KS3 Maths Complete Study & Practice - Foundation PDF 2019, 2021

CGP CGP P...

CGP CGP P G C Conquer the crucial bits of Key Stage Three Mathematics KS3 Maths with CGP! Yep, this brilliant all-in-one book covers all the most important Maths from KS3... Clear, friendly notes for every key topic... Perfect if you’re working at a foundation level! Foundation Level Hundreds of essential questions... You bet — including mixed-topic tests to keep you sharp Plus a whole practice exam... Spot-on for testing yourself at the end of KS3 You could choose someone else’s dreary KS3 Maths book... Complete...but really — why, why would you do that? ☻ Study & Practice Everything you need for the whole course! P G C www.cgpbooks.co.uk  C G P CGP The definitive study and The best that money can buy... practice books for Key Stage 3...from the country’s leading producer of revision material. “Nothing gets my heart racing like my journey to work, knowing I’ll soon have a CGP book in my hands.” n books Liam Stephenson “CGP sell more revisio Spanish Teacher than anyone else.” Kenilworth ools use “Over 90% of all sch pro du cts as an CGP revision par t of the ir cou rses.” integral the most “CGP books are by far and concise, relevant, clear n bo oks you informative revisio “What did I do before I used CGP books? will find any wh ere.” Well, I used to tell my students facts and hope they were convinced. CGP has allowed me to teach with confidence once more.” www.cgpbooks.co.uk Michael Abrams Physics Teacher Pilling 0119 - 18641  P G C Key Stage Three Mathematics Foundation Level This CGP book is packed with crystal-clear notes and examples for every key topic on the KS3 Maths curriculum — perfect if you’re working at a foundation level. But it doesn’t end there. We’ve also included plenty of practice questions and mixed-topic tests (with answers) to make sure you’ve got to grips with it all. Once you’ve worked through that lot, you should be ready for the practice exam at the end of the book — it’ll test you on everything you’ve learned at KS3! Complete Study & Practice Everything you need for the whole course!  Contents Section One — Numbers Section Three — Ratio, Proportion and Rates of Change Calculating Tips 2 Ordering Numbers and Place Value 5 Ratios 65 Add, Subtract, Multiply and Divide 7 Proportion Problems 66 Addition and Subtraction 9 Percentage Increase and Decrease 68 Adding and Subtracting Decimals 10 Warm-Up and Practice Questions 69 Multiplying by 10, 100, etc.  11 Metric and Imperial Units 71 Dividing by 10, 100, etc.  12 Conversion Factors 72 Multiplying Without a Calculator 13 Reading Timetables 74 Dividing Without a Calculator 14 Warm-Up and Practice Questions 75 Warm-Up and Practice Questions 15 Maps 77 Negative Numbers 17 Scale Drawings 79 Special Types of Number 18 Speed 80 Prime Numbers 19 Warm-Up and Practice Questions 82 Multiples, Factors and Prime Factors 20 Revision Summary 84 LCM and HCF 21 Warm-Up and Practice Questions 22 Fractions, Decimals and Percentages 24 Fractions 25 Section Four — More Fractions 27 Geometry and Measures Percentages 28 Warm-Up and Practice Questions 29 Symmetry 85 Rounding Numbers 31 Quadrilaterals 86 Accuracy and Estimating 33 Triangles and Regular Polygons 87 Powers 34 Congruence 88 Square Roots and Cube Roots 35 Similarity 89 Warm-Up and Practice Questions 36 Warm-Up and Practice Questions 90 Revision Summary 38 Perimeter and Area 92 Areas 93 Area of Compound Shapes 94 Circles 95 Section Two — Algebra and Graphs Circle Questions 96 Warm-Up and Practice Questions 97 Algebra — Simplifying 39 3D Shapes 99 Algebra — Multiplying 40 Nets and Surface Area 100 Formulas 42 Volume 102 Making Formulas From Words 43 Warm-Up and Practice Questions 103 Solving Equations 44 Lines and Angles 105 Warm-Up and Practice Questions 46 Measuring Angles 106 Number Patterns and Sequences 48 Five Angle Rules 107 Warm-Up and Practice Questions 50 Parallel Lines 109 X and Y Coordinates 52 Interior and Exterior Angles 110 Straight Line Graphs 53 Warm-Up and Practice Questions 112 Plotting Straight Line Graphs 56 Transformations 114 Warm-Up and Practice Questions 57 Enlargements 116 Reading Off Graphs 59 Warm-Up and Practice Questions 117 Travel Graphs 60 Triangle Construction 119 Conversion Graphs 61 Constructions 121 Warm-Up and Practice Questions 62 Warm-Up and Practice Questions 122 Revision Summary 64 Revision Summary 124  Section Five — Section Six — Exam Practice Probability and Statistics Mixed Practice Tests 148 Probability 125 Practice Exam: Equal and Unequal Probabilities 126 Practice Paper 1 158 Listing Outcomes 127 Practice Paper 2 168 Warm-Up and Practice Questions 128 Venn Diagrams 130 Types of Data 132 Line Graphs and Pictograms 133 Answers 178 Warm-Up and Practice Questions 134 Index 190 Bar Charts 136 Pie Charts 138 Warm-Up and Practice Questions 139 Mean, Median, Mode and Range 141 Frequency Tables 142 Averages from Frequency Tables 143 Scatter Graphs 144 Warm-Up and Practice Questions 145 Revision Summary 147 Published by CGP From original material by Richard Parsons. Editors: Alex Billings, Ellen Burton, Dan Chesman, Sarah George, Shaun Harrogate, Tom Miles, Claire Plowman, Rosa Roberts and Caley Simpson With thanks to Sammy El-Bahrawy and Glenn Rogers for the proofreading. With thanks to Alastair Duncombe for the reviewing. MFS31 ~ 0720 - 21187 Clipart from Corel® Text, design, layout and original illustrations © Coordination Group Publications Ltd. (CGP) 2019. All rights reserved. 800 1712 712 www.cgpbooks.co.uk 2 Section One — Numbers Calculating Tips KS3 Maths — you’re going to love it. There’s plenty of learning to be done and plenty of fun to be had. But first, here are some tips that will help you on your way to maths glory. Don’t Be Scared of Wordy Questions For the ‘real-life’ questions you’ve first got to work out what the question’s asking you to do. 1) Read the question carefully. Think what bit of maths you might need to answer it. 2) Underline the BITS YOU NEED to answer the question — you might not have to use all the numbers they give you. 3) Write out the question IN MATHS and answer it, showing all your working clearly. Ben’s Bakery is branching out into mini pastries. All the mini pastries will weigh 50% as much as a normal pastry. If a croissant normally weighs 200 g, how much will a mini croissant weigh? The “50%” tells you this is a percentages question (covered on page 28). You need 200 g (the normal weight) and 50% (the percentage). It doesn’t matter what the shop is called or what the pastry is. You want to work out 50% of 200 g, so: 50% of 200 g = 0.5 × 200 = 100 g Don’t forget the units in your final answer — this is a question about weight in grams, so the units will be g. Working with Units When you have a question involving units it’s usually best to do the calculations without the units. Then you can add the units back in at the end — careful you don’t forget about them. Money questions always crop up — you’ll have to keep an eye on the units and the decimal places. Nicola buys a new jacket for £30. After 2 months she sells the jacket for a 5% profit. How much profit does she earn on the jacket in a) pounds, and b) pence? 