then, we compare the fractions using the theory that if the denominators are equal,
the fraction with the larger numerator is greater.
Let us consider two fractions, 1 and 5 .

4 12
1

=
4
5

=
12

1
Now look at .

4
Making the denominator equal to 12.

1×3= 3
4 3 12
The second fraction is 5 .

12
Now, the denominators of both the fractions are equal.
Compare the numerators.
3 is less than 5.

35
So, < .

12 12

16 Chapter 1 Fractions

Example 17
Compare using > or < signs.
Compare 15 and 4 .

25 5

Solution 1:

15 ÷ 5 = 3
25 5 5

So, 3 < 4 . Therefore, 15 < 4 .
55 25 5

Answer: 15 < 4
25 5

Solution 2:

4 × 5 = 20
5 5 25

So, 15 < 20 . Therefore, 15 < 4 .
25 25 25 5

15 4
Answer: <

25 5

Ordering Fractions with Unequal Denominators

Let us learn to order two or more multiple denominator fractions in increasing and
decreasing order.

Example 18

Order the fractions in increasing order.
1, 3, 4 , 5
4 8 24 12

The denominator of all the fractions are multiples of 4.

New Mathematics Connection 17

We equate all the denominators to the highest value of the denominator, which is 24.

1×6= 6 4=4
4 6 24 24 24
3×3= 9 5 × 2 = 10
8 3 24 12 2 24

Now, order the fractions.
6 , 9 , 4 , 10
24 24 24 24

So, 4 < 6 < 9 < 10 .
24 24 24 24

Answer: 4 < 1 < 3 < 5
24 4 8 12

Example 19

Order the fractions in decreasing order.
2, 1, 4, 3
7 3 42 21

Equate the denominators to the highest value, which is 42.

2 × 6 = 12 4=4
7 6 42 42 42

1 × 14 = 14 3×2=6
3 14 42 21 2 42

So, 14 > 12 > 6 > 4 .
42 42 42 42

Answer: 1 > 2 > 3 > 4
3 7 21 42

18 Chapter 1 Fractions

Try It

1. Fill in the box with >, < or = signs.

(a) 1 4 (b) 1 5
5 10 4 16

(c) 5 4 (d) 6 7
12 3 7 42

(e) 5 3 (f) 14 2
8 32 42 7

2. Fill in the box with > or < signs.

(a) 5 12 (b) 15 7
8 16 20 10

(c) 1 1 (d) 9 4
2 4 21 15

(e) 1 4 (f) 6 18
8 6 12 24

(g) 5 4 (h) 15 16
10 10 5 8

3. Arrange the following fractions in increasing order.

(a) 1 48 24 40 (b) 15 2 3 5
80 16 32 64 30 5 15 45

(c) 1 3 3 4 (d) 2 378
6 12 24 48 4 8 16 32

(e) 1 3 12 7 (f) 6 7 3 5
3 6 18 12 12 24 8 6

New Mathematics Connection 19

4. Arrange the following fractions in decreasing order.

(a) 2 3 5 6 (b) 7 8 6 9
11 22 44 88 6 12 12 36

(c) 2 8 3 4 (d) 3 2 3 4
9 18 54 36 7 21 14 28

(e) 7 9 8 5 (f) 6 7 9 4
12 6 24 36 20 10 10 5

5. Fill in the box with >, < or = signs.

(a) 6 4 (b) 9 3 (c) 4 5
24 16 24 16 27 18
8 (f) 6
(d) 12 10 (e) 7 32 22 5
20 24 30 33

Go to Exercise 5

Fun Facts

Eight hours is one third of a day.

20 Chapter 1 Fractions

Contents

1 Factors and Multiples 1
2
CHAPTER A Factors 6
B Greatest Common Divisor (GCD) 10
C Lowest Common Multiple (LCM) 15
D Word Problems Using GCD and LCM
19
2 Fractions 21
23
CHAPTER A Comparing and Ordering Unrelated Fractions 26
B Operations on Unrelated Fractions 30
C Mixed Operations on Unrelated Fractions 31
D Operations on Mixed Numbers
E Mixed Operations on Mixed Numbers 33
F Solving Word Problems on Fractions
35
and Mixed Numbers 37
40
3 Decimals
45
CHAPTER A Understanding Three-Position Decimals 47
B Comparing and Ordering Decimals
C Conversion of Decimals to Fractions

and Vice Versa
D Rounding Decimal Numbers

4 Decimal Operations 49
51
CHAPTER A Multiplication of Decimals 53
B Mixed Operations on Decimals 56
C Solving Word Problems 61
D Solving Word Problems using Thai Method
63
5 Percentage and Its Applications 64
66
CHAPTER A Finding Percentage 68
B Interpreting Percentage 70
C Application of Percentage in Trade 72
D Problems on Multiple Selling and Buying
E Finding Simple Interest 73
F Finding Simple Interest for a Period less
75
than a Year 77
80
6 Rations and Proportions 83

