chapter 4 introduction to geometry 4a p

1

Chapter 4 Introduction to Geometry

4A p.2

4B p.15

4C p.28

Chapter 5 Percentages (I)

5A p.39

5B p.51

5C p.65

5D p.76

Chapter 6 Estimation in Numbers and Measurement

6A p.85

6B p.101

For any updates of this book, please refer to the subject homepage:

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For mathematics problems consultation, please email to the following address:

[email protected]

For Maths Corner Exercise, please obtain from the cabinet outside Room 309

2

F1A: Chapter 4A

Date Task Progress

Lesson Worksheet

○ Complete and Checked ○ Problems encountered ○ Skipped

(Full Solution)

Book Example 1

○ Complete ○ Problems encountered ○ Skipped

(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Consolidation Exercise


(Full Solution)

Maths Corner Exercise 4A Level 1


Teacher’s Signature

___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 4A Multiple Choice

○ Complete and Checked ○ Problems encountered ○ Skipped Teacher’s Signature

___________ ( )

E-Class Multiple Choice Self-Test

○ Complete and Checked ○ Problems encountered ○ Skipped Mark:

_________

3

Book 1A Lesson Worksheet 4A (Refer to §4.1)

4.1A Points, Lines and Planes

Refer to the figure.

(a) A is a point.

(b) l is a straight line.

(c) AB is a line segment.

(d) ABCD represents the white plane.

Example 1 Instant Drill 1

In the figure, L is a straight line.

(a) Name three fixed points on the straight

line L.

(b) Name all the line segments on the straight

line L.

Sol (a) A, B and C are on L.

(b) AB, BC and AC are on L.

In the figure, L is a straight line.

(a) Name all the fixed points on the straight

line L.

(b) Name all the line segments in the figure.

Sol (a) are on L.

(b) The line segments are

.

1. Refer to the figure.

(a) Name all the line segments on L.

(b) Among all the line segments in (a),

which of them is the longest?

○○○○→→→→ Ex 4A 1–3

2. By observation only, find the number of

plane(s) in each of the following figures.

(a)

(b)

(a) The figure is formed by

plane/planes.

(b)

○○○○→→→→ Ex 4A 4

A

C

D

B L

D E F L

G

A B C L

A B

C D

l A line is made up of an

infinite number of points.

A plane contains an infinite number of lines.

� A line segment is a part of

a straight line.

4

4.1B Angles

(a) When two lines meet each other at a

point, they will form an angle.

(b) The angle in the figure can be named as

(i) x,

(ii) ∠B,

(iii) ∠ABC (or ∠CBA).


Use the vertex and the symbol ‘∠’ to name

angles x and y in the figure.

Sol x = ∠ABC (or ∠CBA or ∠B)

y = ∠ECD (or ∠DCE)

Use the vertex and the symbol ‘∠’ to name

angles a, b and c in the figure.

Sol a = ∠

b =

c =

○○○○→→→→ Ex 4A 5, 6

Types of Angles:

Name Acute angle Right angle Obtuse angle

Size of angle

Larger than 0° but

smaller than 90° Equal to 90°

Larger than 90° but

smaller than 180°

Example

Name Straight angle Reflex angle Round angle

Size of angle

Equal to 180° Larger than 180° but

smaller than 360° Equal to 360°

Example

P

Q

R

a b

S c

C

y

B

A D E

x

It is not good to name y as ∠C because ∠C

can refer to ∠BCE, ∠DCE or ∠BC .

arm

angle

arm

vertex

B

A

C x

5


Refer to the figure. Determine the types of the

following angles according to their sizes.

(a) ∠BAC

(b) ∠ABC

(c) ∠BCD

Sol (a) ∠BAC = 60°

∴ ∠BAC is an acute angle.

(b) ∠ABC = 90°

∴ ∠ABC is a right angle.

(c) ∠BCD = 180°

∴ ∠BCD is a straight angle.

By observation only, determine the types of the

following angles according to their sizes.

(a) a

(b) b

(c) c

(d) d

Sol (a) a = ( )°

∴ a is a/an angle.

(b) b is a/an angle.

(c) c is a/an angle.

(d) d = ( )°

∴ d is a/an angle.

3.

By observation only, determine the types

of the marked angles in the figure

according to their sizes.

4. Determine the types of the following

angles according to their sizes.

32°, 260°, 97°, 180°, 125°, 45°, 360°

○○○○→→→→ Ex 4A 7–9

A B

C

D

M

F

E

N

a

b

c d

A

B C

D

180°

60°

60° lies in between

0° and 90°.

6

We can use a protractor to measure the size of an angle,

or to construct an angle with a given size.

5. Follow the steps below. Use a protractor to measure the size of ∠ABC in the figure.

∠ABC = °

6. Refer to the figure. Find the sizes of the

following angles.

(a) ∠AOB

(b) ∠AOC

(c) ∠AOD

(d) ∠DOC

7. Use a protractor to measure the following

angles.

(a)

(b)

○○○○→→→→ Ex 4A 10 ○○○○→→→→ Ex 4A 11

J

K L

P Q

R

C

D A O

B

A

B C

Use the outer scale or the inner scale of a protractor appropriately.

Measure obtuse ∠PQR first. Then

reflex ∠PQR = 360° – obtuse ∠PQR

Step 1111: Place the centre of the protractor on the vertex B.

Step 2222: Place the base line along BC.

Step 3333: Read the inner scale where 0° lies on BC and get

∠ABC = °.

7

8. Follow the steps below. Use a protractor to construct ∠ABC = 140°.

9. Use a protractor to construct the following

angles.

(a) 40°

10. Use a protractor to construct the following

angles.

(a) reflex ∠ABC = 310°

(b) 95° (b) reflex ∠PQR = 290°

○○○○→→→→ Ex 4A 12

A B

∵ 310° = 360° – 50° ∴ Construct an angle of 50°

(i.e. acute ∠ABC) first.

Then reflex ∠ABC is 310°.

290° = 360° – ( )°

Step 1111: Draw a line segment AB. (This step has been done for you.)

Step 2222: Place the base line of the protractor along AB and the centre on B.

Step 3333: Mark point C at 140° by using

the inner scale.

Step 4444: Join BC. Then ∠ABC = 140°.

8

4.1C Parallel and Perpendicular Lines

In the same plane,

(a) straight lines that will never intersect are

called parallel lines,

e.g. In the figure, AB // CD.

(b) two straight lines that meet each other at a right

angle (i.e. 90°) are called perpendicular lines.

e.g. In the figure, PQ ⊥ RS.


Use symbols to write down all pairs of parallel

lines and 2 pairs of perpendicular line segments

observed in the figure. Mark the answers on the

figure.

Sol

Parallel lines: FA // EC, AC // FE.

Perpendicular line segments:

GB ⊥ AC, GD ⊥ EC.

(or GB ⊥ AB or GB ⊥ BC or GD ⊥ CD or

GD ⊥ DE)

Use symbols to write down all pairs of parallel

lines and 1 pair of perpendicular line segments

observed in the figure. Mark the answers on the

figure.

Sol

○○○○→→→→ Ex 4A 15, 16

T

Q

P

S

R

A B C

D

E F

G

T

Q

P

S

R

A B C

D

E F

G

A B

C D

P

R Q

S

� Use arrows in the same direction to indicate parallel lines.

Use different numbers of arrows to indicate 2 groups of parallel lines.

9

�� ‘Explain Your Answer’ Question

11. Refer to the figure. It is given that ∠ACB is a straight

angle. Cathy claims that ∠DCE is a right angle. Do

you agree? Explain your answer.

∵ ∠ACB = ( )°

∴ ∠DCE = ( )° – ( )° – ( )°

=

i.e. ∠DCE (is / is not) a right angle.

∴ The claim is (agreed / disagreed).

�� Level Up Questions

12. In each of the following, find the sizes of all the angles and arrange them in ascending order

of size.

9

5 of a straight angle,

3

4 of a right angle,

6

1 of a round angle

13.

Refer to the figure.

(a) Draw and name all the line segments formed by joining any two points in the figure.

How many line segments have been drawn?

(b) Among all the line segments in (a), write down

(i) 1 pair of parallel lines, (ii) 1 pair of perpendicular lines.

A

B

C

D

50° 40° A B

C

D E

Remember to write down the reason.

10

4 Introduction to Geometry

Level 1


(a) Name all the fixed points on the straight line L.

(b) Name all the line segments on the straight line L.

2. (a) Name all the line segments in the figure.

(b) Write down the shortest line segment.


(a) How many line segments are there in the figure?

(b) Use a ruler to measure the length of the line segment AC.

4. By observation only, find the number of plane(s) and the number of curved surface(s) in each of the

following figures.

(a) (b) (c)

Consolidation Exercise 4A ��

B

A

CD

F

LE

P

O

R

Q

11

5. Name the acute angle in the figure in three different ways.

B

A

C

6. Use the vertex and the symbol ‘∠’ to name angles p and q in each of the following figures. (a) (b) (c)

7. Refer to the figure below.

a

b

d

c

e

f

Determine the types of the angles a, b, c, d, e and f according to their sizes.

8. Arrange the following angles in ascending order of sizes.

∠A is a right angle. ∠B is a reflex angle. ∠C = 175° ∠D = 84°

9. Determine the types of the following angles according to their sizes.

126°, 89°, 27°, 190°, 345°, 162°, 45°, 92°

Type Acute angle Obtuse angle Reflex angle

Size

12

10. Refer to the figure. Find the sizes of the following angles.

(a) ∠AOD

(b) ∠BOC

(c) ∠COD

11. Use a protractor to measure the following angles.

(a) (b) (c) (d)

12. Use a protractor to construct the following angles.

(a) 72° (b) 145° (c) 280°

Level 2

13. Complete the following table.

Angle 3 right angles ____ right angles ____ straight angles

Size 180°

Type round angle

14. In each of the following, find the sizes of all the angles and arrange them in descending order of size.

(a) 3

2 of a right angle,

5


8

1 of a round angle.

(b) 5


10

9 of a round angle,

2

7 of a right angle.

15. Without using a protractor, find the marked angle in each of the following clocks.

(a)

(b)

180170

0 0

10

16020

15030 140

40 13

050 12

0

60

110

70

100

80

80

100

70

110

60

120

50

130

40

140

30

150

20

160

10

170

180

90

A BO

D

C

13

16. In the figure, ABCDEFGH is a trapezium. BCDP is a square and DEFHP is a rectangle.

(a) Use arrowheads to indicate all the parallel lines in the figure.

(b) Use the symbol for right angle to indicate all the perpendicular

lines in the figure.

17. (a) In the figure, draw a line

(i) passing through D and perpendicular to AB,

(ii) passing through E and parallel to BC.

(b) Are the lines drawn in (a) parallel to each other?

18. In the figure, all points and lines are in the same plane. There are 5 lines intersecting at the same point.

Name these 5 lines and their point of intersection.

B

A

C

D

FE

H

I

G

19. Refer to the figure. Find the sizes of the following angles.

(a) ∠AOB

(b) ∠COD

14

Consolidation Exercise 4A (Answer)

1. (a) B, D, F (b) BD, BF, DF

2. (a) OP, PQ, QR, OR (b) QR

3. (a) 10 (b) 3.5 cm

4. (a) 1 curved surface (b) 5 planes

(c) 1 curved surface, 3 planes

5. ∠B, ∠ABC, ∠CBA

6. (a) p = ∠AMD, q = ∠MNF (or ∠ANF)

(b) p = ∠ABC, q = reflex ∠ADC

(c) p = ∠FGH, q = reflex ∠EHG

7. a: right angle, b: reflex angle, c: acute angle,

d: obtuse angle, e: straight angle,

f: round angle

8. ∠D, ∠A, ∠C, ∠B

9. Type Acute angle

Obtuse angle

Reflex angle

Size

89°, 27°,

45°

126°,

162°, 92°

190°,

345°

10. (a) 130° (b) 140° (c) 90°

11. (a) 70° (b) 100°

(c) 300° (d) 90°

13. Angle

3 right angles

2 right angles

2 straight angles

Size 270° 180° 360°

Type

reflex angle

straight angle

round angle

14. (a) 60°, 108°, 45°;

5


3

2 of a right angle,

8

1 of a round angle

(b) 288°, 324°, 315°;

10

9 of a round angle,

2

7 of a right angle,

5

8 of a straight

angle

15. (a) 120° (b) 150°

17. (b) yes

18. AB, AG (or AE), AH (or AF), AI (or AC), AD;

point of intersection: A

19. (a) 36° (b) 66°

15

F1A: Chapter 4B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)



(Full Solution)

Maths Corner Exercise 4B Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 4B Multiple Choice


___________ ( )



_________

16

Book 1A Lesson Worksheet 4B (Refer to §4.2)

4.2A Circles

(a) All points on a circle are at the same

distance from a fixed point (centre).

(b) The figure shows a circle with centre O.

(i) Line segment OA is a radius.

(ii) Line segment BE is a diameter.

(iii) Curved line CD, which is a

part of the circumference,

is called an arc.

1. Refer to the circle with centre O in the figure. Match the

left column with the right column.

• line segment OR

(a) Radius • • line segment OS

(b) Diameter • • line segment RQ

(c) Arc • • line segment POR

• curved line PT

2. In the figure, O is the centre. Use a ruler to measure the diameter

of the circle.

○○○○→→→→ Ex 4B 1

O

X

Y

Z

W

O

P

Q

T

S

R

Which line segment in the figure is a diameter?

Diameter

= 2 × radius

Circles or arcs can be drawn by using a

pair of compasses.

17

4.2B Triangles

A triangle is enclosed by three line segments.

The triangle in the figure can be denoted by △ABC.

I. Types of Triangles

According to the sizes of angles:

Name Acute-angled triangle Right-angled triangle Obtuse-angled triangle

Nature All angles are acute

angles. One of the angles is a

right angle. One of the angles is an

obtuse angle.

Example

(The largest angle is an acute angle.)

(The largest angle is a right angle.)

(The largest angle is an obtuse angle.)


Use a protractor to identify △ABC and state

whether it is an acute-angled triangle, an

obtuse-angled triangle or a right-angled

triangle.

Sol The largest angle in △ABC is an obtuse

angle.

∴ △ABC is an obtuse-angled triangle.

