EXAM-STANDARD PROBLEM SUMS

EXAM-STANDARDPROBLEM SUMS

Mathematics

with Terry Chew

PrimaryPrimary

This book is conceived, researched and carefully crafted for students looking to prepare for or embark on the journey to PSLE in addition to acquiring further knowledge in Mathematical Olympiad type questions.

It comprises: 25 lessons on topics spanning both the school based curriculum and MO 3 lessons on non-routine questions to encourage out-of-the-box thinking

and problem-solving Classic examples explained in depth and detail yet in an easy-to-under-

stand manner Exercises to provide for practice and application Mathematical stories, anecdotes and articles that pique students’ interest

and fuel their thirst for history and information as well as fun facts on common and popular Maths concepts

Answers with full work solutions for self-assessment Supplementary reference section to provide extensive resource materials

for lessons covered

Terry ChewBest-selling author of

Maths Olympiad: Unleash theMaths Olympian in You! and

Academic Director of Terry Chew Academy

Title Page_Conquer Exam-standard Maths Prob Sums with Terry Chew P6.pdf 1 23/1/2019 11:24:52 AM

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ALL RIGHTS RESERVEDAll 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 the prior permission of the publishers.

ISBN-13 978-981-4672-06-1 ISBN-10 981-4672-06-8

Printed in Singapore

|Conquer| Exam-Standard Mathematics Problem Sums with Terry ChewPrimary 6

First Edition 2019

© Singapore Asia Publishers Pte Ltd and Terry Chew teachers@work is an imprint of Singapore Asia Publishers Pte Ltd

Published and Distributed by: Singapore Asia Publishers Pte Ltd219 Henderson Road #10-04 Henderson Industrial Park Singapore 159556 Tel : +65 6276 8280 Fax : +65 6276 8292 Email: [email protected]: www.sapgrp.comFacebook: Singapore-Asia-Publishers

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Foreword

Dear parents,

If your child is scratching his/her head over a rather unusual Mathematics question, chances are it is a varied version of a Mathematical Olympiad type of question. Another name often attached to this type of questions is ‘non-routine problems’.

Contrary to the myth and belief that Mathematical Olympiad is tough and tricky, there is actually a substantial overlap between standard syllabus Mathematics and Mathematical Olympiad – although it must be emphasised, and is apparent, that Maths Olympiad type of questions delve much deeper and broader.

Looking at the recent trend of PSLE questions, more Mathematical Olympiad type of questions are being infused to invoke the critical thinking faculty of students. Little surprise, then, that to separate the‘A*’ students from the‘A’ students, the toughest of PSLE Mathematics questions come from or are heavily influenced by Mathematical Olympiad.

The beauty and elegance of Mathematical Olympiad questions lie in the fact that they are trickier in nature, while the topics are also much broader. Most importantly, they require young minds to think out of the box, thus allowing them to become more creative at problem solving.

The objectives of this series are three-fold:

• to serve as an indicator of the current trend and the type of questions often encountered; • to infuse teaching with a strong element of Mathematical Olympiad style problems and

methods, thus extending the students’ thinking skills and giving them the flexibility as well as the versatility in approaching such problems; and

• to inspire students and make the learning of Mathematics more wholesome by including a short story or mathematical amusement at the end of each lesson.

I am most privileged and feel extremely honoured to be able to continue serving students in the field of Mathematics, be it via content from the syllabus or the problems of the Mathematical Olympiad type. To the parents’ greatest comfort, many students who had undergone the training via the questions in the series improved by leaps and bounds. The Mathematical amusements also engage them greatly. I am sure your child will benefit as such, too!

For related courses and workshops, please visitTerryChew.com.sg

Terry Chew

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Contents

Lesson 1 Algebra.......................................... 1

Story An Algebraic Approach to Average .. 10

Lesson 2 Angles(1).....................................11

Story Telephone Number Problem .......... 19

Lesson 3 Angles(2).................................... 21

Story Squaring the Circle ....................... 29

Lesson 4 Average....................................... 31

Story An Interesting Speed Problem ....... 40

Lesson 5 Fractions(1)–FindingtheTotal.. 41

Story The King of Rate ......................... 48

Lesson 6 Fractions(2)–TransferringwithChangedQuantities.................... 49

Story A Problem on Escalator ................ 55

Lesson 7 Fractions(3)–TransferringwithUnchangedQuantities................. 57

Story Socks and Shoes ......................... 63

Lesson 8 Fractions(4)–SettingEquations.. 65

Story The Lucas Problem ...................... 72

Lesson 9 Ratio............................................ 73

Story Cat and Mice ............................... 83

Lesson 10 Percentage.................................. 85

Story The Fibonacci Spiral ..................... 95

Lesson 11 Fractions(5)–Percentage.......... 97

Story Candle Light Problem ................. 103

Lesson 12 Fractions(6)–Ratio.................. 105

Story A Geometry Problem ...................111

Lesson 13 Speed(1)–Encountering..........113

Story Algebra for Percentage ............... 122

Lesson 14 Speed(2)–CatchingUp.......... 123

Story Think Algebra ............................ 131

Lesson 15 Speed(3)–InvolvingRatio....... 133 Story Problems on Time ..................... 141

