276563235-Pearson-International-Mathematics-for-the-MYP-Year-2

Exercise 13:02 b reflection c rotation d rotation and translation
b
1 a translation
2a

c

3 a i ∠L ii ∠R iii ∠H b i SR ii HG iii EF c Yes
4a A bP cF dO eH fG
5a b c

6a b c d

ef gh

7 a yes b yes c yes

8 a The figure has been produced by a combination of rotation, reflection and translation.

b i ∠S ii ∠M iii ∠D c i NO ii RQ iii AH

9a b c yes

10 a b c

480 INTERNATIONAL MATHEMATICS 2

Prep Quiz 13:03 4 ∠EFG 5 ∠ADC 6 WX 7 TS 8 VU 9 ∠QTS 10 ∠WVU

1 EF 2 AD 3 HG

Exercise 13:03

1 a shape C b shape F c shape H d shape L

2 a ∆B ≡ ∆C b shape D ≡ shape F c ∆M ≡ ∆L

3 a DB b CB c DC d ∠CDB e ∠DBC f ∠BCD

4 a CB b CF c BE d ABED ≡ CBEF e yes f yes

5 a A and C b yes c yes d Shapes B and C are not congruent because sides do not match.

They are not the same shape.

6 Two circles will be congruent if they have the same radius.

Exercise 13:04

1 b i ∆TUS ii ∆MLN and ∆IWJ iii ∆RPQ c ∆GHK

2 a ∆YZX; corresponding angles are ∠P and ∠Y, ∠Q and ∠Z, ∠R and ∠X;

corresponding sides are PQ and YZ, PR and YX, QR and ZX.

b ∆TUS; corresponding angles are ∠S and ∠F, ∠T and ∠D, ∠U and ∠E;

corresponding sides are ST and FD, TU and DE, US and EF.

3 a angles: ∠A and ∠D, ∠B and ∠E, ∠C and ∠F sides: AB and DE, BC and EF, AC and DF

b angles ∠X and ∠M, ∠Y and ∠L, ∠Z and ∠N sides: XY and ML, XZ and MN, YZ and LN

c angles: ∠L and ∠E, ∠M and ∠D, ∠N and ∠F sides: LM and ED, LN and EF, MN and DF

d angles: ∠A and ∠R, ∠B and ∠Q, ∠C and ∠P sides: AB and RQ, AC and RP, BC and OP

e angles: ∠M and ∠X, ∠N and ∠G, ∠L and ∠A sides: LM and AX, LN and AG, MN and XG

4 a The circle drawn will be identical to the one shown.

b The circle drawn will be identical to the one shown.

c 5 a 5 cm

3 cm 5 cm 5 cm

b 5 cm 5 cm
c 6 cm

3 cm 3 cm 3·5 cm 3·5 cm
d 5 cm

5 cm

5 cm

6 a ∆ABC and ∆BAD; ∆ADC and ∆BCD; ∆AED and ∆BEC
b ∆ABD and ∆BAD; ∆AED and ∆BEC

5 cm

481ANSWERS

Diagnostic Test 13: Geometric Constructions and Congruent Figures

1a bA cB

B

A BA
2a bc

3a 1 2 3 4
C C
A BA
b1 2 BA 60° B
3 BA 5

4

A A A A 120°
2 3 4 A
c1
5

A AB AB AB A B
bC
4a c A
A B B
C
A
C

5 a A and C b A and C B

6 a MN b ML c NL d ∠MNL e ∠NLM f ∠GEF
b CB c AD d ∠BCD e ∠BDC
7 a ∆ABD ≡ ∆CBD b
Answers will vary, but the identical process would
8a be used to construct each triangle.

c Answers will vary but the identical process would be used to construct each quadrilateral. (See Appendix A:07E.)

13A Revision Assignment

1 a 6x + 7y + 6 b 6x2 + 5x − 6 2 a −x2 + 2x b 4a2 − 12a + 3ab + 2b

3 a p = −7 b a = − 7-- c b = −5 -1- 4 a 34 b −54 c x = −2
5 a 105 b 73 9 c 47 3 6 a x=1 b y=2

7 132 cm3 8 a 23·2 kg b 2·4 L of A and 1·6 L of B 9 a 8·4 kg b 0·4%

482 INTERNATIONAL MATHEMATICS 2

10 11 a x = 27 b x = 6·6 (to nearest mm)
5·5 cm
78°
4·2 cm

6·2 cm

13B Working Mathematically

1 a 38 + b 904 − c 564 + d 1632 − 2 12 L, 33 L
925 6 (1, Emma); (2, Sarah); (3, Ben); (4, Kate)
36 63 537 707

74 841 1101 57

3 5 4 $1190

Chapter 14: Statistics

Exercise 14:01

1 a Alexis b Caroline’s c not a representative sample d more time-consuming

2 a Fellow students are all in a particular age group. b Survey people in the street.

3 a His sample is limited to students of his own school, who would all have a similar range of ages and experiences.

b He has not listed all sports. The order of sports could also influence the result.

c He could use stratified random sampling.

d Student response. (Each person could be asked to write down their favourite sport.)

4 They survey different samples of people.

5 a no b Roll the dice a large number of times.

c The number of times a person rolls a dice is limited by the time available. We would get closer to the real

probability if we threw the dice millions of times. In this sense, it is like a sample.

6 The accuracy of the survey would depend on the size of the sample and the sample is limited to the readers of the

magazine.

7 a To what is MUNCH–O’s preferred? b Which 5 dentists? c 20% more than what? d Longer than what?

8 Grade 8, 13; Grade 9, 16; Grade 10, 19; Grade 11, 16; Grade 12, 16

9 15–25 years, 137; 25–45 years, 341; 45–65 years, 304; 65 years and over, 218. 10 Student research.

Exercise 14:02 b 24 2 a Shape Tally Frequency b 34
c3 c star
1 a Outcome Frequency d6 circle |||| || 7 d triangles
e 12 square |||| | 6
23 oval |||| 4
39 star |||| |||| 10
46 triangle |||| || 7
54
62 Total 34

Total 24

3 a Outcome Tally Frequency b4 4 a 10 c Outcome Tally Frequency d7
c2 b4 e8
1 |||| | 6 d No 4 | 1 f 15
2 |||| 4 e 39 5 ||| 3
3 |||| || 7 d6 6 |||| 4
4 |||| ||| 8 d 19 7 |||| || 7
5 |||| 5 8 |||| | 6
6 |||| | 6 9 |||| | 6
10 ||| 3
Total 36
f 21
5a 4 b 3 c 9 Total 30
6a 1 b 2 c 7 g 162

483ANSWERS

7a b

66
55
44
33
22
11
Frequency
Frequency

0 1 2 345 0 7 8 9 10 11 12
Outcome Outcome

8 a Outcome Freq. b 39 9 a 33, 27 b Outcome Tally Freq. c 12
c 16 d 18
0 3 27 || 2
10 6 28 |||| 4
20 7 29 |||| | 6
30 10 30 |||| |||| |||| |||| 20
40 5 31 |||| |||| 10
50 8 32 |||| | 6
33 || 2
Total 39

Total 50

e 10 a Responses Tally Freq.
20
Strongly agree (A) |||| |||| |||| |||| |||| ||| 28
16 46
12 Agree (B) |||| |||| |||| ||||
8 21
5
4

0 27 28 29 30 31 32 33
Outcome
Frequency |||| |||| |||| |||| |||| |
Frequency
Disagree (C) |||| |||| |||| |||| |

Strongly disagree (D) ||||

b 28 f 50
c 5% 40
d 74% 30
e 26% 20
10

0 A BCD
Response

Exercise 14:03

1 a i 3 ii 3 iii 40 iv 5 v3 4a Outcome fx b Outcome fx c Outcome fx
vi 5 vii 7 viii 0·4 ix 6 x5
12 55 18 18
b i 5 ii 7 iii 10 iv 17, 20 28 6 30 19 57
3 15 7 21 20 20
v 50 vi 8 vii 4 viii 1·1, 1·3 4 24 8 64 21 84
5 15 9 45 22 110
ix 17 x 148, 151 10 30 23 69
Total 64 24 24
c i 4 ii 7 iii 25 iv 20 v 50 Mean = 3·2 Total 195 25 50
Mean = 7·8
vi 9 vii 4 viii 1·2 ix 17 x 150 Total 432
Mean = 21·6
d i 3·8 ii 7·6 iii 26·7 iv 19·4 v 49·6

vi 9·2 vii 4·6 viii 1·2 ix 17 x 149·6

2 a 30·3 b 74·2 c i 40 ii 11

d2

3 a no b the mean

484 INTERNATIONAL MATHEMATICS 2

4 a Outcome fx b Outcome fx c Outcome fx

12 55 18 18
28 6 30 19 57
3 15 7 21 20 20
4 24 8 64 21 84
5 15 9 45 22 110
10 30 23 69
Total 64 24 24
Mean = 3·2 Total 195 25 50
Mean = 7·8
Total 432
5 a 4 b 8 c 22 Mean = 21·6 b 8 c 22

6a 3

7 a Outcome Tally Freq. fx b i 5 ii 1 iii 2 iv 1·94˙ or 1 -1---7-
c
0 18
5 5
0 ||| 3 8 Frequency
1 |||| 5 9 4
2 |||| 4 8
3 ||| 3 5 3
4 || 2
5| 1 35 2

Totals 18 1

d highest column

0 0 1 234 5
Outcome

8 a Outcome Tally Freq. fx e 20 Number of Heads
18
0 | |||| 1 0 16Frequency 0 1 234 5 6 7
1 |||| || |||| |||| 7 7 14 Outcome
2 |||| |||| |||| 14 28 12
3 |||| |||| 20 60 10
4 |||| |||| Totals 15 60 8
5 |||| d 14 5 25 6
6 || 2 12 4
2
b3 c 14 64 192
0
9 a Outcome Freq. fx b2
c2
0 3 0 d 2·5
1 4 4 e 12
2 6 12
3 5 15
4 3 12
5 1 5
6 2 12

Totals 24 60

485ANSWERS

10 a Barbara’s results Rob’s results

Outcome Tally Freq. Outcome Tally Freq

3 || 2 3| 1
4 0 4 |||| 4
5 |||| 4 5 ||| 3
6 |||| 4 6 |||| || 7
7 |||| |||| 9 7 |||| | 6
8 |||| |||| ||| 13 8 |||| | 6
9 |||| |||| 10 9 |||| ||| 8
10 |||| ||| 8 10 |||| |||| |||| 15

Total 50 Total 50

b

Barbara’s results Rob’s results
15 15
10
10 5
5
Frequency
Frequency

0 3 4 5 6 7 8 9 10 0 3 4 5 6 7 8 9 10

Outcome Outcome
c Barbara i 7 ii 8 iii 8 iv 7·74; Rob i 7 ii 10 iii 8 iv 7·76 d Rob’s results were more impressive.

Exercise 14:04

1 Class Tally Frequency 2 Class Tally Frequency

31–40 |||| 5 0–2 |||| 5
41–50 |||| | 6 3–5 |||| |||| || 12
51–60 |||| |||| | 11 6–8 |||| |||| || 12
61–70 ||| 3 9–11 |||| |||| ||| 13
71–80 |||| | 6 12–14 |||
81–90 ||| 3 15–17 ||| 3
91–100 | 1 3
Total
48

Total 35

3 Class Class centre Tally Freq. 4 Outcome Tally Freq. Class Tally Freq.

1–5 3 |||| |||| 10 0 0 2–4 |||| 4

6–10 8 |||| |||| || 12 1 0 5–7 |||| |||| |||| |||| 20

11–15 13 |||| |||| 9 2| 1 8–10 |||| | 6

16–20 18 |||| 4 3 0 Total 30
4 ||| 3
21–25 23 ||| 3

26–30 28 || 2 5 |||| 5 In the second table the value of

Total 40 6 |||| || 7 the individual scores is lost.
7 |||| ||| 8

8 |||| 4

9| 1

10 | 1

Total 30

5 Class Class centre Frequency a Using three classes does not show the spread of scores realistically
as it makes the first two classes appear very similar. If the classes
0–2 1 25 0–1, 2–3, 4–5 etc had been used, it would show how different
3–5 4 22 they are.
6–8 7 13

Total 60

b It would be difficult to use more than 5 classes in this case since there are only 9 outcomes. Having groups with the
same number of outcomes would not be possible if more than 5 outcomes are used.

