numbers and fundamental arithmetic

Summer Adventure in Mathematics Foundation (New Syllabus Edition)2 3

Numbers and Fundamental Arithmetic

1. Types of numbers

(a) Integers (E.g. 0, 10, 153)

Positive integers can be categorized into

(i) odd numbers (e.g. 1, 3, 5, …) and even numbers (e.g. 2, 4, …);

or

(ii) 1, prime numbers (e.g. 2, 3, 5, 7, …) and composite numbers (e.g. 4, 6, 8, 9, …).

(b) Decimal numbers (E.g. 1.45, 10.7)

(c) Fractions: (i) Proper fractions, e.g. 12

, 34

, 29

.

(ii) Improper fractions, e.g. 52

, 134

, 158

.

(iii) Mixed numbers, e.g. 2 12

, 3 14

, 178

.

2. Conversions among decimal numbers and fractions

(a) From fractions to decimal numbers, e.g. by long division, 34

3 4 0 75= ÷ = .

(b) From decimal numbers to fractions, e.g. 0 74 74100

3750

. = =

3. Operations among numbers

(a) Addition and subtraction

Among fractions, e.g.

12

13

36

26

56

+

= +

=

1 is neither a prime number nor a composite number.

Mathematics Tips

← numerator← denominator

34

Mathematics Tips

Make the f ract ions have a common denominator.

Mathematics Tips

1

Numbers and Fundamental Arithmetic2 3

(b) Multiplication

(i) Among decimals, e.g. 1 2 2 4 2 88. . .× =

(ii) Among fractions, e.g. 38

34

932

× =

(c) Division

(i) Among decimals, e.g. 3 5 0 5 35 5 7. .÷ = ÷ =

dividend divisor both of them × 10

(ii) Among fractions, e.g. 916

32

916

23

38

÷ = × =

(d) Order of operations

Operations inside brackets come first. Then, multiplication and division have to be done before addition or subtraction.

E.g. 3 5 6 3 30 33+ × = + =

( )3 5 6 8 6 48+ × = × =

4. Divisibility

When a number (first number) is divided by another (second number) and no remainder is obtained, we say that the first number is divisible by the second number.

(a) Divisibility test of 10

Check whether the last digit of a number is 0.

(b) Divisibility test of 5

Check whether the last digit of a number is 0 or 5.

(c) Divisibility test of 2

Check whether the last digit of a number is an even number.

(d) Divisibility test of 3

Check whether the sum of all digits of a number is divisible by 3.

Count the tota l number o f decimal places for the f inal answer.

1.2× 2.4

2 44 8

2.8 8

← 1decimal places

← +1

← 2

Mathematics Tips

Convert the number (divisor) to a whole number by shifting the decimal point of both numbers to the right first.

Mathematics Tips

It can be changed to multiplication by reversing the numerator and denominator of the divisor.

Mathematics Tips


Matching

1. 2 79

• • Composite number

2. 53

• • Prime number

3. 4 • • Mixed number

4. 2 • • Decimal

5. 5.2 • • Improper fraction

Fill in the Blanks

6. Convert 4 23

into an improper fraction.

4 23=

× +=

7. Convert 75

into decimal form. Column form: )5 7

75

=

8. 34

16

34

16

+ =×

×+

×

×=

9. 34

115

34

× = × =

10.

34

1 15

34

34

÷ = ÷

= ×

=

➤

make the fractions have a common denominator

➤

convert mixed numbers to improper fractions first

➤

convert mixed numbers to improper fractions first

➤

change division to multiplication


True or False11. 0 is an even number.

12. 17 is a composite number.

13. 1 + 2 × 3 = 9

Multiple-choice Questions14. Which of the following is a prime number?

A. 6 B. 7 C. 8 D. 9

15. Which of the following is a composite number?

A. 2 B. 3 C. 4 D. 5

Convert the following fractions (16 – 18) into decimals.

16. 114

17. 910

18. 85

Convert the following decimals (19 – 21) into fractions.