Use a protractor to identify △DEF and state

whether it is an acute-angled triangle, an

obtuse-angled triangle or a right-angled

triangle.

Sol The largest angle in △DEF is a/an

angle.

∴ △DEF is .

According to the lengths of sides:

Name Scalene triangle Isosceles triangle Equilateral triangle

Nature No two sides are equal. Two sides are equal. All sides are equal.

Example

Note: ‘

’, ‘

’ are used to indicate different pairs of equal sides in the figure.

D

E F

A B

C

The largest angle in △ABC is ∠C. It is an obtuse angle.

The largest angle in △DEF is

∠ . It is a/an angle.

18


Use a ruler to identify △ABC and state

whether it is a scalene triangle, an isosceles

triangle or an equilateral triangle.

Sol In △ABC, AC = CB = 2 cm.

∴ △ABC is an isosceles triangle.

Use a ruler to identify △DEF and state

whether it is a scalene triangle, an isosceles

triangle or an equilateral triangle.

Sol

∴ △DEF is .

For each △XYZ in Nos. 3–4,

(a) use a protractor to identify the triangle and state whether it is an acute-angled triangle, an

obtuse-angled triangle or a right-angled triangle;

(b) use a ruler to identify the triangle and state whether it is a scalene triangle, an isosceles

triangle or an equilateral triangle. 3.

4.

○○○○→→→→ Ex 4B 3, 4

5. Refer to the triangles below.

Name the triangles according to the following types, if any.

(a) Acute-angled triangle(s)

(b) Equilateral triangle(s)

(c) Right-angled triangle(s)

X

Y

Z

Y

X Z

P

Q R

X

Y

Z

D E

F

3 cm

3 cm

3 cm

A B

C

D

E F

Are there any sides of equal length?

AB = ____ cm

Does it have equal length as AC and CB?

19

II. Angle Sum of a Triangle

In a triangle ABC,

a + b + c = 180°

[Reference: ∠ sum of △]

6. Can each set of angles below be the interior angles of a triangle?

(a) 30°, 30°, 40° (b) 100°, 20°, 60°

30° + 30° + 40° =

∵ (= / ≠) 180°

∴ This set of angles (can / cannot)

be the interior angles of a triangle.

○○○○→→→→ Ex 4B 5


Find x in the figure.

Sol x + 50° + 70° = 180° �∠ sum of △

x + 120° = 180°

x = 180° − 120°

= 60°

Find y in the figure.

Sol y + ( ) + ( ) = 180° �∠ sum of △

=

Find the unknown angle in each of the following triangles. [Nos. 7–10]

7.

8.

9. 10.

○○○○→→→→ Ex 4B 6–11 X

Y Z d 25°

45°

D

E F

c

V

T

U

b 35° 35°

P

Q R

125°

30°

a

∠E is a right angle,

i.e. °.

B C

A

50°

70°

x 40°

F E

D

70°

y

20

III. Constructions of Triangles

11. Use a pair of compasses and a ruler to construct △ABC, where AB = 4 cm, BC = 5 cm and

AC = 3 cm.

12. Use a ruler and a protractor to construct △PQR, where PQ = 3 cm, ∠RPQ = 60° and

RP = 4 cm.

13. Use a pair of compasses and a ruler to

construct △ABC, where AB = 3.5 cm,

BC = 4 cm and AC = 4 cm.

○○○○→→→→ Ex 4B 12–14

14. Use a ruler and a protractor to construct

△PQR, where PQ = 4 cm, ∠RPQ = 40°

and RP = 3 cm.

○○○○→→→→ Ex 4B 15–17 A B 3.5 cm

A B 4 cm

Step 1111: Use a ruler to draw a line segment AB of 4 cm.

(This step has been done for you.)

Step 2222: With centre at point A and radius 3 cm, use a

pair of compasses to draw an arc.

Step 3333: With centre at point B and radius 5 cm, use a pair of compasses to draw another arc.

Step 4444: The two arcs drawn have to meet at a point C.

Step 5555: Join AC, then BC. △ABC is drawn.

Step 1111: Use a ruler to draw a line segment PQ of

3 cm. (This step has been done for you.)

Step 2222: Use a protractor to draw ∠TPQ which

measures 60°.

Step 3333: Use a ruler to mark a point R on PT (or PT produced) such that RP = 4 cm.

Step 4444: Join QR, then △PQR is drawn.

Q P 3 cm

21

Regular triangle (Equilateral triangle)

Regular hexagon

60°

60° 60°

4.2C Polygons

(a) A polygon is enclosed by 3 or more

line segments.

(b) Polygons can be named by the number of sides. A polygon

with n sides (n ≥ 3) is called an n-sided polygon.

e.g. A polygon with 5 sides is called a 5-sided polygon or a

pentagon.

(c) If a polygon is both equilateral (equal sides) and

equiangular (equal interior angles), then it is called a

regular polygon.

e.g.

15. Refer to the polygons I, II, III and IV below.

List all

(a) hexagon(s), (b) equilateral polygon(s),

(c) equiangular polygon(s), (d) regular polygon(s).

16. Refer to the polygons I, II, III and IV below.

List all

(a) quadrilateral(s), (b) equilateral polygon(s),


○○○○→→→→ Ex 4B 18

270°

I III IV II

III I II

1.5

1.5

1.5

1.5

1.5

IV

3

3

3

3

�‘ ’ is used to indicate the equal angles in

the figure.

�A diagonal is the line segment joining two non-

adjacent vertices. diagonal side

vertex

A polygon with all interior angles

smaller than 180° is called a convex polygon.

∵ One of the

interior angles of IV is greater

than 180°. ∴ IV is also called

a concave

polygon.

22


17. Refer to the triangles below. By observation only, which one is a right-angled scalene

triangle? Explain your answer.

�� Level Up Question

18. Find x and y in △ABC.

I II III IV

60° x

B

C y

D A

110° 30°

x is an interior angle of △ABD.

x and ∠CBD (= 60°) together form an interior angle of △ABC.

23


Level 1

1. In the figure, O is the centre of the circle.

(a) Use a ruler to measure the radius of the circle.

(b) Find the diameter of the circle.

2. In the figure, OABC is a square. A and C are two points on the circle

with centre O. It is given that AB = 5 cm.

(a) Find the radius of the circle.

(b) Find the diameter of the circle.

3. Use a protractor to identify each of the following triangles and state whether it is an acute-angled

triangle, an obtuse-angled triangle or a right-angled triangle.

(a) (b) (c)

4. Use a ruler to identify each of the following triangles and state whether it is a scalene triangle, an

isosceles triangle or an equilateral triangle.

(a) (b) (c)

5. Can each set of angles below be the interior angles of a triangle?

(a) 85°, 50°, 65° (b) 45°, 75°, 60° (c) 70°, 30°, 70°

Find the unknown angle in each of the following triangles. [Nos. 6–11]

6.

B

a

A

C

38°

75°

7.

B

a

A C40° 30° 8.

Consolidation Exercise 4B ��

O

O

B

A

C

24

9.

B

d

A C

55°

10.

B

e

A

C48°48°

11. B

A C

45°

f

Use a pair of compasses and a ruler to construct the following triangles. [Nos. 12–14]

12.

B

A

C5 cm

3 cm 3 cm

13.

4 cm

5 cm3 cm

P

RQ

14.

2 cm

2 cm 2 cm

X

Y Z

Use a ruler and a protractor to construct the following triangles. [Nos. 15–17]

15.

5 cm

35°

M

NP

16. 3 cm

100°50°

F

EG 17.

B

AC

5 cm4 cm

75°

18. Refer to the polygons A, B, C and D as shown.

List all

(a) quadrilateral(s), (b) equilateral polygon(s),


25

19. In each of following figures, find the number of the diagonals passing through the vertex A.

(a)

B

A

C

D

E

(b)

B

A

C

D

H

F

E

G

Level 2

20. Using a pair of compasses and a ruler, construct the figure on the right.

21. Refer to the triangles as shown.

A

B

C

D

E

F

By observation only,

(a) classify the triangles by the sizes of their angles.

(b) classify the triangles by the lengths of their sides.

(c) (i) which is/are obtuse-angled scalene triangle(s)?

(ii) which is/are acute-angled isosceles triangle(s)?

26

Find the unknown(s) in each of the following triangles. [Nos. 22–25]

22.

B

4y

2y

x

A

CD

52°

23.

24. 25.

26. Construct △ABC, where AB = 3 cm, BC = 3.5 cm and AC = 5 cm.

27. (a) Construct △MNP, where MN = 6 cm, NP = 8 cm, and ∠MNP = 90°.

(b) Measure the length of MP.

28. (a) Using a pair of compasses and a ruler, construct a circle with diameter AC = 7 cm.

(b) (i) Construct △ABC, where AB = 4.5 cm and B is a point on the circumference.

(ii) Is △ABC a right-angled triangle?

29. In the figure, A, B and C are three points on the circle with centre O. The

diameter BOC is 10 cm and AC = 5 cm. Is △OAC an equilateral

triangle? Explain your answer.

30. In the figure, AC and BD are diameters of the circle with centre O.

ABCD is a square. Write down the names and the number of

right-angled isosceles triangles in the figure.

B A

C

O

B

A C

D

O

27

Consolidation Exercise 4B (Answer)

1. (a) 1.4 cm (b) 2.8 cm

2. (a) 5 cm (b) 10 cm

3. (a) acute-angled triangle

(b) right-angled triangle

(c) obtuse-angled triangle

4. (a) isosceles triangle (b) scalene triangle

(c) equilateral triangle

5. (a) no (b) yes (c) no

6. 67° 7. 110° 8. 20°

9. 35° 10. 84° 11. 45°

18. (a) A, B (b) C, D

(c) B, D (d) D

19. (a) 2 (b) 5

21. (a) acute-angled triangle: A, E, F

right-angled triangle: C, D

obtuse-angled triangle: B

(b) scalene triangle: B, C, E

isosceles triangle: A, D

equilateral triangle: F

(c) (i) B (ii) A, F

22. x = 38°, y = 15°

23. m = 10°, n = 75°

24. 20°

25. p = 15°, q = 45°

27. (b) 10 cm

28. (b) (ii) yes

29. yes 30. △AOB, △BOC, △COD, △AOD, △ABC, △BCD, △CDA, △DAB, a

total of eight right-angled isosceles triangles

28

F1A: Chapter 4C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)



(Full Solution)

Maths Corner Exercise 4C Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 4C Multiple Choice


___________ ( )



_________

29

Book 1A Lesson Worksheet 4C (Refer to §4.3A–B, D–E)

4.3A Introduction to Solid Figures

(a) Solids enclosed by planes or curved surfaces are

called solid figures.

e.g. Triangular pyramids and spheres are solid

figures.

(b) A solid figure that is enclosed by polygons only

is called a polyhedron.

e.g. Triangular pyramids are polyhedra, but spheres are not.

(c) A polyhedron with n faces is called an n-faced polyhedron.

e.g. A triangular pyramid has 4 faces. It is called a 4-faced polyhedron (or tetrahedron).

1. Determine which of the following solids is (are) a polyhedron (polyhedra).

○○○○→→→→ Ex 4C 1


Count and write down the numbers of vertices,

edges and faces of the given solid.

Count and write down the numbers of vertices,

edges and faces of the given solid.

Sol The solid has 12 vertices, 18 edges,

8 faces.

Sol The solid has vertices, edges,

faces.

2. Count and write down the numbers of vertices, edges and faces of the following solids.

(a) (b)

○○○○→→→→ Ex 4C 2, 3

I II III IV Is a circle a polygon?

Triangular pyramid Sphere

30

4.3B Euler’s Formula

For a convex polyhedron,

the relation among the numbers of faces (F), vertices (V) and edges (E) is:

F + V – E = 2


Given that a convex polyhedron has 8 faces

and 10 vertices, find the number of edges.

Sol ∵ F + V – E = 2

i.e. 8 + 10 – E = 2

∴ E = 16

i.e. The number of edges is 16.

Given that a convex polyhedron has 8 vertices

and 15 edges, find the number of faces.

Sol ∵ F + V – E = ( )

i.e.

3. Complete the following table on the numbers of faces, vertices and edges of a convex

polyhedron by using Euler’s formula.

Number of faces (F) Number of vertices (V) Number of edges (E)

(a) 4 6

(b) 7 7

(c) 6 5

(d) 13 20 ○○○○→→→→ Ex 4C 4

Count the numbers of faces (F), vertices (V) and edges (E) of each of the following polyhedra

and determine whether Euler’s formula holds in each of these polyhedra. [Nos. 4–5]

4.

By counting,

number of faces (F) =

number of vertices (V) =

number of edges (E) =

∵ F + V – E =

∴ Euler’s formula (holds /

does not hold) for this polyhedron.

5.

○○○○→→→→ Ex 4C 5, 6

31

4.3D Cross-sections

I. Uniform Cross-section

(a) The face obtained by cutting a solid along a certain plane is called a cross-

section of the solid.

(b) If we obtain the same (identical in shapes and sizes) cross-section by cutting

a solid along a certain direction, then it is called a uniform cross-section.

e.g.

II. Non-uniform Cross-section

e.g.


(a) Draw the cross-section of

the given solid when it is

cut horizontally along AB.

(b) Determine if the

cross-section is a uniform cross-section.

Sol (a) The cross-section is:

(b) It is a uniform cross-section.

(a) Draw the cross-section of

the given solid when it is

cut vertically along CD.

(b) Determine if the

cross-section is a uniform cross-section.

Sol (a) The cross-section is:

6. (a) Draw the cross-section of the given

solid when it is cut horizontally along

MN.

(b) Determine if the cross-section is a

uniform cross-section.

7. (a) Draw the cross-section of the given

solid when it is cut vertically along

XY.

(b) Determine if the cross-section is a

uniform cross-section.

○○○○→→→→ Ex 4C 10, 11

non-uniform cross-section

uniform cross-section

A B

C

D

The cross-sections are rectangles of difference sizes.

Cut the solid horizontally

along AB and XY.

X

Y

M N

32

4.3E Nets for Polyhedra

A net can be used to make a model of a polyhedron.

A polyhedron may be formed by more than one net.

e.g.

8. Which of the following polyhedra can be made by folding the

net as shown on the right?

