So you really want to learn

Transcription

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So you really want to learn
Maths
Prep
BOOK 1
Serena Alexander B.A. (Hons.), P.G.C.E.
Edited by Louise Martine B.Sc. (Lon.)
www.galorepark.co.uk
Independent Schools
Examinations Board
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Published by ISEB Publications, an imprint of
Galore Park Publishing Ltd
19/21 Sayers Lane, Tenterden, Kent TN30 6BW
www.galorepark.co.uk
Text copyright © Serena Alexander
Illustrations copyright © Galore Park 2003
Cartoons by Ian Douglass
Technical drawings by Graham Edwards
The right of Serena Alexander to be identified as the author of this work
has been asserted by her in accordance with sections 77 and 78 of the
Copyright, Designs and Patents Act 1988.
The 3-D cartoons used in this book are © www.animationfactory.com.
This publication includes images from CorelDRAW ® 9 which are
protected by the copyright laws of the U.S., Canada and elsewhere.
Used under licence.
Typography and layout by Typetechnique, London W1
Cover design by GKA Design, London WC2H
Printed by CPI, Glasgow
The publishers would like to thank Teresa Sibree for her invaluable help
during the production of this book.
ISBN-13: 978 1 902984 18 6
ISBN-10: 1 902984 18 8
All rights reserved: no part of this publication may be reproduced,
stored in a retrieval system, or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
either the prior written consent of the copyright owner or a licence
permitting restricted copying issued by the Copyright Licensing Agency,
90 Tottenham Court Road, London W1P 0LP.
First published 2003
Reprinted 2004, 2006
Available in the series
Maths Prep Book 1
Maths Prep Book 1 Answer Book
Maths Prep Book 1 Worksheets CD
Maths Prep Book 2
Maths Prep Book 3
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Also available in the So you really want to learn series:
English; History; Latin; French; Maths; Science; Spanish
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Preface
It has been a problem for Maths teachers in prep schools for many, many years
that books suited to the requirements of the Common Entrance Examination
have been hard to find. But help is now at hand. The main aim of this ISEB
endorsed book is to give pupils a good grounding in Mathematics which will
equip them for the 11+ Common Entrance (and other 11+ entrance exams),
before embarking on the rigours of the ISEB 13+ Common Entrance Exam.
Algebra is introduced in this book but the main emphasis is on sound number
work with carefully written pencil and paper methods, geometry, probability
and data handling. There are problem solving exercises throughout in order
to develop reasoning skills. Worked examples are there for reference. Most
chapters finish with an extension question designed to stretch the most able,
then a summary exercise and finally an end of chapter activity. Interspersed
between the chapters are a selection of investigations, puzzles and practical
tasks which should be regarded as a starting point for further discussions.
There is also considerable reference to the history and language of
Mathematics in an attempt to make the subject more relevant for those who
do not have a natural affinity with numbers.
The book has been extensively tested and reviewed by a wide range of Maths
teachers from both prep and senior schools, and has been endorsed by ISEB
(the Independent Schools Examinations Board). The leader of the ISEB 11+
Mathematics setting team (who is also a member of the 13+ setting team),
having reviewed the book, concluded: “I imagine that this text will enjoy
widespread acclaim and become a ‘standard’ work.”
Acknowledgements
It would be impossible to write a textbook without being able to trial exercises
and ideas. The most enormous thanks therefore must go to the boys and girls
of Colet Court and Newton Prep for unwittingly proof reading, testing and
criticising all the sample worksheets that were put before them. Thanks are
also due to my colleagues in both schools for their support and
encouragement, to Teresa Sibree at Marlborough House School, to the
Galore Park team, to Kevin at Typetechnique, and to Justin Kirby at dmc for
his technical support.