CHAPTER A Ratios and Fractions
B Parts / Whole and Ratios
C Two Ratios

7 Volume 85
87
CHAPTER A Finding one Dimension of a Cuboid 90
B Finding the Area of a Face of a Cuboid 95
C Solving Word Problems on Volume
97
8 Circles 98
99
CHAPTER A Circumference of a Circle 100
B Finding the Area of a Circle
C Word Problems on Circles 101
102
9 Triangles 103
104
CHAPTER A Types of Triangles 107
B Components of a Triangle 108
C Perimeter of Triangles 109
D Drawing Triangles
E Finding Area of Triangles 111
F Word Problems on Triangles 113
115
10 Polygon 118
120
CHAPTER A Understanding Polygon
B Type of Polygons: Regular and Irregular
C Finding Area of Polygons
D Word Problems on Polygons

11 Three-Dimensional Geometry 121

CHAPTER A Identifying Two-Dimensional Shapes from 123
Three-Dimensional Solids 125
128
B Components of Three-Dimensional Geometry 129
C Nets for Geometric Solids
D Volume of a Rectangular Prism 131
E Word Problems on Volume and

Capacity of a Rectangular Prism

12 Number Patterns 132

CHAPTER A Understanding Number Patterns 133
B Number Patterns with Increasing Relationship 134
C Number Patterns with Decreasing Relationship 135
D Solving Problems on Patterns 136
E Solving Number Patterns 138
F Word Problems Using Patterns 139

13 Data Analysis and Probability 140

CHAPTER A Reading Data from a Line Graph 142
B Reading Data from a Pie Chart 149
C Drawing Bar Graphs 150
D Drawing Line Graphs 151
E Probability of Events 152

1 Factors and Multiples

Objectives

◆ To describe the meaning of factors, prime numbers and prime factorisation;
◆ To list the factors of a whole number;
◆ To identify the common factors of two numbers;
◆ To write the meaning of Greatest Common Divisor (GCD);
◆ To find the GCD of whole numbers;
◆ To describe the meaning of multiples;
◆ To list the multiples of a whole number;
◆ To identify the common multiples of two numbers;
◆ To write the meaning of Lowest Common Multiple (LCM);
◆ To find the LCM of whole numbers;
◆ To apply knowledge to daily life.

Getting Started

Tom climbs up the staircase in steps of two and Pat climbs up
in steps of three. Can you say in which step both will meet?

I have to imagine a
staircase for it.

It is not necessary. We will find the answer by learning
the concept of lowest common multiple.

New Mathematics Connection 1

A Factors

Let us learn what a factor is and how to find the factors of a number.
Here is a box with eight candies.

Now, let us arrange each row to have the same number of candies.

(a) The eight candies can be arranged as shown.

Number of candies in each row : 1

Number of rows :8

Total number of candies :1×8=8

(b) Number of candies in each row :2
Number of rows :4
Total number of candies :2×4=8

(c) Number of candies in each row :4
Number of rows :2
Total number of candies :4×2=8

(d) Number of candies in each row :8
Number of rows :1
Total number of candies :8×1=8

We can observe that 8 can be written as a product of two numbers
in different ways:

8=1×8 A factor of a number
8=2×4 is an exact divisor of
8=4×2
8=8×1 that number.

From 8 = 2 × 4, we can say that, 8 can be divided exactly by 2 and 4. So, 2 and 4
are exact divisors of 8.

2 Chapter 1 Factors and Multiples

From 8 = 1 × 8, we can say that 8 can be divided exactly by 1 and 8.
Thus, 1, 2, 4 and 8 are exact divisors of 8. They are called the factors of 8.

Example 1

Find all the factors of 4.

Solution:
4 ÷ 1 = 4 with no remainder. Therefore, 1 is a factor of 4.
4 ÷ 2 = 2 with no remainder. Therefore, 2 is a factor of 4.
4 ÷ 3 = 1 with a remainder of 1. Therefore, 3 is not a factor of 4.
4 ÷ 4 = 1 with no remainder. Therefore, 4 is a factor of 4.

Answer: The factors of 4 are 1, 2 and 4.

Example 2 Example 3

Write all the factors of 12. Write all the factors of 36.

Solution: Solution:

12 = 1 × 12 36 = 1 × 36
12 = 2 × 6 36 = 2 × 18
12 = 3 × 4 36 = 3 × 12
We can stop here because 3 and 4 are 36 = 4 × 9
consecutive numbers. 36 = 6 × 6
Answer: The factors of 12 are 1, 2, 3, 4, We can stop here because both
6 and 12. factors are the same.
Answer: The factors of 36 are 1, 2, 3, 4, 6, 9, 12,
18 and 36.

We can observe the following facts about factors:

New Mathematics Connection 3

Prime Numbers
The following table shows the factors and number of factors of the numbers from 1 to 12.

Numbers Factors Number of Factors
1 1 1
2 1, 2 2
3 1, 3 2
4 3
5 1, 2, 4 2
6 1, 5 4
7 1, 2, 3, 6 2
8 1, 7 4
9 1, 2, 4, 8 3
10 1, 3, 9 4
11 1, 2, 5, 10 2
12 1,11 6
1, 2, 3, 4, 6, 12

We find that

(a) Some numbers have exactly two factors.The numbers which have only two factors,
1 and the number itself, are called ‘prime numbers’.