Lesson 16 Speed(4)–MultipleEncounters.. 143 Story Casino Problem ........................ 149

Lesson 17 Volume(1)–InvolvingSolids... 151 Story Area of Circle ............................. 158

Lesson 18 Volume(2)–InvolvingRateandRatio.......................................... 159

Story The Study of Numbers ............... 166

Lesson 19 Volume(3)–InvolvingTransferring.............................. 167

Story The Problem of a Hundred Fowls ... 175

Lesson 20 AreaInvolvingRatio................. 177 Story Fibonacci Numbers .................... 183

Lesson 21 AreaandCircumferenceofQuadrant.................................. 185

Story Circles and Triangles .................. 191

Lesson 22 AreaandCircumferenceofCircle(1)................................... 193

Story The Father of Algebra ................. 199


Story The Great Ramanujan ................ 207


Story The Poincaré Conjecture ............. 215

Lesson 25 AreaandPerimeter–ATouchofAlgebraandPythagoras’Theorem............ 217

Story A Reclusive Mathematician .......... 224

Lesson 26 Non-RoutineProblems(1)...... 227

Lesson 27 Non-RoutineProblems(2)...... 233

Lesson 28 Non-RoutineProblems(3)..... 239Solutions ............................................ S1-S43

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� Lesson 1|Conquer| Exam-standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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Lesson 1

Classic Example �

Simplify.(i) 9 + 6y – 3y – 3(ii) 4m – 3 + m ÷ 8 – 2m

Solution (i) 9 + 6y – 3y – 3 = 9 – 3 + 6y – 3y

= 6 + 3y (ii) 4m – 3 + m ÷ 8 – 2m = 4m – 3 + m __ 8 – 2m

= 4m – 2m + m __ 8 – 3

= 2m + m __ 8 – 3

= 16m + m ________ 8 – 3

= 17m ____ 8 – 3

Ans: (i) 6 + 3y(ii) �7m____ 8 – 3

Classic Example 2

Find the value of each of the following.

(i) If n = 7, find the value of 3n + 8n _______ 11 .

(ii) If c = 9, find the value of 6c + 4c _______ 3 .

Solution (i) Substitute n = 7 into the statement:

3 × 7 + 8 × 7 ___________ 11 = 21 + 56 _______ 11 = 77 ___ 11 = 7

(ii) Substitute c = 9 into the statement:

6 × 9 + 4 × 9 ___________ 3 = 54 + 36 _______ 3 = 90 ___ 3 = 30

Ans: (i) 7(ii) 30

Algebra

L1_Solve EType Math WP 28 ELesso1 1 1/11/2019 5:04:25 PM

2 Lesson 1|Conquer| Exam-standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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Classic Example 3

Mei had $P at first. She spent half of the money on a book and bought 8 pens at $2.40 each. How much money had she left? Express the answer in P.

Solution

book $ ( P – P __ 2 ) = $ P __ 2

pens 8 × $2.40 = $19.20

left $( P __ 2 – 19.20)

Ans: $( P__ 2 – �9.20)


3 Lesson 1|Conquer| Exam-standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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Practice� Simplify the following.

(a) 20 + 10w – 5 – 7w (b) 5b + 6 – 3b – 5 + 10 (c) 10m + 12 – 3m + m – 5 (d) 4n ÷ 3 × 15 + (17n + 3n) ÷ 5n (e) If 8b × 3 = 72, find the value of b. (f) Find the value of m in 9m ÷ 3 = 21.

Ans: (a)

(b)

(c)

(d)

(e)

(f)


PrimaryPrimary

EXAM-STANDARDPROBLEM SUMS

Mathematics

with Terry Chew

© Singapore Asia Publishers Pte Ltd & Terry Chew. ALL RIGHTS RESERVED.

Solutions

Ans Cover_Conquer Exam-standard Maths Prob Sums with Terry Chew P6.pdf 1 18/1/2019 2:15:32 PM

S� Lesson 1|Conquer| Exam-standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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Lesson 1 Algebra� (a) 20 + 10w – 5 – 7w = 20 – 5 + 10w – 7w = 15 + 3w Ans: �5 + 3w (b) 5b + 6 – 3b – 5 + 10 = 5b – 3b + 6 – 5 + 10 = 2b + 11 Ans: 2b + �� (c) 10m + 12 – 3m + m – 5 = 10m – 3m + m + 12 – 5 = 8m + 7 Ans: 8m + 7 (d) 4n ÷ 3 × 15 + (17n + 3n) ÷ 5n