486 INTERNATIONAL MATHEMATICS 2

6 a Frequency Histogram and Polygon b Frequency Histogram and Polygon
15 15

10 10
Frequency Frequency

55

0 18 23 28 33 38 43 48 53 0 4 7 10 13 16 19 22 25
Class centre Class centre

7 a Frequency Histogram and Polygon b Frequency Histogram and Polygon

15 15

10 10
Frequency Frequency

55

0 35·5 45·5 55·5 65·5 75·5 85·5 95·5 105·5 0 1 4 7 10 13 16 19
Class centre Class centre

c Frequency Histogram and Polygon 8 a The class 51–60 was by far the most popular
15 class. Nobody scored less than 31.

10 b Most of the scores were between 3 and 11 with
nobody scoring over 17.

c Marks were low with most scoring in the range
1 to 15 with nobody scoring more than 30.

Frequency

5

0 3 8 13 18 23 28 33
Class centre

487ANSWERS

Exercise 14:05

1 a 19 b 20 − 6 = 14 c 17 d 6 is the only outlier. e 3

f An outlier is any score that lies apart from the main body of scores.

2 a 11 b 16 c 11
e6
d 19 is an outlier.

3 5 7 9 11 13 15 17 19

3 a Mode Median Mean Range b Overall they are smoking much less and, apart from the
three who have died, four have given up smoking.
1986 20 12 12·1 19 Factors could include:
• an increased awareness of the dangers of smoking
1996 0 4 7 22 • decreased health that forces a reduction in smoking
• an awareness of reduced fitness because of smoking
• consideration for others
• restrictions on where people can smoke
• the increasing cost of smoking

4

1 1·2 1·4 1·6 1·8 2 2·2
5a

0 1 2 3 4 5 6 7 8 9 10 11 12

0 1 2 3 4 5 6 7 8 9 10 11 12

b The improvement made by c Mode Median Mean Range
particular individuals can
no longer be seen. Beginning 2 4 4·5 10
End 10 9·5 8·7 11

6 a no relationship b inverse relationship c direct relationship d inverse relationship

e direct relationship f no relationship

7 a i Maths 5, English 7 ii Maths 20, English 19 iii Maths 9, English 13 iv Maths 18, English 14

b i 20 ii 19

c The scatter diagram suggests that students tend to get similar results in Maths and English.

d 16 e 13 f 11 g 12

8 a Mr Justin’s Class Results b From the pattern at the left, it is obvious that students
100 tended to perform as well in both tests.

c equal 12th d 16th e 75 f 69

80 9 Results of Mrs Everingham’s

Class in Maths and English
100
60
Yearly result (%)
English mark
80
40

60
20

20 40 60 80 100 40
Half-yearly result (%) 20

20 40 60 80 100
Maths mark

488 INTERNATIONAL MATHEMATICS 2

10 Answers will vary, but should say that students tended to perform about the same in both tests. Some, like R Tist,
have positions that were not close. Marks tended to be higher in the half-yearly exam than in the yearly exam. No
marks were lower than 27% in either test.

Exercise 14:06 b Test Marks 2 a Test Marks b Test Marks
(5/3 represents 53.) (2/6 represents 26.) (5/3 represents 53.)
1 a Test Marks
(2/6 represents 26.) Stem Leaf Stem Leaf Stem Leaf

Stem Leaf 4 5492 1 56789 4 2459
5 379 2 66689 5 379
1 59768 6 7053 3 03568 6 0357
2 66986 7 42073 4 01777 7 02347
3 38560 8 102018 5 18 8 001128
4 70717 9 12 6 22 9 12
5 18
6 22

median = 3----3----+-----3---5-- median = 7----0----+-----7---2--
2 2

= 34 = 71

3 a Runs Scored at Cricket
(0/8 represents 8 runs.)

Stem Leaf Frequency b Stem Leaf Frequency

(f) (f)

0 864192 6 0 124689 6
1 5891 4 1 1589 4
2 0769 4 2 0679 4
3 632 3 3 236 3
4 6503 4 4 0356 4
5 0 5 0
62 1 62 1

Total 22 Total 22

c range 61, median -2---0----+-----2---6-- = 23, mean = 24·18 (to 2 dec. pl.). The only outlier is 62.
2
d 45·5%

4 a Year 8, Total Score, Darts

Stem Leaf b range 59; mode 14; median 26·5
c 60 is the only outlier. There are no obvious clusters.
0 1112345567888999
1 0112233444445677789 The scores tend to be either good or poor. This means
2 0245678 the distribution has a bulge at each end of the scale.
3 00135799 d 41 is 12·7 above the mean, is 19 below the highest
4 1222344455666789 score and is the 30th highest of the 80 scores.
5 0112223344555
60

5 a 3 × tables b mode = 12 s, median = 27 s, range = 47 s

c There are two modes, 32 s and 59 s, median = (39 + 41) ÷ 2 = 40 s, range = 44 s.

d No outlier is present in the data for either table as no score stands alone at one end of the data.

6 a Quiz Scores

Students Parents b Students: mode = 30, median = 30, range = 32

Leaf Stem Leaf Parents: mode = 42, median = 26·5, range = 36
c the median

06 d Students’ mean Ӏ 30·72

9985 1 16688 Parents’ mean Ӏ 27·57

8865432100 2 012345667

9755430000 3 01122578

75422110 4 01222

489ANSWERS

7 a Pulse Rates

Adults Children b For the adults’ pulse rates, 12 and 92 are outliers. For the children’s
pulse rates, 62 might be considered an outlier (or might not).
Leaf Stem Leaf
c The outlier 12 can be ignored unless you know that the adult was dying
21 at the time. It is outside the range of human pulse rates and so must be
2 an error in measurement.
3
4 d Adults: median = (64 + 68) ÷ 2 = 66 (If the score 12 is ignored, the
median for adults is 68.) There are three modes: 64, 68 and 72.
98765 5 Children: median = 86. There are two modes: 85 and 86.
888444100 6 27
e The adult pulse rates range from 55 to 92 (a range of 37), have a
66222 7 2778 median of 68, modes of 64, 68 and 72 and a mean of 65·625. (For this
100 8 125556668 description the score 12 has been ignored.) The child pulse rates range
2 9 2445899 from 62 to 102 (a range of 40), have a median of 86 and modes of
10 02 85 and 86. Overall, the pulse rates of these children tend to be about
18 beats per minute more than those of these adults.

Diagnostic Test 14: Statistics

1 a Outcome Tally Freq. b Outcome Tally Freq. c Outcome Tally Freq.

1 ||| 3 0 ||| 3 3 || 2
2 |||| || 7 2 |||| 5 4| 1
3 |||| 4 4 |||| |||| 9 5 ||| 3
4 |||| 4 6 |||| 5 6 |||| 4
5 || 2 8 || 3 7 |||| | 6
8 ||| 3
Total 20 Total 25 9 |||| 4
10 || 2

Total 25

2aFrequency b9 Frequency
7 8
6 7 2468
5 6 Outcome
4 5
3 4
2 3
1 2
1
0 12345 0
Outcome 0

c6
5

Frequency 4 3a 6 b7 c 0·7
3 4a 6 b3 c 4, 6
2 5a 2 b4 c7
1 6a 7 b6 c 4·5
7 a 2·5 b4 c7
8a 6 b 5·625 c6
9 a 2·75 b4 c 6·84

0 345678 9 10
Outcome

490 INTERNATIONAL MATHEMATICS 2

10 Class Class Tally Frequency 11 a i 29 cm ii 29·5 cm iii 13 cm b 20 cm
centre

0–4 2 ||| 3
5–9 7 |||| |||| 10
10–14 12 |||| |||
15–19 17 |||| | 8
20–24 22 ||| 6
3

Total 30

12 Performance in Art and Music No. There is not a positive relationship between Art marks
100 and Music marks because many who scored high in one
subject scored low in the other subject. Yet some did just
80 as well in each subject.

Music mark 60

40

20

20 40 60 80 100
Art mark

13 a 35 in the Grade 4 results is an outlier. b mode = 42, median = 48, range = 41

c mode = 79, median = (65 + 68) ÷ 2 = 66·5, range = 45

14A Revision Assignment

1 a 3375 cm3 b 750 cm3 c 2625 cm3

2 a a = 15 (∠s in a right ∠) b b = 40 (∠s in a right ∠) c x = 120 (∠s at a point)

3 5 m; 70 mm 4 3 min, 12 sec; 62·5 m 5 30 km/h
6 a 0·3 ha b 6 cm 7 a y = 2x
b y = 3x + 1 c y = x2

8 a 0·6% b 400% 9 a −11 b −18 c5 d −2

10 a 12 b Range = 5
Mode = 4

10 Mean = 3·4 (to 1 dec. pl.)

Frequency 8

6

4

2

0 123456
Outcome

14B Working Mathematically

1 Digit 0 1 2 3 45 6 7 8 9

Number needed 13 15 15 61 12 5 5 4 4 4

2 1248 3 4 cm2 4 16 (assuming that all are different people). 59

6 a hectare b regular shapes c net of a cube d cross-section

e face f vertex g edge h axes of symmetry

i reflection (or flip) j translation (or slide) k rotation (or turn) l tessellation

491ANSWERS

Chapter 15: Probability

Prep Quiz 15:01

1 A: The result ‘tails’ is just as likely as the result ‘heads’. 2 B: Buses don’t often break down.
3 C: Most people on their birthdays are given a present. 4 B: Not many people score 100 runs when they play cricket.
5 A: A girl being born is just as likely as a boy being born.

Exercise 15:01

1 a For something to be impossible means that it will never happen.
b For something to be certain means that it will always happen.

2 a Event 1: unlikely; Event 2: even chance; Event 3: even chance; Event 4: impossible; Event 5: certain
b Event 1: even chance; Event 2: unlikely; Event 3: impossible; Event 4: certain; Event 5: likely
c Event 1: likely; Event 2: certain; Event 3: impossible; Event 4: unlikely; Event 5: likely

3 a B, A, C b A, C, B c B is the least likely. A and C have the same probability, which is greater than that of B.

Prep Quiz 15:02

1 False. Having failed on the last three attempts, I am not likely to be successful on the fourth attempt. Also, getting tired

and/or despondent has a detrimental effect.
2 True. The total shown on two dice can be 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.
3 True. All numbers between 1 and 10 have 1 digit.
4 True. For each toss, the chance of a head will always be -1- , irrespective of the number of heads previously thrown.
5 False. There is one chance in 1000 that the counter chose2n will be yellow.

Exercise 15:02

1 a 1, 2, 3, 4, 5, 6 b head, tail c A, B, C, D, E d red, blue, green, pink, brown, yellow, orange

2a 2 b 99 c 101 d 10

3a -1- b 0 c 1-- d 1-- e -1- f -1- g1
4a d A 3-- , B 1, C 1--
5a 6 2 2 3 2
6a 42
7a i 1-- ii 1-- b i 50% ii 50%
8a
9a 22 c -2- d 0

-1- b -3- 5

55 b A 1-- , B 4-- , C 3-- c A 1-- , B 0, C -1-
25 8
A 1-- , B 1-- , C 1-- 44
c -2- d0
45 8 e 2--
3
-1- b 1-- 3

36 c --1--- d -2---1- (ie --8---4--- )

-1- b 3-- 20 25 100

44

----1---- b --1---

100 10

e 0. If I don’t buy any tickets, it is impossible for me to win.

f 100. Only if I buy all the tickets can I be certain of winning.