1) This percentages question is asking you to find 5% of £30. 0.05 × 30 = 1.5 2) The question uses pounds (£30), so 1.5 means 1.5 pounds. Answers in pounds should always be given to 2 decimal places. You might need to fill in the 2nd decimal place with a 0, like this: £1.50 3) Part b) asks for the answer in pence. £1 = 100p. So to change pounds into pence — times by 100: 1.50 × 100 = 150p Underline the parts of the question you need... EXAM TIP Don’t be put off by ‘real-life’ questions — once you’ve worked out what calculation you need to do you can ignore the wordy bits. Just remember to include the units in your answer. Section One — Numbers  3 Calculating Tips Here are a few tips to help you out with your calculations. They might seem a bit unusual and a little bit tricky, but learn them you must. They’ll come in really handy later on. BODMAS Brackets, Other, Division, Multiplication, Addition, Subtraction If you’ve got multiplications and/or divisions next to BODMAS tells you the ORDER in which operations should be done: each other (like 12 ÷ 6 × 4), Work out Brackets first, then Other things like squaring, just work from left to right. then Divide / Multiply groups of numbers before Adding or Subtracting them. The same thing applies to additions and/or subtractions next to each other too. 1. Work out 4 + 6 ÷ 2 2. Calculate 10 – 23 1) Follow BODMAS — 4+6÷2 1) The cube is an ‘other’ 10 – 23 do the division first... = 4 + 3 so that’s first: = 10 – 8 2)...then the addition: = 7 2) Then do the subtraction: = 2 If you don’t follow the order of BODMAS, you get: 3. Find (8 – 2) × (3 + 4) 4 + 6 ÷ 2 = 10 ÷ 2 = 5 1) Start by working out the brackets: (8 – 2) × (3 + 4) 2) And now the multiplication: =6×7 = 42 Hidden Brackets in Fractions This is a bit of a funny one — when you have a fraction with calculations on the top or bottom you have to imagine they’re in brackets and do them first. Work out 6 +16 6÷3. 1) Imagine the bottom of the fraction is in brackets. 16 (6 + 6 ÷ 3) 2) Now follow BODMAS to do the calculation. = 16 = 16 8 =2 (6 + 2) Work out 420 ×5 + 2×3. 1) Imagine the top and bottom are both in brackets. (20 × 5) (4 + 2 × 3) 2) Now follow BODMAS to do the calculation. = 100 = 100 10 = 10 (4 + 6) You've got to get your operations in the right order... EXAM TIP BODMAS is very important — make sure you learn what it stands for. Remember that it applies to every calculation, so you need to use it whenever there’s more than one operation. Section One — Numbers 4 Calculating Tips Ever wondered what all those fancy buttons on your calculator do? Well you’re about to find out. Careful though — reaching for your calculator isn’t always the best option. Calculators Make sure you know the important features on your calculator and how to use them. SHIFT (OR 2ND FUNC) SQUARE, CUBE AND ROOTS Press this first if you want to use something written above a button E.g. 4 x2 gives 4 squared = 16. (e.g. the pi (p) button). And 3√ 27 gives the cube root of 27 = 3. FRACTIONS THE ANSWER E.g. for 41 press 1 4. (If you have a button that looks like 3.6 Before you jot down 3.6, a bc instead, use it in the same way.) think about what it means. shift ON E.g. in a money question, it For 1 53 press 1 3 5 (you S¤D might mean £3.60. might have to press shift first). x2 3 x 3 To cancel down a fraction, enter it and press =. STO RCL M+ ( ) BRACKETS Pressing the or S¤D button 7 8 9 Del AC Calculators use BODMAS also switches an answer between (see p.3), so if there’s part a fraction and a decimal. 4 5 6 × ÷ of a question you want the 1 2 3 + – calculator to do FIRST then . x put brackets in to tell it so. MEMORY ( STO, RCL & M+) 0 ×10 Ans = 840 E.g. for 12 × 8 : PI (p) (See page 95.) Press 12 × 8 = and then STO M+ to The calculator stores the number store the bottom line in the memory. The ‘Ans’ button gives the for pi (= 3.141...). If it’s above number you got when you Then press 840 ÷ RCL M+ = , another button as shown here, last pressed the ‘=’ button. and the answer is 8.75. press the shift button first. With great power comes great responsibility — so here is some advice about using your calculator. DON’T reach straight for your calculator. If you put a big calculation into your calculator all in one go you’re quite likely to get the wrong answer. DO show all your working — you’re more likely to get the right answer by doing it in stages than doing it all in one go. Calculators are handy but don’t forget to show your working... Your calculator might be a little bit different to the one shown above — no need to panic though. Just make sure you can do all of the above functions on your own calculator and you’ll be flying. Section One — Numbers  5 Ordering Numbers and Place Value A nice easy page here — it’s all about reading, writing and ordering whole numbers. Split Big Numbers into Columns and Parts 1) First, you need to know the names of all the columns. E.g. for the number 3 232 594: HUNDRED- TEN- MILLIONS THOUSANDS HUNDREDS TENS ONES THOUSANDS THOUSANDS 3 2 3 2 5 9 4 2) You can then split any number up into its parts, like this: 3 000 000 Three million 200 000 Two hundred thousand 30 000 Thirty thousand Line up the 2 000 Two thousand These add together to make 3 232 594. columns so you 500 Five hundred can read the 90 Nine tens (ninety) numbers clearly. 4 Four ones Look at Big Numbers in Groups of Three To read or write a BIG number, follow these steps: 3 232 594 1) Start from the right-hand side of the number → MILLIONS THOUSANDS The rest 2) Moving left, ←, put a space every 3 digits So read as: 3 million, 232 thousand, 594 to break it up into groups of 3. or write fully in words: 3) Now going right, →, read each group of three Three million, two hundred and thirty-two as a separate number, as shown. thousand, five hundred and ninety-four. Ordering Numbers Put these numbers in order from smallest to largest: The digits of a number 53 17 9 1729 754 3 548 88 2321 are the individual 1) First put them into groups, the ones with fewest digits first: values (0-9) that are written in each place 1-digit  2-digit 3-digit  4-digit value column. 9 3 53 17 88 754 548 1729 2321 E.g. in 623, the digits 2) Then just put each separate group in order of size: are 6, 2 and 3. 3 9 17 53 88 548 754 1729 2321 Splitting big numbers into groups of three is key... Once you’ve split a big number into groups of three, just remember to add ‘thousand’ or ‘million’ onto the end of the group to say how big it is, e.g. 163 000 is one hundred and sixty-three thousand. Section One — Numbers 6 Ordering Numbers and Place Value You need to be able to put decimal numbers in order too. Again, it’s all about the columns. After the decimal point, the columns are called decimal places. Split Decimals into Decimal Places 1) The decimal places have names too. E.g. for the number 173.753: HUNDREDS TENS ONES DECIMAL TENTHS HUNDREDTHS THOUSANDTHS 1 7 3. 