Serena Alexander, November 2003
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Contents
Chapter 1: Back to basics
In the beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Roman numerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
From Roman numerals to decimals . . . . . . . . . . . . . . . . . . . . . . . . . 3
Number bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Something special about 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Subtracting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Opposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Activity: Roman numeral investigations . . . . . . . . . . . . . . . . . . . . . 15
Chapter 2: More calculations
Multiplying and dividing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Doubles and near doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Larger numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Approximation and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
More calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Activity: What’s in the box?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 3: Introducing geometry
What is geometry? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
How it began . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Shapes enclosed by three straight lines . . . . . . . . . . . . . . . . . . . . 44
Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Shapes enclosed by four straight lines . . . . . . . . . . . . . . . . . . . . . 45
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Activity : Sorting shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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Chapter 4: More about ten
Multiplying and dividing by ten . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Harder examples: multiplying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Harder examples: dividing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
More about spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Simplifying division calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Powers of ten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
More multiplication by tens and hundreds . . . . . . . . . . . . . . . . . . . 60
Long multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Multiple multiplications! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Simple division: a reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Long division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Rounding up or rounding down? . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Activity: Rudolph’s dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Activity: The Fibonacci sequence . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 5: More about numbers
Rules of divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Factors and multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Index numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Square numbers and triangle numbers . . . . . . . . . . . . . . . . . . . . . 78
The table square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Finding factor pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
More about factors and multiples . . . . . . . . . . . . . . . . . . . . . . . . . 85
Highest common factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Lowest common multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Activity: A one to a hundred square investigation . . . . . . . . . . . . . . 92
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Chapter 6: Symmetry
Line symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Experiments with symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Rotational symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
More about symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Activity: Mirror on the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 7: Introducing fractions
Why fractions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Equivalent fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Using equivalent fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Finding a fraction of an amount . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Adding fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Subtracting fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Fractions with different denominators . . . . . . . . . . . . . . . . . . . . . 112
Adding fractions with different denominators . . . . . . . . . . . . . . . . 113
Addition with mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Subtracting fractions with different denominators . . . . . . . . . . . . . 115
Subtraction with mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . 115
Borrowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Activity: Imperial units
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 8: Introducing decimals, money and the metric system
How it began . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Adding and subtracting decimals . . . . . . . . . . . . . . . . . . . . . . . . . 127
Money, money, money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Ordering decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Multiplying and dividing by 10, 100, 1000 . . . . . . . . . . . . . . . . . . 132
The metric system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Metric units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Adding and subtracting measurements . . . . . . . . . . . . . . . . . . . . 136
Zooming in on the number line . . . . . . . . . . . . . . . . . . . . . . . . . . 137
More about decimals and fractions . . . . . . . . . . . . . . . . . . . . . . . 139
Activity: Mathematics from food! . . . . . . . . . . . . . . . . . . . . . . . . . 143
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Chapter 9: Time, travel and tables
More about months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Dividing up the day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
The 24 hour clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Time as a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Distance, speed and time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Calculating speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Calculating distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Calculating time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Mixed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Activity: Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Chapter 10: Charts and tables
Frequency tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Frequency diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Pictograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Pie charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Drawing pie charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Activity: Statistical surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Chapter 11: Below zero, or negative numbers
Credit and debit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Calculating with negative numbers . . . . . . . . . . . . . . . . . . . . . . . 178
Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Using brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Double negatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Negative co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Drawing axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Transformations on a grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Activity: The balloon game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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Chapter 12: Introducing algebra
Solving puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Here comes x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
More than one x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198
