Mathematics-MYP1-1

7a b 5 c 13 d 7 e 5
PC
3

2 11

22 5
2

B
U

Diagnostic Test 15: Sets

1 i a {1, 2, 3, 6, 9, 18} b {1, 2, 3, 4, 6, 9, 12, 18, 36} c6 d9

ii yes

2 a {2} b 14 c 24 d 37 e 13

3 a Group D teams in the second round

b Teams in the second round that were not in group D

c Teams that were either in group D or in the second round or both

d Teams that were neither in group D nor the second round

4 a b 13

CM
22 13 2

5a U b 63 c 30
18
GD
10 15 20

6 U
N
B
F

7 U

B D
A C

U

528 INTERNATIONAL MATHEMATICS 1

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

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 ∠CAD = 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

529ANSWERS

Index expressions simplified 9:04, D&D p274

sentences (equations) 9:10

Most references are to chapter topics, alternate angles 10:06, 10:07

eg 1:02 refers to Chapter 1: Topic 2 analog time 12:07

(Place value). angles Chapter 10, MT 10

A:01 … to I:01 refers to the appendix adjacent 10:05, MT 10

section of the CD. alternate 10:06

The following abbreviations refer to: at a point 10:05, 11:04

D&D Drag and Drops co-interior 10:06

FS Fun Spot comparing 10:02

Inv Investigation complementary 10:05, MT 10

MT Mathematical Terms corresponding 10:06

RM Reading Mathematics finding sizes of 11:04, D&D p308

Ch Challenge Worksheet maths terms 5 D&D p308

measuring 10:03

naming 10:02, 10:04

of triangle 11:02

24-hour time 12:07, MT 12 sum of polygon 11:05
2D Shapes Chapter 11,
ID Card 4 sum of quadrilateral 11:03, 11:04
2nd F 7:03
3D Shapes Chapter 11, sum of triangle 11:02, 11:04
ID Card 4
supplementary 10:05, MT 10

types 10:04, D&D p308

vertically opposite 10:05, MT 10

applications

abacus 1:02 of algebraic equations 9:11, 9:12
abbreviation
9:01, 9:02 of decimals 4:10
algebraic 9:06
index notation 3:03 of fractions 6:13
in mathematics 1:05, 7:04
accuracy and speed 13:02 of percentages 6:13
acre 2:08
acting out problems approximations 4:09, MT 1, MT 4
acute 10:04, MT 10
angle 11:02, MT 11 area Chapter 13
-angled triangle 1:03, B:01, MT 1
addition 4:02, B:13 definition 3:01, MT 3
of decimals 5:06
of directed numbers 6:04, B:08 of a rectangle 3:02
of fractions 6:05
of mixed numbers 10:05, MT 10 of a triangle 3:03
adjacent angles Chapter 9
algebra 9:02, D&D p274 mathematical terms 13 D&D p426
making sense of D&D p232
mathematical terms 8 D&D p270 problems 13:04
mathematical terms 9 9:02, 9:04
simplification FS 9:04 puzzle D&D p427
skills Ch 5:09, 9:03,
substitution D&D p274 astronomical unit FS 12:10

used in solving 9:11 axes 11:06
problems
9:01, D&D p274 axis of symmetry 11:06, MT 11
algebraic
abbreviations backtracking 2:08
base
3:09
binary 10:03, 13:03
line 1:06, 3:01B, 3:02A,
basic numeral MT 1, MT 3
3:09
binary 3:01, 5:01, MT 9
braces 3:01, 9:05, MT 9
brackets 13:02, 13:06
breadth

530 INTERNATIONAL MATHEMATICS 1

calculators Chapter 7 currency, decimal B:18
cylinder 11:07, MT 11
and directed numbers 5:12

crossword FS 13:06

make words with FS 13:06 dates and the calendar 12:06

mathematical terms 7 D&D p212 day 12:06

problem solving 7:06 decade 12:06

speed test D&D p212 decimals 1:09C, Chapter 4,

using 7:01, MT 7 D&D p28

calendar 12:06 addition 4:02, B:13

capacity 13:07, FS 13:07, applications 4:10, Inv 4:10,

and volume, units of D&D p427 FS 4:10

mathematical terms 13 D&D p426 checkup 1:09C

cardinal numbers 3:04, MT 3 comparing B:12

cards, standard pack RM 2:02 currency B:18

centimetres 12:02 division 4:04, 4:07, B:16,

centre of symmetry 11:06, MT 11 B:17, D&D p118

century 12:06, MT 12 fraction conversions 4:08, B:11,

chain Inv 12:02 D&D p118, MT 4

chance 6:15, 6:16 mathematical terms 4 D&D p114

chart, used to solve multiplication 4:03, 4:06, B:15,

problems 2:05 B:17, D&D p118

check, guess, and refine to operations with D&D p118

solve problems 2:03 percentage conversions 6:10, 6:12, B:22,

answers to equations 9:10 B:23

circle 11:01, MT 14 places 7:03, MT 4

clocks 12:01, 12:06, recurring 4:08

D&D p388 review 4:01

co-interior angles 10:06, 10:07 rounding off 4:09

collinear points 14:01, MT 14 subtraction 4:02, B:14

common factors and terminating 4:08

multiples 3:05, 3:07, MT 3 degree

compasses, pair of 14:03, MT 14 angle size 10:03, MT 10

complementary angles 10:05, MT 10 temperature 12:01

composite number 3:06, MT 3 describing number patterns 8:02

computation, direct, diagonal 11:01, 11:03, MT 11

to solve problems 2:01 diagram, make a (strategy) 2:04

concurrent lines 14:01 diameter 14:03, MT 14

cone 11:07 differences, finite D:01

congruent shapes (11) F:01 digit, binary 3:09

consecutive numbers 9:11, MT 9 digital time 12:07

construction dimensions two and three Chapter 11, MT 12

of a triangle 14:04 direct computation to

of a quadrilateral 14:05 solve problems 2:01

coordinates 5:02, 5:03, 5:05, directed numbers Chapter 5,

5:08, D&D p149, D&D p149, MT 5,

MT 5 MT 9

corresponding angles 10:06, 10:07 addition 5:06, D&D p149

counting numbers 3:04, MT 3 and algebra 9:09

cross-section 11:07, MT 11 division 5:10, D&D p149

cubic 11:07, 11:09, 11:10, mathematical terms 5 D&D p147

RM 11:11, MT 11 multiplication 5:09, Inv 5:09,

root 3:03, 3:08, MT 3 D&D p149

Soma Ch 13:07 on the number place 9:08

cube units 13:05, 13:06 subtraction 5:06, D&D p149

cubit Inv 12:02 directory, street 5:02

INDEX 531

distance, speed and time Inv 12:10, Ch 7:06, flowcharts 10:04

G:02 foot Inv 12:02

distributive property 3:02B, MT 3 fortnight 12:06

diving, scoring fractions Chapter 6,

championship FS 4:10 D&D p28,

divisibility tests 3:07, MT 3 D&D p194, MT 1

division 1:03, B:04, MT 1 addition 6:04, B:08

long 1:04, B:04 applications 6:13

of decimals 4:04, 4:07, B:16, B:17 bar (vinculum) 3:01A, MT 9

of directed numbers 5:10 checkup 1:09B

of fractions 6:07 comparing 6:02, B:06

of indices H:01 decimal conversions 4:08, B:11, E:01

divisor 3:07 division 6:07

dodecahedron RM 11:11, F:05 equivalent 6:03, B:05,

drawing, make a (strategy) 2:04 D&D p194, MT 1

solids 11:09 improper B:07, MT 1

mathematical terms 6 D&D p191

edges, of 3D shapes 11:07, MT 11 multiplication 6:06, B:09
Egyptian numerals 1:01, D&D p28
element MT 15, 15:01 of a number B:10
eliminating possibilities
2:06 of a quantity 6:08
(strategy) 11:01
ellipse D&D p274 percentage conversions 6:10, 6:12, B:19, B:21
equations puzzle 9:10, 9:11, 9:12
equations, solving 11:02, 11:04, MT 11 review 6:03
equilateral triangle 6:03, B:05,
equivalent fractions D&D p194, MT 6 subtraction 6:04, B:08
3:06, MT 1, MT 4,
Eratosthenes, Sieve of MT 7 furlong Inv 12:02

estimation 13:07 gallon 13:07
capacity 4:03, Inv 4:10 geometrical instruments
decimals D&D p447
guess, check and 2:03 mathematical terms Ch 14:03
refine Inv 1:02 14A and 14B
large numbers 1:08 golden rectangle 5:01, MT 5
leading digit 12:04 graphing 9:07, 9:08, MT 5
length 1:08 number line D&D p274
speed test 7:02 number plane RM 8:03,
tactics 11:07, MT 11 ordered pairs B Assignments
3:04, MT 3 graphs 8:04
Euler’s theorem 1:02, 1:07, 9:06, 3:01, 9:05, MT 3,
even numbers MT 1 of patterns MT 9
expanded notation 9:04, 9:05, MT 9 grouping symbols
2:03
expression, algebraic guess, check and refine
(strategy)

faces, of 3D shapes 11:07, 11:08, MT 11 HCF 3:05, C:01, MT 3
factor 3:05, MT 3, MT 7 hectare 13:02, MT 13
7:05 hexagon 11:01
and multiple C:01 hexagonal numbers 3:04
prime 3:06 hexagonal prism and
trees Inv 12:02 11:07
fathom 3:04, Inv 3:04, MT 3 pyramid RM 11:11
Fibonacci numbers D:01 hexahedron 3:05, C:01
finite differences F:03 highest common factor 1:01, MT 1
flip Hindu-Arabic numerals MT 14
horizontal 12:06
hour

icosahedron RM 11:11
improper fractions 6:03, 6:06, 6:08

532 INTERNATIONAL MATHEMATICS 1

inch Inv 12:02 model, make a (strategy) 2:04
index notation 9:06 money, operations with D&D p55
indices 1:07, 9:06, H:01, 1:09C
MT 9 checkup B:18
instruments, measuring 12:01 decimal currency 12:06
integers 5:11, MT 5, MT 9 month 3:05, 7:05, MT 3,
intersecting lines 11:07 multiple MT 7
INV (calculator keys) 7:03 1:03, B:03, MT 1
isometric grid paper 11:09 multiplication 4:03, 4:06, B:15, B:17
isosceles triangle 11:02, 11:04, MT 11 of decimals 5:09, Inv 5:09
of directed numbers 6:06, B:09
kite 11:01, 11:03 of fractions H:01
of indices D&D p55
LCD 6:04, 6:05, MT 6 tables A D&D p55
LCM 3:05, C:01, MT 3 tables B
leading figure estimation 1:08
length 12:02, 12:03 negative numbers Chapter 5, MT 5
12:04 nest of squares FS 1:03
estimation 12:03, Inv 12:03 net of a solid 11:08, MT 11
measuring Inv 12:02, 12:02, notation, expanded 1:07
units of D&D p388 number Chapters 1 and 3
FS 12:10 3:09
light years 5:01 binary 5:04–5:12
line, number directed 1:01
lines 14:01 history of D&D p93
11:07 language and symbols 5:01, 5:04, 9:10,
concurrent 11:06, MT 11 line MT 5
intersecting 11:07, 14:02 D&D p26
of symmetry 14:02 mathematical terms 1 D&D p91
parallel 11:07 mathematical terms 3 8:01, 8:02,
perpendicular 2:05 patterns CW 8:02, MT 8
skew 1:04, B:04 5:03, 5:08, MT 5
list, make a (strategy) plane 3:02, MT 3
long division 6:04, 6:05 properties D&D p28, D&D p93
lowest common 3:05, C:01 puzzles Appendix B
denominator review of skills 9:10
lowest common multiple sentences 3:04, D&D p93
whole MT 1
magic squares FS 1:09, FS 3:09 numeral 3:01B, 3:02A
basic 1:01, D&D p28
mass 1:01, Inv 1:01 Egyptian 1:01
Hindu-Arabic 1:01, D&D p28
measurement Chapter 12, MT 12 Roman 4:08, 6:01, MT 1
numerator
of 3D space 13:05, Inv 13:05