For example: 2, 3, 5, 7, 11

(b) The numbers which have more than two factors are called ‘composite numbers’.
For example: 4, 6, 8, 10

(c) The number 1 has only one factor. So, it is neither a prime number nor a composite
number.

Example 4

Determine whether 13 is a prime number.

Solution:
Let us find the factors of 13.
13 can only be divided by 13 and 1. Therefore, 1 and 13 are the only factors of 13.
Answer: 13 is a prime number.

4 Chapter 1 Factors and Multiples

Prime Factorisation

When a number is expressed as a product of its factors, we say that the number has been
prime factorised.
Consider the number 24.
24 = 2 × 12
We say that 24 has been factorised. This is one of the factorisations of 24.
The others are: 24 = 3 × 8; 24 = 4 × 6

Now, let us learn to find the prime factors of 24.

Method 1: 24 = 3 × 8 24 = 4 × 6
24 = 2 × 12

=2× 2 × 6 =3 × 2 × 4 =2×2×2×3

=2×2×2×3 =3×2×2×2

Method 2:

2 24
2 12
26
33

1

In both methods, we finally reach at only one factorisation 2 × 2 × 2 × 3. In this factorisation, the
only factors 2 and 3 are prime numbers.This is called the prime factorisation of a number.

Example 5

Find the prime factorisation of 40.

Solution:

Method 1: Method 2:

40 = 4 × 10 2 40
2×2×2×5 2 20
2 10

5

Thus, the prime factorisation of 40 is 2 × 2 × 2 × 5.

New Mathematics Connection 5

Try It

1. Find the factors of the following numbers.

(a) 28 (b) 54 (c) 35

(d) 26 (e) 44 (f) 60

2. Circle the prime numbers.

(a) 2, 4, 6 (b) 12, 9, 7 (c) 13, 15, 19

(d) 31, 38, 42 (e) 48, 54, 43 (f) 41, 57, 81

3. Write all the prime numbers less than 50.

4. Find the prime factorisation of the following numbers.

(a) 25 (b) 18 (c) 30

(d) 45 (e) 27 (f) 72

Go to Exercise 1

B Greatest Common Divisor (GCD)

Common Prime Factors
Let us find the common prime factors of 6 and 8.

The factors of 6 are 1, 2, 3 and 6.

The factors of 8 are 1, 2, 4 and 8.
Therefore, 2 is the common prime factor of 6 and 8.

6 Chapter 1 Factors and Multiples

Example 6
What is the common prime factors of 9 and 18?

Solution:

The factors of 9 are 1, 3 and 9.

The factors of 18 are 1, 2, 3, 6, 9 and 18.
1, 3 and 9 are present in both the factors of 9 and 18. Therefore, they are the common factors
of 9 and 18.
Answer: The common prime factor of 9 and 18 is 3.

The Greatest Common Divisor or GCD, of two or more
numbers is the greatest of their common factors. It is
also known as the Highest Common Factor or HCF.

Fun Facts

A number is divisible by 2 if its last digit is 0 or even.

New Mathematics Connection 7

Finding GCD Using Prime Factorisation Method
Let us find the GCD of 15 and 25.

15 = 3 × 5
25 = 5 × 5
Therefore, the GCD of 15 and 25 is 5.

Example 7
Find the GCD of 10, 14 and 18.

Solution:
10 = 2 × 5
14 = 2 × 7
18 = 2 × 3 × 3
Answer: The GCD of 10, 14 and 18 is 2.

Example 8
Find the GCD of 28, 36 and 48.

Solution:
28 = 2 × 2 × 7
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
The GCD is 2 × 2 = 4.
Answer: The GCD of 28, 36 and 48 is 4.

Think Aloud

Is 10 a factor of 5? Is 2 a multiple of 8?

8 Chapter 1 Factors and Multiples

Finding GCD Using Short Division

We can also find GCD of numbers using short division. Divide the given number by the prime number
until there is no prime number divisor.Then, find the product of the common divisors.

Example 9

Find the GCD of 16, 28 and 44.

Solution:
Step 1: Find all the prime factors of 16, 28 and 44.

2 16 28 44 We cannot carry on further because there is
2 8 14 22 no common factors besides 1.

4 7 11

Step 2: Find the product of all the common prime factors.
GCD = 2 × 2 = 4.

Therefore, the GCD of 16, 28 and 44 is 4.

Try It

1. Find the common factors. (b) 38 and 57
(a) 6 and 16 (d) 11 and 121
(c) 12 and 64 (f) 65 and 104
(e) 5 and 15

2. Find the GCD using prime factorisation method.

(a) 16 and 20 (b) 12 and 48

(c) 15, 35 and 50 (d) 25, 45 and 90

3. Find the GCD using short division. (b) 25 and 45
(a) 18 and 30 (d) 12, 24 and 32
(c) 26, 34 and 42

Go to Exercise 2

New Mathematics Connection 9