9. Which of the following nets cannot be folded into the cube

as shown on the right?

○○○○→→→→ Ex 4C 12

II I III

III I II IV

How many faces are there

in the polyhedron formed?

33


10. Susan wants to make a convex polyhedron with 13 faces, 8 vertices and 20 edges. Do you

think this polyhedron can be made? Explain your answer.

Suppose a convex polyhedron has ( ) faces, ( ) vertices and ( ) edges.

Then F + V – E =

∵ F + V – E ( = / ≠ ) 2

∴ This polyhedron (can / cannot) be made.


11. Peter uses 12 sticks to make the skeleton of a 6-faced polyhedron.

(a) How many polymer clays does he need to make this 6-faced polyhedron?

(b) What kind of polyhedron is the skeleton he makes? Sketch the polyhedron.

12. (a) Identify and draw the cross-section of each of the following solids according to the given

instruction.

(i) Solid I is cut vertically along (ii) Solid II is cut horizontally along

the line CD. the line UV.

(b) Which solid(s) above has (have) a uniform cross-section?

I II

C D

U V

Sticks are the edges of the polyhedron; polymer clays are the (vertices / faces) of the polyhedron.

Euler’s formula: F + V – E = 2


34


Level 1

1. Determine which of the following solids is (are) a polyhedron (polyhedra).

A C D

2. Count and write down the numbers of vertices, edges and faces of the following solids.

(a) (b) (c)

3. Refer to the solids A, B and C as shown.

A B C

(a) Which one is a hexahedron?

(b) Which one has the greatest number of edges? How many edges are there in that solid?

4. Complete the following table on the numbers of faces, vertices and edges of a convex polyhedron by

using the Euler’s formula.

Number of faces (F) Number of vertices (V) Number of edges (E)

(a) 8 14

(b) 7 11

(c) 15 13

(d) 9 12

Consolidation Exercise 4C ��

35

Count the numbers of faces (F), vertices (V) and edges (E) of each of the following 3-D figures and

determine whether Euler’s formula holds in each of these figures. [Nos. 5–6]

5. 6.

7. Draw a sketch of the solid in the photo.

8. The figure shows a solid formed by 4 cubes each of side 1 cm. Use

solid lines and dotted lines to draw the 2-D representation of this solid

on isometric grid paper.

9. Draw the 2-D representation of the solid on oblique grid paper.

10. Draw the cross-section of each of the following solids according to the given instruction.

(a) The solid is cut vertically (b) The solid is cut horizontally

along the line AB. along the line CD.

D

C

36

11. Each solid shown is cut vertically along the line AB.

(a) Draw the cross-section obtained.

(b) Determine if the cross-section is a uniform cross-section.

12.

Which of the following figures is/are the net(s) of the given polyhedron M?

A C

Level 2

13. Stacy uses 6 polymer clays and 12 sticks to make the skeleton of a convex polyhedron.

(a) How many faces does the polyhedron have?

(b) What kind of polyhedron is the skeleton she made? Sketch the polyhedron.

14. Peter claims that he can make a convex polyhedron with 11 faces, 8 vertices and 15 edges. Do you

agree? Explain your answer.

15. The figure below shows solids consisting of cubes each of side 1 cm. Draw the 2-D representation of

the solids on isometric grid paper.

(a) (b)

37

16. The figure shows a solid consisting of cubes each of side 2 cm.

Draw the 2-D representation of the solids on oblique grid paper.

17. (a) Identify and draw the cross-section of each of the following solids according to the given

instruction.

(i) Solid I is cut

horizontally along

the line MN.

(ii) Solid II is cut

vertically along

the line PQ.

(iii) Solid III is cut

horizontally along

the line RS.

I

II

R

S

III

(b) Which of the above cross-sections is (are) a uniform cross-section?

18. Draw the cross-section obtained when the given solid as shown is

cut along each of the following lines.

(a) AB

(b) MN

(c) PQ

19. The nets given below can be folded to form two different solids. Use solid lines and dotted lines to

draw the 2-D representation of each solid formed on isometric grid paper.

(a)

2 cm

1 cm

1 cm

(b)

38

Consolidation Exercise 4C (Answer)

1. B, C

2. (a) 8 vertices, 12 edges, 6 faces

(b) 12 vertices, 18 edges, 8 faces

(c) 6 vertices, 12 edges, 8 faces

3. (a) A (b) B, 18

4. (a) 8 (b) 6 (c) 26 (d) 5

5. F = 6, V = 8, E = 12; yes

6. F = 7, V = 6, E = 11; yes

11. (b) I: no, II: yes, III: no

12. A, C

13. (a) 8 (b) octahedron

14. no

17. (a) (i) rectangle (ii) hexagon

(iii) octagon (b) solid II, solid III

39

F1A: Chapter 5A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)



(Full Solution)




___________ ( )

Maths Corner Exercise ○ Complete and Checked Teacher’s

40

5A Level 2 ○ Problems encountered ○ Skipped

Signature ___________ ( )




___________ ( )



___________ ( )



_________

41


5.1A Finding the Percentage

If x is to be expressed as a percentage of y, then

the percentage = %100×y

x


What percentage of 45 is 27?

Sol The required percentage

= %10045

27×

= 60%

What percentage of 70 is 21?

Sol The required percentage

= %100) (

21×

= %

1. What percentage of 50 g is 47 g? 2. What percentage of $75 is $105?

○○○○→→→→ Ex 5A 1, 2

3. There are 50 fish and 20 shrimps in a

pond. What percentage of the number of

the fish is that of the shrimps?

4. The price of a shirt is $124 and the price of

a belt is $80. What percentage of the price

of the belt is that of the shirt?

○○○○→→→→ Ex 5A 3, 4

47 g (?%)

50 g

27 (?%)

45

42

Percentage of a part to the whole = %100whole

part×


There are 30 students and 3 of them are girls.

(a) Find the number of boys.

(b) Find the percentage of boys.

Sol

(a) Number of boys

= 30 – 3

= 27

(b) The required percentage

= %10030

27×

= 90%

Alice answered 40 questions and 30 of them

are correct.

(a) Find the number of wrong answers.

(b) Find the percentage of the number of

wrong answers.

Sol (a) Number of wrong answers

= 40 – ( )

=

(b) The required percentage

= %10040

) (×

=

5. There are 15 pencils in a box and 6 of

them are used. What is the percentage of

unused pencils in the box?

6. There are 20 cups. 5 of them are in yellow

and the rest are in green. Find the

percentage of green cups.

○○○○→→→→ Ex 5A 5–7

? unused pencils

(?%)

6 used

pencils

15 pencils

? boys

(?%) 3 girls

30 students

43

5.1B Using Percentage to Find a Quantity

(a) a% of a quantity y = y × a%

(b) Part = whole × percentage of the part to the whole


What is 20% of 70?

Sol 20% of 70

= 70 × 20%

= 70 ×100

20

= 14

What is 80% of 90?

Sol 80% of 90

= ( ) × ( )%

= ( ) ×100

) (

=

7. What is 25% of 40 km?

8. What is 150% of 60 °C?

○○○○→→→→ Ex 5A 8, 9

9. In a bag of 200 candies, 70% of them are

red candies. Find the number of red

candies in the bag.

Number of red candies

=

10. At a clinic, there were 55 patients

yesterday. The number of patients today is

140% of the number of patients yesterday.

Find the number of patients today.

○○○○→→→→ Ex 5A 10–12

?

(20%)

70

? km (25%)

40 km

44

5.1C Finding the Original Quantity from a Given Percentage

If a% of a quantity y is equal to n,

then y can be found by setting up the equation y × a% = n.


10% of k is 13. Find the value of k.

Sol k × 10% = 13

k × 0.1 = 13

k = 13 ÷ 0.1

= 130

16% of y is 8. Find the value of y.

Sol y × ( )% = 8

y × ( ) = 8

y = 8 ÷ ( )

=

11. 90% of a cm is 144 cm. Find the value of

a.

12. 65% of g mL is 78 mL. Find the value of

g.

○○○○→→→→ Ex 5A 13, 14

13. The tuck shop of a school sells 304

chicken wings today, which is 80% of that

sold yesterday. Find the number of chicken

wings sold yesterday.

Let m be the number of chicken wings sold

yesterday.

Then % of m is 304.

i.e. m × ( )% = 304

=

14. Mrs Wong uses 96 g of flour to make some

bread. The weight of flour used is 12% of

the total weight of a bag of flour. Find the

total weight of the bag of flour.

○○○○→→→→ Ex 5A 15−17

144 cm (90%)

a cm

Unit must be included

in the answer.

13

(10%)

k

45

�� ‘Explain Your Answer’ Questions

15. There are 120 coins in a bag. 30% of them are $2 coins. Is it possible that Grace takes 40

$2 coins from the bag? Explain your answer.

Number of $2 coins

=

∵ ______ ( < / > ) 40

∴ It (is / is not) possible that Grace takes 40 $2 coins from the bag.

16. There are 720 dolls in a factory. The manager plans to send 100 dolls to a retailer. If 85% of

the dolls are lost in a fire, does the factory have enough dolls for the retailer? Explain your

answer.


17. 50 students took a mathematics test. 72% of them passed the test. Find the number of students

who failed the test.

18. There are 510 male employees in a company and 85% of the employees in the company are

male. Find the number of female employees.

Pass (72%) Fail

Total number of students = 50


510 male employees

(85%)

? female employees

Total number of employees

46

5 Percentages (I)

Level 1

1. (a) What percentage of 100 is 45?

(b) What percentage of 80 is 120?

2. (a) What percentage of 250 m2 is 140 m2?

(b) What percentage of 60 mL is 72 mL?

3. There are 26 pencils and 65 pens in a box. What percentage of the number of pens is that of

pencils?

4. A tank contains 63 red goldfish and 27 yellow goldfish only. What percentage of the goldfish in the

tank are red?

5. A group of firemen takes a physical fitness test, in which 63 pass the test and 12 fail the test. Find the

percentage of firemen in the group who pass the test.

6. In a book of 320 pages, chapter I has 18 pages and chapter II has 14 pages. Find the percentage of the

total number of pages of these two chapters in the book.

7. There are 70 teachers in a school and 28 of them are male. Find the percentage of female teachers in

the school.

8. Find the result of each of the following.

(a) 60% of 25 (b) 12% of 400

(c) 150% of 8 (d) 30% of 30

9. Find the result of each of the following.

(a) 75% of 96 m2 (b) 12% of 750 mL

(c) 240% of $150 (d) 8% of24

1km

10. A fruit shop sold 180 apples yesterday. The number of apples sold today is 95% of that of yesterday.

Find the number of apples sold today.


47

11. A special drink is made up of 72% orange juice and 28% soda water by volume. Find the volume of

orange juice and that of soda water in a special drink of volume 125 mL.

12. In an English lesson of 75 minutes, 20% of the time is used for a quiz and 36% of the time is for

showing a movie. How much time in total is used for the quiz and the show of movie?

13. John answers 93% of the questions in a test. What percentage of questions are not answered?

14. Yesterday, 6 400 tickets of a concert were available for sale. Up to now, 75% of the tickets have been

sold. Find the number of unsold tickets.

15. In a party, 68% of the 450 participants are female. Find the number of male participants.

16. Find the unknown in each of the following.

(a) 8% of p is 4. (b) 24% of q is 36.

(c) 65% of r is 78. (d) 135% of s is 81.

17. Find the unknown in each of the following.

(a) 86 cm is 43% of x cm. (b) 24 L is 60% of y L.

(c) $7 000 is 280% of $m . (d) 6 kg is 96% of n kg.

18. The income of a salesman is $36 000 this month, which is 80% of the income last month.

Find the income of the salesman last month.

19. Ann spends 2 hours on writing a composition, which is 30% of the total time she spends on doing

homework. Find, in minutes, the total time she spends on doing homework.

20. Among the 220 employees in a company, 15% live on Hong Kong Island. The number of employees

living in Kowloon is 3 times that living on Hong Kong Island. How many employees live in Kowloon?

21. Among the 192 students in a school,

6

1 are boy scouts and the number of girl scouts is 8 less than that

of boy scouts. What percentage of the students in the school are girl scouts?

22. In city A, 270 000 people are university graduates, which are 18% of the population. The population in

city B is 1.6 million. Which city has a greater population? Explain your answer.

48

Level 2

23. There are 15 000 visitors in a theme park. 64% of the visitors are foreigners.

(a) Find the total number of foreign visitors.

(b) If 7 200 foreign visitors are Japanese, what percentage of the foreign visitors are Japanese?

24. All the 45 students in a class vote for the picnic destination. It is known that 20% vote for Stanley, one

third vote for Aberdeen and the rest vote for Tai O. How many students vote for Tai O?

25. Ken has three types of books: comic books, fiction books and magazines. He has 256 books and

37.5% of them are comic books.

(a) Find the number of non-comic books.

(b) Among the non-comic books, 65% are magazines. Find the number of fiction books.

26. In a sports event, a country gets 150 medals, of which 8% are gold medals, 36% are silver medals and

the rest are bronze medals. Find the difference between the number of silver medals and that of bronze

medals.

27. Susan buys 4 dozens eggs. She finds that 62.5% of the eggs are good.

(a) How many eggs are good?

(b) If she buys two more eggs and both of them are good, find the new percentage of eggs that are

good.

28. The height of Frankie is 150 cm. The height of Lucy is 80% of that of Frankie.

(a) Find the height of Lucy.

(b) The height of Mark is 120% of that of Lucy. Among these three people, who is the tallest?

Explain your answer.

29. The favourite sports of some students are shown in the following table.

Football Swimming Cycling

Number of boys 99 72 45

Number of girls 51 66 117

(a) What percentage of the students choose ‘swimming’?

(b) What percentage of the girls choose ‘cycling’?

(c) What percentage of the students choosing ‘football’ are boys?

30. In a photo album, 64% of the photos are portraits and all the remaining 432 are scenic photos.

(a) Find the number of photos in the album.

(b) If 270 photos are black and white, what percentage of the photos in the album are black and

white?

49

31. There are some bottles of drinks in a store. It is given that 45% are soft drinks, 36% are tea and the

remaining 57 are juice.

(a) How many bottles of drinks are there in the store?

(b) If 81 bottles of soft drinks are sugar-free, what percentage of the bottles of soft drinks are sugar-

free?