7 5 3 POINT 2) You can split up decimals into parts too: With decimals, 100.000 One hundred line up the 70.000 Seven tens (seventy) decimal points. 3.000 Three ones These add together to make 173.753. 0.700 Seven tenths 0.050 Five hundredths 0.003 Three thousandths Putting Decimals in Order of Size 1) Do the whole number bit first, then the bit after the decimal point. 2) With numbers between 0 and 1, first group them by the number of 0s at the start. The group with the most 0s at the start comes first. Write these numbers in order, from smallest to largest: 0.1 9.01 0.53 0.0011 0.027 0.023 0.0023 2.6 1) Do the whole-number bit first. Most of these decimals don’t have a whole number bit. 0.1 0.53 0.0011 0.027 0.023 0.0023 2.6 9.01 2) The rest of the decimals are between 0 and 1, so group them by the number of 0s at the start. 2 initial 0s 1 initial 0 no initial 0s 0.0011 0.0023 0.027 0.023 0.1 0.53 2.6 9.01 3) Once they’re in groups, just order them by comparing the first non-zero digits. (If the first non-zero digits are the same, look at the next digit along instead.) 0.0011 0.0023 0.023 0.027 0.1 0.53 2.6 9.01 Ordering decimals can seem strange at first, but stick with it... Ordering whole numbers is the easy part — it gets a bit more complicated when you come to ordering decimals. Learn the rules on this page and you’ll be ordering decimals in no time. Section One — Numbers  7 Add, Subtract, Multiply and Divide These operations are the building blocks of maths — you have to use them all the time, so it’s important that you know how they work. Using Opposite Operations Adding and Subtracting Start off with a number, add any number to it and then subtract the same number — you’ll be back to the number you started with. This comes in handy when checking your answer. Simon has 26p and steals 14p from his sister Emma. How much money does Simon have now? Add the two amounts together to get the total. 26 + 14 = 40p Check your answer by using the opposite operation — you should get the amount you started with. 40 – 14 = 26p Multiplying and Dividing Multiplying and dividing are opposite operations too. Start off with a number, multiply it by any number and then divide by the same number — you’ll be back to the number you started with. Michelle has 3 bags each containing 4 coconuts. She empties all of the coconuts into a box. How many coconuts are in the box? Multiply the two numbers together to get the total. 3 × 4 = 12 coconuts Check your answer by using the opposite operation — you should get the number of bags you started with. 12 ÷ 4 = 3 bags Use opposite operations to check your answers... Many people find subtraction and division harder than addition and multiplication, but with opposite operations it’s easy to check your answer and make sure you haven’t made a mistake. Opposite operations also come up in lots of other places too, so it’s worth learning them now. Section One — Numbers 8 Add, Subtract, Multiply and Divide Here are a couple of handy tricks for addition and multiplication calculations — learn them and they’ll make your life so much easier. Patterns in Calculations Addition You can make addition questions easier by using opposite operations. If you add something to one number you have to subtract the same amount from the other number. 46 + 54 = 100 +4 –4 50 + 50 = 100 Jameela has 18 chocolate mice and Tyrone has 37. How many chocolate mice do they have in total? Adding 2 to 18 makes it an easier 18 + 37 = 55 number to work with. Just remember +2 -2 20 + 35 = 55 to subtract 2 from the other side. Multiplication The same goes for multiplication questions. If you multiply one number by something you have to divide the other number by the same number. 14 × 8 = 112 ×2 ÷2 28 × 4 = 112 ×2 ÷2 56 × 2 = 112 How many 20 g bags of crisps have the same weight as three 120 g bags of crisps? Here the total weight of the crisps ÷2 120 g × 3 ×2 doesn’t matter. You can just use opposite 60 g × 6 operations until you have 20 g ×.... ÷3 ×3 20 g × 18 ION Opposite operations can make calculations easier... RE VIS TASK Try these calculations by making the numbers easier to work with: 13 + 27, 69 + 26, 15 × 6, 22 × 5. Remember to write down what you are doing to each side. (Answers below.) (Ans: 40, 95, 90, 110) Section One — Numbers  9 Addition and Subtraction You ought to know how to do sums with just a pen and paper — so put away those calculators. Adding Add together 342, 231 and 76. 1) Line up the units columns of each number. Line up the 342 units columns 2) Add up the columns from right to left starting 23 1 with the units: 2 + 1 + 6 = 9. + 76 Write a 9 at the bottom of the units column. 342 231 2+1+6=9 3) Add up the next column: 4 + 3 + 7 = 14. + 76 Write a ‘4’ at the bottom of the tens column 49 1 4 + 3 + 7 = 14 — and carry over the ‘1’ to the next column. write 4 and carry the 1 4) Add up the final column, including anything 342 231 carried over: 3 + 2 + 0 + 1 = 6. + 76 649 3 + 2 + 0 + carried 1 = 6 1 Subtracting Zaara the zebra used to weigh 372 kg. 1) Line up the units columns Her current weight is 324 kg. How of each number. much weight has she lost (in kg)? 2) Going right to left, take the bottom 372 Line up the units columns number away from the top number. – 324 2 is smaller than 4, Here the top number (2) is smaller so you can’t do 2 – 4 6 12 than the bottom number (4), so 372 borrow 10 from the next column along. – 324 Borrow 10 from the tens column The ‘7’ in the tens column becomes ‘6’. 6 12 3) You now have 2 + 10 = 12 units, so you 372 can do the subtraction: 12 – 4 = 8. – 324 12 – 4 = 8 8 Write the 8 at the bottom. 6 12 4) Subtracting the numbers in the 372 next column along gives 6 – 2 = 4, – 324 6–2=4 048 so write 4 at the bottom. 3–3=0 So Zaara has lost 48 kg. 5) In the last column, 3 – 3 = 0. Make sure each number is in the right column... EXAM TIP Subtractions can be tricky. Once you’ve done a subtraction question, check your answer by adding it to the number you subtracted. You should get back to the number you started with. Section One — Numbers 10 Adding and Subtracting Decimals The methods for adding and subtracting decimals are just the same as the ones on the last page. But instead of lining up the units, always make sure you line up the decimal points. Adding Decimals 1) Line up the decimal points and put one in the Work out 0.7 + 32.2 + 1.65. space for the answer too. Write in extra zeros 0.7 0 Decimal points lined up to make the decimals the same length. 3 2.20 + 1.6 5 Extra zeros to make the 2) Add up the columns from right to left.  . decimals the same length The first column only has a 5 in it, so write 5 at the bottom of the column. 