introductionpearsondigital.ilongman.com/resources/sec/math/jsmia/... · 2020-03-06 · decimals and...

Introduction

Bridge Programme P6 to S1 (Bilingual version) is specially designed for students facing a change

in learning language medium from Chinese to English in studying Mathematics in S1. It will help

you get prepared before you start your Mathematics lessons in secondary school.

From this book, you can learn most of the mathematical terms covered in primary school in English.

This is very useful for you to understand what your teachers says in Mathematics Lessons.

Key Features

Chinese translations of key terms are included. Selected keywords appear repeatedly in

different part of this book to facilitate cognitive learning.

English pronunciations of useful vocabulary and sentences are provided on our website to

give genuine support for students adapting to a new learning language medium.

Relevant mathematical terms are tabulated in Key Terms / Phrases or underlined for easy

reference and memory reinforcement.

Bilingual arrangement of examples and solutions give students a clear guide for answering

questions in English.

Knowing More focus on some topics which can help students have a better preparation for the

secondary school lessons.

Useful Sentences are available to allow students adequate exposure to different question types.

Exercises are written in English with Chinese translations of difficult vocabulary given in

footnotes to facilitate student’s understanding.

Numerical answers to questions are provided.

How to use?

This book can be distributed to Primary 6 students in the summer holiday for self-study. Students

are suggested to complete this book before their first Mathematics lesson in S1.

CONTENTS 1. Arithmetic Operations .......................................... 1

A Four Basic Arithmetic Operations .............................. 1

B Multiples and Factors ............................................... 3

Key Terms / Phrases ..................................................... 5

Useful Sentences ........................................................ 6

Exercise 1 ..................................................................... 6

2. Decimals and Fractions ....................................... 9

A Conversion between Decimals and Fractions............. 9

B Basic Operations of Decimals and Fractions .............. 11


Useful Sentences ........................................................ 15

Exercise 2 ..................................................................... 15

3. Approximations

4. Basic Algebra and Simple Equations .............. 19

A Introduction to Algebra ............................................. 19

B Simple Equations ..................................................... 21

C Application of Equations ........................................... 22


Useful Sentences. ......................................................... 24

Exercise 4 .................................................................... 24

5. Percentages

6. Shape and Space

7. Perimeters, Areas and Volumes

8. Data Handling .......................................................... 28

A Different Statistical Diagrams ...................................... 28

B Applications of Statistical Diagrams ............................ 29


Useful Sentences. ......................................................... 32

Exercise 8 ..................................................................... 32

Answers ............................................................................ 35

Knowing More

Recurring decimals

Knowing More

Prime Factors

Index Notation

Bridge Programme P6 to S1 (Bilingual version) Junior Secondary Mathematics in Action

9 © Pearson Education Asia Limited 2020

2 Decimals and Fractions 小數和分數

A. Conversion between Decimals and Fractions 小數與分數的互化

16.295 is a decimal, which can be read as sixteen point two nine five.

16.295 是一個小數，讀作十六點二九五。

Fractions can be classified into following three types:

分數可分為以下三類：

Proper fraction

真分數

Improper fraction

假分數

Mixed fraction

帶分數

The numerator is smaller than

the denominator.

分子小於分母

The numerator is greater than

or equal to the denominator.

分子大於或等於分母

An improper fraction which is

written as a sum of a natural

number and a proper fraction.

把假分數寫成自然數與真分數

之和的形式

Read as : three-fifths /

three over five

讀作：五分之三

Read as : seven quarters /

seven over four

讀作：四分之七

Read as : six and two-thirds /

six and two over three

讀作：六又三分之二

numerator

分子 3 5

7 4

integral part

整數部分

6 2 3

Improper fractions can be converted into mixed fractions and vice versa.

假分數可化成帶分數，反之亦然。

e.g. 7 3

14 4

=

denominator

分母 fractional part

分數部分

16.295 thousandths

千分位

tenths

十分位

decimal point

小數點

hundredths

百分位

units

個位

tens

十位



Example 1

Convert 0.45 into a fraction.

把 0.45 化成分數。

450.45

100

9

20

=

=

Example 2

Convert 3

5 and

7

25 into decimals.