of angle size 10:01, 10:03

of length 12:03, G:01, MT 12

mathematical terms 9 D&D p385 oblique pyramid 11:07
obtuse
measuring instruments 12:01 10:04, MT 10
angle 11:02, MT 11
mile Inv 12:02 -angled triangle 11:01
octagon
millennium 12:06 octagonal prism and 11:07
pyramid RM 11:11
minute 12:06 octahedron 3:04, MT 3
odd numbers Inv 3:03
mixed numbers B:07, MT 6 odds and evens 12:01, 12:03, MT 12
odometer
addition 6:05

division 6:07

multiplication 6:06

percentage conversions B:20, B:21

subtraction 6:05

INDEX 533

operations 1:03, MT 1 mixed number
conversions
addition A:01, B:01, MT 1 B:20, B:21
of a quantity 6:11
checkup 1:09A review 6:09
perfect number 3:06
division B:04, MT 1 perimeter 12:05, 14:01, 14:02,
D&D p388, MT 12
indices H:01 perpendicualr lines 14:02, MT 14
pint 13:07
mixed D&D p55 place value 1:01, 1:02, MT 1
plane
multiplication A:01, B:03, MT 1 11:01, 11:05, F:02,
shapes F:04, D&D p346,
order of 3:01 MT 11
number 5:03, 5:08
subtraction A:01, B:02, MT 1 Platonic solids RM 11:11, MT 11
plumb 14:02
opposite angles, vertically 10:06 point
5:01
opposite operations 1:03, MT 1, MT 9 on number line 5:03, 5:08
on number plane 11:06, MT 11
order of operations 3:01, 4:05, D&D p93, symmetry MT 11
polygon 11:05
MT 3, MT 7 angle sum of 11:07, 11:11,
polyhedron (polyhedra) RM 11:11, MT 11
orientation of shapes 11:01, MT 11 Chapter 5, MT 5
positive numbers
origin in the number plane 5:03, MT 5 possibilities, eliminating 2:06
RM 2:10
oval 11:01 (strategy) MT 1, MT 9
posting parcels H:01
palindromic numbers 3:04, MT 3 powers 1:06, 3:01, D&D p28
palm as a measure Inv 12:02
parallel of indices C:01
10:07 of numbers 3:06, MT 3
identifying parallel 10:06, 11:07, 14:02, prime 11:07, MT 11
lines MT 10, MT 14 factors 13:06
11:03 numbers 6:15, 6:16,
lines 11:01 prism D&D p194
11:03 volume of Chapter 2, Inv 8:01,
slides 3:01, 9:05, MT 9 probability RM 8:03, FS 9:12,
parallelogram FS 12:10 FS 6:15, 13:04,
Inv 3:04 problem solving 7:06, B Assignments
properties 9:12
parentheses F:03 in real life 7:06
parsec (large unit) 8:02 with a calculator 9:11, 9:04
Pascal’s triangle 8:04 with algebra 8:03, Ch 5:09, 9:01,
patterns 8:01, Inv 8:01, 8:02, pronumeral 9:02
Ext 8:02, 7:05 D&D p232, MT 8,
by transformations 2:09 pronumerals, and rules MT 9
describing
graphs of 14:03 properties 3:02B
in number F:02 distributive 3:02
11:01 of number 10:03, FS 10:03,
in problems (strategy) MT 14, FS 10:04,
making with 11:07 protractor MT 10
F:01
compasses Chapter 6,
of plane shapes D&D p55,
pentagon D&D p212
pentagonal prism and 6:12, 6:13, 6:14
pyramid 1:09D
pentominos 6:10, 6:12, B:22,
percentage B:23. D&D p194
6:10, 6:12, B:19,
applications B:21, D&D p194,
checkup MT 6
decimal conversions D&D p191

fraction conversions

mathematical terms 6

534 INTERNATIONAL MATHEMATICS 1

pyramid 11:07, MT 11 of integers 5:11
solving FS 9:12 of numbers 3:04, 5:01
square FS 10:03, 14:02,
quadrilateral 11:01 MT 14
angle sum of 11:03, 11:04 universal 15:02–15:04, MT 15
construction of 14:05 shapes Chapter 11
properties of 11:03 11:01–11:06,
6:11 2D D&D p346
quantity, percentage of a 13:07 11:07–11:11,
quart 3D D&D p346
F:01
radius 14:03, MT 14 hidden
rational number 4:01, 6:02, MT 6 mathematical terms D&D p346
reciprocal 6:07, MT 6 11:06
rectangle 11:01 11A & 11B 11:01, F:01, F:02
13:02, Inv 13:02 patterns of 7:03
area of a Ch 14:03 plane 3:06
golden 11:03 shift 11:07
properties Sieve of Eratosthenes F:03
rectangular 11:07, 11:09 skew lines RM 10:04
prism and pyramid 13:06 slide 11:07
prism, volume of 4:08, Ch 4:10, MT 4 smoking and health 11:10, F:05
recurring decimals solids 11:09
refine, guess, check and 2:03 building 11:11
(strategy) F:03 drawing 11:08
reflection 10:04, MT 10 from different views RM 11:11
reflex angle 1:03 nets of
remainder 4:08, E:01, MT 4 Platonic 2:10
repeating decimals 10:04, 11:03, MT 10 solving Chapter 2
revolution a simpler problem 9:11, 9:12
rhombic prism and 11:07
pyramid 11:01, 11:03 (strategy) 4:05
rhombus problems FS 9:12
right 10:04, MT 10 problems with algebra Ch 13:07
angle 11:02, MT 11 problems with 13:05, Inv 13:05
-angled triangle 11:07 Inv 12:02
pyramid Inv 12:02 decimals G:02
rod 1:01, D&D p28 pyramids 1:05, 7:04
Roman numerals 3:03, 3:08 Soma cube 12:01
root, square and cube 11:06, F:03, MT 11 space, measuring 3D Inv 12:10, 7:06, G:02
rotational symmetry F:03 span 11:07
rotation transformation speed 11:01
rounding 1:08 and accuracy 13:02
estimation 4:09, 4:10, MT 4 on speedometer 11:09
of decimals 7:03 time and distance FS 1:10, FS 3:09
with calculator 8:02, 8:03, D:01 sphere FS 1:02
rule, finding the D&D p232 square 1:06, 3:04, MT 3,
rules, and pronumerals 12:03, 14:01 area of a MT 9
ruler grid paper 11:07
magic 11:03
scalene triangle 11:02, MT 11 nest of 3:01A, 3:03, 3:08,
scale on a measuring numbers MT 3
12:01, MT 12
instrument 12:06 prism and pyramid
second 9:01 properties of a
sentence, algebraic Chapter 15 root
set 15:02, MT 15

empty

INDEX 535

standard pack of cards RM 2:02 area of 14:02
construction of 14:04
straight angle 10:04, 11:04, MT 10 types 11:02
triangular
strategies numbers 3:04, 8:03, MT 3
prisms and pyramids 11:07
31 game FS 2:01 triangulation Inv 10:03, Ch 10:06
trundle wheel 12:03, MT 12
problem solving Chapter 2 turn F:03
two-dimensional shapes 11:01
problem solving using

algebra 9:11, 9:12

study hints RM 10:05, Ch 10:06

substitution 5:09, 9:03, MT 9

checking equations by 9:10

subtraction 1:03A, B:02, MT 1

of decimals 4:02, B:14 union of sets 15:03, MT 15
units MT 12
of directed numbers 5:06 FS 12:10
astronomical 13:05
of fractions 6:04, B:08 cubic 4:01
numerical
of mixed numbers 6:05 of capacity and D&D p427
12:02
of negative numbers 5:07 volume 13:01
of length
supplementary angles 10:05, MT 10 square

symbols, grouping 3:02, 3:02B

symbols, maths 3:03, D&D p93

symmetry 11:06, MT 11

table universal set 15:02–15:04, MT 15
make a (strategy)
reading a 2:05 value, place 1:02
RM 9:09
tally 2:05, Inv 10:06 variable 8:02
tangram F:01
tape measure 12:03 Venn diagram 15:02–15:04, MT 15
temperature 12:01
terminating decimals 4:08, MT 4 vertex of an angle 10:01, 10:02, MT 10
terms, like and unlike 9:04, 9:09
tessellation F:02, F:03 vertical lines 14:02
tetrahedron RM 11:11
three-dimensional 11:07 vertically opposite angle 10:05, MT 10
tile patterns F:02, F:03
time 12:01, 12:06–12:10, vertices of 3D shapes 11:07
MT 12
and speed Inv 12:10, Ch 7:06, views of a solid 11:11
G:02
timetables 12:10 volume Chapter 13
transformation F:03
translation F:03 mathematical terms 13 D&D p426
transversal 10:06
trapezium 11:01, 11:03 units of capacity and D&D p427
trapezoidal prism and
11:07 week 12:06
pyramid 2:03 weight 1:01
trial and error 11:01, MT 11 words, making with a
triangle 11:02, 11:04 7:06
calculator
angle sum of working backwards 2:07
Chapter 2,
(strategy) B Assignment
working mathematically and throughout
each chapter

yard Inv 12:02
year 12:06

536 INTERNATIONAL MATHEMATICS 1

Navigating International Maths for the Middle Years 1

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Appendixes

Have you
got what
it takes?

Appendix A Appendix C

A:01 The four operations C:01 HCF and LCM by prime factors (extension)

Appendix B Appendix D

B:01 Addition D:01 Finite differences (finding the rule)
B:02 Subtraction
B:03 Multiplication Appendix E
B:04 Division
B:05 Equivalent fractions E:01 Changing repeating decimals to fractions
B:06 Comparing fractions
B:07 Improper fractions and mixed numbers Appendix F
B:08 Adding and subtracting fractions
B:09 Multiplying a whole number by a fraction F:01 Investigation of plane shapes
B:10 Finding a fraction of a number Fun Spot: Hidden shapes
B:11 Conversions: Fractions and decimals
B:12 Comparing decimals F:02 Plane shapes and patterns
B:13 Adding decimals F:03 Transformations: Reflections, translations and
B:14 Subtracting decimals
B:15 Multiplying a decimal by a whole number rotations
B:16 Dividing a decimal by a whole number F:04 Using plane shapes
B:17 Multiplying and dividing decimals by 10 F:05 Building solids using blocks
B:18 Decimal currency
B:19 Changing percentages to fractions Fun Spot: Making a pop-up dodecahedron
B:20 Changing percentages to whole and mixed
Appendix G
numbers
B:21 Changing fractions and mixed numbers to G:01 Investigation: Measurement extension
G:02 Distance, speed and time
percentages
B:22 Changing percentages to decimals Appendix H
B:23 Changing decimals to percentages
H:01 Operations with indices (extension)
Working Mathematically
Appendix I

I:01 Mass
Investigation: Mass

1

APPENDIX A
A:01 | The Four Operations

The combinations in the following exercises need to be practised until known.