10 a B bC cA

11 a --1--- b --1--- c --3--- d1 e0 f --2---
12 a 13 d --1--- e --3---
1-- 13 13 13
4 13 13
b -1- c -1- f --2---

2 2 13

Prep Quiz 15:03

1 --1--- 2 --9--- 3 --3--- 4 --7--- 5 --2--- (or -1- ) 6 --8--- (or 4-- ) 7 --4--- (or 2-- ) 8 --6--- (or -3- ) 9 0 10 1 11 1
10 5 10 5 10 5 10 5
10 10 10 10

Exercise 15:03

1 a It won’t rain tomorrow. b I won’t win the race. c A six will not be rolled.
d I will catch a cold next year. e That team will beat us. f Alice will not be school captain.
The plate will break. c We will not score 200 runs.
2 a The egg will not break. b I will not win the raffle. f We will not win the soccer match.
d We will toss no heads. e
--3---
3a 98% b 1-- c 60% d 0·01 e0 f 10
4a
5a 1-- b 3 d0 7----9---7-
6a 800
7a 2 -1- c 1 d 98% e 79% f
8a e 3-- f 2-- g 1
9a 1----6- b 2 d 1----5---7-
5 5
17 --7--- c -1---3---3- 200
d --5---1--- (ie 51%) e --5---7--- (ie 57%) f --7--- (ie 28%)
-3- b 10 170 d 3--
100 100 25
4 7-- c -1---7- 5

--3---7--- b 8 20 -4- (ie 80%)

200 1----6---3- c --4---3--- 5

1-- b 200 200

5 -4- c -2-

--4---3--- (ie 43%) 55

100 b 1----8- (ie 72%) c

25

492 INTERNATIONAL MATHEMATICS 2

10 a Yes. The probabilities of the given ticket colours combined do not add up to 1.

b 0·625 c 0·875 d 0·75 e 0·25

11 a -1- b -3- c --1--- d 1----2- e --3--- f -1---0- g --3--- h -1---0- i1
12 a 4
13 a 2----7- 4 13 13 13 13 13 13
53
--6---1--- b --1--- c 4----4- d -4---1- e 170 or 340 stamps, etc. The total number of stamps in this collection
170 must be some multiple of 170 as there have to be 37 French stamps
53 85 85 to choose (or some multiple of 37).

b -1---0---9-

170

14 a -2---2---3- b -5---7---7- c 46% (ie 2----3- ) d 54% (ie 2----7- )

800 800 50 50

e 800 or 1600. The total number of coins in this collection must be a multiple of 800 since the probability with the

largest denominator is P(Chinese) = ----3---- , so there must be at least 3 Chinese coins in the collection (or some multiple of 3).

800

Fun Spot 15:03

1 a -1- b 0 c 1-- d 1-- 2 a -1- b -2- 3 a 1-- b 1-- c 0 d 1--
63 3
6 6 3 3 3

Prep Quiz 15:04

1 1, 2, 3, 4, 5, 6 2 tail, head 3 cleared, not cleared 4 52 54 6 13

7 hearts, diamonds, clubs, spades 8 -1- 9 1-- 10 0

62

Exercise 15:04

1 a Manchester United football team is very likely to win this Saturday.
b Early showers are likely today.
c The drought has been severe and it is very unlikely it will rain tomorrow.
d We believe that our company is very likely to fail.
e This election has Ayun Jun and Danielle with an even chance of victory.
f Everybody overboard. The ship is certain to sink!

2a --4--- b --7--- c --4--- d --7--- e1 f1
3a 11 e 2--
5a 1-- 11 11 11
5 5
--1--- b 4-- c0 4 0·97
10
5 d 4--

b --3--- 5

10

6 a 0·35 b 0·59 c 0·66 7 0·32 8 95%

9 a 1-- b 3-- c 3-- d 1-- e 7--

8 8 8 8 8

10 a 30 b 90 c 75 d 300

Diagnostic Test 15: Probability

1 a B, A, C b A, B, C 2 a head, tail b 1, 2, 3, 4, 5, 6

3a 6 b 52 4 a -3- b 1-- c 3--
5a b1 4
6a 2-- b -3---1- d 4-- 4 4
5
--3--- 36 c0 5
10
c Not choosing a picture card, -1---0- d Not winning the game, --3---
13
10

15A Revision Assignment

1 A and C 2 a 11·18 b 7·94 3 a yes b 10
6 a 37·5% b 87·5%
4 yes, yes 5 6·3 cm (to 1 dec. pl.)
e $10.20 g −18y
7 a 0·045 b 0·125 0 c 6·25% d $13.80 11 y = 10x + 1 g 2----a---2-
8 a 98 g b --2--- 1 2 15
15
9 a $2100 15

10 x

y 11 15 19 71

12 a 2 m b 7x c 4ab d 5x2 e 9a + b f −7a h 28ab
f -5---m--- h -5---x-
i8 j 3a k a2 l 6a2
6 4
13 a 66 b 149 c −9 d −72 -3---x-
14 a m4 b 5a3 c 12x + 12 d 18m − 13 e
b m(m − 3) c 11(2 − 3y) d 3a(3b + 4a) 10
15 a 2(x + 7)

493ANSWERS

15B Working Mathematically

1 a parallel lines b perpendicular lines c concurrent lines d angle ABC or CBA

e acute angle f right angle g obtuse angle h straight angle

i reflex angle j revolution k adjacent angles l 360°

2 14 3 2 836 000 sheets

4 a 365 Days in a Year b 9 Lives of a Cat c 52 Cards in a Standard Pack

d 26 Letters in the Alphabet e 66 Books of the Bible f 12 Months in a Year

g 1000 Years in a Millennium h 64 Squares on a Chess Board

5 a The only reasonable answer is 3 sound systems because $5683.89 divided by 3 is equal to $1894.63.

Dividing $5683.89 by any other reasonable number does not give a whole number of cents.

b $1894.63

6 a As one wheel turns it forces the other wheels to turn because the wheels are interlocked. If a wheel is turning

clockwise, it will force the interlocked wheel to turn anticlockwise.

b There would be about 90 cogs on this wheel. The method used would be to count the number of cogs on a quarter

of the wheel and multiply by 4.

7 a A greater percentage of young men (40%) are disabled through accident than are young women (25%). 60% of

young men and 75% of young women are disabled through sickness.

b Men may be more reckless and would be more likely to be involved in dangerous occupations, while women may

be more susceptible to illness.

Chapter 16: Graph theory

Exercise 16:01

1 A vertex is a train station and the edges are train lines.

2 There are loops and more connections between lines.

3 There is too much information on the map which makes it confusing and it is not as easy to use as the graph.

4 a Vertices are airports and edges are air routes.

b None of the branches are linked.

5a 4 b 3 c 6

6 They all have degree 1.

7 a No, it’s only a graphical representation. b 6, 4 c 15

Exercise 16:02 Park
Ong Little
1 Delhi Metro, Dragonair and the Victoria Line subgraph.
2 Park

Moon Ong Little Moon Little Moon Ong Little Little Little

Ong Ong

Lam Lam Lam Wood Chen Lam Wood Chen Lam Wood Lam Wood

4 n − 1.

5 Airlines operate over large distances and subways are for local transport and so need to connect more places.

6 There are many solutions to these questions; examples below.

aA B bA B cA B dA B eA B

E C E C C CE C
7 D
DD

494 INTERNATIONAL MATHEMATICS 2

Exercise 16:03 3 A possible answer:

2 A possible answer: B3 Administration Block A Block B
R4 B2
R8 R7 R6 R5 B1 B10 B9

Block C

R9 R10 R1 R2 R3 B5 B6 B7 B8 Library Canteen Toilets
B4
2 a By creating a graph, and making sure vertices that are
Exercise 16:04 joined are not the same colour, the result 4 is obtained.

1a b Still 4. The same as Bolivia.

Eq Co V

b 20 hellos Pe G
3 By starting with the vertices and joining Bol Br S

the minimum connectors, this minimum FG
spanning tree is formed. The minimum Ch
distance of wire is 86 metres.
AU
A Pa

11 4

G B 15
E

11

16 Seoul Kangneung

18

15 C

F Weongju
D

Daejeon

Ulsan

Pusan

Kwangju

495ANSWERS

Diagnostic Test 16: Graphs and Networks

1 Graph 1 Graph 2 Graph 3 Graph 4 Graph 5
Degree of vertex A 2 3 2 3 2
Degree of vertex B 1 2 3 3 2
Degree of vertex C 1 2 3 3 1
Degree of vertex D 0 1 2 3 1

2 Jujube Kiwifruit 3 Answers will vary. 4 a Yes b No

Lime
Nectarine Mango

16A Revision Assignment

1 a 5-- b 1-- c 2x2 − 2y d 16mn 2 a $800 b 2-- c 36°
8 3 c 52·81 4 a 90° 3 c 16·2 mm2
6 a 9·66 cm2
3 a −3a b 21b + 6d 8 a 14 b 60°

5 a 16 b -1- litre b 24 m2
2 c --4---
7a 3
b 10·75 15

16B Working Mathematically

1 4 2 18 days 3 10 chickens and 15 goats
5 a 2:00 am
4a 4 b 12 b 11 hours 45 minutes

6 12 years old

496 INTERNATIONAL MATHEMATICS 2

Answers to ID Cards

ID Card 1 (Metric Units) page xiv

1 metres 2 decimetres 3 centimetres 4 millimetres
7 square centimetres 8 square kilometres
5 kilometres 6 square metres 11 cubic centimetres 12 seconds
15 metres per second 16 kilometres per hour
9 hectares 10 cubic metres 19 kilograms 20 tonnes
23 kilolitres 24 degrees Celsius
13 minutes 14 hours

17 grams 18 milligrams

21 litres 22 millilitres

ID Card 2 (Symbols) page xiv

1 add (or plus) 2 subtract (or minus) 3 multiplied by (or times) 4 divided by
8 is less than
5 is equal to 6 is approximately equal to 7 is not equal to 12 is greater than or equal to
16 4 cubed
9 is less than or equal to 10 is not less than 11 is greater than 20 therefore
24 denominator
13 is not greater than 14 is not greater than or equal to 15 4 squared

17 the square root of 25 18 the cube root of 27 19 per cent

21 for example 22 that is 23 numerator

ID Card 3 (Language) page xv

1 6−2=4 2 6+2=8 3 6÷2=3 4 6−2=4 5 6÷2=3
10 6 × 2 = 12
62 76 8 6 × 2 = 12 9 6−2=4 15 6 − 2 = 4
13 62 = 36 20 6 − 2 = 4
11 2 + 6 = 8 12 6 − 2 = 4 14 36 = 6
18 6 + 2 = 8 19 62 = 36
16 6 × 2 = 12 17 (6 + 2) ÷ 2 = 4
23 6 ÷ 2 = 3 24 6 + 2 = 8
21 6 − 2 = 4 22 6 + 2 = 8

ID Card 4 (Language) page xvi

1 square 2 rectangle 3 parallelogram 4 rhombus
7 regular hexagon 8 regular octagon
5 trapezium 6 regular pentagon 11 isosceles triangle 12 equilateral triangle
15 cube 16 rectangular prism
9 kite 10 scalene triangle 19 rectangular pyramid 20 triangular pyramid
23 sphere 24 hemisphere
13 circle 14 oval (or ellipse)

17 triangular prism 18 square pyramid

21 cylinder 22 cone

ID Card 5 (Language) page xvii

1 point A 2 interval AB 3 line AB 4 ray AB
7 number line 8 diagonals
5 collinear points 6 midpoint 11 obtuse-angled triangle 12 vertices
15 180° 16 (a + b)°
9 acute-angled triangle 10 right-angled triangle 19 a° = 60° 20 3 × 180° = 540°
22 circumference 23 semicircle
13 ∆ABC 14 hypotenuse

17 360° 18 a° = b°

21 AB is a diameter. OC is a radius.

24 AB is a tangent. CD is an arc. EF is a chord.

ID Card 6 (Language) page xviii

1 parallel lines 2 perpendicular lines 3 vertical, horizontal 4 concurrent lines

5 angle ABC or CBA 6 acute angle 7 right angle 8 obtuse angle

9 straight angle 10 reflex angle 11 revolution 12 adjacent angles

13 complementary angles 14 supplementary angles 15 vertically opposite angles 16 360°

17 transversal 18 corresponding angles 19 alternate angles 20 cointerior angles

21 bisecting an interval 22 bisecting an angle 23 ∠CAB = 60° 24 CD is perpendicular to AB