把 3

5 和

7

25 化成小數。

Method 1: Expand the fractions

方法一：進行擴分 3 3 2

5 5 2

6

10

0.6

=

=

=

7 7 4

25 25 4

28

100

0.28

=

=

=

Method 2: Use division

..方法二：利用除法

..

33 5

5

0.6

=

=

..

77 25

25

0.28

=

=

Reduce the fraction to its simplest form.

進行約分，把分數化成最簡形式。

9

20

5 3.0

0.6

3 0

25 7.00

0.28

5 0

2 00 2 00



Example 3

Arrange the following numbers in descending order.

把下列各數由大至小排列。

1 3

1 , 1 , 1.1254 8

Method 1: Convert the numbers into fractions

方法一：把各數化成分數 1 1 2 2

1 1 14 4 2 8

= =

125 11.125 1 1

1000 8= =

∵ 3 2 1

1 1 18 8 8

∴ 3 1

1 1 1.1258 4

Method 2: Convert the numbers into decimals

方法二：把各數化成小數 1 1

1 1 1 0.25 1.254 4

3 31 1 1 0.375 1.375

8 8

= + = + =

= + = + =

∵ 1.375 1.25 1.125

∴ 3 1

1 1 1.1258 4

B. Basic Operations of Decimals and Fractions 小數和分數的運算

The following table shows some basic arithmetic operations of decimals.

下表所示為一些基礎的小數運算。

Addition 加法 Subtraction 減法

e.g. 2.08 3.42

5.5

+

=

e.g. 5.4 2.25

3.15

−

=

Multiplication 乘法 Division 除法

e.g. 8.12 3.4

27.608

=

e.g.

3.5 0.4

(3.5 10) (0.4 10)

35 4

8.75

=

=

=

Compare the numerators.

比較分子的值。

.2.08

+ 3.42

5.50

.5.40

– 2.25

3.15

8.12

× 3.4

24 360

+ 3 248

27.608

. 8.75 0.4 3 5.00 3 2 3 0 2 8 20 20



Example 4

Evaluate 2.5 3.6 1.5 .

計算 2.5 3.6 1.5 。

2.5 3.6 1.5 9 1.5

(9 10) (1.5 10)

90 15

6

=

=

=

=

The following table shows some basic arithmetic operations of fractions.

下表所示為一些基礎的分數運算。

Addition 加法 Subtraction 減法

e.g. 1 1 2 1

3 6 6 6

3

6

1

2

+ = +

=

=

e.g. 2 1 6 1

3 2 3 25 15 15 15

51

15

11

3

− = −

=

=

Multiplication 乘法 Division 除法

e.g.

1 1 10 1

33 15 3 15

2

9

=

=

e.g. 5 2 5 9

27 9 27 2

5

6

=

=

3

2

3

1

Multiply both the divisor and the dividend by 10 so that the divisor

becomes a whole number.

把除數和被除數同時乘以 10，使除數變成整數。

. 2.5

× 3.6

. 7 50

+ . 1 50

9.00

.1

If the multiplication and division of fractions involve mixed fractions, we

should first change the mixed fractions into improper fractions.

若分數的乘法和除法中涉及帶分數，我們應先把帶分數化成假分數。



Example 5

Evaluate 1 3 5

2 34 4 6

+ .

計算 1 3 5

2 34 4 6

+ 。

1 3 5 1 15 62 3 2

4 4 6 4 4 5

1 92

4 2

1 12 4

4 2

1 22 4

4 4

36

4

+ = +

= +

= +

= +

=

Example 6

Harry bought a 3

kg4

chocolate cake and ate 3

5 of it. Find the weight of the remaining chocolate

cake.

思朗買了一個重 3

kg4

的巧克力蛋糕，並吃去蛋糕的 3

5。求餘下的巧克力蛋糕的重量。

Weight of the remaining chocolate cake

餘下的巧克力蛋糕的重量

3 31 kg

4 5

3 2 kg

4 5

3 kg

10

= −

=

=

Express 3

34

as an improper fraction. 把 3

34

寫成假分數。

Perform multiplication and division before addition and subtraction.

先乘除，後加減。

Express the fractions in a common denominator.

通分，即把分數以相同分母表示。



Example 7

The selling price of a pen is $6.3 and the selling price of a pencil is $3.7. What is the total selling

price of half a dozen pens and 1

13

dozen pencils?