Two missing items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Numbers and letters don’t mix! . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Writing algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Back to the gobstoppers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Now solve the puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Activity: The shopkeeper’s dilemma . . . . . . . . . . . . . . . . . . . . . . 208
Chapter 13: Calculating with and without a calculator
Using the calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
BIDMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Rules to follow when using a calculator . . . . . . . . . . . . . . . . . . . . 212
Activity: Think of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Chapter 14: More decimals
Multiplying decimals by a whole number . . . . . . . . . . . . . . . . . . . 218
Multiplying decimals by a decimal . . . . . . . . . . . . . . . . . . . . . . . . 219
Dividing decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Foreign exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Conversion graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Activity: How many nets? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Chapter 15: Area
Calculating areas of squares and rectangles . . . . . . . . . . . . . . . . 235
Calculating areas when base and height units are different . . . . . 237
Calculating perimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Calculating perimeters when base and height units are different . . 239
Finding the missing dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Finding the height and length . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Combined shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Calculating the area of a right-angled triangle . . . . . . . . . . . . . . . 247
Activity: Do people with big feet have big hands? . . . . . . . . . . . . 252
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Chapter 16: Triangles, angles and bearings
Drawing triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Angles of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Activity: Non-congruent triangles
. . . . . . . . . . . . . . . . . . . . . . . . 269
Chapter 17: Percentages
Looking at a percentage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Percentages as fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Percentages of a whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Finding a percentage of an amount . . . . . . . . . . . . . . . . . . . . . . . 275
Using percentages as fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Percentages as decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Comparing fractions, decimals and percentages . . . . . . . . . . . . . 278
Finding the percentage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Percentages and shopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Activity: A genius test (for fun only!) . . . . . . . . . . . . . . . . . . . . . . . 284
Chapter 18: Probability
The probability scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
The probability of an event happening . . . . . . . . . . . . . . . . . . . . . 286
Finding the probability of an event not happening . . . . . . . . . . . . 288
Just one event can change the situation . . . . . . . . . . . . . . . . . . . 290
Activity: Probability experiments . . . . . . . . . . . . . . . . . . . . . . . . . 295
Chapter 19: Shapes in 3 dimensions
Isometric drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Looking at nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Volumes of cubes and cuboids . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Finding the third dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Units of volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Activity: Make a litre cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
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Chapter 20. Mean, mode and median
Looking at data: range and average . . . . . . . . . . . . . . . . . . . . . . 312
The mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Finding the total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
The median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
The mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Displaying data: frequency tables and bar charts . . . . . . . . . . . . . 319
Activity: It’s time to go! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
About the author
Serena Alexander has taught Mathematics since 1987, originally in both
maintained and independent senior schools. From 1999 she taught at
St. Paul’s School for Boys, where she was Head of Mathematics at their
Preparatory School, Colet Court, before moving to Newton Prep. She has
been a member of the ISEB setting team for Mathematics, is an ISI inspector
and helps to run regular Mathematics conferences for prep school teachers.
She has a passion for Maths and expects her pupils to feel the same way.
After a lesson or two with her, they normally do!
Note to the reprinted (2004) edition
Please note that, in reprinting this book (2004), we have taken the opportunity
to correct one or two errors that appeared in the first edition. Apart from minor
typographical adjustments, changes have been made to the following
exercises: Ex. 5.2, Q.5 (e); Ex. 7.4, Q.2 (h); Ex. 7.9, Q.7; Ex. 11.12, Q.2 (c)
and (d); Q.3 (d) and (e); Ex. 15.5, Q.8 (b)-(d); Ex. 19.2, Q.5; Ex. 20.5, Q.1 (e);
Ex. 20.7, Q.4 (c).
CHAPTER 07 pp102-120
102
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Maths Prep
Why fractions?
A fraction is a part of something else. Mathematically we use the word fraction
to denote a number between nought and one.
Think of a whole broken into a number of equal parts: 2 parts are called halves,
3 parts are called thirds, 4 parts are called quarters, 5 parts are called fifths etc.
Now that we use the decimal system, and have pocket calculators to help us,
these fractions look a bit old fashioned. However, if we look at thirds, elevenths
and sevenths we find that these involve recurring decimals. For example one third
is 0.3333 (where the 3s on the end recur for ever and ever) and it is therefore
difficult to calculate with them. We will deal with these later.
There will also be times when a calculator is not at hand and so we need to use
fractions for “back of the envelope” calculations.
So fractions are an important part of mathematics and we need to learn how
they work. To begin with, here are two points to note:
1. Fractions where the number on the top is smaller than the number on the
bottom have a value between nought and one and are called proper fractions.
2. “Top heavy” fractions, where the number on the top is bigger than the number
on the bottom, have a value greater than one and are called improper fractions.
For example:
1 2 3 4
5 5 5 5
Proper fractions
A proper fraction
5
5
6 7 8 9
5 5 5 5
One
Improper fractions
An improper fraction
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Exercise 7.1
Many of the following questions are about cakes. For each one draw a circular
cake and use your drawing to answer the questions.
1.
I cut a cake into four equal parts. My brother eats one part, my sister eats
another and I eat one more. What fraction of the cake is left?
2.
I eat half a cake. My cat eats half the remainder. What fraction is left?
3.
How many minutes are there in half an hour? How many in a quarter of an
hour?
4.
We ate one quarter of a cake yesterday. Today we ate a third of what was
left. What fraction of the original cake is left for tomorrow?
5.
At a tea party there was a chocolate cake and a stawberry cake. Half the
chocolate cake was eaten straight away as was three quarters of the
strawberry cake. When the Mums and Dads arrived, they ate half what was
left of the chocolate cake and a quarter of what was left of the strawberry
cake. What fraction of each cake was left?
6.
On Monday we ate half a cake, on Tuesday we ate half what was left and
on Wednesday we ate half the remainder. What fraction of the original cake
was left for Thursday?
7.