ID Card 7 (Language) page xix

1 anno Domini 2 before Christ 3 ante meridiem 4 post meridiem
7 net of a cube 8 cross-section
5 hectare 6 regular shapes 11 edge 12 axes of symmetry
15 rotation (or turn) 16 tessellation
9 face 10 vertex 19 picture graph 20 column graph
23 bar graph 24 scatter diagram
13 reflection (or flip) 14 translation (or slide)

17 coordinates 18 tally

21 line graph 22 sector (or pie) graph

497ANSWERS TO ID CARDS

Index

Most references are to chapter topics, angles

eg 1:03 refers to Chapter 1: Section 3 adjacent 10:01

(Fractions). at a point A:06, 1:06, 10:02,

A:01 to I:01 refers to the appendix MT 10

section of the CD. cointerior A:06, 1:06, MT 10

The following abbreviations are used: complementary A:06, 1:06, MT 10

D&D Drag and Drops construction of 60º,

FS Fun Spot 120º, 30º, 90º F:02

Inv Investigation corresponding A:06, 1:06, MT 10

MT Mathematical Terms obtuse A:06, 1:06

RM Reading Mathematics on parallel lines 10:03, Tech 10:03

Tech Technology Applications reflex A:06, 1:06

CW Challenge Worksheet review of last year’s work A:06, 1:06

right A:06, 1:06

straight A:06, 1:06

supplementary A:06, 1:06

acute angles A:06, 1:06 vertically opposite A:06, 1:06, 10:02

addition annulus 12:04, MT 12

of algebraic fractions 6:10 ant, stupid FS 13:04

of like terms 6:02 anyone for tennis FS 9:07

addition and subtraction arcs 12:01, MT 12

of decimals A:04, 1:04 area Chapter 11, MT 11

of directed numbers A:10, 1:10 in a house Inv 11:01

of fractions A:03, 1:03 of a circle 12:04, Inv 12:04,

of mixed numerals A:03, 1:03 Tech 12:04

adjacent angles A:06, 1:06, 10:01, of a cube 11:05

MT 10 of a kite 11:03

algebra Chapter 6, MT 6 of a parallelogram 11:02, 11:03

addition and subtraction of a quadrilateral 11:02

of like terms 6:02 of a rectangle A:09, 1:09, 11:01,

applications 6:06 11:03

binomial products 6:10 of a rhombus 11:02, 11:03

division of pronumerals 6:04, 6:05 of a square A:09, 1:09, 11:01,

factorising 6:09 11:03

fractions 6:10 of a trapezium 11:02, 11:03

grouping symbols 6:08 of a triangle A:09, 1:09, 11:01,

indices 6:07 11:03

making sense of algebra Inv 6:01 review of last year’s work A:09, 1:09, 11:01

multiplication of units of area 12:04, MT 11

pronumerals 6:03, 6:05 averages 14:03

of magic squares Tech 6:01 axes MT 9, MT 8

review last year’s work A:12, 1:12 of a graph 9:01

algebraic expressions A:12, 1:12, 7:01, lines parallel to axes 8:05

MT 7 back-to-back
stem-and-leaf plot
algebraic fractions 6:10 14:06
bar graph 9:01, 9:04
algebraic sentences A:12 beginnings in number A:01, 1:01
binomial products 6:10, Challenge 6:10
alternate angles 1:06, 10:03, MT 10 bisecting intervals, angles F:01, 13:01, MT 13
boy drinking milk after
analysing data 14:03 FS 6:05
bath
angle sum MT 10

of a quadrilateral 10:05

of a triangle A:07, 1:07, 10:04

498 INTERNATIONAL MATHEMATICS 2

braces A:02, 1:02, A:12, congruence 13:02
brackets 6:08
bridges of Königsberg A:02, 1:02, A:12, and translations 13:02
6:08
CW 1:12 statement 13:03

symbol 13:03

congruent figures Chapter 13, 13:02,

13:03

calculator Chapter 5 and tessellations Inv 13:02
applications 5:06
estimation 5:02 constructing 13:04
fraction bar Inv 5:05
keys 5:03, 5:05, MT 5 congruent triangles G:01
large numbers Inv C:05
make words with your connectivity 16:02
calculator FS 5:07
problem solving 5:07 constant term 8:03, MT 8
speed and accuracy 5:04
using calculators 5:01 constructions F:01–F:04, MT 13
FS 10:03
canary cage falling apart A:09, 1:09 angles of 60°, 120°,
capacity Inv 11:02
car-parking areas A:02, 1:02 30° and 90° F:02
cardinal numbers 9:06, MT 9
categorical data 14:01, MT 14 bisecting intervals and
census 12:01
centre MT 15 angles F:01, 13:01
certain FS 15:02
chance of success 12:01, MT 12 circumcircle of a triangle Tech 13:01
chords Chapter 12
circles 12:04, Inv 12:04, congruent figures
Tech 12:04
area of 12:02, 12:03, MT 12, perpendicular and
Inv 12:02
circumference of 12:01 parallel lines F:03, 13:01
Tech 13:01
parts of 14:04, MT 14 quadrilaterals A:07, 1:07
circumcircle 14:04
class centre 14:04, MT 14 regular polygons in
class group 11:05
class interval 8:03, MT 8 a circle F:04, Inv 13:01
closed solid A:06, 1:06, 10:03,
coefficient MT 10 triangles A:07, 1:07
cointerior angles 14:01
Inv 9:01 continuous data 14:04, MT 14
collecting data 6:02
collecting graphs 9:01, 9:04, MT 9 conversion graphs 9:01, 9:04, MT 9
collecting like terms 3:09, MT 3
column graphs 13:01, MT 13 conversions RM 3:04
commission MT 10
compasses A:06, 1:06, MT 10 metric A:09, 1:09
complement 15:03, MT 15
complementary angles 11:01, MT 11 coordinates A:11, 8:02, MT 8
complementary events A:02, 1:02
composite figures Inv 14:05, 15:08 corresponding angles 10:03, 13:04, MT 10
composite numbers 12:01, MT 12
computer software corresponding sides 10:03, 13:04
concentric circles
cost price MT 3

counting numbers A:02, 1:02

counting the rings FS 2:03

cross-sections A:08, 11:04, MT 11

cube roots A:02, 1:02

cubes, area of 11:05, MT 11

data 14:01, MT 9
analysing 14:03
categorical 9:06, MT 9
collecting 14:01
continuous MT 9
discrete 14:04, MT 9
grouped 14:04, MT 14
quantitative 9:06, MT 9
sorting 14:02
A:04
decimal currency Tech 1:12
decimal equivalents A:04, 1:04
decimals FS 4:06
deer without eyes 3:07, MT 3
depreciation 12:01, MT 12
diameter MT 11
dimensions

INDEX 499

directed numbers A:10, 1:10, MT 6 flip 13:02
discount 3:05, 3:07, 3:08, formulae Chapter 7, 7:05,
MT 3 MT 6, MT 7
discrete data 14:04, MT 14 fraction bar Inv 5:05
distance A:09, 1:09 fraction equivalents Tech 1:12
divided bar graphs 9:01, MT 9 fractions
divisibility tests A:02, 1:02 6:10
division in algebra A:03, 1:03
6:10 Grade 6 review 14:02, MT 14
of algebraic fractions A:04, 1:04 frequency 14:02, MT 14
of decimals A:10, 1:10 distribution table 14:02, 14:03, 14:04,
of directed numbers A:03, 1:03 histogram MT 14
of fractions 6:04, 6:05 14:02, 14:03, 14:04,
of pronumerals polygon MT 14
of a quantity in a given 4:04
C:02, 6:07 geometric constructions Chapter 13, 13:01
ratio 14:05, MT 14
with indices FS H:02 geometrical reasoning,
dot plots 9:04
drawing enlargements non-numerical D:01
drawing graphs
geometry Chapter 10

enlargement 4:06 graph theory Chapter 16
drawing FS H:02
Chapter 7, graphs Chapter 9, MT 9,
equations Tech 7:04, MT 7
7:01 MT 8
inverse operations 8:04
linear and isomorphisms 16:03
pronumerals on both 7:03
7:02 collecting Inv 9:01
sides 7:06
solving 8:03 conversion 9:01, MT 9
solving problems 7:04
straight line graphs A:07, 1:07, 10:06, divided bar 9:01, MT 9
with grouping symbols MT 10
equilateral triangles A:03, 1:03 dot plot 14:05
4:02
equivalent fractions A:01, 5:02 drawing 9:04, 16:03
equivalent ratios 5:02
estimation Inv 16:03, MT 16 features 9:01
MT 6, MT 7
leading digit MT 15 frequency histogram 14:02
eulerian trails A:02, 1:02
evaluate 5:08 frequency polygon 14:02
even chance A:01, 1:01
even numbers 6:08, MT 6 further graphs 8:06
Excel spreadsheets Challenge 6:10
expanded notation MT 6 line 9:01, MT 9
expanding 10:04, MT 10
of inequations 7:07
binomial products
expression misuse of 9:07
exterior angle of triangle
reading 9:02

scatter diagram 14:05

sector 9:01, MT 9

step 9:01, MT 9

straight line 8:03, 8:04,

Inv 8:06

stem-and-leaf plot 14:06

to solve problems 16:04

travel graphs 9:05, MT 9

unusual 9:03

grid references 8:01

grouped data 14:04, MT 14

grouping symbols A:02, A:12, 1:02,

factorising 6:09 6:08, MT 6, MT 7

factors and multiples A:02, 1:02 equations with grouping

fewer bumps and smoother symbols 7:04

rides FS 12:03 heads and tails MT 15
hectare MT 11
Fibonacci numbers A:02 hen feeding chickens
FS 3:10
finite differences Challenge B:01 concrete

500 INTERNATIONAL MATHEMATICS 2

highest common factor A:02, 1:02, 6:09 mazes Inv 2:02
histogram, frequency 14:02
mean 14:03, Inv 14:03,

MT 14

impossible MT 15 measurement
improper fractions A:03, 1:03
index notation A:12, 1:12, 6:07, review of last year’s work A:09, 1:09, 11:01
MT 5
indices Appendix C, MT 6 median 14:03, Inv 14:03
dividing C:02
investigation Tech 2:02 metric conversions A:09, 1:09
large numbers on
calculator Inv C:05 metric units ID Card 1, A:09,
miscellaneous exercises C:04
multiplying C:01 1:09
power of a power C:03
A:02, 7:07, MT 7 mixed numerals A:03, 1:03
inequality sign Chapter 7, MT 7
inequations 7:07 mixing paint Challenge 4:02
7:08
graphing A:10, 1:10 mode 14:03, Inv 14:03
solving Chapter 8, MT 8
integers 3:09, MT 3 multiplication
intercepts 2:02, MT 8
interest, simple 8:06 can be fun Inv 4:04
intersection A:09, 1:09, MT 13
points of 7:01, MT 7 of algebraic fractions 6:10
intervals 12:02
inverse operations 16:03, MT 16 of decimals A:04, 1:04
irrational numbers A:07, 1:07, 10:06,
isomorphism MT 10 of directed numbers A:10, 1:10
isosceles triangles
of fractions A:03, 1:03

of indices C:01

of mixed numerals A:03, 1:03

of pronumerals 6:03, 6:05

with indices C:01

nets A:08, 1:08

nothing but the facts Inv 14:01

number

beginnings in A:01, 1:01

order and structure A:02, 1:02

properties A:02, 1:02

number lines A:02, 1:02, 7:07

number plane Chapter 8, MT 8

kites A:07 coordinates 8:02
area of 11:03
further graphs 8:06

lines parallel to axes 8:05

large numbers on a pictures FS 8:02
calculator
Inv C:05 review of last year’s work A:11, 1:11
latitude 8:01, MT 8
like terms A:12, MT 6 straight line graphs 8:03, 8:04

addition and 6:02 numbered boxes FS 10:07
subtraction of 9:01, 9:04, MT 9,
MT 8 numbers, directed A:10, 1:10
line graphs 8:04, MT 8
8:05 oblique pyramids A:08
linear equations CW 10:07
lines parallel to the axes 8:01, MT 8 obtuse angles A:06, 1:06
logical prisoner MT 3
longitude A:02, 1:02 odd numbers A:02, 1:02
loss Tech 11:04
lowest common multiple open prisms 11:05
luggage volumes
oranges for sale CW 7:06

order of operations A:02, 1:02

ordered pairs A:11, 1:11, 8:02

ordered stem-and-leaf plots 14:06

outcomes 14:02

outliers 14:05

magic squares Tech 6:01 paint mixing Challenge 4:02
maps, reading 8:01 palindromic numbers A:02
marked price MT 3 parabolas 8:06
mass A:09, 1:09