一枝原子筆的售價為 $6.3，一枝鉛筆的售價為 $3.7。問半打原子筆和 1

13

打鉛筆的總售價

是多少？

Total selling price 總售價

1 1$ 6.3 12 3.7 12 1

2 3

4$ 6.3 6 3.7 12

3

$(6.3 6 3.7 16)

$(37.8 59.2)

$97

= +

= +

= +

= +

=

In a decimal, if a digit or a pattern of digits after the decimal point repeats continuously, the decimal is called

a recurring decimal. For example: 0.555…, 0.409 09 …, 0.629 629… are recurring decimals.

當一個小數從小數點後的某個數位開始，有一個數字或一組數字不斷地重複出現，這個小數稱為循環

小數。例如：0.555…、0.409 09 …、0.629 629… 都是循環小數。

In a recurring decimal, the repeated part is called the recurring period. It is indicated by the recurring point(s).

在循環小數中，不斷重複而相同的部分稱為循環節，我們會用循環點標示。

For example:

0.555 555... 0.5•

= 0.409 090... 0.409• •

= 0.629 629... 0.629• •

=

We can use recurring decimals to represent fractions as follows:

我們可以使用循環小數來表示分數，如下所示：

1 8 11

0.333... 0.3, 0.7272... 0.72, 0.407 407... 0.4073 11 27

• • • • •

= = = = = =

Knowing More

Recurring point(s) 循環點

Recurring period 循環節

Recurring decimals 循環小數



decimal 小數 thousandths 千分位 mixed fraction 帶分數

decimal point 小數點 fraction 分數 divisor 除數

units 個位 numerator 分子 dividend 被除數

tens 十位 denominator 分母 recurring decimals 循環小數

tenths 十分位 proper fraction 真分數

hundredths 百分位 improper fraction 假分數

What fraction of the time is spent on studying? 花在學習上的時間佔幾分之幾？

What fraction of 7

8 is

1

2?

1

2 是

7

8 的幾分之幾？

Convert the following mixed fraction into an

improper fraction. 把以下帶分數寫成假分數。

4.085 is read as three point zero seven five. 4.085 讀作四點零八五。

Which digit in 35.62 is in hundredths place? 在 35.62 這個數中，哪一個數字是在百分位？

Exercise 2

1. Consider the number 65.013. Determine whether each of the following is true for the number.

Put a ‘’ or a ‘’ in each of the boxes.

(a) ‘1’ is in the hundredths place. (b) ‘1’ in the number represents 10.

(c) ‘6’ in the number represents 60 000. (d) ‘0’ is in the tenths place.

(e) ‘5’ in the number represents 50. (f) It is equal to 13

65100

.

2. Reduce the following fractions into their simplest forms.

(a) 24

80 (b)

120

135 (c)

126

42

Useful Sentences

Key Terms / Phrases

Pronunciation



3. Convert the following decimals into fractions.

(a) 0.55 (b) 3.75 (c) 15.625

4. Convert the following fractions into decimals.

(a) 98

200 (b)

13

8 (c)

27

25

5. Compare the values of each of the following pairs of fractions. Put a ‘>’ or ‘<’ in each of the boxes.

(a) 4 6

5 7

(b) 11 14

8 11

(c) 5 7

13 17

6. Arrange each of the following sets of numbers in ascending order.

(a) 6.7, 8.03, 0.969, 10.34

(b) 3.6, 3.06, 36, 30.6

7. Arrange each of the following sets of numbers in descending order.

(a) 2

3,

5

6,

7

12,

1

2

(b) 0.5, 1

8,

11

16, 1.15,

11

4, 0.12

Evaluate the following. (8 − 19)

8. 5.4 3.2 2.6− + 9. 19.8 0.33

10. 12.6 0.7 2.25 4 − 11. (6.6 0.3 1.5) 4 −

12. 11 5 7

30 6 10+ − 13.

7 6 10

30 7 12

14. 19 1

1 32 145 7

15. 2 11

3 25 35

−

16. 7 1 1 2

110 2 6 3

− 17. 2 11 5 5

3 12 6 8

− +

18. 3

2.5 3 (2 0.75)4

+ 19. 7 1

0.25 2 112 16

−

ascending order 由小至大排列



Find the result of each of the following. (20 − 22)

20. 6.5 L orange juice is divided into 26 cups equally. How much orange juice

is there in each cup in L?