32. In a group of tourists, 15% are Europeans, where 24 are Britons and the remaining 36 are Germans.

Find the percentage of Britons in the group.

33. In a hospital, 48% of the male patients and 25% of the female patients are smokers. If 65 male patients

and 39 female patients are non-smokers, find the number of patients in the hospital.

34. In a group of students, 35% go to school by bus and 20% go to school by tram. If the difference in the

numbers of students using the two types of vehicles is 54, find the total number of students in the

group.

35. The width of a rectangle is 60% of the length. The perimeter of the rectangle is 80 cm.

(a) Find the length.

(b) Is the area of the rectangle greater than 400 cm2? Explain your answer.

36. In a trip, Carman spent $2 400 more than Dennis. If 35% of the sum of their expenditures was spent by Dennis, how much did Carman spend?

50


1. (a) 45% (b) 150%

2. (a) 56% (b) 120%

3. 40% 4. 70%

5. 84% 6. 10%

7. 60%

8. (a) 15 (b) 48

(c) 12 (d) 9

9. (a) 72 m2 (b) 90 mL

(c) $360 (d) 300

1km

10. 171

11. orange juice: 90 mL, soda water: 35 mL

12. 42 minutes 13. 7%

14. 1 600 15. 144

16. (a) 50 (b) 150

(c) 120 (d) 60

17. (a) 200 (b) 40

(c) 2 500 (d) 6.25

18. $45 000 19. 400 minutes

20. 99 21. 12.5%

22. city B

23. (a) 9 600 (b) 75%

24. 21

25. (a) 160 (b) 56

26. 30

27. (a) 30 (b) 64%

28. (a) 120 cm (b) Frankie

29. (a) %3

230 (b) 50%

(c) 66%

30. (a) 1 200 (b) 22.5%

31. (a) 300 (b) 60%

32. 6%

33. 177 34. 360

35. (a) 25 cm (b) no 36. $5 200

51

F1A: Chapter 5B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

52



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

53

Book 1A Lesson Worksheet 5B (Refer to §5.2)

5.2A Percentage Increase

(a) Increase = new value – original value �

(b) Percentage increase = valueoriginal

increase× 100%


If 10 is increased by 3, find the percentage

increase.

Sol Percentage increase

= %10010

3×

= 30%

If 20 is increased by 5, find the percentage

increase.


= %100) (

) (×

= %

1. If 30 mL is increased by 12 mL, find the

percentage increase.

Percentage increase = %100) (

) (×

=

2. The selling price of a cup was $60 last

month. It is increased by $9 this month.

Find the percentage increase.

○○○○→→→→ Ex 5B 1(a)


If 25 is increased to 40, find the percentage

increase.


= %10025

2540×

−

= %10025

15×

= 60%

If 14 is increased to 21, find the percentage

increase.


= %100) (

) () (×

−

=

3. If 15°C is increased to 18°C, find the

percentage increase.

Percentage increase

=

4. Peter was 140 cm tall last year and he is

147 cm tall this year. Find the percentage

increase.

○○○○→→→→ Ex 5B 1(b)–(c), 3(a), 6

Original value = Increase =

Original value = New value = Increase = ( ) – ( )

Original value = New value = Increase = ( ) – ( )

Original value = 10 Increase = 3

Original value =

New value =

54

New value = original value × (1 + percentage increase)


If 10 is increased by 70%, find the new value.

Sol New value

= 10 × (1 + 70%)

= 10 × 1.7

= 17

If 20 is increased by 40%, find the new value.

Sol New value

= 20 × (1 + ______%)

= 20 × ( )

=

5. The height of a plant is 44 cm. If its height

is increased by 50%, find the new height.

New height

= ( ) × ( ) cm

=

6. The price of a box of cakes was $120 last

month. This month, it increases by 5%.

Find the price of the box of cakes this

month.

○○○○→→→→ Ex 5B 2(a)–(c), 7, 8, 10


If a number n is increased by 6%, the result is

53. Find the value of n.

Sol n × (1 + 6%) = 53

n × 1.06 = 53

n = 53 ÷ 1.06

= 50

If a number y is increased by 5%, the result is

42. Find the value of y.

Sol y × (1 + 5%) = 42

y × ( ) = 42

y =

7. The price of a chair this year is $360,

which is 20% more than the price last year.

Find the price of the chair last year.

Let $y be the price of the chair last year.

y × ( ) =

=

8. The weight of Heidi increases by 10% as

compared to last year. If her weight is

55 kg this year, find her weight last year.

○○○○→→→→ Ex 5B 9

Set up an equation to

find n.

First, let $y be the price of the chair last year. Then set up an equation to find y.

55

5.2B Percentage Decrease

(a) Decrease = original value – new value �

(b) Percentage decrease = valueoriginal

decrease× 100%


If 80 is decreased by 8, find the percentage

decrease.

Sol Percentage decrease

= %10080

8×

= 10%

If 50 is decreased by 40, find the percentage

decrease.


= %100) (

) (×

=

9. If 24 g is decreased by 6 g, find the

percentage decrease.

Percentage decrease = %100) (

) (×

=

10. If the speed of a car decreases by 9 km/h

from 45 km/h, find the percentage

decrease.

○○○○→→→→ Ex 5B 1(d)


If 20 is decreased to 12, find the percentage

decrease.


= %10020

1220×

−

= %10020

8×

= 40%

If 90 is decreased to 36, find the percentage

decrease.


= %100) (

) () (×

−

=

11. If 70 m2 is decreased to 49 m2, find the


Percentage decrease

=

12. Fanny got 80 marks in the first test and

68 marks in the second test. Find the


○○○○→→→→ Ex 5B 1(e)–(f), 4(a), 11

Original value =

Decrease =

Original value = New value = Decrease = ( ) – ( )

Original value = 80

Decrease = 8

Original value = New value = Decrease = ( ) – ( )

Original value = New value =

56

New value = original value × (1 – percentage decrease)


If 36 is decreased by 25%, find the new value.

Sol New value

= 36 × (1 – 25%)

= 36 × 0.75

= 27

If 40 is decreased by 70%, find the new value.

Sol New value

= 40 × (1 – ______%)

= 40 × ( )

=

13. The price of a skirt is $60. If the price

decreases by 20%, find the new price.

New price = $( ) × ( )

=

14. There were 90 members in the Science

Club last year. This year, the number of

members decreases by 30%. Find the

number of members this year.

○○○○→→→→ Ex 5B 2(d)–(f), 12, 13


If a number t is decreased by 40%, the result is

54. Find the value of t.

Sol t × (1 – 40%) = 54

t × 0.6 = 54

t = 54 ÷ 0.6

= 90

If a number k is decreased by 10%, the result is

126. Find the value of k.

Sol k × (1 – %) =

=

15. This year, the number of students of a

tutorial centre is 92, which is 8% less than

the number of students last year. Find the

number of students of the tutorial centre

last year.

16. Oscar spends $44 today, which is 45% less

than the amount he spent yesterday. Find

the amount he spent yesterday.

○○○○→→→→ Ex 5B 14

Set up an equation to solve the problem.

Set up an equation

to find t.

57

5.2C Percentage Change

(a) Change = new value – original value

(b) Percentage change = valueoriginal

change× 100%

Note: (i) When new value > original value (i.e. the value increases),

the sign of the percentage change is positive (+).

(ii) When new value < original value (i.e. the value decreases),

the sign of the percentage change is negative (–).


If 30 is changed to 36, find the percentage

change.

Sol Percentage change

=30

3036 −× 100%

=30

6+× 100%

= +20%


change.


=) (

) () ( −× 100%

=



change.


=40

4012 −× 100%

=40

28−× 100%

= –70%


change.


=) (

) () ( −× 100%

=

17. Complete the table below.

Original value New value Change Percentage change

(a) 12 km 18 km (18 – 12) km

= +6 km

(b) 50 g 42 g

(c) $600 $1 500

○○○○→→→→ Ex 5B 5

‘+’ represents that the value increases.

Original value = , new value =

New value – original value



Original value = New value =

New value – original value

‘–’ represents that the

value decreases.

58

18. A bakery sold 200 cakes yesterday. 120

cakes are sold today. Find the percentage

change in the number of cakes sold.

19. Parco’s savings was $4 000 last month. He

saves $5 440 this month. Find the

percentage change in his monthly savings.

○○○○→→→→ Ex 5B 15, 16

New value = original value × (1 + percentage change)


Kelly’s weight was 50 kg last year. The

percentage change in weight is +7% as

compared to last year. Find her weight this

year.

Sol Her weight this year

= 50 × [1 + (+7%)] kg

= 50 × (1 + 7%) kg

= 50 × 1.07 kg

= 53.5 kg

Gloria’s monthly salary was $8 000 last month.

The percentage change in her monthly salary

is –6% as compared to last month. Find her

monthly salary this month.

Sol Her monthly salary this month

= $( ) × [1 + ( )]

=

20. There were 900 staff members in a factory

last year. The percentage change in the

number of staff members is –1% as

compared to last year. Find the number of

staff members this year.

21. A shop sold 260 hot dogs today. The

percentage change in the number of hot

dogs sold is +4% as compared to

yesterday. Find the number of hot dogs

sold yesterday.

○○○○→→→→ Ex 5B 18

Original value = _____ New value = _____


solve the problem.

59


22. Vincent has $300 originally. If he gives 27% of the money to his sister, does he have enough

money to buy a model car which costs $220? Explain your answer.

If Vincent gives 27% of the money to his sister, then

the money left =

∵ $ ______ ( < / > ) $220

∴ He (has / does not have) enough money to buy the model car.

23. In a test, all scores higher than 80 marks are classified as grade A. Alice got 60 marks in the

first test and got 35% more in the second test. Is her score in the second test classified as

grade A? Explain your answer.


24. Thomas is 5% taller than Samuel and Thomas is 1.68 m tall.

(a) Find the height of Samuel.

(b) If Alex is 25% shorter than Samuel, find the height of Alex.

25. Mandy’s monthly salary this year is $20 140, which has increased by 6% as compared to last

year. Find the increase in her monthly salary.


60

5 Percentages (I)

Level 1

1. Find the percentage increase or decrease in each of the following.

(a) An increase of 15 from 20. (b) An increase from 24 to 36.

(c) An increase from 70 cm to 98 cm. (d) A reduction of 27 from 45.

(e) A decrease from 75 to 66. (f) A decrease from 140 to 21.

2. In each of the following, increase/decrease the quantity by the given percentage in the bracket. Find

the new quantity.

(a) 18°C (percentage increase: 50%) (b) 35 cm (percentage increase: 140%)

(c) 90 min (percentage increase: 6%) (d) 600 mL (percentage decrease: 70%)

(e) $72 (percentage decrease: 25%) (f) 350 g (percentage decrease: 88%)


Original

quantity New quantity Increase

Percentage

increase

(a) 162 cm 72 cm

(b) 8 g 4%


Original

quantity New quantity Decrease

Percentage

decrease

(a) 78 cm2 42 cm2

(b) $120 60%


Original quantity New quantity Percentage change

(a) 320 368

(b) 140 cm 105 cm

(c) 50 kg 120 kg

(d) 67.5 L 54 L


61

6. The weight of a boy was 15 kg a year ago. If his weight becomes 18 kg now, what is the percentage

increase in the weight?

7. The height of a tree is 220 cm. If its height increases by 40%, what is the new height?

8. To join a basketball team, a student should be at least 160 cm tall. Ryan’s height was 125 cm four

years ago and has increased by 30% since then. Can he join the basketball team now? Explain your

answer.

9. In a test, Amy’s score is 15% higher than Betty’s. If Betty’s score is 60, find Amy’s score.

10. Peter’s savings increases by 8% as compared to last month. If his savings this month is $5 400, find

his savings last month.

11. There was 800 mL of orange juice in a bottle yesterday. If the volume of orange juice decreases by

640 mL today, what is the percentage decrease in the volume of orange juice?

12. Edward spends 18% less money this month than last month. If he spent $5 000 last month, how much

does Edward spend this month?

13. The price of a mobile phone has decreased by 16% as compared to last year. If the decrease in price is

$800, find the price of the phone last year.

14. In a forest, the number of monkeys this year is 704, which is 12% less than the number of monkeys

last year. How many monkeys were there in the forest last year?

15. The weight of an apple is 350 g. An orange is 20% lighter than the apple. Find the total weight of the

apple and the orange.

16. 560 babies were born in a hospital last year. If the number of babies born in the hospital is changed by

+84 this year, find the percentage change.

17. Last month, there were 1 400 car accidents. If 980 car accidents occur this month, find the percentage

change in the number of car accidents.

18. Yesterday, there were 150 patients in a clinic. The percentage change in the number of patients today

is −8%, as compared to yesterday. Find the change in the number of patients.

19. The percentage change of the volume of an ice ball is −16%. The new volume is 126 cm3 after the

change. Find the original volume of the ice ball.

62

Level 2

20. The number of visitors to a museum this week increases by 60% to 4 800 as compared to last week.

Find the increase in the number of visitors.

21. Martin’s monthly income increases by 20% this year. The increase is $6 000.

(a) Find Martin’s monthly income last year.

(b) If the increase in Martin’s monthly income remains the same next year, find his monthly income

next year.

22. The amount of rainfall today is 13.2 mm, which is 1.8 mm less than that yesterday.

(a) Find the amount of rainfall yesterday.

(b) Find the percentage decrease in the amount of rainfall from yesterday to today.

23. Billy and Gordon took part in a marathon race these two years. The table below shows their finishing

time.

Last year This year

Billy 48 minutes 40.8 minutes

Gordon 40 minutes 33.6 minutes

(a) Who has a greater decrease in finishing time?

(b) Does the same person in (a) also have a greater percentage decrease in finishing time? Explain

your answer.

24. Last year, 25 000 candidates took a public examination and 32% passed the examination. This year,

the same number of candidates took the examination but the passing rate became 20%.

(a) Find the decrease in the number of candidates passing the examination.

(b) Find the percentage decrease in the number of candidates passing the examination.

25. Kim spent 72 hours on cycling and 48 hours on swimming last month. He reduces the time spent on

cycling and swimming by 25% and 10% respectively this month.

(a) What is his total time spent on cycling and swimming this month?