0.7 0 3 2.20 0+0+5=5 + 1.6 5 3) Add up the numbers in the next column along:.5 5 7 + 2 + 6 = 15. Write the ‘5’ at the bottom 1 7 + 2 + 6 = 15 — and carry over the ‘1’ to the next column. write 5 and carry the 1 0.7 0 4) Add up the next column, including anything 3 2.20 carried over: 0 + 2 + 1 + 1 = 4. + 1.6 5 3 4.5 5 0 + 2 + 1 + carried 1 = 4 1 5) The last column just has a 3, so write this in. There’s only a 3 in the last column Subtracting Decimals 1) At first, this question doesn’t look like it has any decimals in it. Bilal has £5 and spends 91p on a But we need both numbers in pounds pie. How much does he have left? — so it becomes £5.00 – £0.91. Decimal points lined up £ 5.00 – £0. 9 1 0 is smaller than 1, 2) Set it out as usual, making sure you. so you can’t do 0 – 1 line up the decimal points and put one in for the answer. 4 10 There’s nothing to borrow here £ 5.00 so you have to borrow from – £0. 9 1 the next column 3) Look at each column from right to left, taking away the bottom number from 9 Now borrow 10 the top. 4 10 10 from this column £ 5.00 – £0. 9 1 10 – 1 = 9 4) Borrow 10 from the next column along £ 4.09 if you need to, as with whole numbers. 9–9=0 4–0=4 Always line up the decimal points... Decimal questions look tough, but if you remember to line up the decimal points they are exactly the same as whole number questions. Don’t forget about the carried or borrowed numbers either. Section One — Numbers  11 Multiplying by 10, 100, etc. Learn a few simple rules on this page and you’ll be able to multiply any number by 10, 100, 1000, etc. 1) To Multiply Any Number by 10 Move the decimal point ONE place BIGGER E.g. 1.6 × 10 = 1 6 and if it’s needed, ADD A ZERO on the end. 6213 × 10 = 6 2 1 3 0 672.12 × 10 = 6 7 2 1. 2 2) To Multiply Any Number by 100 E.g. 3.5 × 100 = 3 5 0 Move the decimal point TWO places 78 × 100 = 7 8 0 0 BIGGER and ADD ZEROS if necessary. 3.7734 × 100 = 3 7 7. 3 4 3) To Multiply by 1000 or 10 000, the same rule applies: Move the decimal point so many places E.g. 99.67 × 1000 = 9 9 6 7 0 BIGGER and ADD ZEROS if necessary. 1.729 × 10 000 = 1 7 2 9 0 You always move the DECIMAL POINT this much: 1 place for 10, 2 places for 100, 3 places for 1000, 4 for 10 000, etc. 4) To Multiply by Numbers like 20, 300, 8000 etc. MULTIPLY by 2 or 3 or 8 etc. FIRST, then move the decimal point so many places BIGGER ( ) according to how many zeros there are. Calculate 110 × 500. 1) First multiply by 5... 110 × 5 = 550 2)...then move the decimal point 2 places. 550 × 100 = 55000 Moving a decimal point to the right makes a number bigger... Nothing too strenuous on this page. The key thing to remember is that you move the decimal point so many places across according to how many zeros are in the number you’re multiplying by. Section One — Numbers 12 Dividing by 10, 100, etc. This page tells you how to do the opposite of the previous page. Remember — when you multiply by 10, 100, 1000, etc. you move the decimal point to the right... here, you move it to the left. 1) To Divide Any Number by 10 Move the decimal point ONE place E.g. 32.2 ÷ 10 = 3. 2 2 SMALLER and if it’s needed, REMOVE 6541 ÷ 10 = 6 5 4. 1 ZEROS after the decimal point. 4200 ÷ 10 = 4 2 0. 0 = 4 2 0 2) To Divide Any Number by 100 E.g. 333.8 ÷ 100 = 3. 3 3 8 Move the decimal point TWO places SMALLER and REMOVE ZEROS after the decimal point. 160 ÷ 100 = 1. 6 0 = 1. 6 1729 ÷ 100 = 1 7. 2 9 3) To Divide by 1000 or 10 000, the same rule applies: Move the decimal point so many E.g. 6587 ÷ 1000 = 6. 5 8 7 places SMALLER and REMOVE 978 ÷ 10 000 = 0. 0 9 7 8 ZEROS after the decimal point. You always move the DECIMAL POINT this much: 1 place for 10, 2 places for 100, 3 places for 1000, 4 for 10 000, etc. 4) To Divide by Numbers like 40, 300, 7000 etc. DIVIDE by 4 or 3 or 7 etc. FIRST, then move the decimal point so many places SMALLER (i.e. to the left ). Calculate 180 ÷ 200. 1) First divide by 2... 180 ÷ 2 = 90 2)...then move the decimal point 2 places smaller. 90 ÷ 100 = 0.9 Moving a decimal point to the left makes a number smaller... Knowing how to divide by multiples of 10, 100, 1000, etc. will make harder topics like percentages (see page 24) so much easier. So make sure you learn the rules of how it’s done. Section One — Numbers  13 Multiplying Without a Calculator Multiplying with a calculator is easy. The real challenge comes when you don’t have a calculator. There are lots of methods you can use — three popular ones are shown below. Just make sure you can do it using whichever method you prefer... The Traditional Method a) Work out 32 × 18 b) Work out 272 × 52 Split it into separate 3 2 2 7 2   × 1 8 This is 8 × 32 × 52 multiplications, then This is 272 × 2 2 5 16    51 4 4 add up the results in 3 2  0 This is 10 × 32   1 336 1 0 0 This is 272 × 50 columns (right to left). 576 1  4 1 1 4 4 This is 256 + 320 This is 544 + 13 600 This method’s got lots of Other Methods different names — you might Here are a couple more methods you can use. know it as lattice multiplication or Chinese multiplication. Work out 48 × 33. The Grid Method The ‘Gelosia’ Method: 48 × 33 1) Split up each number into its units and tens 1) Arrange the calculation (and hundreds and thousands if it has them). 4 8 as shown and do 4 3 48 = 40 + 8 and 33 = 30 + 3 easy multiplications to fill up the grid... 3 40 8 2) Draw a grid, with the ‘bits’ of the numbers 30 4 8 8 × 3 = 24 round the outside. 3 4 × 3 = 12 1 2 3 2 4 4 × 3 = 12 1 2 3 8 × 3 = 24 40 × 30 2 4 40 8 8 × 30 3) Multiply the bits 30 1200 240 2) Then just add up along the diagonals round the edge to (going right to left) to get the answer. fill each square. 3 120 24 4 8 40 × 3 8×3 1 2 3 2 4 1 2 4) Finally, add up 120 0 2 4 3 2  4 0 1 5 the numbers in   1 2 0 8 4 the squares. + 24 You might need to carry a   1  5  8 4 number over when adding up. RE VIS ION All that multiplication and not a calculator in sight... TASK Work out 25 × 17, 84 × 34 and 19 × 78 using the method you like best. (Answers below.) (Ans: 425, 2856, 1482) Section One — Numbers 14 Dividing Without a Calculator OK, time for some dividing without a calculator — ready for another challenge? Dividing Whole Numbers There are two common ways to do division — long division and short division. Here are some examples of both methods at work. Learn the method you find easier. You’ll find it helpful to write out the first few Short Division multiples of the number you’re dividing by. What is 468 ÷ 9? Multiples of 9: 9×1=9 1) Set out the division as shown. 9 468 9 × 2 = 18 2) Look at the first digit under the line. 9 × 3 = 27 4 doesn’t divide by 9, so put a zero 0 9 × 4 = 36 above and look at the next digit. 9 468 9 × 5 = 45 3) 9 × 5 = 45, so 9 into 46 goes 5 times, 05 carry the remainder with a remainder of 46 – 45 = 1. 9 4 6 18 4) 9 into 18 goes 2 times exactly. 0 5  2 the top line has 1 9 468 the final answer So 468 ÷ 9 = 52 Long Division What is 354 ÷ 8? Multiples of 8: 1) Set out the division as shown. 8 354 8×1=8 2) 3 doesn’t divide by 8. Write a zero 8 × 2 = 16 above the 3 and look at the next digit. 8 × 3 = 24 04 4 8 × 4 = 32 3) 8 into 35 goes 4 times, so put a 4 above the 5. 