Charlotte divides a cake into four pieces and then takes the largest piece.
William divides a cake into four pieces and cannot decide which piece is
the biggest. Who divided their cake into quarters: Charlotte, William or both
of them?
8.
Anthony divides a cake into thirds, and then divides each piece in two.
Mark divides a cake in two and then divides each half into three equal
pieces. Whose cake has the more pieces, Anthony’s or Mark’s?
9.
We eat a third of a cake, and the dog eats half what is
left. What fraction of the original cake remains?
10. We eat half of a cake and the cat eats a third of what is
left. What fraction of the original cake remains?
103
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Maths Prep
Equivalent fractions
Here is a rectangle divided into twelfths:
If we shade in six of those twelfths, we have in fact shaded half of the rectangle:
and we can write this as:
6
12
=
1
2
These are called equivalent fractions: six twelfths is equivalent to one half. We
can also say that one half is a fraction in its lowest terms because the fraction
cannot be rewritten as an equivalent fraction with less parts (or smaller
numbers). It is very important to give fraction answers in their lowest terms.
We may simplify fractions by dividing the top and bottom numbers by the same
factor. We call this cancelling. This will not alter the value of the fraction as
2 3 24 51
= = = = 1 , and so we are dividing the original fraction by 1.
2 3 24 51
Example: Write as fractions in their lowest terms:
(a)
16
48
8
1
(b)
= 24 = 3
(a)
36
42
16
48
=
(b)
36
42
18 6
=
21 7
Do not worry if you do not get the fraction into its lowest terms in one, or even
two, goes. Remember the rules of divisibility and keep cancelling until you can
go no further.
Exercise 7.2
1.
Copy each of these rectangles into your book and shade half of them.
(a)
(b)
(c)
(d)
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2.
Copy each of these rectangles into your book and shade one quarter of them.
(a)
3.
(b)
5.
(d)
What fraction of these rectangles is shaded? Give your answer as a
fraction in its lowest terms:
(a)
4.
(c)
(b)
(c)
(d)
What fraction of these rectangles is shaded?:
(a)
(c)
(e)
(b)
(d)
(f)
Draw four rectangles in your book, each 4 cm by 5 cm.
Shade the following fractions of each: (a)
6.
2
5
(c)
3
10
(d)
3
4
14
24
(b)
24
36
(c)
12
46
(d)
33
121
(e)
68
100
Here are the answers to Helena’s homework. Sometimes she remembers
to cancel her fractions down to their lowest terms, but sometimes she
forgets. Which of these are correct?
(a)
8.
(b)
Write these fractions in their lowest terms:
(a)
7.
1
2
14
21
(b)
24
35
(c)
14
22
(d)
8
25
(e)
12
75
Fill in the missing numbers to make these fractions equivalent:
4
?
33
?
(c)
1
4
= ? = 24 =
9
?
33
?
(d)
2
9
= ? = 27 =
(a)
1
3
= ? = 24 =
(b)
3
5
= ? = 25 =
3
?
12
?
4
?
50
?
105
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Maths Prep
Using equivalent fractions
It is very difficult to compare fractions, or to add or subtract fractions, when they
have different denominators (or bottom numbers). To make it easier, we have
to find the equivalent fractions with the same denominator.
This is rather like finding the lowest common multiple. It is worth spending a little
time making sure that we have the lowest common denominator, as then we
have easier calculations.
Exercise 7.3
Example: Which is the greater
7
9
or
9
11
77
?
9
81
=
11 99
= 99
9
11
1.
7
9
is greater than
7
9
What is the lowest common denominator for these pairs of fractions?
(a)
1
2
and
2
3
(c)
1
4
and
3
8
(e)
2
5
and
2
3
(b)
3
7
and
4
9
(d)
7
8
and
5
6
(f)
7
12
and
2
3
2.
Write each of the fraction pairs in Q1 with the same common denominator.
In each pair of fractions, which is the larger?
3.
Which is the smaller of each of these pairs of fractions?
4.
(a)
3
5
(b)
7
15
and
2
3
and
5
12
(c)
1
3
(d)
7
12
and
3
8
and
5
9
(e)
2
15
(f)
7
9
and
and
2
3
2
3
Arrange these in order of size, smallest first:
4
9
3
7
2
3
1
2
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5.
Arrange these in order of size, largest first:
2
15
1
3
6.
4
9
1
12
In these groups of fractions, which is the odd one out?