(b) What is the percentage change in his total time spent on cycling and swimming?

26. In a city, the unemployment was 200 000 in April and it dropped to 160 000 in May.

(a) Find the percentage change in unemployment from April to May.

(b) If the percentage change from May to June was the same as that in (a), find the unemployment in

June.

(c) Actually, the unemployment in June was 112 000. By what percentage was this figure less than

the result in (b)?

63

27. Simon practises running each day. During the examination week, he reduced the daily practising time

by half an hour. The percentage change was −40%.

(a) How long did he spend on practising each day during the examination week?

(b) After the examination week, he changes his practising time in (a) by +40%. Is the new practising

time the same as the one before the examination week? Explain your answer.

28. A restaurant charges an additional 10% service charge on the total food price. Ellen orders a pizza of

price $87 and a special drink. If she has to pay $132, what is the price of the special drink?

29. Nelson is a salesman and his monthly income is the sum of the fixed basic salary and the commission.

In January, the commission of Nelson was $24 000, which was 75% of his income.

(a) Find his basic monthly salary.

(b) In February, Nelson’s total income increased by 22.5% as compared to January. Find the

percentage increase in the commission.

30. Sand of 400 g is inside a sandglass formed by two cones. There is

w g of sand in the upper cone originally. After a while, 30% of the sand in

the upper cone drops into the bottom cone, and hence the weight of sand in

the bottom cone increases by 20%. Find the value of w.

64


1. (a) 75% (b) 50%

(c) 40% (d) 60%

(e) 12% (f) 85%

2. (a) 27°C (b) 84 cm

(c) 95.4 min (d) 180 mL

(e) $54 (f) 42 g

3. (a) 90 cm, 80% (b) 200 g, 208 g

4. (a) 120 cm2, 35%

(b) $300, $180

5. (a) +15% (b) −25%

(c) +140% (d) −20%

6. 20% 7. 308 cm

8. yes 9. 69

10. $5 000 11. 80%

12. $4 100 13. $5 000

14. 800 15. 630 g

16. +15% 17. −30%

18. −12 19. 150 cm3

20. 1 800

21. (a) $30 000 (b) $42 000

22. (a) 15 mm (b) 12%

23. (a) Billy (b) no

24. (a) 3 000 (b) 37.5%

25. (a) 97.2 hours (b) −19%

26. (a) −20% (b) 128 000

(c) 12.5%

27. (a) 45 minutes (b) no

28. $33

29. (a) $8 000 (b) 30% 30. 160

65

F1A: Chapter 5C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 5C Multiple Choice


___________ ( )



_________

66

Book 1A Lesson Worksheet 5C (Refer to §5.3)

5.3A Profit

(a) Profit = selling price – cost price �

(b) Profit % =

pricecost

profit× 100%


The cost price of a vase is $60. Its selling price

is $84.

(a) Find the profit.

(b) Find the profit per cent.

Sol (a) Profit = $(84 – 60)

= $24

(b) Profit % =$60

$24× 100%

= 40%

The cost price of a packet of peanuts is $20. Its

selling price is $26.

(a) Find the profit.


Sol (a) Profit = $[( ) – 20]

= $( )

(b) Profit % =$20

) $(× 100%

= ( )%


Cost price Selling price Profit Profit %

(a) $15 $18 $15

) $(× 100% =

(b) $80 $124

(c) $130 $195

○○○○→→→→ Ex 5C 1(a), (c), (e), 2

2. Michael bought a set of game cards for

$70 and sold it for $77. Find the profit per

cent.

Profit

= $[( ) – ( )]

=

Profit %

=

3. Harry bought a toy boat for $85 and sold it

to Tom for $136. Find the profit per cent.

○○○○→→→→ Ex 5C 4, 5

Cost price = ____ Selling price = ____

67

Selling price = cost price × (1 + profit %)


The cost price of a jewellery box is $200. It is

sold at a profit of 25%. Find the selling price.

Sol Selling price

= $200 × (1 + 25%)

= $200 × 1.25

= $250

The cost price of a dictionary is $90. It is sold

at a profit of 20%. Find the selling price.

Sol Selling price

= $90 × (1 + ______%)

= $90 × ( )

=

4. The cost price of a tie is $140 and it is sold

at a profit of 55%. Find the selling price.

5. Tina bought a set of stamps for $50 and

sold it to Carmen at a profit of 28%. Find

the selling price.

○○○○→→→→ Ex 5C 8, 9

6. An oven is sold at a profit of 35%. If the

selling price is $324, find the cost price.

Let $y be the cost price.

7. Mr Chan sold a calculator for $168 at a

profit of 20%. Find the cost price.

○○○○→→→→ Ex 5C 12

Let $y be the cost price. Set up an equation to find y.

Cost price = ____ Profit % = ____

68

5.3B Loss

(a) Loss = cost price – selling price �

(b) Loss % =

pricecost

loss × 100%


The cost price of a belt is $150. Its selling price

is $60.

(a) Find the loss.

(b) Find the loss per cent.

Sol (a) Loss = $(150 – 60)

= $90

(b) Loss % =$150

$90× 100%

= 60%

The cost price of a box of chocolates is $70. Its


(a) Find the loss.


Sol (a) Loss = $[( ) – ( )]

=

(b) Loss % =) $(

) $(× 100%

=


Cost price Selling price Loss Loss %

(a) $20 $15 $20

) $(× 100% =

(b) $70 $63

(c) $180 $99

○○○○→→→→ Ex 5C 1(b), (d), 3

9. Jason bought a product for $120 and sold

it for $78. Find the loss per cent.

Loss

= $[( ) – ( )]

=

Loss %

=

10. Angel bought a set of story books for $480

and sold it to Jacky for $144. Find the loss

per cent.

○○○○→→→→ Ex 5C 6, 7

Cost price = ____ Selling price =

69

Selling price = cost price × (1 – loss %)


The cost price of a bag is $90 and it is sold at a

loss of 60%. Find the selling price.

Sol Selling price

= $90 × (1 – 60%)

= $90 × 0.4

= $36

The cost price of a dress is $110 and it is sold

at a loss of 20%. Find the selling price.

Sol Selling price

= $( ) × (1 – ______%)

=

11. The cost price of a wardrobe was $600.

Two years later, it is sold at a loss of 75%.

Find the selling price.

12. A shop bought a pair of jeans for $400 and

sold it at a loss of 30%. Find the selling

price.

○○○○→→→→ Ex 5C 10, 11

13. A bookshelf is sold at a loss of 10%. If the

selling price is $189, find the cost price.

14. A shop sold a pair of earrings for $210 at a

loss of 16%. Find the cost price.

○○○○→→→→ Ex 5C 13, 14


solve the problem.

70


15. Marie bought a gold coin for $2 000. She is willing to sell it if the profit per cent is not less

than 15%. If Ken wants to pay Marie $2 600 for the gold coin, can the deal be made?


If Marie sells the coin at a profit of 15%, then

selling price of the gold coin =

∵ $ ______ ( < / > ) $2 600

∴ The deal (can / cannot) be made.

16. A merchant bought a watch and a handbag for $1 500 and $1 800 respectively. Later he sold

the watch at a profit of 8% and the handbag at a loss of 7%. Did he make a profit or a loss on

the whole? Explain your answer.


17. William bought 25 heaters for $8 000. 9 of them failed the safety test and they cannot be used.

All the remaining heaters were sold for $575 each. Find the overall profit per cent.

18. Janet bought a Geography book for $210. One year later, she sold it to Ricky at a loss of 60%.

One more year later, Ricky sold the book for $50.4. Find Ricky’s profit or loss per cent.


71

5 Percentages (I)

Level 1


((a) has been done for you as an example.)

Selling price Cost price Profit or loss Amount

(a) $280 $250 Profit $30

(b) $360 $420

(c) $4 900 $1 700

(d) $56.5 $38

(e) $840 $960


Cost price Selling price Profit Profit %

(a) $60 $90

(b) $280 $350

(c) $400 $72

(d) $12.8 $0.3


Cost price Selling price Loss Loss %

(a) $50 $42

(b) $300 $180

(c) $65 $39

(d) $7.2 $16.8

4. Alex bought a watch for $640 and sold it at a profit of $400. What is the profit per cent?

5. Harry bought a smart phone for $4 500. One year later, he sold it for $3 150. What is the loss per cent?

6. Cuson buys a photo album and sells it. Find the profit or loss per cent in each of the following.

(a) He buys the album for $360 and sells it for $180.

(b) He buys the album for $180 and sells it for $360.

7. Jason sells a washing machine for $1 890 and makes a loss of $360. Find the loss per cent.

Consolidation Exercise 5C ��

72

8. Samson sells a toy helicopter at a loss of $800. Find the loss per cent in each of the following cases.

(a) The cost price of the helicopter is $3 200.

(b) The selling price of the helicopter is $3 200.

9. Neil bought a painting for $45 000 and sold it at a profit of 28%. How much was the profit?

10. Andrew bought a camera for $2 400 and sold it at a loss of 5%. How much was the loss?

11. Leonard bought a bicycle for $770 yesterday. Today, he sells the bicycle at a loss of 30%. How much

does he sell the bicycle?

12. Laura sold a handbag at a profit per cent of 2.5%. It is given that the cost price of the handbag was $9

600. Find the selling price of the handbag.

13. Jack sells a wallet at a profit of 120% and makes a profit of $840. Find the cost price.

14. Amy bought a concert ticket. Later, she could not attend the concert and she sold the ticket at a loss of

30%. The loss made was $810. How much did she sell the ticket?

15. A boutique sold a dress for $187 at a loss of 15%. Let $x be the cost price.

(a) Express the selling price in terms of x.

(b) Hence, find the cost price.

16. Nancy bought a book last year. She sells the book now for $216 and makes a profit of 80%. How

much did Nancy pay for the book?

17. A shop buys 15 boxes of sponge cakes for $300. Each box contains 8 cakes. All the cakes are sold out

for $5 each.

(a) Find the total income.


18. A hawker buys 20 dozen apples. He finds that 40 apples are rotten and sells all the remaining apples at

$3.6 each. Finally, he makes a loss of $30.

(a) Find the loss per cent.

(b) Find the cost price of each dozen apples.

73

Level 2

19. Queenie buys a rice cooker for $2 500 and a refrigerator for $3 900. She then sells them altogether for

$4 800. Find the loss per cent.

20. Paul bought a wallet for $500 and then sold it for $620. Jason bought a jacket for $1 500 and then sold

it for $1 620. Did they make the same profit per cent? Explain your answer.

21. Chris buys a diamond ring for $80 000. Then he sells the ring.

(a) If the ring is sold at a profit of 15%, find the selling price.

(b) If the ring is sold for a price $2400 higher than the selling price in (a), find the profit per cent.

22. Martin bought a motorbike for $30 000 in 2010 and sold it to Nick at a loss of 20% in 2013.

(a) How much did Nick pay for the motorbike?

(b) If Nike sold the motorbike to Osman for $21 600 in 2014, find the profit or loss per cent.

23. Doris sells a car at a profit of 20%. Find the profit in each of the following cases.

(a) The cost price of the car is $600 000.

(b) The selling price of the car is $600 000.

24. A manufacturer produces 8 000 T-shirts and sells all of them at a total profit of $288 000. If the profit

per cent was 60%, find the cost price of each T-shirt.

25. Katie sells a bag of tea leaves for $5 600. The profit per cent is 40%. Find the profit.

26. A supermarket sells a bottle of tea for $9.8 at a profit of 75%.

(a) Find the cost price of a bottle of tea.

(b) A customer can buy three bottles of tea together at a special price, and the profit per cent

decreases to 25%. Find the special price.

27. Mr Leung bought an antique vase last year. If he sells the vase for $32 400, he will make a loss of

40%.

(a) Find the cost price.

(b) Mr Cheung wants to buy the vase for $48 600. Mr Leung will sell the vase to Mr Cheung if the

loss per cent is less than 15%. Can the deal be made? Explain your answer.

28. In a furniture store, a sofa is sold at $3 600 and the profit per cent is 50%. If the selling price is set

$240 higher, find the new profit per cent.

74

29. A microwave oven is sold for $1 200 and the loss per cent is 4%.

(a) If the microwave oven is sold for $950, what is the loss per cent?

(b) If the microwave oven is sold at a profit of 36%, what is the selling price?

30. A camera is sold at a loss of $1 800. The loss made is 60% of the selling price.

(a) Find the selling price.


31. A shopkeeper sets the price of a jacket to be $792. If the jacket is sold at this price, the profit per cent

is 10%.

(a) Find the cost price of the jacket.

(b) The shopkeeper marks up the price of the jacket such that the profit increases by 25%. Find the

new selling price.

32. Adrian bought two oil paintings A and B for $9 000 and $6 000 respectively. Then he sold painting A

at a loss of 10% and painting B at a profit of 20%.

(a) On the whole, did he make a profit or a loss? Explain your answer.

(b) What is the overall profit or loss per cent?

33. Scott buys 15 bottles of red wine for $200 each and 4 bottles of white wine for $1 000 each. He sells

each bottle of red wine at a profit of 40% and each bottle of white wine at a profit of 5%.

(a) Find the total selling price of the wine.

(b) Find the overall profit per cent.

34. Ken sold an electric kettle and a coffee machine for $180 and $2 160 respectively. It is given that the

kettle is sold at a loss of 60% and the coffee machine is sold at a profit of 60%.

(a) Find the cost prices of the kettle and the coffee machine.

(b) On the whole, did he make a profit or a loss? Explain your answer.

(c) Find the overall profit or loss per cent.

35. One day, Dickson bought 180 packets of potato chips for $700. He then packed the potato chips in

bags of 6 packets and set the price of each bag to be $35.

(a) If all bags of potato chips were sold but for $35 each on that day, find the profit per cent.

(b) Suppose only 12 bags were sold for $35 each on that day and the rest were sold for $14 each on

the next day.

(i) On the whole, did he make a profit or a loss? Explain your answer.

(ii) Find the overall profit or loss per cent.