INDEX 501

parallel lines MT 7, MT 13 applications 15:04

and angles 10:03, Tech 10:03 in games FS 15:03

construction F:03 language of 15:01

parallelograms A:07, 1:07 of complementary events 15:03

area of 11:02 of simple events 15:02

parentheses A:02, 1:02, A:12, problem solving

6:08 involving circumference 12:03

parking areas Inv 11:02 mazes Inv 2:02

patterns and algebra Chapter 6 strategies 2:01, 2:02

and rules review 6:01 using algebra A:12, 1:12

people laughing up their using equations 7:06

sleeves FS 6:03 using ratios 4:03

per annum (p.a.) MT 3 using Venn diagrams 2:03

percentage change 3:07, MT 3 with the calculator 5:07

percentage composition 3:06, MT 3 profit MT 3

percentage equivalents Tech 1:12 pronumerals A:12, 1:12, 7:01,

percentage project Tech 3:05 7:03, 7:05, MT 6,

percentages Chapter 3, MT 3 MT 7

applications 3:05, 3:08 division of 6:04, 6:05

conversions 3:03 multiplication of 6:03, 6:05

estimating 3:02 protractors A:06, 1:06

finding a quantity 3:10 pyramids A:08, 1:08

misuse of Inv 3:08

of a quantity 3:04 quadrants 8:02, MT8
quadrilaterals
review of last year’s work A:05, 1:05, 3:01 A:07, 1:07, 10:05
angle sum A:07, 1:07
perimeter A:09, 1:09, MT 11 parallelogram A:07, 1:07
properties of A:07, 1:07
perpendicular to a line F:03, 13:01 rectangle A:07
rhombus A:07, 1:07
MT 13 square A:07, 1:07
trapezium 9:06, MT 9
pi 12:02, MT 12 quantitative data

pictograms 9:01

picture graphs 9:01, MT 9

pie graphs (charts) 9:01, 9:04, MT 9

place value A:01, 1:01

plane shapes, review of

last year’s work A:07, 1:07 radius 12:01, MT 12
random MT 15, MT 14
plotting points in four MT14
sample 14:03
quadrants Tech 8:02 range Chapter 4, 4:05,
rates MT 4
points of intersection 8:06 Chapter 4
ratio 4:01, MT 4
polygons, frequency 14:02 definition 4:04
dividing quantities 4:02, MT 4
regular F:04, Inv 13:01 equivalent ratios 4:03
solving problems Inv 4:03
power of a power C:03 using Inv 4:04
9:02
powers of numbers A:01, 1:01, MT 6 ratio research 8:01
reading graphs 9:06
prime factors A:02, 1:02 reading maps Chapter 10
reading tables A:07, 1:07
prime meridian 8:01 reasoning in geometry 11:03
rectangles Challenge 4:06
prime numbers A:02, 1:02 MT 7
area of
prisms A:08, 1:08, MT 11 cutting
reduction
open 11:05

rectangular 11:04, MT 9

surface area of 11:05

triangular 11:04, MT 11

volume of 11:04

prisoner, logical CW 10:07

probability Chapter 15,

Tech 15:02, MT 15

an unusual case CW 15:04

502 INTERNATIONAL MATHEMATICS 2

reflection 13:02, MT 13 tables Chapter 9, MT 9
reflex angles A:06, 1:06 reading 9:06
regular polygon 14:02
F:04 tally 12:01, MT 12
constructed in circle A:06 tangents Tech 14:03
revolution A:07 temperatures, world FS 9:07
rhombuses 11:03 tennis, anyone for Inv 13:02
A:06, 1:06 tessellations A:09, 1:09
area of A:01, 1:01 time FS 7:02
right angles 13:02, MT 13 tooth getting dressed up FS E:01
Roman numerals A:04, 1:04 tower of Hanoi CW 2:02
rotation train puzzle 13:02, MT 13
rounding off transformations 13:02, MT 13
translations A:06, 10:03, MT 10
sample 14:01, 14:03 transversals A:07, 1:07
random 14:01, MT 14 trapeziums 11:03
space 15:02, MT 15 9:05, MT 9
stratified random 14:01 area of 16:02, MT 16, 16:04
Chapter 4, 4:06, travel graphs
scale drawing MT 11 tree A:07, 1:07
Inv 4:06 triangles A:07, 1:07, 10:04
scale my room 14:05 11:03
scatter diagrams acute-angled
secrets in a vegetable FS 8:05 angle sum G:01
9:01, MT 9 area of A:07, 1:07, 10:06
garden MT 3 conditions for 10:04
sector graphs H:01 A:07, 1:07, 10:06
selling price 15:02, MT 15 congruency A:07, 1:07
similar triangles 3:09 equilateral A:07, 1:07
simple events exterior angle A:07, 1:07
simple interest FS 12:03 isosceles H:01
smoother ride and fewer A:08, 1:08 obtuse-angled A:02, 1:02
14:02 right-angled 13:02
bumps Inv 14:02 scalene
solid shapes Tech 4:05 similar
sorting data Chapter 5 triangular numbers
sorting it out 5:08 turn
speed, distance and time 15:02, MT 15
spreadsheets Chapter 14 vegetable gardens and
14:06
on a computer 9:01, MT 9 secrets FS 8:05
standard pack of cards 8:03, Inv 8:06
statistics 8:04 Venn diagrams 2:03
stem-and-leaf plots FS 13:04
step graphs 16:02, MT 16 vertically opposite angles A:06, 1:06, 10:02,
straight line graphs 7:05
straight lines, graphing A:12, 1:12, MT 6, MT 10
stupid ant MT 7
subgraphs vertex 16:01
subject of a formula 6:10
substitution 6:02 volume Chapter 11, MT 11
Tech 9:04
subtraction 13:02, MT 13 estimating Inv 11:04
of algebraic fractions A:06, 1:06, MT 10
of like terms MT 11 of irregular solids A:09, 1:09
11:05
sunburnt country CW 11:05 of luggage Tech. 11:04
superimposing
supplementary angles of prisms 11:04, MT 11
surface area
review of last year’s work A:09, 1:09
of a prism
practical applications wedding anniversaries Challenge 14:06
weighing cornflakes Inv 5:02
Working mathematically Chapter 2
world temperatures Tech 14:03

x-coordinates 8:02

y-coordinates 8:02

INDEX 503

Acknowledgments

We would like to thank the following for permission to reproduce photographs, texts and
illustrations.
The following abbreviations are used in this list: t = top, b = bottom, c = centre, l = left, r = right,
British Crime Survey: p.212.
Carey, Kevin: pp.10, 18, 32, 145.
Delhi Metro: By courtesy Delhi Metro, p.419t.
Dragonair.com: p.421b.
Getty Images: pp.237, 369.
Land Transport Authority of Singapore: p.419b.
McSeveny, Alan: pp.172, 311, 334, 353, 354, 356, 376, 381.
Microsoft Corporation: pp.109, 111, screenshots reprinted with permission if Microsoft
Corporation.
National Statistics Office Library & Information Unit: p.204.
National Australia Bank: p.219.
Photolibrary: p.204.
Piemonte, Lisa: pp.110, 267, 293, 372, 405, 408.
Singapore Department of Statistics: p.224.
Statistics Canada: p.240.
ToSingapore.com: p.421t.
Transport for London: pp.420, 422.
University of Michigan: p.219.
Utrecht University Library: p.238.

Every effort has been made to trace and acknowledge copyright. However, should any infringement
have occurred, the publishers tender their apologies and invite the copyright owners to contact
them.

504 INTERNATIONAL MATHEMATICS 2

Appendixes

Appendix A revises the whole
Grade 6 course.

The other
appendixes extend
the Grade 7 work.

x 1 50% ∴ _x cm
2

x = b 6 ×
2
0·5 %
{1,2}
mL area f ix x2

Appendix A Appendix D

A:01 Beginnings in number D:01 Non-numerical geometrical reasoning
A:02 Number: Its order and structure (extension)
A:03 Fractions
A:04 Decimals Appendix E
A:05 Percentages
A:06 Angles E:01 Fun Spot: The Tower of Hanoi
A:07 Plane shapes
A:08 Solid shapes Appendix F
A:09 Measurement
A:10 Directed numbers F:01 Bisecting intervals and angles
A:11 The number plane F:02 Constructing angles of 60°, 120°, 30°
A:12 Algebra
and 90°
Appendix B F:03 Constructing perpendicular and parallel lines
F:04 Constructing regular polygons in a circle
B:01 Challenge: Finite differences
Appendix G
Appendix C
G:01 The minimum conditions for congruent
C:01 Multiplying with indices triangles
C:02 Dividing with indices
C:03 Power of a power Appendix H
C:04 Miscellaneous exercises on indices
C:05 Investigation: Large numbers on a calculator H:01 Similar triangles
H:02 Fun Spot: Drawing enlargements

Appendix I

I:01 The pictogram or picture graph
I:02 The column graph

1

• Much of last year’s work will form a foundation for our work this year.
• Revision exercises based on this review can be found in Chapter 1 of the text.

APPENDIX A
A:01 | Beginnings in Number

A:01A Roman Numerals see

Numerals • LX means 50 + 10. • Large numerals are formed by 1:01
1I • XL means 50 − 10.
5V • XXXIV means placing a stroke above the symbol.
10 X
10 + 10 + 10 + 5 − 1. V = 5000 That’s a
X = 10 000 handful
worked examples M = 1 000 000 of arrows!

50 L 1 392 = CCCXCII
100 C 2 1988 = MCMLXXXVIII
500 D 3 56 000 = LVM
1000 M 4 594 = DXCIV
5 LXXIII = 73
6 MDCXVI = 1616
7 DMMLX = 502 060

A:01B Place Value and Expanded Notation

314 million In the numeral to the left:

952 thousand • the value of the 3 is 300 000 000,
867 units • the value of the 1 is 10 000 000,
100 000 000 • the value of the 4 is 4 000 000,
10 000 000
1 000 000 • the value of the 9 is 900 000,
100 000 • the value of the 5 is 50 000,
10 000 • the value of the 2 is 2000,
1 000
100 • the value of the 8 is 800,
10 • the value of the 6 is 60,
1 • the value of the 7 is 7.

314 952 867

worked examples

1 Write ‘six million, ninety thousand’ as a basic (simple) numeral.
2 Write (8 × 10 000) + (4 × 1000) + (7 × 100) + (0 × 10) + (5 × 1) as a basic numeral.
3 Write (7 × 103) + (5 × 102) + (3 × 10) + 9 as a basic numeral.

Solutions

1 We use zeros as place holders: 6 000 000 + 90 000 = 6 090 000
2 (8 × 10 000) + (4 × 1000) + (7 × 100) + (0 × 10) + (5 × 1) = 84 705
3 (7 × 103) + (5 × 102) + (3 × 10) + 9 = 7539

2 INTERNATIONAL MATHEMATICS 2

A:01C Long Multiplication and Long Division

Multiplication Division
1 Estimate first. 1 Estimate first.
2 Multiply by the ones digit. 2 Use either the contracted form method or the
3 Multiply by the tens digit.
4 Add the two answers. preferred multiples method.
3 Write the answer using a remainder or fraction.