8 × 5 = 40 8 354 4) Take away 8 × 4 = 32 from 35. Write the answer underneath, and –3 2 move the next digit after the 35 down. 34 5) 8 into 34 goes 4 times, so put a 4 above the 4. –3 2 Take away 8 × 4 = 32 from 34. That leaves 2   2 and there are no more digits to bring down. 6) You are left with a remainder so give that in your final answer. So 354 ÷ 8 = 44 remainder 2 Try out both methods of dividing to see which you prefer... A lot of people think that short division is better because it can be quick — but it’s no good getting a quick answer if it’s not the right answer. Take your time and use whichever method you like the most. Section One — Numbers  15 Warm-Up and Practice Questions These warm-up questions will test whether you’ve learnt the facts properly. Go back over any bits you don’t know, then there will be no nasty surprises when it comes to the practice questions. Warm-Up Questions Only use your calculator in question 1. 1) Using BODMAS, work out (11 – 9) × (3 × 2)2. Then try it on your calculator. 2) Write these numbers out fully in words: a) 9 905 285 b) 6 054 203 3) Write < or > between each of the following pairs of numbers to make a true statement: a) 34 47 b) –2 –6 c) 1.4 0.8 d) –1.2 0.4 4) Put these numbers in order from smallest to largest: 13.54 3.42 0.55 0.004 8.63 1.23 0.032 5) Find the missing numbers: a) 96 + 67 = 100 +.... b) 8 × 21 = 2 ×... 6) Work out a) 113 + 645 + 39 b) 1239 – 387 7) Alia bought 3 items with prices £11.74, £7.12 and £0.76. What was the total cost? 8) Carry out the following multiplications and divisions: a) 4.9 × 100 b) 1729 × 10 c) 3.33 ÷ 10 d) 85.21 ÷ 1000 9) Work out a) 55 × 18 b) 150 × 22 c) 502 × 35 10) Work out a) 128 ÷ 8 b) 504 ÷ 7 c) 552 ÷ 12 Practice Questions Once you’ve managed to get through the warm-ups, you can have a go at these practice questions. Here’s a worked example to start you off, and then you’re on your own for some more practice. 1 Keisha has 26 boxes of coloured pencils and each box contains 15 pencils. a) How many pencils are there altogether? 2 6 × 1   5 This is 5 × 26 1 33 0 260 This is 10 × 26 390 390............................... [2 marks] b) Keisha wants to separate the pencils into boxes of 7 pencils. How many boxes would she completely fill? Write out the first 7 × 1=7 few multiples of 7 7 × 2 = 14 7 × 3 = 21 7 × 4 = 28 0 5 5 remainder 5 7 × 5 = 35 7   3 9 40 55.................... boxes [2 marks] Section One — Numbers 16 Practice Questions 2 Use BODMAS to work out the values of the following calculations. a) 5 + 6 × (7 – 5)......................... [2 marks] 57 + 3 b) 4 × 5 - 10......................... [2 marks] 3 Put each set of numbers in order of size, starting with the smallest. a) 48 572 309 6479 517 7.................................................................................................................. [2 marks] b) 0.4 0.303 0.43 0.31 0.44 0.043.................................................................................................................. [2 marks] 4 Jack has £6.37 in his wallet. On his birthday, his grandma gives him £25. a) How much money does he have altogether? £................................ [2 marks] b) He spends £8.90 on a new computer game. How much does he have left? £................................ [2 marks] Section One — Numbers  17 Negative Numbers Numbers less than zero are negative. You can add, subtract, multiply and divide with them. Adding and Subtracting with Negative Numbers Use the number line for addition and subtraction involving negative numbers: numbers get lower in this direction numbers get higher in this direction –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 move this way to subtract move this way to add What is –3 + 5? Start at –3 and move 5 places in the positive direction: So –3 + 5 = 2 –4 –3 –2 –1 0 1 2 3 Work out 3 – 6 Start at 3 and move 6 places in the negative direction: So 3 – 6 = –3 –4 –3 –2 –1 0 1 2 3 4 Find –1 – 5 Start at –1 and move 5 places in the negative direction: So –1 – 5 = –6 –6 –5 –4 –3 –2 –1 0 Use These Rules for Combining Signs + + makes + These rules are ONLY TO BE USED WHEN: + – makes – – + makes   – 1) Multiplying or dividing –  – makes + (invisible + sign) Find: a) –3 × 5 – + makes – so –3 × 5 = –15 b) –18 ÷ –3 – – makes + so –18 ÷ –3 = 6 2) Two signs appear next to each other Work out: a) 3 – –9 – – makes + so 3 – –9 = 3 + 9 = 12 b) 2 – –8 + –12 – – makes + and + – makes – so 2 – –8 + –12 = 2 + 8 – 12 = –2 ION Are you positive you’ve learnt all about negative numbers? RE VIS TIP Always start a calculation by combining signs — if they’re the same it makes + and if they’re different it makes –. Then it’s just a straightforward add, subtract, multiply or divide question. Section One — Numbers 18 Special Types of Number Whole numbers can be put into special groups with different properties. Here are a few of them. Even and Odd Numbers EVEN numbers all divide by 2 ODD numbers don’t divide by 2 2 4 6 8 10 12 14 16 18 20... 1 3 5 7 9 11 13 15 17 19 21... All EVEN numbers END in 0, 2, 4, 6 or 8 All ODD numbers END in 1, 3, 5, 7 or 9 These rules for adding, subtracting and multiplying odd and even numbers are always true. Try them out with some random odd or even numbers. The answer will give you the rule. Adding Subtracting Multiplying odd + odd = even odd – odd = even odd × odd = odd even + even = even even – even = even even × even = even odd + even = odd odd – even = odd odd × even = even even – odd = odd Square Numbers 1) When you multiply a whole number by itself, you get a square number. 1 2 3 1 2 2) They’re called square numbers because they’re like the areas of squares. 1×1=1 3 2×2=4 3) You have to know the first few off by heart. 3×3=9 12 22 32 42 52 62 72 82 92 102 112 122 1 4 9 16 25 36 49 64 81 100 121 144 (1×1) (2×2) (3×3) (4×4) (5×5) (6×6) (7×7) (8×8) (9×9) (10×10) (11×11) (12×12) Cube Numbers 1) When you multiply a whole number by itself, 1 1 then by itself again, you get a cube number. 3 4 1 2 2 3 4 2) They’re called cube numbers because they’re 1×1×1=1 2 3 4 like the volumes of cubes. 2×2×2=8 3×3×3=27 3) It’s useful to know these cube numbers off by heart. 4×4×4=64 13 23 33 43 53 103 1 8 27 64 125 1000 (1×1×1) (2×2×2) (3×3×3) (4×4×4) (5×5×5) (10×10×10) Learn the first twelve square numbers... It’s handy to be able to recognise square and cube numbers, so make sure you learn the ones on this page. Don’t forget that even numbers always divide by two — useful when simplifying calculations. Section One — Numbers  19 Prime Numbers There’s one more special type of number you need to know about — the prime numbers... PRIME Numbers Don't Divide by Anything Prime numbers are all the numbers that only come up in their own times table: 2 3 5 7 11 13 17 19 23 29 31 37... The only way to get ANY PRIME NUMBER is: 1 × ITSELF E.g. The only numbers that multiply to give 3 are 1 × 3 The only numbers that multiply to give 19 are 1 × 19 Show that 18 is not a prime number. Just find another way to make 18 other than 1 × 18: 3 × 6 = 18 18 divides by other numbers apart from 1 and 18, so it isn’t a prime number. Five Important Facts 2 3 5 7 1) 1 is NOT a prime number. 11 13 17 19 2) 2 is the ONLY even prime number. 