19. 0.4 20. 2.25 21. 1.05

Calculate the following expressions (22 – 23) and leave the answers in fractions.

22. 34

12

+ 23. 59

35

×

Calculate the following expressions (24 – 25) and leave the answers in decimals.

24. 2.35 + 3.14 25. 0.8 × 1.2➤

check whether 0 can be divided by 2 or not

➤

check whether 17 is only divisible by 1 and itself or not

➤

remember the order of operations


Scoring Tips 1 : Consider the total cost of the tickets first.

43. The figure shows the poster of the theme park. Four students, Peter, Paul, Susan and Mary decided to go to the park to celebrate Mary’s birthday. Peter, Paul and Susan agree to share the cost so that Mary does not need to pay for her ticket. How much should Peter pay? Scoring Tips 1

44. (a) Calculate the following expressions.

(i) 100 - 99

(ii) 98 - 97

(iii) 100 - 99 + 98 - 97 + 96 - 95 + ... + 2 – 1

(iv) 300 - 297 + 294 - 291 + 288 - 285 + ... + 6 - 3

(b) (i) Calculate 888 - 880 + 872 - 864 + 856 - 848 + ... + 24 - 16 + 8.

(ii) Is the value of 896 - 888 + 880 - 872 + 864 - 856 + ... + 32 - 24 + 16 - 8 the same as that in (b)(i)? Explain your answer.

Xxx Park

Original price:

$166 per person

* If 4 tickets are bought together, the price of each ticket will be reduced by $16.

EYA


Besides the divisibility tests for 2, 3, 5 and 10, there are divisibility tests for other numbers. For example:• Divisibility test of 4 Check whether the last two digits of a number form a number divisible by 4.

• Divisibility test of 8 Check whether the last three digits of a number form a number divisible by 8.

• Divisibility test of 9 Check whether the sum of all digits of a number is divisible by 9.

Determine whether 6714 is divisible by 4.

Solution:\ 14 is not divisible by 4.

\ 6714 is not divisible by 4.

Follow-up45. Determine whether 9528 is divisible by 4.

46. Determine whether 7240 is divisible by 8.

47. Determine whether 4005 is divisible by 9.

1. composite number 合成數 2. decimal number 小數

3. decimal place 小數位 4. denominator 分母

5. dividend 被除數 6. divisibility 整除性

7. divisible 可整除 8. divisor 除數

9. even number 偶數 10. fraction 分數

11. improper fraction 假分數 12. integer 整數

13. mixed number 帶分數 14. numerator 分子

15. odd number 奇數 16. prime number 質數

17. proper fraction 真分數


Section A (16 marks, each question carries 1 mark)

1. Which of the following numbers is divisible by 3?

A. 31 259 B. 62 754

C. 96 325 D. 125 678

2. It is given that Y is a factor of 120. Which of the following number(s) must be the common factor of Y and 120?

I. 2

II. 60

III. Y

A. I only B. III only

C. I and II only D. II and III only

3. In the figure, the radius of the circle is 5 cm. Find the area of the circle. (Take π = 3.14.)

5 cm

A. 15.7 cm2 B. 31.4 cm2

C. 62.8 cm2 D. 78.5 cm2

4. Find the average of 23, 27, 29 and 25.

A. 23 B. 25

C. 24 D. 26

5. Convert 12% into a fraction.

A. 112

B. 125

C. 325

D. 425

6. Solve − =x2

2 8 .

A. x = 3 B. x = 5

C. x = 12 D. x = 20

7. Which of the following is a negative number?

A. (-2) + (-3) B. (-2) - (-3)

C. (-2)(-3) D. --23

8. In the figure, the marked angle is

A. an acute angle. B. a right angle.

C. an obtuse angle. D. a reflex angle.

9. Which of the following points lies on the y-axis?

A. (-2, 0) B. (1, 1)

C. (0, 3) D. (2, -2)

10. Simplify 3x - 2x + y.

A. x + y B. 2x

C. 2y D. 2x + y

Marks: 71Revision TestHKEP

numbers and fundamental arithmetic

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