Example Example (long division) Preferred multiples
Extended form Contracted form

531 × 28 r 10 28 r 10
48
31) 878 31) 878
4 248 ← 8 × 531
21 240 ← 40 × 531 − 62↓ − 620 20

25 488 258 258
248 − 248 8

10 10 28
Answer: 28 remainder 10 or 28 -1---0-

31

A:01D Powers of Numbers

five squared five cubed ■ The small raised number (the index) tells how
many times that number is used in the product.
52 = 5 × 5 53 = 5 × 5 × 5
= 25 = 125

worked examples

1 Write 25 in expanded form and write its basic numeral.
2 Write 10 × 10 × 10 × 10 as a power of ten.
3 What is the basic numeral of 8 × 104?

Solutions 2 10 × 10 × 10 × 10 3 8 × 104
= 104 = 8 × (10 × 10 × 10 × 10)
1 25 = 8 × 10 000
=2×2×2×2×2 = 80 000
= 32

A:01E Expanded Notation Using Powers ■ 6 ten-thousands
9 thousands
Ten-thousands Thousands Hundreds Tens Units 8 hundreds
10 000 7 tens
1000 100 10 1 5 units

10 × 10 × 10 × 10 10 × 10 × 10 10 × 10 10 1 69 875
104 103 102 101 1

6 9 8 75

69 875 = (6 × 10 000) + (9 × 1000) + (8 × 100) + (7 × 10) + (5 × 1)
= (6 × 104) + (9 × 103) + (8 × 102) + (7 × 10) + (5 × 1)

3APPENDIX A

A:01F Estimation

Estimate before you calculate.

We usually estimate an answer by rounding off each number involved to its ■ 5314
leading digit and then performing the operation. ↑

leading digit

worked examples

Use leading digit estimation to estimate the answer for each of these.

Question Rounding off Estimate

1 618 + 337 + 159 + 409 600 + 300 + 200 + 400 1500
2 38 346 − 16 097 40 000 − 20 000 20 000
3 3250 × 11·4 3000 × 10 30 000
4 1987 ÷ 4 2000 ÷ 4
500

A:02 | Number: Its Order and Structure

see A:02A Order of Operations

Mathematicians have agreed on an order of operations so that confusion does not occur.

1:02 Order of Operations ■ Order of

1 Do operations within grouping symbols. Operations

2 Do multiplications and divisions as they appear (from left to right). 1 ( )

3 Do additions and subtractions as they appear (from left to right). 2 × and ÷

3 + and −

Where grouping symbols occur inside other grouping symbols, deal with the innermost grouping

symbols first.

Parentheses Brackets Braces

() [] {}

worked examples

1 4 + 20 ÷ (4 + 1) 2 12 − 6 × 2 3 50 − (25 − [3 + 19])
= 4 + 20 ÷ 5 = 12 − 12 = 50 − (25 − 22)
=4+4 =0 = 50 − 3
=8 = 47
5 6 × 9 + 25
4 80 ÷ -1- of 40 =6×3+5 6 -5---0----+-----5--
= 23 20 ÷ 4
2
= 5----5-
= 80 ÷ 20 5
=4

= 11

4 INTERNATIONAL MATHEMATICS 2

A:02B Using Number Properties

• Multiplying any number by 1 leaves the number unchanged,

eg 347 × 1 = 347.

• Multiplying any number by zero gives the answer zero

eg 84 × 0 = 0.

• Adding zero to any number leaves it unchanged

eg 36 + 0 = 36.

• When adding two numbers, the order does not change the answer

eg 4 + 1 = 1 + 4.

• When multiplying two numbers, the order does not change the answer

eg 7 × 9 = 9 × 7.

• When adding more than two numbers, we may add them in any order

eg 4 + 169 + 96 = (4 + 96) + 169 = 100 + 169 = 269.

• When multiplying more than two numbers, we may multiply them in any order

eg 4 × 836 × 25 = (4 × 25) × 836 = 100 × 836 = 83 600.

• Multiplication can be distributed over addition or subtraction

eg 8 × (17 + 3) = 8 × 17 + 8 × 3, 4 × (100 − 3) = 4 × 100 − 4 × 3,

7 × 99 = 7 × 100 − 7 × 1, 17 × 102 = 17 × 100 + 17 × 2.

A:02C The Number Line and Using Inequality Signs

To graph a number on the number line, we place a large dot on the point which represents that
number.

worked examples

1 Graph the odd numbers less than 8. 2 On the number line graph the set {0, 4, 7}.

–1 0 1 2 3 4 5 6 7 8 –1 0 1 2 3 4 5 6 7 8

If one number is to the left of another on the number line, then it is smaller than that number,
eg 3 < 5, −1 < 0. The number to the right is larger, eg 7 > 4, 0 > −1.

A:02D Special Groups of Whole Numbers

Name of group Pattern Diagram or Comment or rule
explanation

Cardinal numbers 0, 1, 2, 3, . . . Zero + counting Zero is included.
numbers
01 234567

Counting 1, 2, 3, 4, . . . Zero is not included.
numbers
, , , , , , ...
01 234567

Even numbers 2, 4, 6, 8, . . . All dots are in pairs. (2 × n)

,, , , ...
01 234567

Odd numbers 1, 3, 5, 7, . . . One dot not paired. (2n − 1)

,, , ...
01 234567

5APPENDIX A

Name of group Pattern Diagram or Comment or rule
explanation
Square numbers 1, 4, 9, 16, . . . A counting number times itself,
,, , , ... eg 1 × 1, 2 × 2, 3 × 3, etc. (n2)
Triangular 1, 3, 6, 10, 15,
numbers ... ,, , 1, 1 + 2, 1 + 2 + 3,
, ... 1 + 2 + 3 + 4, . . .
Hexagonal 1, 7, 19, 37,
numbers ... The sum of the first n counting
numbers.
Fibonacci 1, 1, 2, 3, 5, 8,
numbers ... ,, 1, 1 + (6 × 1), 1 + (6 × 1) + (6 × 2),
Examples , ... 1 + (6 × 1) + (6 × 2) + (6 × 3), . . .
Palindromic 929, 7337
numbers +6 +12 +18 +24 +30

ie 1, 7, 19, 37, 61, 91, . . .

Except for the first two in the pattern, each term is the
sum of the two terms before it.

The number reads ‘Able was I ere I saw Elba’
the same forwards is a palindromic sentence.
or backwards.

A:02E Factors and Multiples ■ 3 × 8 = 24. Here 3 and 8
are factors of 24, and 24 is
• A factor of a counting number divides it exactly. a multiple of both 3 and 8.
• A multiple of a counting number is found when you multiply

it by another counting number.

worked examples

1 List all factors of 102. 2 List the first nine multiples of 7.

To do this we look for numbers that 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21,

multiply together to give 102. 7 × 4 = 28, 7 × 5 = 35, 7 × 6 = 42,

1 × 102 = 102 7 × 7 = 49, 7 × 8 = 56, 7 × 9 = 63.

2 × 51 = 102 The first nine multiples of 7 are:
3 × 34 = 102 7, 14, 21, 28, 35, 42, 49, 56, 63.
6 × 17 = 102

Note: 4, 5, 7, 8, 9, 10 and 11 ■ You need not check numbers higher than 11.
do not divide 102 exactly. 102 ÷ 11 Ӏ 9·27

The factors of 102 are: When your answer (9·27) is smaller than your divisor (11)
1, 102, 2, 51, 3, 34, 6, 17. you have gathered all factors.

• The highest common factor of two numbers is the largest number that is a factor
of both. This is called the HCF.

• The lowest common multiple of two numbers is the smallest number that is a multiple
of both. This is called the LCM.

6 INTERNATIONAL MATHEMATICS 2

worked examples

1 Find the highest common factor 2 Find the lowest common multiple
(HCF) of 102 and 153. (LCM) of 15 and 9.

Number Factors Number Multiples

102 1, 102, 2, 51 , 3, 34, 6, 17. 15 15, 30, 45 , 60, 75, 90, . . .

153 1, 153, 3, 51 , 9, 17. 9 9, 18, 27, 36, 45 , 54, . . .

The common factors are red. There would be other common multiples
The highest common factor is 51. but the lowest common multiple can be
seen in the table to be 45.

A:02F Prime and Composite Numbers

• A prime number is a counting number that has exactly two factors, itself and 1.
7 has only two factors, 7 and 1, so 7 is a prime number.

• A composite number has more than two factors.
25 has three factors, 25, 1 and 5, so 25 is a composite number.

• The number 1 has only one factor, 1, so it is neither prime nor composite.

worked examples This would look more
like a tree if it were
Use a factor tree to write 252 as a product of prime numbers. turned upside-down.

252

4 63 252 = 2 × 2 × 3 × 3 × 7
2 2 3 21 = 22 × 32 × 7

2 23 3 7

7APPENDIX A

A:02G HCF and LCM by Prime Factors

When we express numbers as products of their prime factors it is sometimes easier to find their
HCF and LCM.

worked examples

For the numbers 400 and 1080 find:
1 the HCF, and 2 the LCM.

400 1080 From these factor trees we find:
20 ϫ 20 20 ϫ 54
4ϫ5ϫ5ϫ4 4ϫ5ϫ6ϫ9 Number Product of prime factors
2ϫ2ϫ5ϫ5ϫ2ϫ2 2ϫ2ϫ5ϫ2ϫ3ϫ3ϫ3 400
2×2×2×2×5×5
1080 or 24 × 52
2×2×2×3×3×3×5
or 23 × 33 × 5

The two numbers have 2 × 2 × 2 × 5 in common.
2 × 2 × 2 × 5 = 23 × 5 or 40.

∴ HCF = 40

The highest common factor of 400 and 1080 is 40.

The LCM must contain the whole of both numbers, so it will contain the whole of 400
(ie 2 × 2 × 2 × 2 × 5 × 5) as well as any other prime factors of 1080 not already written down.

LCM = (2 × 2 × 2 × 2 × 5 × 5) × (3 × 3 × 3)
= 24 × 33 × 52
= 10 800

The lowest common multiple of 400 and 1080 is 10 800.

■ Note:
The HCF is found by using the smallest common power of each prime factor of the numbers.
The LCM is found by using the highest power of each prime factor of the numbers.

A:02H Divisibility Tests

If 4 divides a number and leaves no remainder, then the number is divisible by 4. Here are some
simple divisibility tests you should know.

Divisor Divisibility test Example Learn these!
2 The number is even, They’re useful.
3 ie it ends in 0, 2, 4, 6 or 8. 4136 is divisible by 2
The sum of the digits is as it is even.
4 divisible by 3.
5 30 012 is divisible by 3
The number formed by the last as (3 + 0 + 0 + 1 + 2) is
two digits is divisible by 4. divisible by 3.
The last digit is 5 or 0.
76 112 is divisible by 4
as 12 is divisible by 4.

11 225 is divisible by 5
as it ends with a 5.

8 INTERNATIONAL MATHEMATICS 2

Divisor Divisibility test Example

6 The number is divisible by 10 002 is even and its By using the tests in
both 2 and 3 digit sum is divisible by 3. the table, we can

8 The number found by the last 963 216 is divisible by 8 construct other tests.
three digits is divisible by 8. as 216 is divisible by 8.
■ • Divisibility
9 The sum of the digits is 142 128 is divisible by 9 by 12 is
divisible by 9. as (1 + 4 + 2 + 1 + 2 + 8) the same as
is divisible by 9. divisibility by
10 The last digit is 0. both 3 and 4.
814 170 is divisible by 10
11 The sum of the odd-numbered as it ends with 0. • Divisibility by
digits is equal to the sum of the 50 is the same
even-numbered digits or differs 7 081 426 is divisible by as divisibility by
from it by a multiple of 11. 11 as (7 + 8 + 4 + 6) − both 2 and 25.
(0 + 1 + 2) is 22 and this
25 The last two digits are 00, 25, is divisible by 11.
50 or 75.
80 925 is divisible by 25
100 The last two digits are 00. as it ends in 25.

81 700 is divisible by 100
as it ends in 00.

A:02I Square and Cube Roots

• The number which when squared gives 225 is called the
square root of 225.

• The number which when cubed gives 512 is called the
cube root of 512.

If the square of 15 is 225, then the square root of 225 is 15.

■ If 152 = 225, then 225 = 15 .