75

Consolidation Exercise 5C (Answer)

1. (b) loss, $60 (c) profit, $3 200

(d) profit, $18.5 (e) loss, $120

2. (a) $30, 50% (b) $70, 25%

(c) $472, 18% (d) $12.5, 2.4%

3. (a) $8, 16% (b) $120, 40%

(c) $26, 60% (d) $24, 70%

4. 62.5% 5. 30%

6. (a) loss per cent: 50%

(b) profit per cent: 100%

7. 16%

8. (a) 25% (b) 20%

9. $12 600 10. $120

11. $539 12. $9 840

13. $700 14. $1 890

15. (a) $0.85x (b) $220

16. $120

17. (a) $600 (b) 100%

18. (a) 4% (b) $37.5

19. 25% 20. no

21. (a) $92 000 (b) 18%

22. (a) $24 000 (b) loss per cent: 10%

23. (a) $120 000 (b) $100 000

24. $60 25. $1 600

26. (a) $5.6 (b) $21

27. (a) $54 000 (b) yes

28. 60%

29. (a) 24% (b) $1 700

30. (a) $3 000 (b) 37.5%

31. (a) $720 (b) $810

32. (a) profit (b) profit per cent: 2%

33. (a) $8 400 (b) 20%

34. (a) kettle: $450, coffee machine: $1 350

(b) profit (c) profit per cent: 30%

35. (a) 50% (b) (i) loss (ii) loss per cent:

4%

76

F1A: Chapter 5D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)

Book Example 20


(Video Teaching)



(Full Solution)

Maths Corner Exercise 5D Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 5D Multiple Choice


___________ ( )



_________

77

Book 1A Lesson Worksheet 5D (Refer to §5.4)

5.4 Discount

(a) Discount = marked price – selling price �

(b) Discount % =

price marked

discount× 100%


The marked price of a magazine is $30. Its


(a) Find the discount.

(b) Find the discount per cent.

Sol (a) Discount = $(30 – 27)

= $3

(b) Discount % =$30

$3× 100%

= 10%

The marked price of a book is $40. Its selling

price is $16.

(a) Find the discount.

(b) Find the discount per cent.

Sol (a) Discount = $[40 – ( )]

= $( )

(b) Discount % =$40

) $(× 100%

= ( )%


Marked price Selling price Discount Discount %

(a) $60 $42 %100$60

) $(× =

(b) $44 $11

(c) $25 $14

○○○○→→→→ Ex 5D 1, 2

2. The marked price of a cheese cake is $170.

It is now sold for $136. Find the discount

per cent.

Discount

= $[( ) – ( )]

=

Discount %

=

3. The marked price of an umbrella is $95. It

is now sold for $57. Find the discount per

cent.

○○○○→→→→ Ex 5D 6, 7

Marked price = ____ Selling price = ____

78

Discount = marked price × discount %


A clock marked at $70 is sold at a discount of

40%. Find the discount.

Sol Discount = $70 × 40%

= $70 × 0.4

= $28

A calendar marked at $90 is sold at a discount

of 30%. Find the discount.

Sol Discount = $90 × ( )%

=

4. A teapot marked at $45 is sold at a

discount of 10%. Find the discount.

5. A chicken marked at $52 is sold at a

discount of 25%. Find the discount.

6. A pencil case marked at $50 is sold at a

20% discount. Find the discount.

7. A bag of rice is sold at a 30% discount. Its

marked price is $98. Find the discount.

○○○○→→→→ Ex 5D 3(a), (b), 4, 5

8. Kelly buys a pillow at a 5% discount and

saves $100. Find the marked price.

9. Annie buys a school bag at a 35% discount

and saves $63. Find the marked price.

○○○○→→→→ Ex 5D 3(c), (d), 10

Let $y be the marked price. Set up an equation to find y.

i.e. $28 is saved.

79

Selling price = marked price × (1 – discount %)


A crystal ball marked at $100 is sold at a

discount of 5%. Find the selling price.

Sol Selling price

= $100 × (1 – 5%)

= $100 × 0.95

= $95

A plate marked at $80 is sold at a discount of

40%. Find the selling price.

Sol Selling price

= $80 × (1 – ______%)

=

10. A jacket marked at $200 is sold at a

discount of 55%. Find the selling price.

11. A box of biscuits marked at $45 is sold at a

60% discount. Find the selling price.

○○○○→→→→ Ex 5D 8, 9

12. A dress is sold at a discount of 80%. If the

selling price is $280, find the marked

price.

13. All the items in a toy shop are sold at a

45% discount. If the selling price of a box

of puzzles is $121, find the marked price.

○○○○→→→→ Ex 5D 11–13

Marked price = ____ Discount % = ____

Set up an equation to solve the problem.

80


14. In shop A, a microwave oven marked at $720 is sold at a 15% discount. In shop B, the same

model of microwave oven marked at $700 is sold at a 12% discount. If Sara wants to buy the

microwave oven at a lower selling price, which shop should she choose? Explain your

answer.

Selling price of the microwave oven in shop A =

Selling price of the microwave oven in shop B =

∵ $ ______ ( < / > ) $ ______

∴ She should choose shop (A / B).

15. In a supermarket, the marked price of each brand A apple is $4 and it is sold for $3. The

marked price of a pack of 4 brand B apples is $24 and each pack is sold at a discount of $9. If

Kelly wants to buy 4 apples, which brand of apples have a greater discount per cent? Explain

your answer.


16. In an electrical appliance shop, Fiona bought a digital camera at a discount of 30% and saved

$210. Later, she bought a digital photo frame marked at $500 at a 40% discount. How much

did she pay in total?


81

5 Percentages (I)

Level 1


Marked price Selling price Discount

(a) $80 $64

(b) $9.5 $5.7

(c) $250 $30

(d) $31.2 $4.8


Marked price Selling price Discount %

(a) $250 $150

(b) $480 $192

(c) $1 600 $720


Marked price Selling price Discount Discount %

(a) $350 40%

(b) $760 25%

(c) $90 30%

(d) $810 60%

4. A calculator marked at $160 is sold at a discount of 15%. Find the discount.

5. The marked price of a fitness band is $720. In a sale, Terry buys the band and saves $144, find

(a) the selling price, (b) the discount per cent.

6. The marked price of a dinnerware set is $600 and the selling price is $420. Find

(a) the discount, (b) the discount per cent.

7. In a computer fair, a printer is sold for $520 at a discount of $280. Find

(a) the marked price, (b) the discount per cent.

Consolidation Exercise 5D ��

82

8. A TV set marked at $5 200 is sold at a discount of 30%. What is the selling price?

9. All the items in a shopping mall are sold at 20% off in the Christmas sale. If a briefcase is sold for

$560, find the marked price of the briefcase.

10. David buys a box of cookies at 60% discount and saves $90. How much is the box of cookies sold for?

11. A handbag is sold at a discount of 45%. The selling price is $990. Find

(a) the marked price, (b) the discount.

12. In a book store, the marked prices of three books are $110, $120 and $130. The book store offers the

customers to buy the three books together at a special price of $216. Find the discount per cent.

13. In a casual wear store, the marked price of a T-shirt is $180. Customers who buy a T-shirt at the

marked price can buy another T-shirt at a special price of $108. Kenneth buys two T-shirts at one time

from the store. Find the discount per cent.

14. In a store, each pair of socks of marked price $90 is sold at a discount of 10%.

(a) Find the selling price of each pair of socks.

(b) Mrs Lee buys three pairs of socks. She claims that she enjoys a discount of 30% on the whole.

Do you agree? Explain your answer.

15. In a supermarket, the marked price of a carton of milk is $15. If a customer buys 4 cartons of milk at

one time, he can get one of them for free. Mr Chow buys 4 cartons of milk at one time. Find the

discount per cent.

Level 2

16. A tablet computer is sold at a discount of 18%. Derek buys it with two $1 000 notes and the change is

$770. Find the marked price of the tablet computer.

17. A chair in a department store is sold at $500. The marked price is 60% higher than the selling price.

Find the discount.

18. In shop A, the marked price of a toaster is $360 and it is sold at 40% discount. In shop B, a customer

can buy the same toaster at a discount of 20% and saves $55. If Keith wants to buy the toaster at a

lower price, which shop should he choose? Explain your answer.

83

19. In a boutique, a tie is marked at $420 and a leather belt is marked at $280. Calvin buys the tie at a

30% discount and the leather belt at a 20% discount.

(a) Find the overall discount.

(b) Calvin claims that he buys the two items at an overall discount per cent of 25%. Do you agree?


20. A computer is sold at a discount of 15%. Elsa buys the computer and uses a $300 gift coupon in

payment. If she saves a total of $1 500 from the 15% discount and the gift coupon, find the marked

price of the computer.

21. In a furniture store, the marked price of a computer desk is $1 500. Now, the desk and an office chair

are sold together at a special price of $1 080, and the overall discount per cent is 40%. Find the

marked price of the office chair.

22. A manufacturer makes a hairdryer at a cost of $140. The marked price of the hairdryer is set to be

$560 and it is sold at $308 finally.

(a) Find the discount per cent.


23. The marked price of a watch is $50 000. Philip buys the watch at a discount of 40% and then sells it at

a profit of 40%.

(a) How much profit does he make?

(b) Is the profit made the same as the discount offered to him? Explain your answer.

24. A stationery shop bought a box of pencils for $80. Its marked price is set 75% above its cost price.

Later, the box of pencils is sold at a discount of 40%.

(a) Find the marked price and the selling price of the box of pencils.

(b) Does the shop make a profit or a loss? Explain your answer.

(c) Find the profit or loss per cent.

25. Cindy buys a jacket for $960. It is then sold at a discount of 32% and a loss of 15% is made.

(a) How much does Cindy sell the jacket?

(b) Find the marked price of the jacket.

(c) If Cindy sells the jacket at 15% discount instead, can she make a profit? Explain your answer.

26. Jordon buys 600 bags of rice crackers at $15 each. He sets the marked price of each bag of rice crackers to be $36. He sells n bags at the marked price and all the rest at 75% off. If a loss of 10% is made on the whole, find the value of n.

84

Consolidation Exercise 5D (Answer)

1. (a) $16 (b) $3.8

(c) $220 (d) $36

2. (a) 40% (b) 60%

(c) 55%

3. (a) $210, $140 (b) $570, $190

(c) $300, $210 (d) $1 350, $540

4. $24

5. (a) $576 (b) 20%

6. (a) $180 (b) 30%

7. (a) $800 (b) 35%

8. $3 640 9. $700

10. $60

11. (a) $1 800 (b) $810

12. 40% 13. 20%

14. (a) $81 (b) no

15. 25% 16. $1 500

17. $300 18. shop A

19. (a) $182 (b) no

20. $8 000 21. $300

22. (a) 45% (b) 120%

23. (a) $12 000 (b) no

24. (a) marked price: $140, selling price: $84

(b) profit (c) profit per cent: 5%

25. (a) $816 (b) $1 200

(c) yes 26. 100

85

F1A: Chapter 6A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

86


6.1A Significance of Estimation

Estimation is the process of finding an approximate value of a quantity or an expression.

The value obtained is called an estimated value.

e.g. Last New Year’s Eve, about 270 000 citizens celebrated New Year in Tsim Sha Tsui.

1. When expressing the value in each of the following cases, determine whether an estimated value or

an exact value is more appropriate to use.

(a) the total number of pages of a student handbook (b) the UV index of a day

(c) the number of spectators of a dragon boat race (d) the price of a bicycle

○○○○→→→→ Ex 6A 1

6.1B Estimation Strategies

I. Reformulation Strategy

Do quick estimation by replacing the numbers with approximate values.

(1) Rounding Off

e.g. (a) 53.7 = 50 (round off to the nearest ten)

(b) 53.8 = 54 (round off to the nearest one)

(c) 1.952 4 = 2.0 (round off to 1 decimal place)

2. Round off each of the following numbers

to the nearest one.

(a) 32.1 (b) 147.8

(a) 32.1 = (round off to the nearest one)

(b) 147.8 = 3. Round off each of the following numbers

to 1 decimal place.

(a) 194.64 (b) 239.17 ○○○○→→→→ Ex 6A 2

(2) Front-end Method

Consider the left-most digit of the number only. All other digits are replaced by 0.

e.g. (a) 1 357 ≈ 1 000 (b) 2.58 ≈ 2

4. By front-end method, find the estimated

value of each of the following numbers.

(a) 501 (b) 17

(a) 501 ≈

(b) 17 ≈ 5. By front-end method, find the estimated

value of each of the following numbers.

(a) 8 631.4 (b) 1.606

○○○○→→→→ Ex 6A 3

1 3 5 7

1 0 0 0

2. 5 8

2. 0 0

87

Example 1 Install Drill 1

Estimate the value of 314 + 568

(a) by front-end method,

(b) by rounding off each number to the nearest

ten.

Sol (a) 314 + 568

≈ 300 + 500

= 800

(b) 314 + 568

≈ 310 + 570

= 880

Estimate the value of 76.9 – 20.3


(b) by rounding off each number to the nearest

one.

Sol (a) 76.9 – 20.3

≈ ( ) – ( )

=

(b) 76.9 – 20.3

≈ ( ) – ( )

=

6. Estimate the value of 851 + 224 + 159


(b) by rounding off each number to the

nearest ten.

(a) 851 + 224 + 159

≈ ( ) + ( ) + ( )

=

(b) 851 + 224 + 159

≈ ( ) + ( ) + ( )

=

7. Estimate the value of 1 253 + 4 309 + 287



nearest hundred.

8. Estimate the value of

32.96 + 54.03 – 18.07



nearest one.

9. Estimate the value of 12.4 ÷ 2.8 × 3.6



nearest one.

○○○○→→→→ Ex 6A 6, 7

Replace each number in the expression with an

approximate value first.

88

(3a) Rounding Down

Step 1111: Identify the digit to be rounded down.

Step 2222: Replace all the digits to the right of the digit with zeros.


Round down each of the following numbers to the

place value given in brackets.

(a) 263 (ten)

(b) 4.798 (1 decimal place)

Sol (a)

263 ≈ 260

(b) 4.798

≈ 4.7

Round down each of the following numbers to the


(a) 5 074 (hundred)

(b) 21.8 (one)

Sol (a) 5 074

≈

(b) 21.8

≈

10. Round down each of the following

numbers to the nearest ten.

(a) 145

(b) 77

(a) 145 ≈

(b) 77 ≈ 11. Round down each of the following

numbers to 1 decimal place.