Exercise A:01

1 Have someone time you, as you do each line four times. Use scrap paper
Record the times in a table like the one on the right, like this until you
and graph the results.
a 3104728596 can break the
2+ 20 second barrier

b 0219357864 for each line
10 +

c 3402817596 Time for each trialTime (in seconds)
3+ Table 1 2 3 4
2 + 20 s 18 s 16 s 15 s
d 1426037859 10 +
4+ 3+
4+
e 5206415398 5+
5+ 6+
7+
f 2401637958 8+
6+ 9+

g 0143728659 Graph for table 2+
7+ 20

h 2381075946 15
8+
10
i 4026583197
9+ 5

2 INTERNATIONAL MATHEMATICS 1 0
1234
Trial number

2 Try to do each of these in 2 minutes. +1963 Add the number on
a +1905826473 2 the top to the
1 7 13
9 5 number on the left
0
5
8

b +1905826473

2

6+9=9+6

6 4+7=7+4

4 5+8=8+5

7 The answer to
3 17 — 7
3 Try to do each line in 20 seconds. goes here.

a 10 8 9 b 9 10 8 7
18 − 17 −

c 7 9 6 10 8 d 8 5 10 6 9 7
16 − 15 −

e 10 6 8 4 5 9 7 f 5 10 4 7 9 3 6 8
14 − 13 −

g 5 2 9 4 8 3 6 10 7 ■ If 10 + 9 = 19,
12 − then
19 − 10 = 9
h 2 8 5 1 9 6 3 10 7 4 and
11 − 19 − 9 = 10

i 1 10 2 5 7 3 9 4 0 8 6
10 −

4 Each box should be completed in 10 seconds.

a 5−3 b 6−6 c 6−5 d 9−9 e 9−8
7−5
8−2 3−1 7−4 6−4 9−6
8−4
6−3 9−7 4−2 7−3 9−5

9−2 8−5 8−3 9−4

8−7 7−6 9−3 8−6

3APPENDIX A

5 Have someone time you as you do each part four times. Record the times in a table like the one
on the right, and graph the results.

a 2 1 5 0 3 10 4 9 6 8 7 11 Table Time for each trial
2× 1234

b 1 0 3 2 5 10 6 4 9 8 7 11 2 × 24 s 21 s 15 s 13 s
3× 3×

10 ×

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

d 0 8 2 5 10 4 3 1 6 7 9 11 6×
5× 7×

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

f 1 0 3 2 5 10 6 4 9 8 7 11 Time (in seconds) Graph for table 2 ×
6× 25
20
g 0 2 1 5 10 3 7 4 8 6 9 11 15
7×

h 2 1 5 0 3 10 4 9 6 8 7 11 10
8× 5

i 1 0 3 2 5 10 6 4 9 8 7 11 0
9× 1234
Trial number

j 0 2 1 5 10 3 7 4 8 6 9 11 ■7×4=4×7
11 ×

6 Try to do this in 3 minutes.
× 0 2 1 5 10 3 7 4 8 6 11 9
3
7
5
9
8
6

4 INTERNATIONAL MATHEMATICS 1

APPENDIX B

The Diagnostic Checkups in Section 1:08 will show you which number skills you need to revise.

Before each practice exercise you will find:
• explanations
• rules
• worked examples

Read these carefully before attempting the exercise.

Using the Four Operations

B:01 | Addition

worked examples

1 515 2 19208 10 units 10 hundreds 1 ten-thousand
307 7537 = 1 ten = 1 thousand 9 thousands
096 2 hundreds
292823 328161910 0 tens
8 units
1901 65435

192 09

10 tens 10 thousands
= 1 hundred = 1 ten thousand

Exercise B:01

1 a 63 b 93 c 75 d 92
50 88 69 48
82 7 18 67
8 46 34 90

e 735 f 806 g 472 h 655
41 40 88 788
71 391
686 93 693 849
317 486
d 5 843
2 a 18 550 b 48 657 c 69 435 92 549
9 275 66 982 88 647 87 104
14 361 93 156
43 827 h 99 999
g 845 329 156 085
e 73 186 f 168 936 693 246 565 117
264 092 39 214 816 094 318 628
181 925 135 298
9 135 610 293
88 150

5APPENDIX B

3 a In 1984 the Australian Commonwealth parliament had 224 members while the state
parliaments’ memberships were: Victoria 125, New South Wales 144, Queensland 82,
Western Australia 91, South Australia 69 and Tasmania 54. How many members of
parliament were there altogether in 1984?

b The Australian Armed Forces in 1984 had 10 482 officers, 57 961 other ranks, 1306 cadets,
1817 apprentices and 76 junior recruits. How many people were in the armed forces in 1984?

c In 1980, 94 500 settlers arrived in Australia. There were 118 740 in 1981, 107 170 in 1982
and 78 400 in 1983. How many settlers arrived in Australia in those four years?

d In 1980, 11 450 former settlers permanently left Australia. There were 11 280 in 1981, 13 350
in 1982 and 16 920 in 1983. How many former settlers permanently left Australia in those
four years?

B:02 | Subtraction

worked examples

1 612 815 ■ Method 1 ■ Method 2 2 56122815
− 119 117 Equal additions Decomposition −1 9 1 7

4368 4368

Which method
do you use?

Exercise B:02

1 a 839 b 574 c 750 d 617
− 256 − 139 − 506 − 583

e 4605 f 3511 g 8000 h 8436
− 1821 − 1076 − 1960 − 7737

2 a 56 453 b 80 473 c 375 314 d 680 935
− 19 216 − 1 984 − 218 109 − 48 566

e 1 650 000 f 8 654 000 g 5 000 000 h 9 500 000
− 940 000 − 1 672 000 − 1 456 000 − 2 754 000

3 a Of the permanent defence force of 71 642 in 1984, 4673 were female. How many were male?
b The population of Australia in 1983 was 15 378 600, to the nearest hundred. If the
population was 15 543 600 twelve months later, what was the approximate increase in
population during those twelve months?
c The area of Australia is 7 682 300 square kilometres. The area of the United States of America
is 9 372 614 square kilometres. By how much is the area of the United States of America
greater than the area of Australia?

6 INTERNATIONAL MATHEMATICS 1

d Spiros was surprised to discover that of the 111 280 tonnes of butter produced in Australia
last year, 93 884 tonnes were produced in Victoria. How many tonnes were produced
altogether in the other states last year?

B:03 | Multiplication

worked examples

1 123 035 To multiply by 70, 2 8 653 ■ ‘Long’
×7 0 write down the ‘0’ × 45 multiplication
and multiply by 7.
91350 43 265 (8653 × 5)
346 120 (8653 × 40)

389 385

8653 × 45 = (8653 × 5) + (8653 × 40)

Exercise B:03

1 a 872 b 8046 c 2559 d 1746
×6 ×7 ×8 ×9

e 495 f 6140 g 7086 h 7823
× 20 × 60 × 50 × 90

2 a 172 b 793 c 4658 d 2009
× 26 × 57 × 41 × 63

e 5186 f 13 814 g 65 046 h 16 287
× 45 × 78 × 93 × 59

3 a Bulli football club paid each of its first grade team members $2937 in fees. How much money
did the club pay altogether if there were 24 members of the team?

b If 65 046 kilograms of metal are required to make one kilometre of cable, how many
kilograms of metal are required to make 25 kilometres of cable?

c In 1999 the average monthly production of crude oil was 1913 megalitres. What was the total
production of crude oil in 1999?

d From 1 July 1999 to 30 June 2000, average weekly milk production was 73 million litres.
How much milk was produced during that financial year (52 weeks)?

7APPENDIX B

B:04 | Division

worked examples

406 Once the 4 is placed ■ Method 1 ■ Method 2
above the 6, we must Contracted Preferred
1 4 ) 1 6 224 place a figure above form multiples
the 2 and 4 as well.
106 r 40 106 r 40
Estimate the answer
before you begin. 2 91 ) 9686 91 ) 9686
Say:
‘How many fours − 91 − 9100 100
in 1600?’ 586
586
− 546
40 − 546 6

40 106

Answer: 106 remainder 40 or 106 4----0-

91

Exercise B:04

1 a 8 ) 880 b 6 ) 397 c 7 ) 843 d 4 ) 1111
e 3 ) 7418 f 8 ) 38 168 g 9 ) 1469 h 7 ) 68 142

2 a 23 ) 8104 b 59 ) 6183 c 32 ) 7894 d 19 ) 3856
e 500 ) 7265 f 36 ) 60 660 g 42 ) 96 143 h 91 ) 14 965

3 a 23 009 books are to be packed into 19 crates so that each crate contains the same number of
books. How many books will be placed in each crate?

b How many trips must a truck make to carry 1246 sheep to the stock yards if the truck can
carry 54 sheep on each trip?

c 36 450 mL of milk must be placed into bottles so that each bottle contains 600 mL of milk.
How many bottles can be filled?

d A tank that holds 10 000 litres is filled by means of a hose in 24 hours. Find the amount
added each hour if in each hour the same amount is added.

Fractions ■ -3- × 2 = -6-
4 × 2 8
B:05 | Equivalent Fractions
3-- ×3 = --9---
-3- = -6- = --9--- 4 ×3 12
4 8 12

8 INTERNATIONAL MATHEMATICS 1

■ -1- × 2 = 2--
2 × 2 4

-1- = 2-- = -4- 1-- × 4 = -4-
248 2 × 4 8

--6--- ÷2 = 3--
10 ÷2 5

--6--- = 3--
10 5

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

■ 1 = 2-- = -4- = 5-- = -8- = 1----0- = -1---2-
2 4 5 8 10 12

There are many different ways to write the number one.

worked examples

1 --3--- = -.---.----. 2 -6- = .----.---. 3 Simplify 2----0- This process is
10 100 84 50 sometimes called

Multiply top and Divide top and -2---0- = -2---0- ÷ 10 ‘reducing’ or
bottom by 10. bottom by 2. 50 50 ÷ 10 ‘cancelling’.