21 23 27 29 3) The first four prime numbers are 2, 3, 5 and 7. 31 33 37 39 4) Prime numbers end in 1, 3, 7 or 9 (2 and 5 are the only exceptions to this rule). 41 43 47 49 5) But NOT ALL numbers ending in 1, 3, 7 or 9 are primes, as shown here: 51 53 57 59 (Only the circled ones are primes.) 61 63 67 69 How to FIND Prime Numbers — a very simple method 1) All primes (above 5) end in 1, 3, 7 or 9 — ignore any numbers that don’t end in one of those. 2) To find which of them ACTUALLY ARE primes you only need to divide each one by 3 and 7. If it doesn’t divide exactly by either 3 or 7 then it’s a prime. This works for primes up to 120. Find all the prime numbers in this list: 51, 52, 53, 54, 55, 56, 57, 58, 59 1) Get rid of anything that doesn’t end in 1, 3, 7 or 9: 51, 52, 53, 54, 55, 56, 57, 58, 59 2) Now try dividing 51, 53, 57 and 59 by 3 and 7: 51 ÷ 3 = 17 so 51 is NOT a prime number 53 ÷ 3 = 17.666... and 53 ÷ 7 = 7.571... so 53 is a prime number 57 ÷ 3 = 19 so 57 is NOT a prime number 59 ÷ 3 = 19.666... and 59 ÷ 7 = 8.428... so 59 is a prime number So the prime numbers in the list are 53 and 59. RE VIS ION A number is prime if it only divides by 1 and itself... TIP Remember — 1 is not a prime number, it’s the only exception to the rule. Section One — Numbers 20 Multiples, Factors and Prime Factors Make sure you know about prime numbers (p.19) before you try to find prime factors. Multiples and Factors The MULTIPLES of a number are just the values in its times table. Find the first 5 multiples of 12. You just need to find the first 5 numbers in the 12 times table: 12 24 36 48 60 The FACTORS of a number are all the numbers that divide into it exactly. Here’s how to find them: Find all the factors of 28. 1 × 28 1) Start off with 1 × the number itself, then try 2 ×, 2 × 14 Increasing by then 3 × and so on, listing the pairs in rows. 1 each time 3×— 4×7 2) Try each one in turn. Cross out the 5×— row if it doesn’t divide exactly. 6×— 7×4 3) Eventually, when you get a number repeated, stop. So the factors of 28 are: 1, 2, 4, 7, 14, 28 4) The factors are the numbers you haven’t crossed out. Finding Prime Factors — The Factor Tree Any whole number can be written as a string of prime numbers all multiplied together — this is called a prime factorisation. The easiest way to find a prime factorisation is using a factor tree. Find the prime factorisation of 40. 1) Start with the number at the top, 40 and split it into factors as shown. 4 10 2) Every time you get a prime, ring it. 2 2 3) Keep going until you can’t go further 2 5 (i.e. you’re just left with primes), then write the primes out in order. So 40 = 2 × 2 × 2 × 5 = 23 × 5 4) (Optional) Group numbers that are the You could split 40 into 2 and 20 or 5 and 8 instead same into powers (i.e. 2 × 2 × 2 = 23). — you’ll always get the same prime factorisation. The prime factorisation of a number is always the same, no matter how you split it up. Every number has a unique prime factorisation — no two are the same. ION Any number can be written as a product of prime factors... RE VIS TASK If this is all new to you, have another read through it — especially prime factorisation. Then scribble down a two-digit number and try to find its prime factorisation. Section One — Numbers  21 LCM and HCF Here are two big fancy names for you — but don’t be put off, they’re both easy. LCM — ‘Lowest Common Multiple’ ‘Lowest Common Multiple’ sounds a bit complicated, but all it means is this: The SMALLEST number that will DIVIDE BY ALL the numbers in question. METHOD: 1) LIST the MULTIPLES of ALL the numbers. 2) Find the SMALLEST one that’s in ALL the lists. 3) That’s the LCM. Find the lowest common multiple (LCM) of: a) 3 and 4 b) 9 and 12 Multiples of 3: 3, 6, 9, 12, 15,... Multiples of 9: 9, 18, 27, 36, 45, 54, 63,... Multiples of 4: 4, 8, 12, 16,... Multiples of 12: 12, 24, 36, 48, 60, 72,... So the LCM of 3 and 4 is 12. So the LCM of 9 and 12 is 36. HCF — ‘Highest Common Factor’ ‘Highest Common Factor’ — all it means is this: The BIGGEST number that will DIVIDE INTO ALL the numbers in question. METHOD: 1) LIST the FACTORS of ALL the numbers. 2) Find the BIGGEST one that’s in ALL the lists. 3) That’s the HCF. Find the highest common factor (HCF) of 12 and 30. Factors of 12 are: 1, 2, 3, 4, 6, 12 Factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 So the highest common factor (HCF) of 12 and 30 is 6. List the multiples or factors when finding LCMs and HCFs... Be careful when you’re listing the factors of a number — make sure you use the proper method (as shown on p.20). This way you’re much less likely to miss one or make a silly mistake. Section One — Numbers 22 Warm-Up and Practice Questions It’s time to face the music — try these warm-up questions to see how much you’ve learnt. When you feel like you’re ready you should move on to the tougher practice questions. Warm-Up Questions Try these without 1) Work out: a) –6 + 11 b) –5 – 10 c) –3 × –6 using a calculator.d) 21 ÷ –7 2) On Friday the temperature in Negaton was –5 °C and the temperature in Tiverville was –19 °C. What was the difference in temperature between Negaton and Tiverville? 3) Which of the following numbers are odd? 53, 61, 123, 74, 90, 872, 712, 7305, 5012 4) Find the first two numbers that are both square numbers and cube numbers. 5) Write down all the prime numbers from this list: 49, 63, 38, 73, 77, 18, 39, 83 6) Find: a) the first 8 multiples of 9, b) all the factors of 36. 7) Express 60 as a product of its prime factors. 8) Find the lowest common multiple (LCM) of 3 and 7. 9) Find the highest common factor (HCF) of 32 and 48. Practice Questions Take a look at the worked question below and see if you can follow the working. Once you’ve read through it carefully you can have a go at the practice questions on the next page. 1 The factors of a number are all the numbers that divide into it. a) Find all the factors of the following numbers. 1, 2, 11, 22 22...................................................................................................................................... 1, 41 41...................................................................................................................................... 1, 3, 11, 33 33...................................................................................................................................... 1, 2, 5, 10, 11, 22, 55, 110 110...................................................................................................................................... [4 marks] b) Which of the four numbers above is prime? Give a reason for your answer. 41 is the only prime number. It is the only number from the list............................................................................................................................................... where the factors are just 1 and itself................................................................................................................................................ [2 marks] c) What is the HCF of 22, 33 and 110? Look at the lists above to find the highest number which is a factor of 22, 33 and 110. 