If the cube of 8 is 512, then the cube root of 512 is 8. ■ means ‘the square root of’.
■ If 83 = 512, then 3 512 = 8 . 3 means ‘the cube root of’.

A:03 | Fractions

A:03A Basic Concepts see

• --3--- means 3 of 10 equal parts. 1:03

10

• 10 tenths make 1 whole. 3 ← numerator

• A mixed numeral has a ‘whole number’ 10 ← denominator

part and a ‘fraction’ part, eg 2 --3--- .

10

• An improper fraction has a numerator that is bigger than its denominator, eg 2----3- .

10

9APPENDIX A

A:03B The Fraction Bar

The fraction bar is like a division sign.

worked examples

1 1----0- = 10 ÷ 2 2 -9- = 9 ÷ 4 3 -8---7- = 87 ÷ 10
4 = 2 1-- 10 = 8 --7---
2
4 10
=5

Note: -9- = 8-- + -1- = 2 -1-
444 4

8----7- = 8----0- + --7--- = 8 --7---
10 10 10 10

A:03C Mixed Numerals and Improper Fractions

To write a mixed numeral as an improper fraction, express the whole number with the same
denominator as the fraction part.

worked examples

1 3 -1- = -3- ×2 + -1- 2 5 --3--- = 5-- × 10 + --3--- 3 1 3-- = -1- ×4 + -3-
1 ×2 2 1 × 10 10 1 ×4 4
2 10 4

= 6-- + -1- = -5---0- + --3--- = 4-- + 3--
22 10 10 44

= 7-- = 5----3- = 7--
2 10 4

A:03D Simplifying Fractions

A fraction may be simplified by casting out equal factors from both the numerator and
denominator. (This process is called cancelling.)

worked examples

1 --8--- = --8--- ÷ 2 2 2----0- = -2---0- ÷ 10
10 10 ÷ 2 50 50 ÷ 10
--8--- and -4- are
= --8---4--- = -2---0---2-
105 505 10 5

= 4-- = -2- the same!
5 5

A:03E Equivalent Fractions

The size of a fraction is unchanged if both the numerator and denominator are multiplied by the
same number.

10 INTERNATIONAL MATHEMATICS 2

worked examples

Complete the following equivalent fractions:

1 2-- = ----- 2 3-- = --------
5 10 4 100

Multiply top and Multiply top and

bottom by 2. bottom by 25. They’re
the same!
2-- × 2 = --4--- -3- × 25 = --7---5---
5 × 2 10 4 × 25 100

3 4-- = -----
1 10

Multiply top and

bottom by 10.

4-- × 10 = 4----0-
1 × 10 10

A:03F Addition and Subtraction (1)

To add or subtract fractions that have the same denominator, add or subtract the numerators.

worked examples

1 --3--- + --4--- = 3-----+-----4-- 2 --1---9--- – ----6---- = -1---9----–-----6- 3 -3- + 7-- = 3-----+-----7--
10 10 10 100 100 100 88 8
= --7--- = --1---3--- = 1----0-
10 100 8
= 1 2-- or 1 1--

84

A:03G Addition and Subtraction (2)

To add or subtract fractions that have different denominators, use equivalent fractions to give each
the same denominator.

worked examples

1 -3- + --3--- 2 --9--- – 3-- 3 3-- + -2-
5 10 10 4 45

= -3- × 2 + --3--- = --9--- × 2 – -3- × 5 = 3-- × 5 + -2- × 4
5 × 2 10 10 × 2 4 × 5 4 × 5 5 × 4

= --6--- + --3--- = -1---8- – -1---5- = -1---5- + --8---
10 10 20 20 20 20

= --9--- = --3--- = -2---3- or 1--3---
10 20 20 20

A:03H Comparing Fractions

To compare fractions, use equivalent fractions to give them the same denominator.

11APPENDIX A

worked examples

1 Which fraction is 2 Arrange in order, from smallest

smaller, 3-- or --6--- ? to largest: { --8--- , 1----7- , -3- }.

4 10 10 20 4

-3- × 5 = -1---5- --8--- × 2 = 1----6-
4 × 5 20 10 × 2 20

--6--- × 2 = -1---2- 1----7- = -1---7-
10 × 2 20 20 20

∴ --6--- is the smaller. 3-- × 5 = 1----5- ■ When denominators
10 4 × 5 20 are the same, compare
numerators.
In order they are 3-- , --8--- , 1----7- .

4 10 20

A:03I Addition and Subtraction of Mixed Numerals

worked examples

1 3 -2- + 1 3-- 2 10 -1- − 1 1--

54 52

= (3 + 1) + ( 2-- + 3-- ) = (10 − 1) + ( -1- − -1- ) ■ -1- is bigger
2
54 52
× 4 × 5 than
=4 + ( 2-- × 4 + -3- × 5 ) = 9 + ( -1- − -1- ) -1- , so use
5 4
52 5

= 4 + ( --8--- + 1----5- ) = 8 + (1 -1- − -1- ) one whole.
52
20 20 × 2 × 5
× 2 × 5
= 4 2----3- (ie 4 + 1 --3--- ) =8 + ( -6- − -1- )
5 2
20 20

= 5 --3--- = 8 + ( 1----2- − --5--- )

20 10 10

3 5 − 2 -3- = 8 --7---

10

= (5 5 − 3--
5
− 2)

= 3 − 3--
5
= 2 + − -3- )
(1 5

= 2 2--
5

A:03J Multiplication

To multiply fractions, we may multiply numerators and multiply denominators.

worked examples

1

■ -3- has been ■ 2-- of the shaded part ■ ( 2-- of 3-- ) or ( -2- × -3- ) is --6--- .
54 5 4 20
4 5

shaded. has been cut out.

12 INTERNATIONAL MATHEMATICS 2

Conclusion: 2-- × -3- = -25----××-----43- = --6--- or --3---
5 4 20 10

If a number will divide into a numerator and a denominator, this should be done. It is easier to

do it before multiplying, rather than doing it when you simplify the answer.

2 --3--- × --7--- 3 -3- × 2-- 4 --9--- × 1----5-
10 10 53 10 16
(No number will exactly = -1--3- × --2-- = ---9---- × 1----5---3-
(divide top and bottom.) 5 31 210 16
= -1----×-----2- = --9-----×----3----
= 1----30----××-----17---0-- 5×1 2 × 16

= --2---1--- = 2-- = -2---7-
100 5 32

A:03K Multiplication of Mixed Numerals

To multiply mixed numerals, write them as improper fractions.

worked examples

1 4 × -3- 2 1 -7- × 3 3 3 3-- × 1 -1-

5 8 43

= 4-- × -3- = -1---5- × -3- = -5--1---5- × 4----1

15 81 14 21

= 1----2- = 4----5- = -5-

5 8 1

= 2 2-- = 5 -5- =5

5 8

A:03L Division

There are two methods for doing division involving fractions. The method on the right is more
popular.

When the fractions have the same To divide by a fraction, invert the
denominator, you can divide the fraction and multiply (ie multiply
numerators to get the answer. by the reciprocal).

eg 3-- ÷ 5-- = -6- ÷ 5-- = 6 ÷ 5 = 1 -1- eg 3-- ÷ -2- = -3- × -5- = -1---5-- = 1 -7-
4888 5 4542 8 8

Mixed numerals should first be changed to improper fractions.

worked examples

1 --9--- ÷ -1- 2 4 ÷ 3-- 3 1 -1- ÷ 3 3--

10 2 5 24

= ---9---- × 2----1 = 4-- × 5-- = -3- ÷ -1---5-

510 1 13 24

= -9- = -2---0- = -1--3- × --4---2---

5 3 12 155

= 1 4-- = 6 -2- = -2-

5 3 5

13APPENDIX A

A:03M A Fraction of a Quantity

When finding a fraction of a quantity, it is sometimes wise to change to a smaller unit. To find one
quantity as a fraction of another, make sure the quantities have the same units.

worked examples

1 Find -3- of 2 km. 2 What fraction is 40 cm of 2 m?

5 Fraction = --4---0----c---m-----
200 cm
1-- of 2 km = -1- of 2000 m
= --4---
55
20
= 2000 ÷ 5 m
= 1--
= 400 m
5
∴ -3- of 2 km = 3 × 400 m

5

= 1200 m or 1·2 km

A:04 | Decimals

see A:04A Our Number System is Based on Ten ■ 0·8
means
5 hundreds 500· 8 of 10 0·65
7 tens 70· means
3· equal parts. 65 of 100
1:04 3 units 0·1 equal parts.
1 tenth 0·09
9 hundredths 0·004
4 thousandths 0·0002
2 ten-thousandths

573·1942

5 7 3·1 9 4 2

100 10 1 --1--- ----1---- -----1------ --------1--------
57 3 10 100 1000 10 000

· 1942

If the ‘large cube’ represents one whole, the ‘flat’ represents 0·1, the ‘long’ represents 0·01 and the
‘short’ represents 0·001, then the number represented below is 2·152.

14 INTERNATIONAL MATHEMATICS 2

A:04B Conversions: Fractions and Decimals

The following examples show the connection between fractions and decimals.

worked examples

1 --7--- = 0·7 2 2 --1---3--- = 2·13 3 -----9------ = 0·009
1000
10 100

1 zero 1 figure 2 zeros 2 figures 3 zeros 3 figures
after the after the after the

point point point

To write a decimal as a fraction, use the method above to write the decimal as a fraction with a
denominator of 10, 100, 1000, etc, then simplify.

worked examples

Change each to a fraction or mixed number in simplest form.

1 0·85 2 0·025 3 1·8 4 9·04

= --8---5--- ÷5 = ----2---5---- ÷ 25 = 1 --8--- ÷ 2 = 9 ----4---- ÷4
100 ÷5 1000 ÷ 25 ÷ 2 ÷4
10 100

= 1----7- = --1--- = 1 -4- = 9 --1---

20 40 5 25

To write a fraction as a decimal, we may use either of two methods.

1 Express the denominator of the fraction as If a decimal keeps repeating,
a power of ten, then change to a decimal. it’s called a repeating
or or recurring decimal.

2 Divide the numerator by the denominator.

worked examples

Method 1

Change each fraction to a decimal. ■ 0·3333333333 … = 0·3˙
0·1717171717 … = 0·1˙ 7˙
1 3-- = -3- ×2 0·1081081081 … = 0·1˙ 08˙
5 5 ×3
15APPENDIX A
= --6---

10

= 0·6

2 --3---3--- = --3---3--- ×5
200 200 ×5

= --1---6---5---

1000

= 0·165

Method 2 When a decimal
does not repeat,
3 5-- = 5 ÷ 8
it’s called a
8 terminating decimal.

= 0·625 ■ 0·6, 32·75
= 8 ) 5 · 02040 8·11, 0·0067

= 0·625

4 --3--- = 3 ÷ 11

11

= 0·2727…
= 11 ) 3 · 080308030
= 0·2˙ 7˙

A:04C Comparing Decimals

■ To compare decimals, Whole numbers may be written as decimals by placing
give them the same number zeros after a decimal point
of decimal places. eg 7 = 7·00, 130 = 130·0, 2 = 2·000
eg
The number line can be used to compare two decimals.
0·3 0·300 The decimal to the left will be the smaller one. The
0·33 0·330 decimals 0·5, 2 and 3·1 have been graphed on the
0·303 0·303 number line.

‘Ascending order’ means 0 0·5 1 1·5 2 2·5 3 3·5
‘smallest to largest’.

A:04D Addition and Subtraction

We place tenths under tenths, hundredths under hundredths, and thousandths under thousandths.