∴ --3--- × 10 = --3---0--- ∴ 6-- ÷2 = -3- = 2--
10 × 10 100 8 ÷2 4 5

Exercise B:05

1 Complete the following to make equivalent fractions:

a --1--- = ---ٗ----- b 1-- = ٗ---- c 1-- = -ٗ---- d 3-- = -ٗ--- e --7--- = ---ٗ-----
10 100 24 4 20 48 10 100

f --5--- = -ٗ---- g 1-- = ٗ---- h 3-- = -ٗ---- i -3- = -ٗ---- j --2--- = ---ٗ-----
10 50 18 4 20 5 10 10 100

k -1- = -ٗ---- l --3--- = ---ٗ----- m 1-- = -ٗ---- n 5-- = -ٗ---- o 2-- = -ٗ----
5 10 10 100 3 12 8 16 5 10

p --4--- = ---ٗ----- q 2-- = ---ٗ----- r --7--- = -ٗ---- s --9--- = ---ٗ----- t -4- = -ٗ----
10 100 5 100 20 40 10 100 5 10

APPENDIX B 9

2 Use the diagram on the right to complete the following:

a --5--- = ٗ---- b -1---0- = ٗ---- c -1---5- = -ٗ--- d 2----0- = -ٗ---
20 4 20 4 20 4 20 4

e --4--- = -ٗ--- f --8--- = -ٗ--- g -1---2- = -ٗ--- h -1---6- = ٗ----
20 5 20 5 20 5 20 5

i --2--- = -ٗ---- j --6--- = -ٗ---- k 1----0- = -ٗ---- l -1---4- = -ٗ----
20 10 20 10 20 10 20 10

3 Write each fraction in its simplest form:

a --4--- b 4-- c --6--- d --2--- e --8---
10 8 12 10 20

f 1----8- g --5--- h --7---0--- i --6--- j --1---0---
20 10 100 10 100

k --2--- l 1----5- m -1---0- n 3-- o --8---
12 20 16 9 10

p 1----0- q -1---4- r 1----0- s 2----0- t 1----2-
12 16 10 25 60

u 3----3- v -1---0---0- w --2---0--- x -2---4---0- y --6---2---5---
44 800 100 360 1000

B:06 | Comparing Fractions

Use equivalent fractions to make the denominators the same.
Then compare the numerators.

worked examples

Place in order, from smallest to largest:

1 { --3--- , 1, --1--- } 2 { 1-- , -2- , --6--- }

10 10 2 5 10

Give each fraction the same denominator. 5

■ 1 = -1---0- , 1-- = --5--- , 2-- = --4--- 4 4
3 3
10 2 10 5 10 2 2
1 1
{ --3--- , -1---0- , --1--- } { --5--- , --4--- , --6--- } ascending descending

10 10 10 10 10 10

In ascending order these become:

{ --1--- , --3--- , 1} { 2-- , 1-- , --6--- }

10 10 5 2 10

10 INTERNATIONAL MATHEMATICS 1

Exercise B:06

1 Use this number line to find which of the two numbers is smaller.

0 5 1 1150 2
10

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

10 10 10 2 10 10 10 10 2

f --4--- , --9--- g 2-- , 1-- h -2- , --1--- i 4-- , --7--- j -1- , 2--

10 10 55 5 10 5 10 25

2 Which of the three fractions is smallest? d 3-- , -1- , -2-

a --4--- , --3--- , --7--- b 4-- , 2-- , 3-- c --2--- , -1- , --3--- 525

10 10 10 555 10 2 10 h -2- , -1- , --7---

e 1-- , -1- , 3-- f --7--- , -1- , -1- g 3-- , --6--- , 1-- 5 2 10

244 10 2 5 4 10 2 d { --9--- , --7--- , 1-- }

3 Arrange in order, from smallest to largest: 10 10 2

a { --2--- , --1--- , --7--- } b { -1- , 1-- , 3-- } c {1, -4- , -3- } h {2 --1--- , 1 -1- , --9--- }

10 10 10 244 55 10 2 10

e { --7--- , 1 --1--- , -1- } f { 1-- , -1- , --1--- } g { 2-- , -3- , -1- } l { --2---3--- , -1- , --3--- }

10 10 5 2 4 10 542 100 5 10

i { -3- , 1-- , 3-- } j { -3- , --9--- , 3-- } k { --7--- , --2--- , 1-- }

425 4 10 5 40 10 4

B:07 | Improper Fractions and

Mixed Numbers

An improper fraction has a numerator A mixed number is one that
that is greater than its denominator. has a whole number part and
a fraction part, eg 2 -1- , 1 --3---- .
eg -5- , 1----3-- .
2 10
2 10

worked examples

1 Write 5 as an improper 8 seems
fraction in quarters. 3
Now 1 = 4 quarters
∴ 5 = 20 quarters improper to me!

= 2----0- 2 2 = 8
3 3
4

2 Write 2 2-- as an 3 Change -8- to a

3 3

improper fraction. mixed number.

2 -2- = 2 + -2- 8-- = 6-- + -2- OR 8-- = 8 ÷ 3 ■ -2---5- means
4
33 333 3 25 ÷ 4.

= 6-- + 2-- = 2 + 2-- = 2 2-- 11APPENDIX B

33 3 3

= -8- = 2 -2-

3 3

Exercise B:07

1 Write each mixed number as an improper fraction.

a 1 -1- b 1 -1- c 1 --1--- d 1 -1- e 1 -1- f 1 1--

2 4 10 3 5 8

g 2 --3--- h 1 -3- i 2 1-- j 3 -2- k 4 -3- l 1 --9---

10 4 2 3 5 10

m 5 -4- n 2 -3- o 2 --7--- p 5 --7--- q 4 3-- r 3 -3-

5 4 10 10 8 5

s 1 5-- t 7 --1---7--- u 4 -1---3- v 2 ----1---- w 8 7-- x 6 3--

8 100 20 100 8 4

2 Write each improper fraction as a mixed or whole number.

a -5- b 1----3- c 7-- d -5---7- e 2----0- f -4---4-

4 10 4 10 5 5

g -2---1- h 9-- i 6----1- j -1---1- k -7- l -1---1-

8 4 10 4 5 8

m 1----5- n 9-- o 9-- p -8---9- q 2----4- r -1---5-

2 5 8 10 3 4

s 3----6- t -1---3- u 1----3- v 1----5- w -1---7- x -1---9-

4 4 5 8 4 8

3 Simplify by writing each as a mixed or whole number.

a -1---2---0- b 7----3- c 4----9- d 6----5- e 1----2---7- f 8----1-

4 2 3 5 10 8

g -2---9- h -1---7- i -1---0---0- j -1---0---0- k 4----6- l -1---4---0-

24 10 3 5 22 7

m 1----8- n -3---6- o -1---2---2- p -1---8---5- q -1---1---0- r -2---7---4-

10 8 4 10 20 100

4 a Thirty-one children were each given one-quarter of an apple. How many apples were given

away.
b When Karen was injured working in the factory, each of her friends donated -1- of a day’s pay

5

to help her. How many days’ pay were given if 33 friends contributed money?

c Ray used 1-- of a page for each graph he drew. How many pages would he use to draw

4

15 graphs?

d Ruby used 6 --3--- pages to do many similar division questions. If each question took --1--- of a
page, how m1a0ny questions did she do? 10

B:08 | Adding and Subtracting Fractions

When fractions have the same denominator, we can add them by adding numerators;
we can subtract them by subtracting numerators.

worked examples

1 --7--- + --6--- = 7-----+-----6-- = 1----3- = 1 --3--- 7 6
10 10 10 10 10 10 10

0 24 68 1 1120 1130 1140
10 10 10 10

12 INTERNATIONAL MATHEMATICS 1

Write answers Write answers
as mixed in simplest form
numbers.

9
10

2 --9--- − --3--- = -9----–-----3- = --6--- = -3- 3
10 10 10 10 5 10

0 24 68 1 1120 1140
10 10 10 10

Exercise B:08

1 a --5--- + --4--- b --8--- – --5--- c --9--- – --8--- d --1--- + --6--- e 2-- + -3-
10 10 10 10 10 10 10 10 88
= .----.---.---+-----.---.---. = -.---.---.---–----.---.---. = -.---.---.---–----.---.---. = -.---.---.---+-----.---.---. = -.---.---.---+-----.---.---.
10 10 10 10 8

= ----- = ----- = ----- = ----- = -----
10 10 10 10 8

f --7--- + --2--- g --3--- + --6--- h --4--- + --3--- i 5-- + 2-- j 2-- + -1-
10 10 10 10 10 10 88 55

k --7--- – --4--- l --3--- – --2--- m --4--- – --3--- n -5- – -4- o -2- – 1--
10 10 10 10 10 10 88 55

p ----3---- + ----8---- q --1---7--- + ----2---- r --3---5--- + --2---6--- s --5---1--- + --1---6--- t --8---2--- + --1---7---
100 100 100 100 100 100 100 100 100 100

u --1---8--- – --1---5--- v --4---1--- – --1---8--- w --5---5--- – --3---4--- x --9---8--- – --8---9--- y --3---1--- – --1---8---
100 100 100 100 100 100 100 100 100 100

2 Give answers as mixed numbers or in simplest form. d --8--- + --9--- e --3--- + --7---
10 10 12 12
a --3--- + --2--- b -3- + -3- c --6--- + --5---
10 10 88 10 10 i --7--- – --1--- j 1----1- – --3---
10 10 10 10
f --9--- – --3--- g --8--- – --4--- h 7-- – 5--
16 16 10 10 88 n --3--- + --9--- o --9---5--- + ----9----
20 20 100 100
k --6--- + --6--- l ----7---- + ----7---- m --9--- + --9---
10 10 100 100 10 10 s -1---5- – --3--- t -1---0- – --3---
12 12 10 10
p -5- – -1- q 1----3- – --4--- r 5-- – -1-
44 10 10 88 x --9---5--- + --9---5--- y --6---1--- + --8---9---
100 100 100 100
u --5--- + --9--- v -4- + 3-- w -3- + 1--
12 12 55 44

13APPENDIX B

3 Where there are only ‘+’ and ‘−’, work from left to right.

a --2--- + --1--- + --3--- b --7--- + --2--- + --1--- c --8--- + --5--- + --1--- d --6--- + --3--- + --1---
10 10 10 10 10 10 10 10 10 10 10 10

e --9--- – --2--- – --1--- f --8--- – --3--- – --4--- g --1---7--- – ----8---- – ----3---- h 7-- – 1-- – 3--
10 10 10 12 12 12 100 100 100 888

i 1----1- + --6--- – --9--- j --9--- + --4--- – --7--- k 7-- – 5-- + 3-- l -4- – 3-- + -2-
15 15 15 10 10 10 888 555

B:09 | Multiplying a Whole Number

by a Fraction

5 × -3- means ‘5 lots of 3-- ’ or 3-----+-----3-----+-----3----+-----3-----+-----3- or 5-----×-----3--
4 44 4

worked examples

1 8 × --3--- = -8----×-----3- 2 10 × -2- = 1----0----×-----2-- means Simplify
10 10 55 ‘lots of’ your

(eight lots of --3--- ) (ten lots of 2-- ) answers.
10 5

= 2----4- = 2----0-
10 5

= 2 --4--- =4

10

= 2 2--

5

Exercise B:09

1 Give the simplest answer for:

a 2 × --3--- b 3 × -1- c 7 × ----3---- d 7 × --2--- e 3 × --3---

10 4 100 15 10

f 3 × --1--- g 6 × --1--- h 8 × ----9---- i 5 × 1-- j 11 × ----7----

12 10 100 8 100

k 3 × --7--- l 7 × 3-- m 2 × -3- n 5 × -3- o 7 × --3---1---

10 4 5 4 100

p 8 × --7--- q 9 × -5- r 6 × --9--- s 7 × 4-- t 10 × -3-

10 8 10 5 8

u 11 × -7- v 7 × --2---1--- w 8 × -3- x 100 × 3-- y 9 × --8---

8 100 4 4 15

B:10 | Finding a Fraction of a Number

To find -7- of a number, find -1- of the number and then multiply by 7.