(a) 31.48

(b) 22.638 ○○○○→→→→ Ex 6A 4

12. Round down each number in the following

expressions to the nearest hundred and then

estimate the values of the expressions.

(a) 5 187 + 3 249 – 1 065

(b) 2 541 – 1 594 + 315

(a) 5 187 + 3 249 – 1 065

≈

13. Round down each number in the following

expressions to the nearest one and then


(a) 2.5 × 3.2 – 1.6

(b) 37.9 + 12.5 ÷ 4.83

○○○○→→→→ Ex 6A 9, 11

Step 1111: The digit to be rounded down is 6.

Step 2222:

Note: Estimated value < exact value

2 6 3

2 6 0

4. 7 9 8

4. 7 0 0

5 0 7 4

5 0__ __

2 1 . 8

2 __.__

Multiplication and division first, then addition and subtraction.

89

(3b) Rounding up

Step 1111: Identify the digit to be rounded up.

Step 2222: Add 1 to the digit and replace all the digits to the right of the digit with zeros.


Round up each of the following numbers to the


(a) 607 (ten)

(b) 2.845 (1 decimal place)

Sol (a)

607 ≈ 610

(b) 2.845

≈ 2.9

Round up each of the following numbers to the


(a) 3 159 (hundred)

(b) 48.6 (one)

Sol (a) 3 159

≈

(b) 48.6

≈

14. Round up each of the following numbers

to the nearest hundred.

(a) 281

(b) 3 462

(a) 281 ≈

(b) 3 462 ≈ 15. Round up each of the following numbers to

the 1 decimal place.

(a) 60.83

(b) 14.082 ○○○○→→→→ Ex 6A 5

16. Round up each number in the following

expressions to the nearest ten and then


(a) 254 – 159 + 315

(b) 42 + 73 × 19

(a) 254 – 159 + 315

≈

17. Round up each number in the following

expressions to the nearest one and then


(a) 4.728 + 7.936 – 3.173

(b) 10.48 – 5.45 ÷ 1.46

○○○○→→→→ Ex 6A 8, 10

Step 1111: The digit to be rounded up is 0.

Step 2222:

Note: Estimated value > exact value

6 0 7

6 1 0

2. 8 4 5

2. 9 0 0

3 1 5 9

3__ __ __

4 8 . 6

4__.__

90

(4) Choosing a Clustered Value

If the numbers involved in estimation are close in value, we may choose an approximate

value to replace each of them. The chosen value is called a clustered value.

e.g. 5.6 + 5.7 + 6.1 + 6.4 + 5.9

≈ 6 + 6 + 6 + 6 + 6

= 6 × 5

= 30

18. By choosing a suitable clustered value,

estimate the value of 51 + 49 + 52 + 47. 51 + 49 + 52 + 47

≈ ( ) + ( ) + ( ) + ( )

=

19. By choosing a suitable clustered value, estimate

the value of 18 + 23 + 19 + 21 + 22.

20. By choosing a suitable clustered value,

estimate the value of

23.9 + 27.3 + 24.6 – 25.1.

○○○○→→→→ Ex 6A 12

21. By choosing a suitable clustered

value, estimate the value of

(309 + 297 + 315 + 286 + 302) ÷ 15.

○○○○→→→→ Ex 6A 13

(5) Using Compatible Numbers

Suppose a pair of numbers is involved in estimation. Sometimes, we can make the

calculation easier by replacing the two numbers with suitable approximate values. These

two approximate values are called compatible numbers.

e.g. (a) 6.4 ÷ 2.9 ≈ 6 ÷ 3 = 2 � 6 and 3 are compatible numbers in division.

(b) 8.62 × 0.33 ≈ 9 ×3

1= 3 � 9 and

3

1are compatible numbers in multiplication.

Estimate the value of expression in the brackets first.

6.4 ≈ 6

2.9 ≈ 3

8.62 ≈ 9

0.33 ≈3

1

� ∵ These five numbers are close to 6.

∴ Choose 6 as the clustered value.

91

22. Estimate the value of 23.7 ÷ 5.68 by using

compatible numbers. 23.7 ÷ 5.68

≈ ( ) ÷ ( )

=

23. Estimate the value of 35.6 ÷ 7.22 by using

compatible numbers. 35.6 ÷ 7.22

≈

24. Estimate the value of 97.4 × 0.48 by using

compatible numbers.

25. Estimate the value of 11.91 × 0.74 by using

compatible numbers. ○○○○→→→→ Ex 6A 14, 15

II. Translation Strategy

Change the order of operations of the approximate values for easy calculation.


Using translation strategy, estimate the value of 454

× 8 ÷ 5.

Sol 454 × 8 ÷ 5

≈ 450 ×××× 8 ÷÷÷÷ 5 � Use Reformulation Strategy. = 450 ÷÷÷÷ 5 ×××× 8 �

= 90 × 8

= 720

Using translation strategy, estimate the value of

11.69 × 25 ÷ 2.95.

Sol 11.69 × 25 ÷ 2.95

≈ ( ) × ( ) ÷ ( )

=

26. Using translation strategy, estimate the

value of 30.06 × 11.17 ÷ 5.84. 27. Using translation strategy, estimate the

value of5

1÷ 4.72 × 103.06.

○○○○→→→→ Ex 6A 16

0.48 ≈)(

1 0.74 ≈

)(

)(

450 ÷ 5 is easier to calculate

than 450 × 8.

92

III. Compensation Strategy

Obtain an estimation by rough calculation first. Then, make an adjustment

(or compensation) to increase the accuracy of the estimation.


Using compensation strategy, estimate the value of

2.3 + 4.6 + 1.7 + 5.2.

Sol Step 1111 Estimation by rough calculation

(sum of the integral parts):

2 + 4 + 1 + 5

= 12 Step 2222 Adjustment

(sum of the decimal parts): 0.3 + 0.6 + 0.7 + 0.2

= (0.3 + 0.6) + (0.7 + 0.2)

≈ 1 + 1

= 2 ∴ The required estimated value

= 12 + 2

= 14


6.54 + 9.25 + 3.69 + 2.47.

Sol Step 1111 Estimation by rough calculation

(sum of the integral parts):

( ) + ( ) + ( ) + ( )

= Step 2222 Adjustment

(sum of the decimal parts):

∴ The required estimated value

=

28. Using compensation strategy, estimate the

value of 2.73 + 4.95 + 8.26.


value of 6.83 + 14.77 + 5.21 + 15.31.

○○○○→→→→ Ex 6A 17(a)

0.3 and 0.6, 0.7 and 0.2 are two pairs of compatible numbers in

addition.

93



323 × 4.

Sol Step 1111 Estimation by rough calculation:

300 × 4

= 1 200 Step 2222 Adjustment: 23 × 4

≈ 25 × 4

= 100 ∴ The required estimated value

= 1 200 + 100

= 1 300


4 079 × 5.

Sol Step 1111 Estimation by rough calculation:

( ) × 5

= Step 2222 Adjustment: ∴ The required estimated value

=


value of 7 × 129.


value of 2 026 × 8.

○○○○→→→→ Ex 6A 17(b)

Front-end method

323 – 300

25 and 4 are compatible numbers.

94

6.1C Application of Estimation

Example 7 Install Drill 7 The price of a can of coke is $3.8. Estimate the

total price of 30 cans of coke.

Sol Total price = $3.8 × 30

≈ $4 × 30

= $120

The price of a box of biscuits is $9.8. Estimate the

total price of 12 boxes of biscuits.

Sol Total price = $( ) × ( )

≈

32. Tim buys 68 gift coupons for $1 428. Estimate

the price of a gift coupon. 33. The hourly wage of Paul is $41. If he works 49

hours in a week, estimate his total income in

that week.

○○○○→→→→ Ex 6A 19, 20

Example 8 Install Drill 8 The daily numbers of goods sold in a shop last

week are as follows.

72, 53, 48, 51, 49, 52, 50

Estimate the total number of goods sold last week.

Sol Total number of goods sold

= 72 + 53 + 48 + 51 + 49 + 52 + 50

≈ 70 + (50 × 6)

= 70 + 300

= 370

The monthly expenses on transportation of

Mr Siu in the past six months are as follows.

$385, $497, $371, $402, $392, $413

Estimate the total expense on transportation of Mr

Siu in the past six months.

Sol Total expense on transportation

=

34. Miss Li buys four clothes. Their prices are

$207, $204, $119 and $198. Round up the

prices to the nearest $10 to estimate the total

price of the clothes.

35. The daily distances travelled by a car in the

past five days are 81.8 km, 77.2 km,

89.5 km, 74.6 km and 115.4 km. Round down

the distances to the nearest 1 km to estimate

the total distance travelled by the car in the

past five days. ○○○○→→→→ Ex 6A 18, 21

∵ All the numbers are

close to 50 except 72. ∴ Choose 50 as the

clustered value.

Use compatible numbers.

Rounding off

95


36. Use a suitable estimation strategy to determine whether the estimated value of

25.4 + 24.8 + 25.2 + 24.6 + 48 ≈ 150 is reasonable and explain your estimation strategy. to do the estimation. 25.4 + 24.8 + 25.2 + 24.6 + 48

≈

∴ It (is / is not) reasonable.

37. In a restaurant, each of 6 students orders a lunch set. The prices of the lunch sets are $41, $33, $42,

$35, $33 and $39.

(a) Round down the prices to the nearest $10 to estimate the total price they need to pay.

(b) If the total price is more than $200, they will get a 10% discount. Using the result of (a),

determine whether they can get the discount. Explain your answer.


38. Ben buys 5 reference books in a book fair. Their prices are $393, $207, $195, $206 and $208. Estimate

the average price of these 5 books.

39. (a) By front-end method, estimate the value of 20.6 + 7.21 + 4.98 + 10.17.

(b) Using compensation strategy, find an estimated value which is more accurate than that in (a).

Why do we choose rounding down method to do the estimation?

96

6 Estimation in Numbers and Measurement

Level 1

1. When expressing the value in each of the following cases, determine whether an estimated value or an

exact value is more appropriate to use.

(a) the cost price of a smart phone

(b) the number of leaves of a tree

(c) the number of passengers in a lift

(d) the population of a country

2. Round off each of the following numbers to the place value given in brackets.

(a) 3 448 (hundred) (b) 427 (ten) (c) 16.46 (one)

3. By front-end method, find the estimated value of each of the following numbers.

(a) 18.44 (b) 667 (c) 3 573

4. Round down each of the following numbers to the place value given in brackets.

(a) 6.58 (one) (b) 366.5 (ten) (c) 12 580 (thousand)

5. Round up each of the following numbers to the place value given in brackets.

(a) 3.22 (one) (b) 141.3 (ten) (c) 426 (hundred)

6. Round off each number in the following expressions to the nearest ten and then estimate the values of

the expressions.

(a) 173 + 336 − 204 + 38 (b) 483 ÷ 77 × 42

7. By front-end method, estimate the values of the following expressions.

(a) 143 + 86 + 68 − 234 (b) 6.25 × 5.36 ÷ 10.68

8. Round up each number in the following expressions to the nearest ten and then estimate the values of

the expressions.

(a) 231 + 356 + 165 + 193 (b) 44 − 237 ÷ 26

9. Round down each number in the following expressions to the nearest one and then estimate the values

of the expressions.

(a) 23.8 + 37.2 + 16.5 + 44.4 (b) 24.6 × 5.3 − 45.8

10. Estimate the value of 146.7 − 32.3 + 46.5 by rounding down each number to

(a) the nearest one,


97

(b) the nearest ten.

11. Estimate the value of 7.57 − 9.22 ÷ 3.08


(b) by rounding up each number to 1 decimal place,

(c) by rounding off each number to the nearest one.

By choosing suitable clustered values, estimate the values of the following expressions.

[Nos. 12–13]

12. (a) 72 + 69 + 71 + 67 (b) 34.3 + 34.1 − 33.8 + 33.6 + 34.2

13. (a) (211 + 209 − 212) ×

3

1 (b) (88 + 92 + 91 + 87) ÷ 6

Estimate the values of the following expressions by using compatible numbers. [Nos. 14–15]

14. (a) 83

161 (b) 41.8 ÷ 7.1

15. (a) 17.91 × 0.444 4 (b) 0.667 × 54.2 ÷ 1.9

16. Using translation strategy, estimate the values of the following expressions.

(a) 131 – 59 + 267 – 144 (b) 5

1 ÷ 19.3 × 498

17. Using compensation strategy, estimate the values of the following expressions.

(a) 11.31 + 3.62 – 5.77 – 4.09 (b) 8 × 324

18. The length of a book is 19.5 cm. Estimate the total length of 16 books.

19. The scores of 6 students in a test are shown below:

60, 61, 58, 56, 63, 90

Estimate the total score of the students.

20. The weight of a box of oranges is 5 633 g. If 1 ounce equals 28.3 g, estimate the weight of the box of

oranges in ounces.

21. A theatre has 180 seats. The numbers of students of 4 classes are 38, 42, 45 and 33 respectively.

Round up the numbers of students to the nearest ten to estimate whether the theatre can accommodate

all the students.

Level 2

Use suitable estimation strategies to determine whether the estimated value of each of the following

98

numerical expressions is reasonable and explain your estimation strategy. [Nos. 22–23]

22. (a) 0.26 × 8.3 ≈ 4

(b) 726.6 × 23.4 ÷ 7.3 ≈ 2 000

23. (a) 30.2 × 4.8 ≈ 150

(b) 20.7 + 19.8 + 19.1 + 20.3 – 60.42 ≈ 40

Use suitable estimation strategies to estimate the values of the following numerical expressions and explain

your estimation strategy. [Nos. 24–27]

24. 15.01 + 14.88 + 14.9 + 15.2

25. 26.8 − 4.36 + 23.4 − 34.6

26. 6.92 × 99.7 + 3304 ÷ 11.2

27. 60 – 31.8 × 6.04 ÷ 4.1

28. (a) By front-end method, estimate the value of 12.28 + 43.3 + 6.97 + 20.7 − 21.3.

(b) Using compensation strategy, find an estimated value which is more accurate than that in (a).

29. Estimate whether the sum of 16.3, 20.6, 32.8, 56.7 and 73.2

(a) is smaller than 230,

(b) is larger than 180.