(a)
3
6
12
24
5
9
13
26
15
30
(c)
3
9
6
16
9
24
15
40
75
200
(b)
8
12
12
18
50
75
22
36
32
48
(d)
16
56
4
15
10
35
8
28
24
84
Mixed numbers
All the fractions we have looked at so far have been smaller than one, and these
are called proper fractions. But sometimes when we add fractions together we
get an answer that is greater than one, and these are called improper fractions,
because the numerator (that’s the number on top) is bigger than the
denominator (that’s the number on the bottom) and the fraction is “top heavy”.
We meet these improper fractions when we need to mix numbers with fractions.
Consider the following:
“It is a quarter past four.”
“I am eleven and three quarters.”
“I have grown two and a quarter centimetres.”
We call these numbers “mixed numbers” because they are a mixture of a
number and a fraction. If we need to turn mixed numbers into fractions, we end
up with improper fractions.
(i) Example: Write
23
7
as a mixed number.
23
7
=37
2
(23 ÷ 7 = 3 remainder 2 )
2
(ii) Example: Write 3 5 as an improper fraction.
2
35=
17
5
(5 # 3 + 2 = 17 )
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Maths Prep
Exercise 7.4
1.
Write these mixed numbers as improper fractions:
3
(c) 3 5
2
3
(d) 12 5
(a)1 4
2
(b) 7 7
2.
1
(g) 6 5
7
(h) 15 4
(e) 2 7
(f) 4 12
4
3
(i) 10 10
3
1
(j) 9 9
4
(k) 8 9
5
(l) 5 12
Write these improper fractions as mixed numbers:
(a)
(b)
13
4
23
8
(c)
(d)
21
6
29
10
(e)
(f)
19
9
27
2
(g)
(h)
42
7
13
4
(i)
(j)
93
12
101
5
(k)
(l)
84
11
65
13
Finding a fraction of an amount
Earlier we coloured half of a rectangle.
If the rectangle had 24 squares then we would have coloured 12 of them.
Finding a fraction of an amount is another way of thinking of division.
Example:
1
3
of 24 = 24 ÷ 3
=8
Exercise 7.5
1.
Find
1
3
of 24
6.
Find
1
3
of 12
2.
Find
1
6
of 30
7.
Find
1
9
of 18
3.
Find
1
2
of 10
8.
Find
1
8
of 16
4.
Find
1
4
of 16
9.
Find
1
2
of 8
5.
Find
1
8
of 24
10.
Find
1
5
of 15
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Exercise 7.6
Just as we can find one third, or one quarter, or one eighth of a number, so we
might wish to find two thirds, or three quarters, or four fifths.
2
3
Example:
of 24 = 24 ÷ 3 x 2
=8x2
= 16
1.
Find
3
4
of 24
6.
Find
2
3
of 12
2.
Find
5
6
of 30
7.
Find
4
9
of 18
3.
Find
3
5
of 10
8.
Find
3
4
of 16
4.
Find
3
8
of 16
9.
Find
3
5
of 25
5.
Find
5
6
of 24
10.
Find
7
10
of 30
Which is bigger?:
11.
1
4
of 24 or
1
3
of 21
12.
3
4
of 16 or
1
2
of 20
13.
14.
2
3
5
6
of 24 or
of 36 or
3
5
2
3
of 20
of 45
Adding fractions
I can divide a rectangle up into four equal quarters.
From this is it easy to see that
1
4
1
2
+4=4
1
=2
1
4
1
3
+2=4
1
4
3
+ 4 =1
109
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Maths Prep
Exercise 7.7
Add these fractions. You may need to add another stage so that your answer is
in its lowest terms.
1.
1
4
+4
1
4.
1
8
+8
1
7.
1
8
+8
3
2.
1
5
+5
1
5.
2
9
+9
2
8.
1
6
+6
3.
1
3
+3
1
6.
2
8
+8
3
9.
3
10
5
2
+ 10
Some of these next additions may give you an answer of more that one (i.e. an
improper fraction). Write your final answer as a mixed number.
4
5
Example:
3
7
+5=5
2
= 15
10.
3
4
+4
3
13.
5
8
+8
7
16.
5
6
+6
5
11.
4
5
+5
4
14.
7
9
+9
5
17.
4
5
+5
12.
2
3
+3
2
15.
7
8
+8
3
18.
9
10
3
6
+ 10
Exercise 7.8
These next fractions are already mixed numbers. You need to add the whole
numbers and the fractions. Make sure that your final answer does not contain
an improper fraction.