88

(To find -1- of a number, divide it by 8.)

8

14 INTERNATIONAL MATHEMATICS 1

1 -3- of 100 = ٗ worked examples ‘170 × 30’
and
5 2 --7--- × 30 = ٗ
‘170 of 30’
1-- of 100 = 100 ÷ 5 10 mean the

5 --1--- of 30 = 30 ÷ 10 same.

= 20 10

-3- of 100 = 3 × 20 =3

5 --7--- of 30 = 7 × 3

= 60 10

= 21

Exercise B:10

1 a 1-- of 20 b -1- of 66 c -1- of 30 d 1-- of 24 e --1--- of 90

4 6 5 8 10

f -3- of 20 g -5- of 66 h -4- of 30 i 3-- of 24 j --7--- of 90

4 6 5 8 10

k --3--- of 200 l 3-- of 40 m -5- of 16 n --3--- of 100 o 2-- of 15

10 4 8 10 3

p --1--- of 80 q 1-- of 60 r 1-- of 66 s ----3---- of 600 t --7--- of 100

10 5 3 100 20

u 1-- × 40 v 1-- × 24 w --3--- × 30 x -2- × 60 y -7- × 8

8 4 10 3 8

Decimals and Money

B:11 | Conversions: Fractions and Decimals

The examples below show the connection between fractions and decimals.

worked examples

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

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

point point

■ If a fraction has a denominator of 10, 100, 1000, etc, it can

easily by changed to a decimal. If the fraction has some other In section 6:08, we
do this another way.
denominator, it may be possible to change the denominator

using equivalent fractions, eg -3- = -3- × 25 = --7---5--- = 0·75.
4 4 × 25 100

Exercise B:11

1 Write these fractions as decimals.

a --1--- b --2---3--- c --9--- d ----9---- e --8--- f --2---5---

10 100 10 100 10 100

15APPENDIX B

g -----1------ h --3--- i ----6---7---- j --2--- k --1---3---5--- l --9---5---

1000 10 1000 10 1000 100

m --6---3--- n -----6------ o --4--- p --3---1--- q ----1---1---- r --5---

100 1000 10 100 1000 10

s --9---9---9--- t --4---8--- u --3---7---5--- v --6--- w --6---8--- x --8---4---

1000 100 1000 10 100 100

2 Write these mixed numbers as decimals. f 7 --3---

a 1 --7--- b 3 --4---3--- c 6 ----3---- d 4 --4--- e 9 ----4---- 10

10 100 100 10 100 l 20 ----2----

g 2 --4---1---2--- h 5 ----1---- i 8 ----9---6---- j 10 ----7---- k 12 --7---1--- 100

1000 100 1000 100 100 r 41 --1---4---7---

m 79 --8---7---5--- n 14 ----8---- o 15 --7---3--- p 9 ----2---2---- q 5 ----5---- 1000

1000 100 100 1000 100 3·25 = 312050
= 341
3 Write these decimals as fractions or mixed numbers in simplest form.

a 0·1 b 0·57 c 0·3 d 0·03 e 0·9

f 0·2 g 0·42 h 0·81 i 0·8 j 0·729

k 9·61 l 11·03 m 8·107 n 26·5 o 73·6

p 0·07 q 5·04 r 7·9 s 0·632 t 16·25

u 0·805 v 1·125 w 0·346 x 75·2 y 94·625

B:12 | Comparing Decimals

When comparing two decimals, you can place zeros at the end of one so that the two
decimals have the same number of decimal places. This makes them easier to compare.

worked examples

1 Which is smaller? b 0·5, 0·49 2 Put in order, smallest to largest:
a 0·11, 0·2 0·12, 0·8, 0·5

Solutions

1 a 0·2 = 0·20 b 0·5 = 0·50 2 0·8 = 0·80, 0·5 = 0·50

∴ 0·11 is smaller ∴ 0·49 is smaller ∴ in order we have

than 0·2 (since than 0·5 (since 0·12, 0·5, 0·8 (since 12 is less than 50,

11 is less than 20). 49 is less than 50). which is less than 80).

Exercise B:12

1 By referring to the number line below, say which is smaller.

0 0·5 1 1·5 2 2·5 3 3·5 The decimal to the
left on the number line
a 0·6, 0·9 b 0·5, 0·1 c 1, 0·9 d 2·5, 1·8
e 1·1, 1 f 3·4, 2·8 g 0·9, 2·2 h 2·4, 2·1 is smaller.
i 0·2, 0·1 j 2, 0·3 k 3·5, 2·9 l 0·8, 1·6

16 INTERNATIONAL MATHEMATICS 1

2 Which is smaller? ■ 0·1 = 0·10
0·2 = 0·20
a 0·8, 0·3 b 0·93, 0·15 c 0·61, 0·51 d 0·09, 0·15 0·3 = 0·30
g 0·9, 0·33 h 0·09, 0·6 0·4 = 0·40
e 0·8, 0·14 f 0·22, 0·3 k 0·5, 0·88 l 0·2, 0·19 0·5 = 0·50
0·6 = 0·60
i 0·03, 0·2 j 0·4, 0·39 0·7 = 0·70
0·8 = 0·80
3 Put in order, smallest to largest: 0·9 = 0·90

a 0·2, 0·5, 0·3 b 0·8, 0·1, 0·3 c 0·9, 0·5, 0·2
f 0·1, 0·09, 0·08
d 0·3, 0·41, 0·21 e 0·33, 0·5, 0·4 i 0·6, 0·3, 1
l 1·84, 1·9, 1·79
g 0·6, 0·55, 0·3 h 0·5, 1, 0·7

j 0·31, 0·4, 1 k 1, 0·39, 0·4

B:13 | Adding Decimals

The column in which a figure is placed determines its size. Therefore in adding (or subtracting)
decimals, we must be sure to put figures in the same column, under one another.

To add (or subtract) decimals use the PUP rule: Place Points Under Points.
An empty space can be filled by a zero.

5·85

0·60 ■ 0·6 = 0·60 ‘PUP’

7·00 7 = 7·00 Point

Under
· Point

worked examples

1 2·74 + 0·4 2 5·18 + 3 3 7·2 + 16 + 4·1
=2·74 = 5·18 = 7·2
+ 01· 4 0 + 3·00 16·0
14·1
3·14 8·18
27·3

Exercise B:13

1 a 3·8 b 9·15 c 16·03 d 9·7
+ 6·3 + 8·31 + 9·97 + 18·6

e 8·2 f 18·31 g 19·9 h 47·8
0·6 9·45 8·6 11·3
1·8 3·8 9·5
17·88
i 8·65 k 8·83 l 3·4
1·6 j 16·3 7.5 9.18
9·3 9·15
12·08 16·02
10·9

17APPENDIX B

m 8·06 n 4·9 o 10 p 23·8
0·9 5 9·04 4·99
0·04 6·15 7·95 6

2 a 0·15 + 0·58 b 3·5 + 9·6 c 23·1 + 9·8
d 6·4 + 10·6 e 0·8 + 2·5 + 1·8 f 3·8 + 10·6 + 9·1
g 0·85 + 1·15 + 0·65 h 0·95 + 8·75 + 1·11 i 2·4 + 12
j 6 + 0·6 k 5 + 0·5 l 8·2 + 8·33
m 8·15 + 6 + 1·8 n 6·7 + 7·15 + 4 o 8·15 + 0·8 + 0·08
p 0·9 + 9 + 0·09 q 60 + 9·3 + 0·7 r 18·4 + 3·21 + 10
s 0·06 + 6 + 6·66 t 8·3 + 9·81 + 2·4 u 3·85 + 1·4 + 1·85
v 6·21 + 9·9 + 1·86 w 5·1 + 6 + 2·85 x 9·35 + 8·9 + 6·95

B:14 | Subtracting Decimals

To subtract (or add) decimals use the PUP rule: Place Point Under Points.
An empty space can be filled by a zero.

worked examples

1 18 − 9·18 2 9·2 − 1·25 9·2 = 9·20 6·0 = 6 8·50 = 8·5
= 18·00 = 9·20
− 9·18 − 1·25 5 = 5·0

8·82 7·95 Subtraction: 10·00 = 10
18 = 18·00 point under point.

Exercise B:14

1 a 14·95 b 90·8 c 48·5 Fill the
− 9·21 − 49·3 − 28·9 empty
spaces
d 9·65 e 19·85 f 23·4 with zeros.
− 3·4 − 1·6 − 1·5
m5
g 12·6 h 8·5 i 19·8 − 4·1
− 4·24 − 3·15 − 3·85
q 160·05
j 15 k 10 l6 − 45
− 3·15 − 4·8 − 3·9
d 6·92 − 1·08
n 16·4 o 8·04 p 63·18 h 91·45 − 6·7
− 3·89 − 6·4 − 9·5 l 6·8 − 0·37
p 15 − 1·8
2 a 18·6 − 3·4 b 23·41 − 8·62 c 4·95 − 0·85 t 23·8 − 0·09
e 45·83 − 9·1 f 6·85 − 5·8 g 43·6 − 13
i 8·4 − 0·35 j 10·4 − 3·85 k 8·9 − 4·15
m 9 − 0·3 n 6 − 1·25 o 9 − 0·08
r 6·08 − 1·6 s 19·45 − 6·3
18 q 20·9 − 1·35
INTERNATIONAL MATHEMATICS 1

u 76 − 19·8 v 154 − 3·72 w 9·08 − 3 x 17·85 − 1·9

3 a Mrs Leong bought 9·5 metres of ribbon. This was cut from a roll that had 32·8 metres of
ribbon on it. How much was left?

b Mrs Hooker owned 80 hectares of land in Orange. She sold 8·7 hectares as a primary school
site. How many hectares did she have left?

c Brian began to train in order to lose weight. He reduced his original weight of 86 kilograms
by 9·2 kilograms. What did he weigh in the end?

d I am 37 km from Parkes post office. If I live 3·2 km this side of the post office, how far am I
from home?

B:15 | Multiplying a Decimal by a

Whole Number

1 1·6 × 5 worked examples 8·0
=8
Solutions 2 0·181 × 3
3·00
1 1·6 2 0·181 =3
×5 ×3
1·50
8·0 0·543 = 1·5

One figure Three figures
after the point after the point

Exercise B:15

1 a 1·8 b 3·7 c 2·4 d 5·9
×3 ×5 ×6 ×4

e 0·6 f 1·31 g 0·9 h 2·14
×7 ×4 ×5 ×3

i 0·152 j 1·82 k 9·31 l 1·811
×3 ×4 ×2 ×6

2 a 0·5 × 9 b 0·6 × 8 c 0·7 × 6 d 0·4 × 9
e 0·8 × 8 f 0·6 × 9 g 0·7 × 9 h 0·9 × 9
i 0·11 × 6 j 0·97 × 3 k 0·85 × 5 l 0·91 × 3
m 7·2 × 4 n 3·6 × 5 o 8·5 × 8 p 1·4 × 9
q 6·04 × 7 r 9·09 × 2 s 3·04 × 6 t 8·05 × 8

3 a For one dollar Luke can buy 3·2 metres of hose. How many metres of hose can he buy for
five dollars?

b In one second Alana can walk 1·4 metres. How far would she walk in 8 seconds?
c Her younger sister, Naomi, can walk 1·2 metres in one second. How far would she walk in

8 seconds?
d If Alana and Naomi started at the same place and walked in opposite directions for 8 seconds,

how far apart would they be?