To add or subtract decimals, use the PUP rule: Place Points Under Points.

worked examples

1 3·7 + 1·52 2 63·85 − 2·5 3 8 − 1·625 PUP!
= 3·70+ = 63·85 − = 8·000 −
11· 5 2 2·50 1·625

5·22 61·35 6·375

16 INTERNATIONAL MATHEMATICS 2

A:04E Multiplication with Decimals

When multiplying, we must have the same number of figures after the decimal point in
the answer as we had altogether after decimal points in the question.

worked examples

1 0·006 × 0·5 2 38·2 × 0·11 ■ Multiply as though It helps to estimate
= 0·0030 = 4·202 there were no decimal your answer first.
= 0·003 points, then use the
4 1·3 × 19·1 rule to place the point
3 (0·05)2 = 24·83 in your answer.
= 0·05 × 0·05
= 0·0025

A:04F Multiplying and Dividing by 10, 100, 1000

1 When we multiply a decimal by 10, 100 or 1000, we move the decimal point 1, 2
or 3 places to the right.

2 When we divide a decimal by 10, 100 or 1000, we move the decimal point 1, 2 or
3 places to the left.

worked examples

1 0·60 × 100 2 34·180 × 1000 3 0·075 × 10 ■ Zeros may have to
= 60 = 34 180 = 0·75 be put in to fill an
empty space.
4 81·6 ÷ 100 5 056· ÷ 1000 6 0·045 ÷ 10
= 0·816 = 0·056 = 0·0045

A:04G Division of a Decimal by a Whole Number

To divide a decimal by a whole number use the PUP rule: Place Points Under Points.

worked examples

1 48·9 ÷ 3 2 1·5 ÷ 5 3 0·123 ÷ 4 ■ In each case we
have kept POINT
1 6·3 0·3 0·030 7 5 UNDER POINT.

= 3 ) 418·9 = 5 ) 1·5 = 4 ) 0·1233020

A:04H Division of a Decimal by a Decimal

To divide one decimal by another decimal we change the question so that we are
dividing by a whole number. We achieve this by moving the decimal point to the right
in both numbers.

17APPENDIX A

worked examples

1 0·8136 ÷ 0·04 1 0·8136 ÷ 0·04 2 875 ÷ 0·05

= -0---·--8---1---3---6-- Move the decimal point = 875·00 ÷ 0·05
0·04
two places to the right = 87 500 ÷ 5

0----·--8---1---3---6-- × 100 OR in each to make the = 17 500
0·04 × 100
= second number whole.

= -8---1---·--3---6-- = 81·36 ÷ 4 3 3·612 ÷ 1·2
4
= 20·34 = 36·12 ÷ 12
= 20·34
= 3·01

A:04I Decimal Currency

The rules for decimals also apply to amounts of money.

1 $362 + $3.42 worked examples 4 $90 ÷ 8

Solutions 2 $100 − $41.63 3 $8.37 × 8

1 $362.00 + 2 $100.00 − 3 $ 8.37 × 4 $1 1 . 2 5
$ 3.42 $ 41.63 8
8 ) $910 .2040
$365.42 $ 58.37 $66.96
■ 25c = $0.25

A:04J Rounding Off

see This is an important skill used in estimation.

To round off (or approximate) a number correct to a given place, we round up if the

next figure is 5 or more, and round down if the next figure is less than 5.

A:01F

worked examples

Round off: 1 96 700 000 to the nearest million. ■ Notice to Customers
2 0·085 to the nearest hundredth.
3 86·249 correct to one decimal place. With the abolition of 1- and 2-cent

Solutions pieces it is necessary to round off

1 96 700 000 has a 6 in the millions column. to the nearest 5 cents using the
The number after the 6 is ‘5 or more’ (ie 7),
∴ 96 700 000 = 97 000 000, correct to the following rules.
nearest million.
Amounts ending in:
2 0·085 has an 8 in the hundredths column.
The number after the 8 is ‘5 or more’ (ie 5), 1c or 2c round down
∴ 0·085 = 0·09, correct to the nearest hundredth.
eg $7.62 $7.60

3c or 4c round up

eg $7.63 $7.65

6c or 7c round down

8c or 9c round up.

18 INTERNATIONAL MATHEMATICS 2

3 86·249 has a 2 in the first decimal place.
The number after the 2 is ‘less than 5’ (ie 4),
∴ 86·249 = 86·2, correct to one decimal place.

A:05 | Percentages

A:05A Changing Percentages to Fractions and Mixed Numerals

A percentage is a convenient way of writing a fraction that has a denominator of 100. see
‘Per cent’, written %, means ‘per 100’ or ‘for every 100’.
1:05
worked examples

1 9% = ----9---- 2 64% = --6---4--- ÷4 Write the percentage
100 ÷4 as a fraction and
100 simplify.
= 1----6-
(9 for every 100) 100% = 1
25 200% = 2
3 125% = 1----2---5-
4 14 1-- % = -1---4---21--- × 2
100 2 100 × 2

= 1 --2---5--- ÷ 25 = --2---9---
÷ 25 200
100

= 1 -1-

4

A:05B Changing Fractions to Percentages

To express a fraction as a percentage, multiply by 1----0---0-- %.

1

worked examples

1 -3- = -3- × 1 2 1 -3- = -1---1- × 1
44 88
= -3- × -1---0---0- %
41 = 1----1- × 1----0---0- % That is the
= 3-- × 1----0---0-2%5 81 same as
14 1
= 75% = 1----1- × -1---0---0- 2%5 multiplying by 1.
28 1
3 --3---7--- = --3---7--- × -1---0---0-1% ■ We could also have changed
300 3300 1 = -2---7---5- %
= 3--3--7- % 2
= 12 1-- %
= 137·5%
3
4 4 -3- % = 2----3- × 1----0---0- 2%0 the denominator to 100,
5 15 1
eg --3--- = --3---0---
10 100
-4---6---0-
= 1 % = 30%

= 460%

19APPENDIX A

A:05C Changing Percentages to Decimals

To change 93% to a decimal, divide 93 by 100. eg 93· ÷ 100 = 0·93

worked examples

1 47% = 47· ÷ 100 2 30% = 30· ÷ 100 ■ Hundredths column
= 0·47 = 0·30 or 0·3
13% = 0·13 13 ÷ 100 = 0·13
3 325% = 325· ÷ 100 4 300% = 300· ÷ 100 150% = 1·50 150 ÷ 100 = 1·50
= 3·25 =3
4% = 0·04 4 ÷ 100 = 0·04
5 12·7% = 12·7 ÷ 100 6 0·3% = 00·3 ÷ 100
= 0·127 = 0·003

A:05D Changing Decimals to Percentages

To express a decimal as a percentage, multiply by 100%.

1 0·87 = 0·87 × 100% worked examples
= 87%
2 1·30 = 1·30 × 100%
3 5 = 5·00 × 100% = 130%
= 500%
4 0·825 = 0·825 × 100%
= 82·5%

■ Add zeros when they are
needed. eg 0·8 = 0·80

A:05E Conversions You Should Know

Percentage 1% 5% 10% 12 1-- % 20% 25% 33 1-- % 50% 100%
0·2 0·25 0·5 1
2 3 1
1-- -1- -1-
Decimal 0·01 0·05 0·1 0·125 5 4 0·3˙ 2

Fraction ----1---- ----5---- --1--- -1- -1-
100 100 10 8 3

A:05F Using Percentages

To find a percentage of a quantity, write the percentage as a decimal (or fraction) and
multiply by the quantity.

worked examples

1 8% of 560 L 2 70% of 680 g 3 5% of $800 ■ 24 out of 32
= 0·08 × 560 = 0·70 × 680 = ----5---- × -8---0---0-8 is the same as
= 44·80 = 476·00 1100 1 75 out of 100.
= 44·8 L = 476 g = $40

20 INTERNATIONAL MATHEMATICS 2

4 John scored 24 marks out of 32. What is this as a percentage? 75% 24
32
Fraction of marks obtained = -2---4-
32

Percentage of marks obtained = -2---4-3× -1---0---0- %
324 1

= -3---40---0- %
= 75%

A:06 | Angles

A:06A Naming Angles see

An angle can be named in several ways. Two ways are shown here. 1:06

Angle Method 1 Method 2 • In Method 1, we go from the letter on
one arm to the vertex and then to the
A ∠ABC ∠B letter on the other arm.
or or
BC Bˆ • Method 2 can only be used when there
∠CBA is no chance of confusion.

A:06B Measuring Angles with a Protractor

Place the protractor as shown below and read off the angle.
You need to choose the correct reading from the two on the arm of the angle.

50136001207101081000 90 8100071010601250013014041050301620017100 This angle measures 110°, as
it is bigger than a right angle.
108011700162001503104040
108011700162001503104040180
0

This angle measures 40°, as it 50136001207101081000 90 8100071010601250013014041050301620017100
is smaller than a right angle.

180
0

21APPENDIX A

A:06C Types of Angles

Angles may be classified into different types depending on their size.

Type acute right obtuse straight reflex revolution
Size 90° 180° 360°
less than between 90° between
90° and 180° 180° and
360°

Diagram

A:06D Special Angles Diagram Explanation
Adjacent angles:
Name Common 1 have a common arm
Adjacent arm 2 have a common vertex
3 and lie on opposite sides of the
Complementary Vertex
common arm.
Supplementary 44°
46° Any two angles whose sum is 90° are
Adjacent complementary angles.
complementary 128° 52° eg 46° + 44° = 90°

Any two angles whose sum is 180° are
supplementary angles.
eg 128° + 52° = 180°

These are formed when a right angle is
cut into two parts. The angles add up
to 90°.

Adjacent These are formed when a straight angle
supplementary is cut into two parts. The angles add
up to 180°.

22 INTERNATIONAL MATHEMATICS 2

Name Diagram Explanation
Angles at a If angles meet at a point, then their
point 92° 125° sum is 360° or one revolution.
143° eg 92° + 125° + 143° = 360°
Vertically
opposite 33° 147° When two straight lines intersect, the
147° 33° vertically opposite angles are equal.
A
A transversal is a line that cuts two or more other lines.
B In the diagram AB is a transversal.

Corresponding x There are four pairs of corresponding
angles.
These are equal when the lines crossed

x by the transversal are parallel.

Alternate There are two pairs of alternate angles.
These are equal when the lines crossed
Co-interior by the transversal are parallel.

x There are two pairs of co-interior
x angles.

These are supplementary when the
lines crossed by the transversal are
parallel. (They add up to 180°.)

23APPENDIX A

A:07 | Plane Shapes

see A:07A Naming Plane Shapes

Plane shapes are shapes that lie in one plane, or flat surface. They have area but not volume.
Below are some of the common plane shapes.

1:07

This is the family of quadrilaterals.

rectangle

trapezium parallelogram

quadrilateral square

rhombus

kite

Each member of this family is a special kind of each of the shapes on its left,
eg a rhombus is a special parallelogram, trapezium, kite and quadrilateral.

Other plane shapes include:

triangle pentagon hexagon octagon circle oval (ellipse)

Shapes that have only straight sides are called polygons.
The ID cards at the front of this book (pages xv to xx) should be used often to help you remember
the language of Mathematics.

A:07B Constructing Triangles

When we construct triangles, we make use of a ruler, compasses and protractor.

worked examples

Example 1

Construct a triangle that has sides of length 6 cm, 5 cm and 4 cm.

24 INTERNATIONAL MATHEMATICS 2

Solution 1

1 234

5 cm 4 cm
6 cm
6 cm 6 cm 6 cm

Step 1 Draw an interval 6 cm long.
Step 2 Set the compasses to 5 cm and from one end of the interval draw an arc.
Step 3 Set the compasses to 4 cm and from the other end of the interval draw an arc to cut the

arc that has already been drawn.
Step 4 Join the point where the arcs meet to both ends of the interval to obtain the triangle.

Example 2

Construct a triangle that has sides of 4 cm and 5 cm which form an angle of 60°.

Solution 2

1 234

60° 60° 4 cm
4 cm
60° 60°
5 cm
5 cm 5 cm

Step 1 Use the protractor to draw a 60° angle.
Step 2 Measure 5 cm along one arm of the angle.
Step 3 Measure 4 cm along the other arm of the angle.
Step 4 Join the ends of the arms to form the triangle.

A:07C The Angle Sum of a Triangle

The sum of the angles of any triangle is 180° (or one straight angle).

A:07D Types of Triangles and Their Properties

Name Diagram Properties

Equilateral • All sides are equal.
triangle • All angles are equal at 60°.

Isosceles • Two sides are equal.
triangle • Angles opposite equal sides are

equal. (These are called base
angles.)

25APPENDIX A