5
7
18 + 2 8 = 3
Example:
=4
=4
3
3
4.
38 +28
4
1
5.
19 + 19
2
6.
18 + 18
1.
14 + 2 4
2.
15 + 15
3.
2 3 + 13
2
5
3
12
8
4
8
1
2
1
5
7.
26 +26
8
2
8.
45+5
4
4
3
9.
110 + 3 10
3
3
9
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Exercise 7.9: Subtracting fractions
Subtract these fractions. You may need to add another stage so that your
answer is in its lowest terms.
3
6
Example:
1
2
-6=6
1
=3
1.
3
4
-4
1
4.
7
8
-8
1
7.
7
8
-8
3
2.
4
5
-5
1
5.
7
9
-9
2
8.
7
9
-9
3.
5
6
-6
1
6.
7
8
-8
3
9.
9
10
1
3
- 10
Now take a fraction from a mixed number. To do this you may need to turn the
mixed number into an improper fraction.
Example:
1
5
7
5
2 6 - 6 = 16 - 6
2
= 16
1
= 13
3
1
13. 2 8 - 8
4
3
14. 1 9 - 9
1
2
15. 2 8 - 8
10. 1 4 - 4
11.
25 - 5
12. 1 3 - 3
5
7
16. 2 6 - 6
1
4
17. 1 5 - 5
7
3
1
2
3
5
3
7
18. 3 10 - 10
Taking a fraction from a mixed number
111
112
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Maths Prep
Exercise 7.10
These next fractions are both mixed numbers. You need to subtract the whole
numbers and the fractions.
5
7
5-7
8
13 - 7
= 8
5
(in the middle stage the 1
8
6
=8
13
has to become
)
8
3
=4
1
3
7.
56 -26
3
4
8.
45-5
1
3
9.
5 10 - 2 10
2 8 - 18 = 1
Example:
3
1
4.
3 8 - 18
4
1
5.
4 9 - 19
1
2
6.
4 8 - 18
1.
2 4 - 14
2.
4 5 - 15
3.
2 3 - 13
1
2
5
3
3
7
Now a try a mixture of adding and subtracting
3
1
13. 3 8 - 2 8
1
3
14. 5 9 - 3 9
2
4
15. 1 8 + 1 8
10. 2 4 + 1 4
11.
45 -25
1
5
16. 5 9 - 4 9
2
7
17. 2 8 + 1 8
7
12. 2 5 + 4 5
3
1
5
5
7
1
3
18. 3 10 - 2 10
Fractions with different denominators
1
1
3
1
1
We saw that 4 + 2 = 4 . This is because
calculation like this:
1
4
2
+2=4+4
3
=4
Fractions with different
denominators
1
2
2
= 4 . We could have written the
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Exercise 7.11
Add these fractions. You may need to add another stage so that your answer is
in its lowest terms.
1
1
1
3
1
1
1.
14 + 2
4.
18 + 4
7.
18 + 12
2.
1
5
+ 10
3
5.
1
9
+3
1
8.
13 + 6
3.
1
6
+3
1
6.
2
9
+3
2
9.
3
10
1
5
1
+ 12
These need an extra stage of working.
5
3
5
3
2 8 + 14 = 3 8 + 4
Example:
5+6
8
11
=3 8
3
=48
=3
3
1
13. 3 8 + 1 4
3
4
3
14. 2 9 + 3
10. 2 4 + 1 2
4
11. 3 5 + 110
1
1
7
12. 1 6 + 3
3
1
3
1
2
5
16. 3 8 + 2 2
17. 2 3 + 1 6
2
15. 3 9 + 1 3
7
4
18. 2 10 + 1 5
Adding fractions with different denominators
In the last section it was quite simple to add the fractions because in each pair
one denominator was a factor of the other. It is possible to add fractions with
different denominators, but you need to put in that middle stage when they have
both been written as equivalent fractions with a common denominator.
If we are asked
3
10
7
+ 12
we cannot directly add tenths to twelfths. We have to write both fractions as their
equivalent fractions, but with a lowest common denominator. Turn the page to
see how we do it.
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114
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Maths Prep
The lowest common multiple of 10 and 12 is 60, and
3
10
Example:
3
10
5
+ 12 =
3#6
18
= 10 # 3 = 60 and
5
12
5#5
25
= 12 # 5 = 60
18 + 25
60
43
= 60
Exercise 7.12
Add these fractions, remembering to put the answer in its simplest form. If the
13
1
answer is an improper fraction (e.g. 12 ) write it as a mixed number: 112
1.