19APPENDIX B

B:16 | Dividing a Decimal by a

Whole Number

0·6 0·6 0·6 0·6 0·6 × 4 so 0·6
2·4 = 2·4
4 ) 2·4

When dividing a decimal by a whole number, place the point in the answer above the
point in the question.

worked examples

1 6·2 0· 3 0·0 4 1 ■ In each case we have
kept Point Under Point.
1 3 ) 418·6 2 6 ) 1·18 3 3 ) 0·112 3

Exercise B:16

1 a 5 ) 15·5 b 4 ) 8·52 c 3 ) 7·11 d 6 ) 37·2

e 8 ) 0·96 f 5 ) 5·05 g 4 ) 38·4 h 7 ) 15·4

i 2 ) 2·16 j 9 ) 2·79 k 5 ) 5·45 l 6 ) 6·24

2 a 0·72 ÷ 3 b 0·303 ÷ 3 c 0·726 ÷ 2 d 0·145 ÷ 5
e 4·88 ÷ 8 f 12·04 ÷ 4 g 45·6 ÷ 6 h 6·72 ÷ 6
i 4·56 ÷ 8 j 1·82 ÷ 7 k 9·27 ÷ 9 l 74·9 ÷ 7
m 41·64 ÷ 4 n 10·12 ÷ 2 o 15·05 ÷ 7 p 56·79 ÷ 9

3 a Rachel walked 10·5 metres in 7 seconds. If she walked at a constant speed, how far did she
travel in one second?

b Three brothers inherited 54·6 hectares of land. If the land was divided equally, how many
hectares did each inherit?

c Six tug-of-war ropes of equal length are to be made from a length of rope 57·6 metres long.
How long will each rope be?

d Each member of a four-girl relay team completed her lap in the same time. If the four laps
took 137·2 seconds, how long did each swimmer take?

B:17 | Multiplying and Dividing Decimals

by 10

Investigation

1 0·8 × 10 = 8·0 2 3·05 × 10 = 30·50 Zeros at the end after the
=8 = 30·5 decimal point can be put in

3 0·00125 × 10 = 0·01250 or taken off as you like
eg 305·00 = 305.
20 0·00125 × 10 = 0·0125

0·08 0·18 5·68

4 10 ) 0·80 5 10 ) 1·80 6 10 ) 56·80

From the examples above we see that:
• when we multiply a decimal by 10, we move the decimal point 1 place to the right
• when we divide a decimal by 10, we move the decimal point 1 place to the left

worked examples

1 0·6 × 10 2 34·180 × 10 3 0·013 × 10 4 0·035 × 10
=6 = 341·8 = 0·13 = 0·35

5 30·7 ÷ 10 6 56· ÷ 10 7 0·045 ÷ 10 8 845·6 ÷ 10
= 3·07 = 5·6 = 0·0045 = 84·56

Exercise B:17

1 a 0·5 × 10 b 0·37 × 10 c 0·9 × 10 d 0·08 × 10
e 3·85 × 10 f 0·815 × 10 g 17·125 × 10 h 0·071 × 10
i 0·91 × 10 j 8·5 × 10 k 0·01 × 10 l 2·375 × 10
m 0·005 × 10 n 15·8 × 10 o 1·382 × 10 p 7·8 × 10

2 a 3·8 ÷ 10 b 52 ÷ 10 c 78·6 ÷ 10 d 84·15 ÷ 10
e 0·04 ÷ 10 f 108·5 ÷ 10 g 0·74 ÷ 10 h 15·7 ÷ 10
i 7·2 ÷ 10 j 8 ÷ 10 k 0·7 ÷ 10 l 6·7 ÷ 10
m 0·125 ÷ 10 n 635 ÷ 10 o 3·6 ÷ 10 p 845·1 ÷ 10

B:18 | Decimal Currency

The rules for decimals also apply to money, eg 5 dollars 63 cents = $5.63.

worked examples

1 $201 + $1.20 2 $50 − $3.50 3 $1.85 × 6 4 $70 ÷ 8

Solutions 2 $50.00 3 $1.85 $ 8. 7 5
− 3.50 ×6
1 $201.00 4 8 ) $7 0.6040
+ 1.20 $46.50 $11.10

$202.20

21APPENDIX B

Exercise B:18

1 Write each amount as dollars using a decimal.

a 8 dollars 15 cents b 16 dollars 50 cents c 146 dollars 95 cents
f 2 dollars 35 cents
d 37 dollars 15 cents e 93 dollars 5 cents i 18 dollars 65 cents

g 100 dollars 50 cents h 40 dollars 60 cents

2 a $1.45 + $6.50 b $6.80 + $9.20 c $3.45 + $6.15 d $2.90 + $9.55
e $20 + $1.65 f $3.85 + $25 g $14.30 + $8 h $165 + 19.35
i $36.40 j $84.00 k $145.30 l $29.40
8.35
11.35 9.15 9.45 56.85
0.60 63.40 16.80 17.00
101.75 67.95
6.90

3 a $8.70 b $10.00 c $7.60 d $76.00
− 3.45 − 7.65 − 5.45 − 9.40

e $8.60 − $3.15 f $9.85 − $3.15 g $6.50 − $6.35 h $8.75 − 4.80
i $20 − $1.35 j $40 − $23.40 k $50 − $7.30 l $100 − $40.70
m $465 − $71.15 n $96.10 − $9.95 o $30 − $27.95 p $63.80 − $36.45

4 a $4.05 × 3 b $3.15 × 4 c $6.05 × 2 d $7.25 × 5
e $7.45 × 7 f $8.60 × 9 g $3.85 × 6 h $9.35 × 8
i $17.00 × 6 j $18.00 × 5 k $95.00 × 4 l $103.00 × 6
m $16.70 × 8 n $78.40 × 9 o $37.45 × 7 p $103.00 × 11

5 a $8.25 ÷ 3 b $9.30 ÷ 2 c $8.00 ÷ 5 d $3.80 ÷ 4
e $18.45 ÷ 9 f $36.00 ÷ 6 g $40.40 ÷ 8 h $63.70 ÷ 7
i $20 ÷ 8 j $21 ÷ 6 k $52 ÷ 4 l $81 ÷ 4
m $136.20 ÷ 6 n $43.40 ÷ 7 o $736.00 ÷ 80 p $30 ÷ 8

Percentages

B:19 | Changing Percentages to Fractions

worked examples

1 7% = ----7---- 2 53% = --5---3--- % 100
100 100

(7 for every 100) (53 for every 100)

3 90% = --9---0--- ÷ 10 4 64% = --6---4--- ÷4
100 ÷ 10 100 ÷4

= --9--- = 1----6- ■ % has a ‘one’
10 25 and two ‘zeros’.

22 INTERNATIONAL MATHEMATICS 1

Exercise B:19

1 Write these as hundredths:

a 13% b 1% c 5% d 3% e 11%
h 2% i 10% j 77%
f 9% g 15% m 58% n 14% o 30%
r 8% s 50% t 81%
k 97% l 4% w 43% x 6% y 29%

p 63% q 35% e 7%
j 5%
u 90% v 12% o 76%
t 80%
2 Write each percentage as a fraction in its simplest form. y 16%

a 10% b 50% c 2% d 4%

f 8% g 25% h 18% i 64%

k 85% l 52% m 60% n 75%

p 24% q 15% r 28% s 30%

u 46% v 66% w 92% x 55%

B:20 | Changing Percentages to Whole

and Mixed Numbers

To write a percentage as a fraction or mixed number, write it as a fraction and simplify.

worked examples

1 100% = -1---0---0- 2 400% = -4---0---0- 100% = 1
100 100 so 125%
must be greater
=1 =4 than 1.

3 125% = -1---2---5- 4 306% = 3----0---6-
100 100

= 1 1----2---5- ÷ 25 = 3 ----6---- ÷ 2
100 ÷ 25 100 ÷ 2

= 1 -1- = 3 --3---

4 50

Exercise B:20

1 Change these percentages to whole or mixed numbers in simplest form.

a 200% b 500% c 100% d 300% e 600%
j 350%
f 150% g 170% h 250% i 220% o 185%
t 478%
k 106% l 138% m 175% n 144%
23APPENDIX B
p 410% q 635% r 365% s 880%

2 Write each as a fraction, whole number or mixed number in simplest form.

a 173% b 84% c 375% d 450% e 66%

f 800% g 550% h 425% i 302% j 900%

k 205% l 700% m 181% n 980% o 4%

p 623% q 475% r 16% s 111% t 1000%

B:21 | Changing Fractions and

Mixed Numbers to Percentages

To change fractions to percentages, first change the denominator of the fraction to 100.

worked examples

1 --3---7--- = 37% 2 --3--- = --3--- ×5 ■ 1 = 100% Know
100 20 20 ×5 ∴ 7 = 700% these
(% means facts.
‘per 100’) = --1---5---
100

= 15%

3 4 = 4-- × 100 4 7 --3--- =7+ --3--- × 10
1 × 100 10 10 × 10

= 4----0---0- = -7---0---0- + --3---0--- ■ 1 = 100%
100 100 100
∴ -1- = 50%
= 400% = 730%
2

and -1- = 25%

4

Exercise B:21

1 Write each as a percentage.

a ----7---- b --5---0--- c --1---8--- d --7---3--- e ----8----

100 100 100 100 100

f --9---0--- g --7---7--- h --2---0--- i --9---9--- j --4---0---

100 100 100 100 100

k --6---7--- l --8---0--- m ----1---- n --1---9--- o --2---2---

100 100 100 100 100

p --8---7--- q --7---5--- r --3---0--- s --2---5--- t ----3----

100 100 100 100 100

2 Write each as a percentage. c --9--- d --1--- e --7---

a -1- b --3--- 10 10 10

2 10 h -1- i 1----1- j -4-

f -2- g --3--- 5 20 5

5 20 m --8--- n -3- o 2----4-

k --3--- l 1-- 25 4 25

50 4 r 1----7- s --8--- t 2----1-

p --2--- q 3-- 50 10 50

25 5

24 INTERNATIONAL MATHEMATICS 1

3 Change each to a percentage.

a1 b3 c5 d2 e8

f 1 -1- g 3 -1- h 4 1-- i 6 1-- j 7 1--

2 2 2 2 2

k 2 --3--- l 5 --7--- m 6 --1--- n 1 --3--- o 2 --9---

10 10 10 10 10

p 1 -1---3- q 3 -1- r 5 --7--- s 2 -3- t 6 --3---

50 4 20 5 25

B:22 | Changing Percentages to Decimals

To change a percentage to a decimal, we can write it first as hundredths and then write
it as a decimal.

worked examples

1 7% = ----7---- 2 93% = --9---3--- Very interesting!