11..................... [1 mark] Section One — Numbers  23 Practice Questions 2 Factor trees allow you to find the prime factors of a number. a) Complete the factor tree. 24 12 2.................................... [2 marks] b) Write down the prime factorisation of 24............................................................. [1 mark] 3 In a football league, teams get 3 points for a win, –3 points for a loss and 0 points for a draw. The table below shows the points for 5 teams in the league. Team Number of points Ringford Rovers –3 Woodgreen Wanderers 3 Fleeting Footers 12 Steventon Scorers –6 Trybridge Tigers –9 a) Which of these teams is lowest in the league?................................................................................... [1 mark] b) The Steventon Scorers play the Woodgreen Wanderers and win. How many points do the Steventon Scorers have now?........................................ [1 mark] c) How many points do the Ringford Rovers need to equal the score of the Fleeting Footers?........................................ [1 mark] Section One — Numbers  24 Fractions, Decimals and Percentages Fractions, decimals and percentages are three different ways of describing when you’ve got part of a whole thing. They’re closely related and you can convert between them. This table shows the really common conversions which you ought to know straight off: Fraction Decimal Percentage 1 0.5 50% Fractions with a 1 on the 2 1 1 1 1 top (e.g. 2 , 3 , 4 , etc.) 0.25 25% 4 are called unit fractions. 3 0.75 75% 4 0.3333... and 0.6666... 1 0.333333... 33 1 3% are known as ‘recurring’ 3 2 2 decimals — the same 3 0.666666... 66 3 % pattern of numbers carries on repeating itself forever. 1 0.1 10% 10 2 0.2 20% 10 1 0.2 20% 5 2 0.4 40% 5 The more of those conversions you learn, the better — but for those that you don’t know, you must also learn how to convert between the three types. These are the methods: Divide × by 100 Fraction Decimal Percentage 7 E.g. 20 is 7 ÷ 20 = 0.35 e.g. 0.35 × 100 = 35% Fraction The awkward one Decimal ÷ by 100 Percentage Converting decimals to fractions is a bit more awkward. The digits after the decimal point go on the top, and a power of 10 on the bottom — with the same number of zeros as there were decimal places. 6 3 7 0.6 = 10 0.3 = 10 0.7 = 10 etc. These can often be cancelled down 12 0.12 = 100 78 0.78 = 100 5   0.05 = 100 etc. — see the next page. N Practise converting between fractions, percentages and decimals IO RE VIS TIP Learn the conversions in the table at the top of the page off by heart. Work out any other conversions using the four methods for converting — decimals to fractions is the tricky one. Section One — Numbers  25 Fractions This page tells you how to deal with fractions without your calculator. 1 4 Equivalent Fractions 4...is equivalent to... 16 1) Equivalent fractions are equal in size... 2)...but the numbers on the top and bottom are different. 3) To get from one fraction to an equivalent one — MULTIPLY top and bottom by the SAME NUMBER: ×2 ×5 ×100 1 =2 3 = 15 1 = 100 2 4 4 20 5 500 ×2 ×5 ×100 Cancelling Down ÷3 1) You sometimes need to simplify a fraction by ‘cancelling down’. 3 =1 2) This means DIVIDING top and bottom by the SAME NUMBER. 15 5 3) To get the fraction as simple as possible, ÷3 you might have to do this more than once: ÷10 ÷2 20 = 2 = 1 40 4 2 ÷10 ÷2 Ordering Fractions Ordering fractions — first you just have to make the bottom numbers (denominators) of the two fractions equal by finding equivalent fractions. Then compare the size of the top numbers (numerators). Which is bigger, 2 3 3 or 4 ? 1) Look at the bottom numbers of the fractions: ×4 ×3 3 and 4. 2 8 3 = 9 3 = 12 4 12 2) Think of a number they will both go into — try 12. 3) Change each fraction (make equivalent fractions) ×4 ×3 so the bottom number is 12. 4) Now check which is bigger by looking at their 9 is bigger than 8, top numbers. 3 2 so 4 is bigger than 3. Always cancel down fractions as far as possible... EXAM TIP A question might ask you to simplify a fraction or give your answer in its simplest form. Don’t be put off — all it means is cancel the fraction down until you can’t cancel any further. Section One — Numbers 26 Fractions When dealing with fractions, always simplify your answer as much as possible. Multiplying 1) Multiply the top numbers to find the numerator... 2)...and multiply the bottom numbers to find the denominator. 3 4 3 # 4 12 E.g. 5 # 7 = 5 # 7 = 35 Dividing 1) Turn the 2nd fraction UPSIDE DOWN... 2)...and then multiply, as shown above. 3 1 3 3 3#3 9 E.g. 4 ' 3 = 4 # 1 = 4 # 1 = 4 Adding and Subtracting 1) If the bottom numbers are the same, add or subtract the TOP NUMBERS ONLY, leaving the bottom number as it is. 2 + 1 = 2+1 = 3 = 1 5 - 3 = 5-3 = 2 E.g. adding: 6 6 6 6 2 and subtracting: 7 7 7 7 2) If the bottom numbers are different, you have to make them the same using equivalent fractions (see previous page). Work out 1 + 5 4 12. 1) Look at the denominators of the fractions: 4 and 12 1 1 = 3 2) They are both factors of 12 so change 4 so that the denominator is 12. 4 12 3) Now add the fractions — 3 + 5 = 8 =2 simplify your answer if possible. 12 12 12 3 What fraction of this page have you learnt? When adding or subtracting fractions, remember that the denominators of the fractions always have to be the same. When you’re multiplying or dividing, the denominators don’t have to be the same. Section One — Numbers  27 More Fractions One last page on fractions — starting with what happens when you have a fraction greater than 1. Mixed Numbers Mixed numbers have an integer part and a fraction part. Improper fractions are where the top number is bigger than the bottom number. 1. 4 as an improper fraction. Write 3 5 1) Think of the mixed number as an addition: 3 54 = 3 + 54 15 + 4 2) Turn the integer part into a fraction: 3 + 54 = 155 + 54 = 5 = 195 23 2. Write 3 as a mixed number. 1) Divide the top number by the bottom. 23 ÷ 3 = 7 remainder 2 2) The answer gives the whole number part. 23 = 7 2 3) The remainder goes on top of the fraction. 3 3 You can add, subtract, multiply and divide mixed numbers by turning them into improper fractions and then using the rules on the previous page. 5 1 12 22 34 6 1 2 16 5 11 2 Adding: 1 7 + 3 7 = 7 + 7 = 7 = 4 7 Subtracting: 5 3 - 1 3 = 3 - 3 = 3 = 3 3 1 1 7 1 7 2 2 2 8 2 3 6 1 Multiplying: 3 2 × 4 = 2 × 4 = 8 Dividing: 3 ÷2 3 = 3 ÷ 3 = 3 × 8 = 24 = 4 Finding a Fraction of Something Multiply the ‘something’ by the TOP of the fraction, and divide it by the BOTTOM. It doesn’t matter which order you do these two steps in — just start with whatever’s easiest. 