2
5
+4
1
4.
5
12
+8
3
7.
5
6
+4
3
2.
1
7
+5
3
5.
2
15
+9
4
8.
5
7
+5
3.
2
3
+5
1
6.
3
10
+8
3
9.
3
5
+4
3
3
Addition with mixed numbers
To add mixed numbers, first we add the whole numbers together and then the
fractions.
Example:
1
11
3 8 + 2 12 = 5
3 + 22
24
25
= 5 24
1
= 6 24
Exercise 7.13
1
2
5.
25 +27
1
3
6.
1
7.
1 6 + 3 10
8.
46 +24
1.
15 + 3 3
2.
27 +35
3.
15 + 4 8
4.
58 +35
2
3
1
1
2
2
2
5
3
13
11
4
2
9.
64 +36
3
5
10. 7 9 + 2 15
1
3
11. 3 8 + 3 10
5
3
12. 6 15 + 4 12
5 12 + 3 3
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Subtracting fractions with different denominators
Subtraction of fractions follows the same first steps as addition. Find the lowest
common denominator, find the equivalent fractions and then subtract:
3
5
Example:
1
-3=
9-5
15
4
= 15
Exercise 7.14
1.
4
5
-3
2
4.
2
3
-7
2
7.
2
5
-9
2
2.
2
3
-4
1
5.
7
8
-6
5
8.
4
9
-3
3.
5
8
-6
1
6.
4
5
-3
1
9.
5
7
-3
1
2
Subtraction with mixed numbers
If we are coping with mixed numbers, we treat them as we did when adding,
except that we subtract instead!
3
1
3 7 - 14 = 2
Example:
12 - 7
28
5
= 2 28
Exercise 7.15
4
1
1.
25 -3
2.
33 -25
3.
26 - 8
2
5
1
5
1
2
4.
13 - 7
5.
2 8 - 16
6.
35 -23
4
5
7.
45 -39
5
1
8.
19 - 3
7
3
1
9.
3 7 - 15
4
2
2
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Maths Prep
Borrowing
As in any other subtraction there will be times when the first subtraction is not
possible (e.g. 5 – 9). As usual you need to borrow from the next number on the
left. Remember, however, that you are not borrowing ten units – you are
borrowing 8 eighths, 12 twelfths, 16 sixteenths or whatever.
1
3
1
3
3 3 - 14 = 2 3 - 4
Example:
=2
4-9
12
16
4-9
12
7
= 112
1
=2
Now you borrow 12 twelfths
from the 2 and then add it
to the 4 twelfths to make
16 twelfths.
Exercise 7.16
1
1
7.
56 -24
1
3
8.
2 5 - 13
1
2
9.
4 8 - 15
1.
3 5 - 13
2.
4 3 - 14
3.
3 8 - 13
4.
28 - 6
5.
36-6
6.
2 7 - 15
1
3
13. 1 3 - 8
2
2
14. 2 7 - 3
3
2
15. 2 4 - 1 6
5
1
10. 1 5 - 3
5
1
11. 3 4 - 1 8
4
1
2
2
1
2
7
2
1
5
1
5
16. 4 9 - 2 6
3
7
17. 1 7 - 4
3
4
18. 5 6 - 1 8
12. 5 7 - 2 5
2
A fraction borrowing from a whole number
1
3
5
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Exercise 7.17
To answer each of these questions, write out the fraction calculation.
1.
I buy a packet of ‘Gummos’. One third of them was stuck together; what
fraction of the packet was not stuck together?
2.
It is 5 miles from my house to my friend’s house. One half of the way is by
road and one third by cycle track. The last part is across a field. What fraction
of the journey is across the field?
3.
I am twelve and a half and my little sister is seven and three quarters.
What is the difference between our ages?
4.
I am eleven and three quarters now. How old will I be in two and a half
years’ time?
5.
In a bag of ‘Gummos’, one quarter of the sweets is red and one third is
green. The rest are orange. What fraction of the sweets is orange?
6.
A ball of string is 4 metres long. I cut off one and two fifths metres from the
ball of string. How many metres are left?
7.
My friends and I ate three quarters of a tub of ice cream. When they had
gone I ate another one tenth of the tub. What fraction of the tub of ice
cream was left?
8.