21. Wendy is 35 years old and Jacky is 15 years old. What fraction of Wendy’s

age is Jacky’s age?

22. A bag of coconuts weighs 10.5 kg. If we sell 5 bags of coconuts for $756,

how much does one kilogram of coconuts cost?

Solve the following problems. Show your working steps clearly. (23 – 26)

23. Each box of apples costs $35.8. It costs $4.3 more than each box of oranges. Nelson pays $250 for

5 boxes of oranges. How much change should he get?

24. A bag of peanuts weighing 4

25

kg costs $30. Mary buys a bag of peanuts that weighs 2

43

kg.

How much should she pay?

coconut 椰子



25. Red roses cost $50.4 per dozen, yellow roses cost $81.6 per dozen.

How much do 5 red roses and 8 yellow roses cost?

26. Linda uses 1

7 of a bag of flour to make some bread and

3

5 of the

rest of it to make some biscuits. What fraction of the flour is left?



Answers

Exercise 1 Arithmetic Operations

1. 12 + 70 = 82 2. 2000 – 106 = 1894

3. 97 – 48 + 23 = 72 4. 30 × 5 – 38 = 112

5. (12 + 4) × 8 = 128 6. (35 + 17) ÷ 13 = 4

7. (30 – 14) + 20 = 36 8. 28 ÷ 4 + 16 × 4 = 71

9. 50 – 45 ÷ 9 = 45 10. 10

11. 70 12. 3317

13. 160 14. 177

15. 36 16. 41

17. 34 18. False

19. False 20. True

21. True

22. (a) 56 (b) 72

(c) 150 (d) 120

23. (a) 6 (b) 4

(c) 27 (d) 6

24. 87 25. $15

26. $36 27. 2 kg

28. $152 29. $80

30. 16

31. 1:45 p.m., 5:30 p.m., 9:15 p.m.

Exercise 2 Decimals and Fractions

1. (a) (b) (c)

(d) (e) (f)

2. (a) 3

10 (b)

8

9 (c) 3

3. (a) 11

20 (b)

33

4 (c)

515

8

4. (a) 0.49 (b) 3.125 (c) 1.08

5. (a) < (b) > (c) <

6. (a) 0.969 < 6.7 < 8.03 < 10.34

(b) 3.06 < 3.6 < 30.6 < 36

7. (a) 5 2 7 1

6 3 12 2

(b) 1 1 1

1 1.15 1 0.5 0.124 16 8

8. 4.8 9. 60

10. 9 11. 1.92

12. 1

2 13.

1

6

14. 7

180 15.

101

81

16. 16

45 17.

49

72

18. 5

16

19. 1

3

20. 0.25 L 21. 3

7

22. $14.4 23. $92.5

24. $50 25. $75.4

26. 12

35

Exercise 4 Basic Algebra and

Simple Equations

1. Alegebraic

expression Equation

(a) 5 4b+

(b) 7 3 17c+ =

(c) 23 8 1d d+ =

(d) 2 100x −

(e) 2

6 73

m n− =

2. a + 10 3. b – 7

4. 5c 5. 4

d

6. a + 5.5 7. b – 10



8. 3y 9. 50

k

10. 25m2 11. 7

12. 173 13. 19

14. 60 15. 9

16. 5 17. 5

18. 35 19. 7

20. 3.9 21. 1.5

22. 4.61 23. x + 25 = 59

24. y + 1 = 15 25. p – 7 = 12

26. 130z = 3380 27. 100 – 4d = 16

28. 2[12 + (12 – a)] = 42

29. 35 30. 65

31. $5.4 32. 32

33. $3120

Exercise 8 Data Handling

1. (a) broken-line graph (b) broken-line graph

(c) bar chart (d) bar chart

(e) broken-line graph

2.

3. (a) Monday, 240 mm (b) Saturday, 540 mm

(c) 2140 mm

4. (a) $220 000 (b) September

(c) March and April

5. (a) Cooking (b) 17

(c) Book store A, 46

introductionpearsondigital.ilongman.com/resources/sec/math/jsmia/... · 2020-03-06 · decimals and...

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