9 9 E.g. 20 of £360 = £360 ÷ 20 × 9 = £162 In other words, £162 is 20 of £360. You can write one number as a fraction of another number just by putting the first number over the second and cancelling down. This works if the first number is bigger than the second number too — you’ll just end up with a fraction greater than 1. E.g. 32 is 32 = 8 44 = 11 44 11 of 44, and 44 is 32 8 of 32. Don’t be put off by mixed number calculations... EXAM TIP Mixed number calculations are just the same as normal fraction calculations once you convert to improper fractions — don’t forget to turn your answer back into a mixed number. Section One — Numbers 28 Percentages These simple percentage questions shouldn’t give you much trouble. Especially if you remember: 1) ‘Per cent’ means ‘out of 100’, so 20% means ‘20 out of 100’ = 20. 100 2) If a question asks you to work out the percentage OF something you can replace the word OF with a multiplication (×). ‘Percentage Of ’ Questions If you have a calculator, these questions are easy. Find 18% of £4. 18 1) Translate the question into maths. 18% of £4 = 100 × 4 18 ÷ 100 × 4 2) Work it out. = £0.72 = 72p If you don’t have a calculator, you’re going to have to do a bit more work. Find 35% of 120 kg. 1) First find 10% by dividing by 10. 120 ÷ 10 = 12 2) Then find 5% by dividing 10% by 2. 12 ÷ 2 = 6 3) Make 35% by adding up 10% and 5%. 35% = 10% + 10% + 10% + 5% = 12 + 12 + 12 + 6 = 42 kg Writing One Number as a Percentage of Another To write one number as a percentage of another, divide the first number by the second, then multiply by 100. Give 36p as a percentage of 80p. Divide 36p by 80p, then multiply by 100: (36 ÷ 80) × 100 = 45% Farmer Littlewood measured the width of his prized pumpkin at the start and end of the month. At the start of the month it was 80 cm wide and at the end of the month it was 1.4 m wide. Give the width at the end of the month as a percentage of the width at the start. 1) Make sure both amounts are in the same units 1.4 m = 140 cm — convert 1.4 m to cm. 2) Divide 140 cm by 80 cm, then multiply by 100: (140 ÷ 80) × 100 = 175% Make sure you know the percentage basics really well... There are more percentage questions coming up later in the book (p.68), so make sure you learn the basics before you move on. If anything is unclear, have another read through these examples. Section One — Numbers  29 Warm-Up and Practice Questions It’s about time you had a bit of practice on fractions, decimals and percentages. Have a go at these warm-up questions first — they’ll be a good indication of how much you’ve learnt. Warm-Up Questions Try questions 1-5 without using a calculator. 1) Turn the following decimals into fractions and reduce them to their simplest form. a) 0.8 b) 0.3 c) 0.04 d) 0.15 1 2 17 2) Which is greater: a) 4 or 30%? b) 50% or 5 ? c) 0.6 or 25 ? 20 9 3) Cancel these down as far as possible: a) 28 b) 36 4 2 4) Which is bigger, 5 or 3 ? 2 1 3 5 1 9 1 2 3 1 1 1 5) a) 5 # 6 b) 7 ' 8 c) 8 + 8 d) 2 5 - 1 5 e) 8 + 2 f) 3 + 4 6) Find: a) 39% of 505 b) 15% of 340 7) Give: a) 18 m as a percentage of 60 m b) 1.2 km as a percentage of 480 m Practice Questions Have a look at the practice questions below — I’ve written in the answers for you. Don’t be too jealous though, there’s plenty more to keep you going on the next page. Enjoy. 1 Order the following fractions from smallest to biggest: 2 1 1 3 2 6 All of the fractions need to have the same denominator. 2 and 3 both go into 6, so put 6 on the bottom and find the equivalent fractions. Then use the top number to put them in order. You multiply the bottom by 4 3 1 Write your answer with the 6 6 6 original fractions in the question, 2 to make 6, so you need to multiply the top by 2 as well. not the equivalent fractions. You multiply the bottom by 3, 1 1 2  so multiply the top by 3 too. 6 2 3..................................... [2 marks] 18 2 Lucy got 20 in her maths test. Saqib got 85% in his. Which student got the higher score? You need to write both scores in the same way — it's easiest to write Lucy's score as a percentage. First convert the 18 fraction to a decimal. 20 is 18 ÷ 20 = 0.9 Then convert the 0.9 × 100 = 90% decimal to a percentage. Lucy......................... [2 marks] Section One — Numbers 30 Practice Questions 3 Fill in the missing numbers. a) 5 = 4 4 [1 mark] b) 3 = 2 5 5 [1 mark] 4 Callum has 15 hamsters. 6 of his hamsters have brown fur. a) What percentage of his hamsters have brown fur?......................... [1 mark] b) 20% of his hamsters have white fur. How many white-furred hamsters does he have?......................... [1 mark] 5 Helen has 21 marbles. 2 of them are blue, 1 are red and the rest are green. 7 3 How many green marbles does Helen have?......................... [2 marks] Section One — Numbers  31 Rounding Numbers You need to be able to use 3 different rounding methods. We’ll do decimal places first, but there’s the same basic idea behind all three. If you’re rounding to Decimal Places (d.p.) 2 d.p. the last digit is the second digit after To round to a given number of decimal places: the decimal point. 1) Identify the position of the ‘last digit’ from the number of decimal places. 2) Then look at the next digit to the right — called the decider. 3) If the decider is 5 or more, then round up the last digit. If the decider is 4 or less, then leave the last digit as it is. 4) There must be no more digits after the last digit (not even zeros). What is 21.84 correct to 1 decimal place? 2 1. 8 4 = 21.8 LAST DIGIT to be written DECIDER The LAST DIGIT stays the same (1st decimal place because because the DECIDER is 4 or less. we’re rounding to 1 d.p.) What is 39.7392739 to 2 decimal places? 39. 7392739 = 39.74 LAST DIGIT to be written DECIDER The LAST DIGIT rounds UP (2nd decimal place because because the DECIDER is 5 or more. we’re rounding to 2 d.p.) Watch Out for Pesky Nines If you have to round up a 9 (to 10), replace the 9 with 0, and add 1 to digit on the left. Round 48.897 to 2 d.p.: The question asks for 2 d.p. so 90 you must put 48.90 not 48.9. 48.897 48.89 48.90 to 2 d.p. LAST DIGIT DECIDER EXAM Always take extra care when the last digit is a 9... TIP Make sure you always give your answer to the number of decimal places asked for. Section One — Numbers 32 Rounding Numbers Here are the other two rounding methods — they’re each slightly different in their own way. Significant Figures (s.f.) The 1st significant figure of any number is the first digit which isn’t a zero. The 2nd, 3rd, etc. significant figures follow straight after the 1st — they’re allowed to be zeros. 0.002309 506.07 SIG. FIGS: 1st 2nd 3rd 4th 1st 2nd 3rd 4th To round to a given number of significant figures: 1) Find the last digit — if you’re rounding to, say, 3 s.f., then the 3rd significant figure is the last digit. 2) Use the digit to the right of it as the decider, just like for d.p. 3) Once you’ve rounded, fill up with zeros, up to but not beyond the decimal point. Round 1276.7 to 2 significant figures. LAST DIGIT is the 2nd sig. fig. Need two zeros to fill up to decimal point. 1276.7 = 1300 DECIDER is 5 or more LAST DIGIT rounds UP To the Nearest Whole Number, Ten, Hundred etc. You might be asked to round to the nearest whole number, ten, hundred, thousand or million