100 100

= 0·07 = 0·93

3 10% = --1---0--- 4 425% = 4----2---5- hundredths column

100 100 13% = 0·13 13 ÷ 100 = 0·13
150% = 1·50 150 ÷ 100 = 1·50
= 0·10 = 4 --2---5---
4% = 0·04 4 = 100 = 0·04
= 0·1 100

= 4·25

Exercise B:22

1 Write each percentage as a decimal.

a 6% b 5% c 2% d 9% e 1%
h 83% i 65% j 88%
f 17% g 24% m 90% n 60% o 40%
r 70% s 11% t 95%
k 20% l 50%
d 86% e 427%
p 43% q 3% i 342% j 886%
n 390% o 180%
2 Express each percentage as a decimal. s 100% t 106%

a 112% b 97% c 204%

f 114% g 293% h 147%

k 250% l 160% m 420%

p 5% q 200% r 193%

25APPENDIX B

B:23 | Changing Decimals to Percentages

To change a decimal to percentage, we can write it first as a fraction, then change it to
a decimal, or we can multiply the decimal by 100%.

1 0·08 = ----8---- worked examples 0·3 = 0·30
=30%
100 2 5·9 = 5·90
= -5---9---0- 0·4 = 40%
= 8%
100 0·3 = 30%
3 0·93 = --9---3---
= 590% 0·2 = 20%
100
OR 4 0·93 = 0·9 3 × 100% 0·1 = 10%
= 93% = 93%

■ When multiplying by 100, move
the decimal point 2 places right.

Exercise B:23

1 Write each decimal as a percentage.

a 0·15 b 0·33 c 0·92 d 0·52 e 0·38
h 0·19 i 0·25 j 0·75
f 0·83 g 0·55 m 0·08 n 0·09 o 0·01
r 0·99 s 0·42 t 0·21
k 0·03 l 0·05
c 0·9 d 0·8 e 0·2
p 0·67 q 0·86 h 2·64 i 2·95 j 3·24
m 2·61 n 1·93 o 3·88
2 Change to percentages. r 3·1 s 1·3 t 2·6

a 0·4 b 0·6 c 0·09 d 0·28 e 1·62
h 0·32 i 0·84 j 1·97
f 1·77 g 1·06 m 0·7 n 0·54 o 0·98
r 0·03 s 0·2 t 0·78
k 1·02 l 4·34

p 1·7 q 2·5

3 Express each as a percentage.

a 0·4 b 1·53

f 1·11 g 0·6

k 0·07 l 1·46

p 2·69 q 0·72

26 INTERNATIONAL MATHEMATICS 1

Appendix B | Working Mathematically assignment

1 A boy is playing with a pile of matches. He notices that when he divides them into piles of B
2, 3 or 5 there is always one left over. What is the smallest number of matches that he
could have?

2 A domino is made by joining two squares and placing a number of dots from 0 to 9 in each
square so that each domino is different. How many dominoes are there in a complete set?

These are
the same
domino.

3 A girl has a box containing a total of 8 spiders and beetles. She counted the legs and foundMillions of tonnes
that there were 54 altogether. Find how many of each were in the box. (You will need to
know that spiders have 8 legs and beetles have 6 legs.)

4 What is the largest area that could be covered by a rectangle with a perimeter of 40 cm?
5 Jason cut an apple into 20 equal parts. If he ate -1- of the apple, how many parts remained?

4

6 Exports for Hinesland

12

10

8

6

4

2

2000 2001 2002 2003
Coal Steel

a What amount of coal was exported in 2000?
b What amount of steel was exported in 2003?
c In which year was the smallest amount of steel exported?
d In which year was the largest amount of coal exported?
e What was the average amount of steel exported for the four-year period? (Answer to the

nearest million tonnes.)

27APPENDIX A

APPENDIX C
C:01 | HCF and LCM by Prime Factors

(Extension)

ep quiz 1 List all factors of 30. 2 List all factors of 54.

pr C:01 3 What is the highest common factor of 30 and 54?

4 List all multiples of 4 that are less than 40.

5 List all multiples of 3 that are less than 40.

6 What is the lowest common multiple of 4 and 3?

7 Use a factor tree to write 48 as a product of its prime factors.

Write the following products of prime numbers in index notation.

8 2×2×2×2×2 9 2×2×2×3×3×3×3

10 3 × 3 × 5 × 5 × 5 × 7

When we express numbers as products of their prime factors, it is sometimes easier to find their
highest common factor (HCF) and their lowest common multiple (LCM).

worked example 1

Find the highest common factor of 400 and 1080.

Solution 1080

400

20 × 20 20 × 54

4×5 5×4 4×5 6×9

2×2×5×5×2×2 2×2×5×2×3×3×3

From the factor trees:

Number Product of prime factors

400 2 × 2 × 2 × 2 × 5 × 5
or 24 × 52

1080 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 = 40, so the highest common factor of 400 and 1080 is 40.

28 INTERNATIONAL MATHEMATICS 1

worked example 2

Find the lowest common multiple of 400 and 1080. 2 × 2 × 2 × 3 × 3 × 3 × 5 or 23 × 33 × 5

Solution

We write each number as a product of its primes:

400 2 × 2 × 2 × 2 × 5 × 5 or 24 × 52 1080

LCM = (2 × 2 × 2 × 2 × 5 × 5) × (3 × 3 × 3) These are the prime factors of
1080 not already
This is 400 written written in the
as a product of first part.
its primes.

LCM = 24 × 33 × 52 = 10 800, so the lowest common multiple of 400 and 1080 is 10 800.

worked example 3

Find the highest common factor and the lowest common multiple of 12 150 and 39 375, if
12 150 = 2 × 35 × 52 and 39 375 = 32 × 54 × 7.

Solution 3 × 3 × 5 × 5 is contained in both,
∴ the HCF is 32 × 52 or 225.
12 150 = 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5

39 375 = 3 × 3 × 5 × 5 × 5 × 5 × 7

2 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 × 5 × 7 contains both numbers,
∴ the LCM is 2 × 35 × 54 × 7 or 2 126 250.

The highest common factor of 12 150 and 39 375 is 225 and the lowest common multiple of
these two numbers is 2 126 250.

Exercise C:01

1 Use this table to answer the questions.

Number Products of prime factors a Find the highest common
factor of:
144 2 × 2 × 2 × 2 × 3 × 3 24 × 32 i 144 and 324
ii 324 and 1890
324 2 × 2 × 3 × 3 × 3 × 3 22 × 34 iii 1890 and 4900
iv 4900 and 1960
1890 2 × 3 × 3 × 3 × 5 × 7 2 × 33 × 5 × 7 v 1960 and 3375
vi 3375 and 8232
4900 2 × 2 × 5 × 5 × 7 × 7 22 × 52 × 72 vii 8232 and 1568
viii 1960 and 8232
1960 2 × 2 × 2 × 5 × 7 × 7 23 × 5 × 72 ix 1890 and 3375
x 1568 and 3375
3375 3 × 3 × 3 × 5 × 5 × 5 33 × 53
29APPENDIX C
8232 2 × 2 × 2 × 3 × 7 × 7 × 7 23 × 3 × 73

1568 2 × 2 × 2 × 2 × 2 × 7 × 7 25 × 72

b Find the lowest common multiple of:

i 144 and 324 ii 324 and 1890 iii 1890 and 4900 iv 4900 and 1960
vii 8232 and 1568 viii 1960 and 8232
v 1960 and 3375 vi 3375 and 8232

ix 1899 and 3375 x 1568 and 3375

2 Complete the table below and then answer the questions.

Number Products of prime factors a Find the highest common
factor of:
18 i 18 and 36
ii 36 and 24
36 iii 24 and 300
iv 300 and 1050
24 v 1250 and 1050
vi 2475 and 1250
300 vii 2475 and 2310
viii 1050 and 2475
1050 ix 1250 and 2310
x 24 and 2475
1250

2475

2310

b Find the lowest common multiple of:

i 18 and 36 ii 36 and 24 iii 24 and 300 iv 300 and 1050
vii 2475 and 2310 viii 1050 and 2475
v 1250 and 1050 vi 2475 and 1250

ix 1250 and 2310 x 24 and 2475

3 a Find the highest common factor of 1155 and 2079.
b Find the lowest common multiple of 1155 and 2079.
c What is the highest common factor of 264 and 1386?
d What is the lowest common multiple of 264 and 1386?

4 a Two people are jogging around an oval.

ch allenge They start together and one takes 168 seconds These questions have
to complete exactly one lap while the other something to do
with LCMs.
takes 189 seconds. How long after they start

will it take before they again meet at the

C:01 starting point?

b Two cannons are fired together, then one is

fired every 72 minutes while the other is fired

every 108 minutes. How long after the first shot

will they again be fired together?

c Judy is told that she may purchase chairs for

$44 each and tables for $231 each, as long as

she pays exactly the same amount for chairs

as for tables. What is the least amount she

needs to spend to take advantage of

these prices?

30 INTERNATIONAL MATHEMATICS 1

APPENDIX D
D:01 | Finite Differences (Finding the Rule)

worked example

Using the pronumerals shown, find the rule I really have trouble
that would give the following table of values. finding the rule
for these.
x1 2345
y 9 16 23 30 37 This method
usually works.

Solution 7 Step 1 Make sure the values of x increase by just one at a time.
7
xy 7 Step 2 Find the difference between the values for y.
19 7 (16 − 9 = 7, 23 − 16 = 7, 30 − 23 = 7, 37 − 30 = 7)
2 16
3 23 Step 3 The difference is always 7, so the rule will take the form
4 30 y = 7 × x + ∆ or y = 7 × x − ∆.
5 37
Step 4 Since y = 9 when x = 1, 9 = 7 × 1 + ∆.
So ∆ = 2.
∴ The rule is y = 7 × x + 2.

■ The difference
between the two
numbers

Discussion

If we are given a rule, we can test it to see if the difference between y values remains the same.

1 y=3×x+4 2 y=6×x+8 3 y=2×x−1

x y difference x y difference x y difference

17 3 1 14 6 11 2
2 10 3 2 20 6 23 2
3 13 3 3 26 6 35 2
4 16 4 32 47

Conclusion: Rules like y = 3x + 4 have equal differences between successive y values.

31APPENDIX D

Now let’s test other rules.

4 y = x2 5 y = 12 ÷ x 12 Ϭ x
can be written as
x y difference x y difference
12
11 3 1 12 6 x
24 5 26 2
39 7 34 1
4 16 43

Conclusion: Be careful! Some rules do not have equal differences between successive
y values.

