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ABOUT THIS BOOK As a seasoned secondary level Mathematics teacher who has taught for more than 10 years, I have been exposed to, and therefore am very familiar with, the changes in the secondary mathematics curriculum over the years. It is evident that our curriculum focuses on 4 fundamental and important themes: Algebra, Numbers, Geometry and Statistics. While schools differ in their pace and the sequence in which the topics are being taught, their objective remains largely similar: to ensure their students are ready to take on the demands of the upper secondary mathematics syllabus and be ready for national examinations. It is this same desire to help my students that this series of books is conceived. I have written three books to cover all the topics in the entire lower secondary mathematics course: Algebra, Numbers, Functions and Graphs, Geometry and Statistics. The examples and questions are paced to help students understand concepts, recognize common steps, see through typical trick questions and apply mathematical knowledge taught and learned. While the progressive practices ensure familiarity and exposure to a wide variety of question types, there are opportunities to exercise higher order thinking through the challenging questions provided in every topic. Detailed solutions are included to guide and help students through critical thinking processes. While my students will certainly benefit more from my personal teaching of these questions, I am sure many others will also benefit from working through the same questions systematically and studying the solutions diligently. Alex Lim CONTENTS Numbers, Functions and Graphs 1 Primes and Prime Factorisation 1 2 Highest Common Factor (HCF) and Lowest Common Multiple (LCM) 9 3 Squares, Cubes, Square roots and Cube roots 27 4 Operations on Integers 38 5 Operations on Rational Numbers 62 6 Real numbers, Using calculator and Number line 83 7 Approximation and Estimation 98 8 Ratio, Rate and Speed 112 9 Percentage 131 10 Functions and Graphs (I) 144 11 200 Functions and Graphs (II) 12 Map Scales 242 13 Direct and Inverse Proportion 260 14 Set Language and Notation 277 Solutions S1 - S87 1 Primes and Prime Factorisation A prime number is an integer bigger than 1 which has only two factors, 1 and the number itself. Examples : 2, 3, 5, 7, 11, 13, 17, 19, … A composite number is an integer bigger than 1 which has more than 2 factors. Examples : 4, 6, 8, 9, 10, 12, 14, 15, 16, … Example 1 Express 42 as a product of its prime factors. Solutions: Method 1 (Factor tree) Method 2 (Ladder method) 42 2 6 × 7 × 3 × 2 3 7 Divide by the smallest prime number, in order. 7 42 21 7 1 Thus 42 = 2 × 3 × 7 Example 2 Express 105 as a product of its prime factors. Solutions: Method 1 (Factor tree) 105 5 35 × 3 × 7 × 3 Thus 105 = 3 × 5 × 7 © Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre 1 Topic 1 Method 2 (Ladder method) 3 5 7 105 35 7 1 Let’s Practise Prime factors 1 Express the following numbers as a product of their prime factors. a 45 b 74 c 110 d 128 e 146 f g 320 h 4345 i 7115 j 9600 l 50 400 k 11 350 © Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre 2 Topic 1 258 Let’s Practise Mixed practice 2 Find the largest multiple of 25 that is less than 427. 3 Given that x is a prime number and y is a composite number, state “True” or “False” for each statement. (a) 7y is a prime number. (b) x + x + x is a composite number. 4 Find the sum of the first three prime numbers that end with the digit 1. 5 Find a possible pair of prime numbers such that their sum is 90. © Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre 3 Topic 1 6 Find a possible pair of prime numbers such that their difference is 2. 7 Find the smallest number that is divisible by all prime numbers between 1 and 11. 8 Express, in index notation, 150 as a product of its prime factors, using a factor tree. 9 Express, in index notation, 156 as a product of its prime factors, using a factor tree. 10 Express, in index notation, 432 as a product of its prime factors, using a factor tree. © Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre 4 Topic 1 11 Express, in index notation, 896 as a product of its prime factors, using ‘ladder method’. 12 Express, in index notation, 2376 as a product of its prime factors, using ‘ladder method’. 13 Express, in index notation, 3432 as a product of its prime factors, using ‘ladder method. 14 Find the prime factorisation of 245. 15 List the prime factors of 2100. © Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre 5 Topic 1