In a class of children, one quarter has only one pet and two thirds have
two pets. If no-one has more than two pets, what fraction of the class has
no pets at all?
9.
I leave home at a quarter past ten and travel for an hour and a half. What
time do I arrive?
10. My father buys a load of seed potatoes but when he gets them home he finds
that one tenth of them is rotten and two fifths have already started sprouting.
What fraction of the potatoes is OK? (As you know, rotten potatoes do not sprout.)
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Maths Prep
Exercise 7.18: Extension questions
1.
Number patterns and series can be seen in fractions too. Copy and
complete these:
1
2
=2
1
1
2
+3=
1
2
+3+4=
2
2
3
Add the next three lines of the pattern and calculate the answers.
2.
3.
Add the next two lines to this pattern and calculate the answers:
1
3
=3
1
1
3
+5=
1
3
+5+7=
3
3
5
Look at this series. Write the three next lines and calculate:
1
2
=2
1
1
2
+4=
1
2
+4+8=
1
1
1
If you continue the series, will the sum ever be equal to one?
Is there a point when the answer is so close to one that you can assume
that it is equal to one? (For discussion).
4.
Study the series below. If you continue the series, will you ever get the
1
2
answer 10 or 5 ?
1
10
= 10
1
1
10
+ 100 =
1
10
+ 100 + 1000 =
1
1
1
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Exercise 7.19: Summary exercise
1.
What fraction of a metre is 25 cm?
2.
Find
3.
Write these fractions in their lowest terms:
(a)
4.
of 36.
12
26
(b)
35
50
(c)
100
75
(d)
125
100
Fill in the missing numbers so that these fractions are equivalent:
3
4
5.
2
3
9
= = 24 =
24
= 120
Which of these is the larger?
7
9
or
5
7
3
4
5
6.
Write a fraction that is greater than
7.
Calculate the answers to these addition sums:
(a)
8.
1
+4
(b)
3
4
2
+3
1
4
(c) 3 6 + 1 9
5
7
(d) 3 8 + 10
Calculate the answers to these subtraction sums:
(a)
9.
3
5
and less than 6 .
3
7
1
-4
(b)
7
8
5
-6
1
4
(c) 1 6 - 5
1
7
(d) 3 4 - 110
One half of my class has brown hair, one third has blonde hair, one sixth
has black hair and the rest have ginger hair. What fraction of the class has
ginger hair?
10. A snail is at the bottom of a bucket which is 40 cm high. The snail climbs
at the rate of 10 cm every 50 minutes. After every 50 minute climb the snail
rests for ten minutes. During the rest the snail slips down one tenth of the
total distance that he has climbed from the bottom of the bucket. How long
will it take him to climb out of the bucket?
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Maths Prep
End of chapter activity: Imperial units
In the next chapter you will be revising the metric units that we use for
measurements of lengths and weight. However before we had metric units we
used imperial units and many of these are still in use today. Your parents might
still weigh themselves in stones and pounds, and measure their height in feet
and inches. When we travel in this country, we look at distances in miles.
When we are calculating we usually work in one unit or the other, but we need
to know the equivalent units for some calculations. Here are the commonly used
approximations:
1 metre = 39 in = 3 ft 3 ins
1 kg = 2.2 lb
1 kilometre = mile
1 litre = 1.75 pints
1 foot = 0.3 m
1 pound = 0.45 kg
1 mile = 1 km
1pint = 0.6 litres
Look at your own ruler. You might find that it has centimetres on one side and
inches on the other, but it is marked so that you cannot read one unit off from
the other. You are now going to make four “rulers” so that you can measure each
of the four comparative units in the tables above.
Feet ➞ Metres
Metres ➞ Feet
1.
First you will need to work out the equivalent of 1 ft, 2 ft, 3 ft etc. in metres,
and 1 m, 2 m, 3 m, etc. in feet. Go up to 5 m and 16 feet.
2.
Take a long piece of paper, say 1 m long. You may need to stick several
strips together. Choose a scale. If your paper is 1 m long and you are
looking at units from 1 to 5 metres then you will need to take 20 cm for 1 m.
0m
3.
1m
2m
3m
4m
Match up 0 feet on the other side and then mark off the equivalent feet. You
will need to do this carefully so the equivalent amounts line up correctly.
0m
1m
2m
3m
4m
0ft 1ft 2ft 3ft
4.
Now make a similar “ruler” to compare miles and kilometres, litres and pints
and pounds and kilograms.

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