Exercise D:01

1 For each of the rules below, find the differences between successive y values.

a y=4×x−4 b y=9×x+5 c y = 11 × x − 10

x y difference x y difference x y difference

10 ٗ 05 ٗ 11 ٗ
24 ٗ 1 14 ٗ 2 12 ٗ
38 ٗ 2 23 ٗ 3 23 ٗ
4 12 3 32 4 34

d y = 3x + 2

x y difference This side
looks shorter!
15 ٗ
28 ٗ 3x = 3 ϫ x
3 11 ٗ 7x = 7 ϫ x
4 14

e y = 7x − 12

x y difference

22 ٗ
39 ٗ
4 16 ٗ
5 23

f y = 2x − 5 g y = 5x + 9

x3456 x 1234

y1357 y 14 19 24 29

difference ٗ ٗ ٗ difference ٗٗٗ

2 Use the rule given to find the y values and the differences.

a y = 2x + 1 b y = 5x − 5 c y = 3x − 2

x y difference x y difference x y difference

32 1ٗ ٗ 1ٗ ٗ 1ٗ ٗ
2ٗ ٗ 2ٗ ٗ 2ٗ ٗ
3ٗ ٗ 3ٗ ٗ 3ٗ ٗ
4ٗ 4ٗ 4ٗ

d y = 6x + 3 e y = 10x − 3
x0123 x 1234
y y

difference ٗ ٗ ٗ difference ٗ ٗ ٗ

3 Find the differences between successive y values and use the differences to discover the rule.

ax 1234 bx 2 345

y 11 15 19 23 y 6 13 20 27

difference ٗ ٗ ٗ difference ٗ ٗ ٗ
Rule: y = ٗ x + 7 Rule: y = ٗ x − 8

cx012 3 Once you have
y 4 6 8 10
ofound the ,you
difference ٗ ٗ ٗ ocan find the ,
Rule: y = ٗ x + ∆
since y = 4 and
x = 0.

dx 1 2 3 4 ex 0 1 2 3
y 6 10 14 18 y5678

difference ٗ ٗ ٗ difference ٗ ٗ ٗ

Rule: y = ٗ x + ∆ Rule: y = ٗ x + ∆

4 Use the method of finding differences to discover the rule for each table below.

a x123 4 b x234 5 c x1234

y 3 6 9 12 y 7 8 9 10 y3579

d a01 23 e a3 456 f a123 4
b 7 9 11 13 b 5 10 15 20 b 0 4 8 12

g x 1234 h x23 45 i x0 123
y 10 17 24 31 y 1 7 13 19 y 8 18 28 38

j s 10 11 12 13 k s 5678 l s 4567

m 0369 m 13 15 17 19 m 26 31 36 41

5 Farmer Powell’s water was getting low because of the drought, so

he had water delivered in a truck. As the water was being pumped

from the truck into the tank, he measured the height of water in

the tank. After 1 minute the height of water was 23 cm. After 2

minutes it was 38 cm. After 3 minutes it was 53 cm. After 4 minutes

it was 68 cm. Draw a table of values showing the connection between

minutes passed (m) and height of water (H). Find the rule that gives

the height (H) of the water in the tank as the minutes (m) pass.

6 Let’s consider what happens when the values of x do not increase by one at a time.

a y = 2x + 5 b y = 3x − 4

x1357 x2468

yy

difference ٗ ٗ ٗ difference ٗ ٗ ٗ 33

Can you explain why the difference in each does not equal the number multiplying the x?

APPENDIX E
E:01 | Changing Repeating Decimals to

Fractions

Here is a clever shortcut method for writing a repeating decimal as a fraction. Follow the steps
carefully.

worked example

1 0·2˙ 6˙ = 2----6----–-----0- Step 1 (numerator) 2 0·327˙ = -3---2---7----–-----3---2-
99 Subtract the digits before the 900
repeating digits from all the digits. two 0s one 9
= 2----6- Step 2 (denominator)
99 Write down a 9 for each repeating = -2---9---5-
900

digit and then a zero for each non- = --5---9---
180
repeating digit in the decimal.
Step 3 That’s pretty
Simplify the fraction if possible. nifty!

3 8·60˙ 7˙ = 8-6---0---7----–-----6-
990

one two 9s
0

= 86----0---1-
990

Exercise E:01

1 Convert these repeating decimals to fractions.

a 0·7˙ b 0·1˙ c 0·3˙ d 0·8˙
h 0·9˙
e 0·2˙ f 0·6˙ g 0·5˙ l 0·8˙ 2˙
p 0·3˙ 0˙
i 0·1˙ 5˙ j 0·8˙ 4˙ k 0·1˙ 2˙ t 0·04˙ 2˙
x 0·91˙ 0˙
m 0·7˙ 3˙ n 0·5˙ 8˙ o 0·0˙ 7˙
d 0·1256˙
q 0·14˙ 8˙ r 0·73˙ 2˙ s 0·18˙ 6˙ h 4·9˙
l 9·05˙ 2˙
u 0·06˙ 3˙ v 0·94˙ 7˙ w 0·25˙ 4˙

2 Convert these repeating decimals to fractions.

a 0·321˙ 7˙ b 0·450˙ 8˙ c 0·052˙ 8˙

e 1·6˙ f 7·0˙ 4˙ g 9·8˙ 6˙

i 8·30˙ 4˙ j 7·61˙ 5˙ k 3·87˙ 5˙

3 Why is 0·9˙ equal to 1?

34 INTERNATIONAL MATHEMATICS 1

APPENDIX F
F:01 | Investigation of Plane Shapes

Exercise F:01 Experimenting with shapes is
very interesting! But be
Equipment needed: square grid paper, scissors patient, the problems are
1 Copy the following figures onto a piece of not always easy.
square grid paper.

12 3

a Name the shapes.
b Cut the shapes out and, by moving them about,

trace them in different orientations.
c Make two of shape 2. What shapes can be made by joining the two together along an edge?
d Cut shape 3 along its longest diagonal and use the two triangular pieces to make a

parallelogram. Can more than one parallelogram be made?

2 On the remaining grid paper from question 1, draw two triangles
like the one shown. What shapes can be made if the two triangles
are joined along:
a the shortest edge? b the middle-sized edge?
c the longest edge?

3 On a piece of square grid paper, copy the figure shown and
cut it out.
a Name the figure.
b Cut the figure into a rectangle and two identical triangles.
c Use the two triangular pieces to make a rhombus.
d Use the two triangular pieces to make a parallelogram.
e Cut each of the triangular pieces in half and use the four triangular pieces to make
the rectangle.

35APPENDIX F

4 Mark out a 8 cm square on a piece of square grid 2 3
paper and carefully copy the shapes shown on the 4
right. Cut them out. These seven shapes form the 1
ancient Chinese puzzle called the tangram. 7 5
a Name the two shapes that are not triangles.
b Which two shapes make number 5? 6
c Which two shapes make number 6?
d Use three smaller pieces to make shape 1.
Can this be done in more than one way?
e Use pieces 3, 4 and 7 to make a parallelogram.
f Use pieces 3, 4 and 7 to make a trapezium.
g Use pieces 3, 4 and 7 to make a rectangle.
h Make a square with the five smallest pieces.
i Use the seven pieces of your tangram to make
each of the pictures on the right.

5 Draw a square of side 4 cm on a piece of square grid paper. Mark in the diagonals, then cut out
the square and cut along the diagonals. Use the 4 triangular pieces obtained to make the
following shapes.
a bc d

6 On a piece of square grid paper, draw a rectangle 6 cm long and 3 cm wide. Mark in the
diagonals, then cut out the rectangle and cut along the diagonals. Use the 4 pieces obtained to
make the following shapes.
abc

7 Make one square and four triangles like these The square in (c) And if it’s bigger,
drawn below. must be bigger than it must have a
the square in (b). different side
D CG Note: GE = BC length.
EF = 2 × AB

A BE F

a Use two of the triangles to make a rectangle.
b Use four triangles to make a square.
c Use the five pieces to make a square.

36 INTERNATIONAL MATHEMATICS 1

8 The diagram shows how two pieces are made from a square ABCD. E is the midpoint of the
side AB. Starting with any square ABCD, follow the steps shown to make the pieces shown in
step 4 of the diagram.

1A B 2 E B 3A E B 4A EE B

A

D C D C D C D CC

Using the two pieces make:

a a triangle b a trapezium c a parallelogram

d a quadrilateral that has all its sides different

e a pentagon that has a pair of parallel sides

9 The diagram shows how a square is dissected into three pieces. E and F are the midpoints of
the sides AB and BC respectively. Follow the steps in the diagram to make the three pieces
shown in step 4 of the diagram.

1A B 2A E B 3 4

F

D CD C

a Use the two larger pieces to make a triangle.
b Use the three pieces to make a triangle.
c Make a rectangle, parallelogram and trapezium from the three pieces.

10 A pentomino is formed by joining 5 identical squares together
so that each square is joined to the one alongside it by an edge.
Two of the 12 possible pentominoes are shown.
a Find the other ten pentominoes.
b Use three of the pieces to form a 5 × 3 rectangle.
c Use four of the pieces to form a 5 × 4 rectangle.

11 The word ‘congruent’ means ‘identical’. challenge
Divide the shape given into:
a 2 congruent shapes F:01
b 3 congruent shapes
c 4 congruent shapes

12 Repeat question 11 for this shape:

37APPENDIX F

fun spot Fun Spot F:01 | Hidden shapes

2:04 Many different shapes can be hidden in one design.

A shape may appear in any position, even flipped over.

How many times does each of the following shapes appear in the design above?
a bc d

38 INTERNATIONAL MATHEMATICS 1

F:02 | Plane Shapes and Patterns

Plane shapes are often used to make patterns. When a single shape is repeated to make a pattern,
leaving no gaps, the pattern is called a tessellation.

Examples

12

This pattern is not a tessellation as This pattern is a tessellation as one
two different shapes are used. shape is used to cover the entire surface.

Exercise F:02 (Practical)

Equipment needed: square grid paper, scissors, pencils and erasers

Pattern 1 Pattern 2 Pattern 3

1 Which of these patterns above is a tessellation?

2 a Name the shape used in pattern 1.
b Copy this pattern (8 units by 6 units) onto a piece of grid paper and extend it until it is
16 units by 12 units.

3 a Name the shapes used in pattern 2.
b Copy this pattern (9 units by 6 units) onto a piece of grid paper and extend it until it is
18 units by 12 units.

4 a Name the shapes used in pattern 3.
b Copy this pattern (8 units by 6 units) onto a piece of grid paper and extend it until it is
16 units by 12 units.

5 On a piece of square grid paper, use the shapes given below to make a tile pattern.

39APPENDIX F

6 In the pattern, find the

following shapes: I can’t see the shapes

a hexagon for the triangles!

b trapezium

c parallelogram

d rhombus

7 F Draw two squares and copy the figures shown.
The points A, B, C, D, E and F are midpoints of

A B the sides of the squares.
C E a In figure 1, find the following shapes:

i kite ii rhombus

b In figure 2, find and sketch the following shapes:

Figure 1 D i kite ii rhombus iii square
Figure 2
iv parallelogram v regular octagon

8 Sometimes patterns are formed by shapes overlapping. In the following example, some patterns
involving overlapping squares are given.

AB C

a In which pattern does it appear that the squares are transparent?
b In which pattern does the square to the right appear to be on top?
c Repeat the three types of patterns shown above using circles.
d Use square grid paper to make a diamond and repeat the three types of patterns shown

above using the diamond.

9 When we look at tile patterns we can often see other shapes within the pattern. Copy the tile
pattern given, and then colour it in using the shape given as the basic tile shape.
ab

40 INTERNATIONAL MATHEMATICS 1