30. The prices of 6 books are as follows.

$248, $252, $246, $250, $253, $249

Estimate the average price of the 6 books.

31. The area of Hong Kong Island is about 78.3 square kilometres. If 1 square mile equals 2.59 square

kilometres, use compatible numbers to estimate the area of Hong Kong Island in square miles.

≈ .

5

132.59Take

32. A toy shop offers a ‘Buy 2 Get 1 Free’ discount, which allows Susan to buy 7 dolls by paying the total

price of 5 dolls only. If the marked price of each doll is $69.8, estimate the average price of the dolls

bought by Susan.

99

33. In a bookshop, all books are sold at a discount of 19%. If each comic book is marked at $14.9,

estimate the total selling price of 39 comic books.

34. The table below shows the time spent by Jimmy to finish four tasks.

Task Task 1 Task 2 Task 3 Task 4

Time (min) 122 148 204 156

(a) Estimate the total time by

(i) rounding up each number to the nearest 10 min,

(ii) rounding down each number to the nearest 10 min.

(b) Using the results of (a), determine whether Jimmy can finish all the tasks within 10 hours.


35. In each of the last four months, Peter saved $628, $336, $204 and $425 respectively. Estimate whether

Peter has enough money to buy a toy sold at $1 500. Explain your answer.

36. The table below shows the weights and the quantities of some new products in a store. Round up the

weight of each product to the nearest 1 kg and estimate the total weight of all the products.

Product I II III IV V

Weight (kg) 3.8 14.2 7.7 5.1 2.6

Quantity 3 1 1 4 2

37. In a company, a machine runs four different programs. The cost of running each program is

shown as follows.

Program A B C D

Cost ($) 402.1 597.5 100.3 98.6

Yesterday, the machine ran program A 5 times, program B 3 times, program C 2 times and program D

4 times.

(a) Estimate the total cost of running the programs yesterday by

(i) rounding up each number to the nearest $10,

(ii) rounding down each number to the nearest $10.

(b) A worker of the company reported that the total cost of running the programs yesterday was $4 938. Based on the results of (a), did the worker make a mistake? Explain your answer.

100


1. (a) exact value (b) estimated value

(c) exact value (d) estimated value

2. (a) 3 400 (b) 430 (c) 16

3. (a) 10 (b) 600 (c) 3 000

4. (a) 6 (b) 360 (c) 12

000

5. (a) 4 (b) 150 (c) 500

6. (a) 350 (b) 240

7. (a) 40 (b) 3

8. (a) 970 (b) 42

9. (a) 120 (b) 75

10. (a) 160 (b) 150

11. (a) 4 (b) 4.6 (c) 5

12. (a) 280 * (b) 102 *

13. (a) 70 * (b) 60 *

14. (a) 2 (b) 6

15. (a) 8 (b) 18

16. (a) 200 (b) 5

17. (a) 5 (b) 2 600

18. 300 cm *

19. 390 *

20. 200 ounces *

21. yes

22. (a) no (b) yes

23. (a) yes (b) no

24. 60 *

25. 11 *

26. 1 000 *

27. 12 *

28. (a) 56 (b) 62

29. (a) yes (b) yes

30. $250 *

31. 30 square miles

32. $50 *

33. $480 *

34. (a) (i) 650 min

(ii) 610 min

(b) no

35. yes

36. 65 kg

37. (a) (i) $4 470 (ii) $4 330

(b) yes

101

F1A: Chapter 6B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

102



___________ ( )



_________

103

Book 1A Lesson Worksheet 6B (Refer to §6.2A, BI, C)

6.2A Approximate Nature and Degree of Accuracy of Measurement

� All measurements are approximations only.

� The higher the degree of accuracy of a measuring tool, the more accurate is the

measured value obtained (i.e. a result closer to the exact value).

6.2B Direct Measurement

I. Choosing Appropriate Measuring Tools and Units

Choose an appropriate measuring tool for measurements, and a suitable unit to express

the measured values properly.

e.g. Use a measuring cup to measure the volume of a bowl of soup, and the measuring

unit is mL.

1. Choose an appropriate measuring tool and unit for each of the following measurements.

(a) Measurement Measuring tool Unit

(i)

(ii)

(iii)

the weight of a calculator •

the temperature of milk •

the volume of a flask •

• thermometer • • m3

• bathroom scale • • mL

• mechanical scale • • °C

• measuring cup • • g

(b) Measurement Measuring tool Unit

(i)

(ii)

(iii)

the thickness of a tablet •

the length of a blackboard •

the sleeping time •

• clock • • hour

• measuring tape • • m

• stopwatch • • km2

• ruler • • mm

2. Choose an appropriate unit for each of the following, and circle the answer.

(a) the length of Castle Peak Road

( cm / m / km )

(b) the area of a poster

( mm2 / cm2 / km2 )

(c) the volume of a chocolate bar

( mm3 / cm3 / m3 ) ○○○○→→→→ Ex 6B 1

104

6.2C Indirect Measurement

I. Benchmark Strategy

Estimation is based on a known reference, which is called benchmark.


Refer to the figure. The height of the boy is about

1.6 m. Estimate the height of the flagpole.

Sol From observation, the height of the flagpole is

about twice the height of the boy. ∴ Height of the flagpole ≈ 1.6 × 2 m

= 3.2 m

The figure above shows the face of a cartoon

character. It is given that the diameter of its eye is 3

cm. Estimate the width of its face.

Sol From observation, the width of its face

is about ( ) times the diameter of its

eye.

∴ Width of its face

≈ ( ) × ( ) cm

= cm

3. The length of the pen

in the figure is about

15 cm. Estimate the

length of the desk.

Length of the desk

≈ ( ) × ( ) cm

= cm


Estimate the length

of the bed. Length of the bed

≈

5.

Refer to the figure. The height of the girl

is about 1.2 m. Estimate the height of the

wall.

6.

The figure shows the result after Peter

poured 200 mL of water into an empty

vase. Estimate the capacity of the vase. ○○○○→→→→ Ex 6B 2–5

30 cm

The height of the wall is about ( ) times the height of the girl.

105

II. Decomposition-Recomposition Strategy

Decompose a large quantity into some smaller parts first, and estimate the quantity

of each smaller part. Then, recompose the results to obtain an approximate value of

the original quantity.

Example 2 Install Drill 2 In a building of 20 floors, there are 50 residents on

the 8th floor. Estimate the number of residents in

the building.

Sol Number of the residents in the building

≈ 20 × 50

= 1 000 In a school of 30 classes, there are 35 students in

S2C. Estimate the number of students in the school.

Sol Number of students in the school

≈ ( ) × ( )

=

7. There are 15 rows in a passage, and one of

them contains 20 words. Estimate the total

number of words in the passage. 8. There are 55 bookshelves in a library. John

estimates that there are 240 books on a

bookshelf. Estimate the total number of

books in the library. ○○○○→→→→ Ex 6B 8

III. Measurement by Grouping Small Objects

Group a certain number of small objects together and obtain the measurement, and

then estimate the measurement of each small object by division.

Example 3 Install Drill 3 If the total weight of 100 candies is measured to be

50 g, estimate the weight of each candy.

Sol Weight of each candy ≈ (50 ÷ 100) g

= 0.5 g If the total weight of 200 plastic beans is measured

to be 30 g, estimate the weight of each plastic bean.

Sol Weight of each plastic bean

≈ [( ) ÷ ( )] g

= g

9. If the total thickness of 12 workbooks is

measured to be 3 cm, estimate the thickness of

each workbook. 10. If the total volume of 25 oil droplets is

measured to be 8 mL, estimate the volume of

each oil droplet. ○○○○→→→→ Ex 6B 6, 7

IV. Using Formulae

106

We can use formulae to estimate quantities such as areas, volumes, speeds, etc.

First obtain the measured values of the quantities required in the relevant formulae,

and then do the estimation.

Example 4 Install Drill 4 The length and the width of a rectangular postcard

are measured to be 14 cm and 10.5 cm respectively.

Estimate the area of the postcard.

Sol Area of the postcard

≈ 14 × 10.5 cm2

= 147 cm2 The radius of a circle is measured to be 10 cm.

Using the formula

‘circumference = 2 × π × radius’,

estimate the circumference of the circle.

(Take π = 3.14.)

Sol Circumference of the circle

≈ 2 × ( ) × ( ) cm

= cm

11. The length, the width and the height of a

rectangular box are measured to be

14.1 cm, 10 cm and 2 cm respectively.

Estimate the volume of the box. Volume of the box

≈ 12. The base and the height of a triangular carpet are

measured to be 60 cm and 38 cm respectively.

Estimate the area of the carpet.

13. A MTR train travels from Central to Tsuen

Wan. The distance travelled is 16 km and

the time used is 0.5 h. Using the formula

‘speed =time

distance’, estimate the average

speed of the train.

14. The diameter of a sphere is measured to be

7 cm. Using the formula

‘surface area of a sphere = 4 × π × (radius)2’,

estimate the surface area of the sphere.

= .

7

22πTake

○○○○→→→→ Ex 6B 9, 10

Area of a rectangle

= length × width

Volume of a rectangular box

= length × width × height

Area of a triangle

=2

1× base × height

107


15. A water tap leaks 20 droplets of water in one minute. Mable uses a cup of capacity 300 mL to collect

the water leaking from the tap. After 2 hours, the cup is fully filled with water. Is it possible that the

volume of each water droplet is more than 0.1 mL? Explain your answer. Number of water droplets collected in 2 hours

=

Volume of each water droplet

≈

∵ ______ mL ( < / > ) 0.1 mL

∴ It (is / is not) possible that the volume of each water droplet is more than 0.1 mL.

16. It is given that a book has 167 pages. Gary counts that there are 300 words on one of the pages, and

then claims that the whole book has 50 000 words. Do you agree? Explain your answer.


17.

The figure above shows a notice board in a classroom. The diameter of the circle is 20 cm.

(a) Estimate the length and width of the notice board. (Give the answers in m.)

(b) Estimate the area of the notice board. (Give the answer in m2.)


108

6 Estimation in Numbers and Measurement

Level 1


Appropriate

measuring tool

Appropriate

measuring unit

(a) Temperature of a bowl of soup

(b) Weight of a watermelon

(c) Volume of a bottle of milk

(d) Length of an exercise book

2. In the figure, building A is the Oriental Pearl Tower in Shanghai. If the

height of building B is 155 m, estimate the height of the Oriental Pearl Tower.

3. In the figure, some buses are parked on a horizontal ground. If the distance between P and Q is 15.1 m,

estimate the width of each bus.

P Q

4. A book consists of 120 pieces of paper. The weight of each piece of paper is 4.98 g. Estimate the

weight of the book.

5. The figure shows some liquid in a measuring cup. If the volume of the liquid is 1 L, estimate the

capacity of the measuring cup.

6. The weight of 16 marbles is about 728 g. Estimate the weight of each marble.


109

7. (a) Estimate the area of the figure shown on the right.

(b) Suggest a way to reduce error in the measurement

in (a).

8. Estimate the total number of words (excluding punctuation marks) in the following passage.

9. The relation between the circumference (C cm) of a circle and its radius (r cm) can be expressed by

the formula below:

rC π2=

Estimate the circumference of a circle with radius 3 cm. (Take π = 3.14.)

10. A car travels 180 km and its average speed is 60 km/h. Use the formula ‘speed =time

distance’ to

estimate the time spent by the car to travel 180 km.

11. Emma measures the volume of a cup of tea with a cylinder.

(a) Suggest an appropriate degree of accuracy for the measurement.

(b) Suggest a way to reduce error in the measurement.

110

Level 2

12. In the figure, the dimensions of the photo on the bulletin board are 3 cm × 6 cm.

6 cm

x cm

3 cm

y cm

(a) Estimate the values of x and y.

(b) Hence, find the approximate area of the board.

13. The figure shows a box of coins. Estimate the capacity of the box.

14. The photo below shows some sweets. After counting, it is known that there are about 5 sweets in the

rectangular frame as shown. Estimate the total number of sweets in the whole photo and explain your

estimation strategy.

15. The weight and the volume of 12 ice bricks are about 54 g and 60 cm3 respectively.

(a) Estimate the weight of each ice brick.

(b) Estimate the volume of each ice brick.

16. Frank spends $180 to buy a box of game cards. The box consists of 5 sets of cards. By counting, he

finds that there are 24 cards in one of the sets.

(a) Estimate the total number of the cards bought by Frank.

(b) Estimate the price of one game card.

111

17. In a library, each bookshelf has 8 layers. By counting, a layer of a bookshelf holds 45 books.

(a) Estimate the number of books in the bookshelf.

(b) There are 420 such bookshelves in the library. Ken claims that there are about 150 000

books in the library. Do you agree? Explain your answer.

18. On the map as shown, the distance between positions

P and Q is 1.5 cm. The actual distance between the two

positions is about 150 km. Estimate the actual distances

between the following positions.

(Give the answers correct to the nearest 10 km.)

(a) O and P

(b) O and R

19. The relation between the area (A cm2) of a circle and its radius (r cm) can

be expressed by the formula below:

2

πrA =

The lengths of the minute hand and the hour hand are 10.5 cm and 7 cm

respectively.

(a) Estimate the area of the clock face.

(b) Hence, estimate the area of the shaded region on the clock.

= .π

7

22Take

Q

P

R

O

112


1. (a) thermometer, °C

(b) mechanical scale, kg

(c) measuring cup (or cylinder), mL (or

cm3)

(d) ruler, cm

2. 465 m *

3. 2.5 m *

4. 600 g *

5. 3 L *

6. 45.5 g *

7. (a) 14 cm2 *

(b) use a graph paper with smaller grids

8. 225 *

9. 18.84 cm

10. 3 h

11. (a) correct to the nearest mL/correct to the

nearest 0.1 mL

(b) use a cylinder with smaller scale

interval/the line of sight is vertical to

the surface of the markings

12. (a) x = 9 *, y = 12 *

(b) 108 cm2 *

13. 48 cm3

14. 40 *

15. (a) 4.5 g (b) 5 cm3

16. (a) 120 (b) $1.5

17. (a) 360 (b) yes

18. (a) 350 km * (b) 250 km * 19. (a) 346.5 cm2 (b) 192.5 cm2

chapter 4 introduction to geometry 4a p

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