= 4n × 15 _______ 3 + 20n ____ 5n 5

= 20n + 4 Ans: 20n + 4 (e) 8b × 3 = 72 24b = 72 b = 3 Ans: 3 (f) 9m ÷ 3 = 21

9m ___ 3 3

= 21

3m = 21 m = 7 Ans: 7

2 (a) 7k – k ÷ 2 = 7 × 8 – 8 ÷ 2 = 56 – 4 = 52 Ans: 52

(b) 6 __ a + 2a – 4

= 6 __ 3 + 2 × 3 – 4

= 2 + 6 – 4 = 4 Ans: 4

(c) 2a + 1 __ 2 a + 3 __ 4

= 2 × 1 __ 2 + 1 _____ 2 × 1 __ 2

+ 3 __ 4

= 1 + 1 + 3 __ 4 = 2 3 __ 4

Ans: 2 3 __ 4 (d) 12r + 14 + 17 – 6r = 12 × 4 + 14 + 17 – 6 × 4 = 48 + 31 – 24 = 55 Ans: 55 (e) 12 + 28y ÷ 4 – 3y = 12 + 28 × 7 ÷ 4 – 3 × 7 = 12 + 49 – 21 = 40 Ans: 40 (f) a + b = 20 + 4c + 21c – 19

= 1 + 25c Ans: � + 25c

3 (a) 6k + 5 – 2k = 4k + 5 Ans: (4k + 5) years (b) Average of 3 numbers → x + 53 + 87 _______ 2

= x + 70 Sum of 3 numbers → 3 × (x + 70)

= 3x + 210 Ans: 3x + 2�0

4 nS

M

n

2 × 64 + 2n = $(128 + 2n) Ans: $(�28 + 2n)

5 3m – 4 + 2m + 5 + m – 9 + 5m + 1 = 3m + 2m + m + 5m – 4 + 5 – 9 + 1 = 11m – 7

Ans: ��m – 7 _______4 cm

6 1 + 2 + 3 ________ 1 × 2 × 3 = 6 __ 6 = 1

Ans: �

Solution_Solve EType Math WP 28 1 1 12/19/2018 9:34:33 AM

References|Conquer| Exam-Standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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References

Lesson 3: Squaring the CircleTerry Chew, Maths Olympiad: The Next Lap, Singapore Asia Publishers, 2012, pp. 141.

Lesson 5: The King of RateTerry Chew, Maths Olympiad: Unleash the Maths Olympian in You, Advanced, Singapore, Singapore Asian Publications 2015, pp. 167.

Lesson 6: A Problem on Escalator Henry Ernest Dudney, 536 Puzzles and Curious Problems, Charles Scribner’s Sons, New York 1967.Asia-Pacific Mathematical Olympiad for Primary Schools, 2008, Problem 29.

Lesson 8: The Lucas ProblemHenry Ernest Dudney, 536 Puzzles and Curious Problems, Charles Scribner’s Sons, New York 1967.National Mathematical Olympiad of Singapore: 2011, Problem 28, NUS High of Math and Science, Singapore 2011.

Lesson 9: Cat and MiceBoris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, Charles Scribner’s Sons, New York 1972. Terry Chew, Math Olympiad: The Next Lap, Singapore Asia Publishers, Singapore 2012, pp. 34.

Lesson 10: The Fibonacci SpiralUta C. Merzbach, Carl B. Boyer, A History of Mathematics, New Jersey, John Wiley and Sons 2011

Lesson 12: A Geometry ProblemRichard Elwes, Mathematics 1001: Absolutely everything that matters in mathematics, Firefly Books, New York 2010Henry Ernest Dudney, 536 Puzzles and Curious Problems, Charles Scribner’s Sons, New York 1967.

Lesson 19: The Problem of a Hundred FowlsTerry Chew, Math Olympiad: The Next Lap, Singapore Asia Publishers, Singapore 2012, pp. 45.

Lesson 22: The Father of AlgebraAmir D. Aczel, A Strange Wilderness: The Lives of Great Mathematicians, Sterling Publishing, New York 2011.J J O’Connor and E F Robertsonhttp://www-history.mcs.st-and.ac.uk/Biographies/Diophantus.html

Lesson 24: The Poincaré ConjectureGrigori Perlman Documentary,https://www.youtube.com/watch?v=Ng1W2KUHI2sMasha Gessen, Perfect Rigor: A Genius and the Mathematical Breakthroughof the Century, New York 2009.Poincaré Conjecture,https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture

Reference_Solve EType Math WP 281 1 2019-2-14 12:44:37

References|Conquer| Exam-standard Mathematics Problem Sums with Terry Chew Primary 6© Singapore Asia Publishers Pte Ltd & Terry Chew

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Lesson 25: A Reclusive MathematicianMasha Gessen, Perfect Rigor: A Genius and the Mathematical Break through of the Century, New York 2009.Poincaré Conjecture,https://en.wikipedia.org/wiki/Poincar%C3%A9 conjectureHistory of Poincare Conjecture by John Morgan,https://www.youtube.com/watch?v=Utf-uwArrq0&list=PLz9X7oPAiY5QHVfAdpYgzQGm2x9dn_DSIPoincaré Conjecture Documentary,https://www.youtube.com/watch?v=Ng1W2KUHI2s

Reference_Solve EType Math WP 282 2 12/19/2018 9:34:05 AM

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