1

Year 9 Higher – Number – Presumed knowledge

This is material that you have met before. Some of it might be easy, some a little more demanding.

The notes at the start are designed to assist you, but you may need to track down more help. Use

revision guides, friends, teachers, websites – anything you like. As you answer each one of the

questions make sure you do the following:

(i) show your working. Your teacher needs to know that you really understand this work and the best

way to do this is to set your work out neatly, demonstrating your thinking. This may just mean

writing down the question as well as the answer if it is a simple calculation.

(ii) take a minute to make sure that you have read the question carefully and that your answer

makes sense.

(iii) ask yourself if you are really happy that you understand what you are doing. If you have any

doubt DO SOMETHING!!! (the easiest thing may be to ask your teacher next lesson).

Contents

Page 2.............................................................Four rules for positive and negative integers

Page 3.............................................................Fraction Arithmetic – addition and subtraction

Page 4.............................................................Fraction Arithmetic – multiplication and division

Page 5.............................................................Decimal Arithmetic – addition and subtraction

Page 5 and 6...................................................Decimal Arithmetic – multiplication

Page 7.............................................................Decimal Arithmetic – division

Page 8.............................................................Ratio

Page 9.............................................................Rules for positive, integer indices

Page 10...........................................................Rounding (decimal places)

Page 11...........................................................Rounding (significant figures)

Page 12...........................................................Estimation

Page 13...........................................................Factors, primes and multiples

Pages 14 and 15..............................................Prime Factorisation, HCF and LCM

Pages 16, 17 and 18........................................Fraction/Decimal equivalence

Pages 19 and 20..............................................QUESTIONS

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1) Four rules for positive and negative integers

You need to be able to add and subtract positive and negative integers (whole numbers). This is all

about moving in the correct direction along a number line. Make sure you start in the right place,

make sure you move in the right direction, make sure you move the right distance.

Adding a positive number: move UP the number line Adding a negative number: move DOWN the number line Subtracting a positive number: move DOWN the number line Subtracting a negative number: move UP the number line EXAMPLES (a) 5 + (-7)

You need to START at 5 on the number line. You’re adding a negative, so you move DOWN the number line. You need to move 7 places.

So: 5 + (-7) = -2 (b) (-4) – (-9)

You need to START at -4 on the number line. You’re subtracting a negative, so you move UP the number line. You need to move 9 places.

So: (-4) – (-9) = 5 (c) (-1) + (-3)

You need to START at -1 on the number line. You’re adding a negative, so you move DOWN the number line. You need to move 3 places.

So: (-1) + (-3) = -4 Notice that in this case “two minuses definitely do NOT make a plus!”

You also need to be able to MULTIPLY and DIVIDE with positive and negative integers. So long as you

know your tables this is easy (but you do NEED to know your tables).

There is a simple rule for multiplying and dividing with negative numbers... If the signs are the same then the answer is ALWAYS POSITIVE If the signs are different then the answer is ALWAYS NEGATIVE Examples (a) 4 x (-5) = -20 (negative since signs are different – one negative, one positive) (b) (-3) x (-9) = 27 (positive since signs are the same – both negative) (c) 24 ÷ (-8) = -3 (negative since signs are different – one positive, one negative) (d) (-35) ÷ (-5) = 7 (positive since signs are the same – both negative)

3

2) FRACTION ARITHMETIC You need to be able to add, subtract, multiply and divide “simple” fractions and mixed numbers. You need to be able to give your answer “in its simplest form”. You need to be able to change between mixed numbers and top-heavy fractions. You need to be able to find other fractions that are equivalent to a given fraction. Addition and Subtraction The absolute CRUCIAL thing here is that the fractions need to be written with the same denominator. Firstly you need to choose this denominator – you need a “common multiple” of the given denominators, usually the “lowest common multiple”. Once you’ve chosen this new denominator, you write each of the original fractions as an equivalent fraction with that denominator. You then simply add or subtract the numerators, simplifying your answer if necessary. Unless the question tells you otherwise you can leave an answer larger than 1 as a mixed number or a top-heavy fraction. For addition and subtraction of mixed numbers, you can EITHER deal with the “whole number parts” on their own (normally easier, but sometimes tricky) or write the mixed numbers as top-heavy fractions.

DON’T EXPECT TO BE ABLE TO DO THESE QUICKLY!!! Examples (try to set your answers out like this)

a) 15

13

15

3

15

10

5

1

3

2 b)

12

1

12

8

12

9

3

2

4

3

c) 14

35

14

174

14

10

14

74

7

5

2

131

7

53

2

11

OR

14

35

14

83

14

52

14

21

7

26

2

3

7

53

2

11

d) 15

81

15

72

15

12

15

52

5

4

3

113

5

41

3

13

OR

15

81

15

23

15

27

15

50

5

9

3

10

5

41

3

13

You can’t have a top-heavy fraction in a mixed

number – so this needs to be changed to 14

31

You need to deal with this negative

fraction – take 7 away from 15 to get 8

4

Multiplication

For simple fractions, just multiply tops and multiply bottoms. For numbers given as mixed numbers, change them to top-heavy fractions first. It is always a good idea to look to “cancel” numbers from top and bottom at the start in order to keep the numbers small. Always make sure that your answer is in its simplest form, although unless the question specifically asks for a certain form, you may leave answers greater than one as mixed numbers or top-heavy. Examples (try to set your answers out like this)

a) 21

10

7

5

3

2 No need to simplify at all

b) 10

1

20

2

5

2

4

1 You COULD have divided top and bottom by two at the beginning:

10

1

5

2

4

1

c) 24

91

8

13

3

7

8

51

3

12 , or if you prefer

24

193

d) 18

35

9

10

4

7

9

11

4

31 , or if you prefer

18

171

Division

If necessary write any mixed numbers as top-heavy fractions. Turn the SECOND fraction “upside-down” – this is also known as taking the “reciprocal” of the second fraction. Now multiply, following the rules above, and you have your answer. Examples (try to set your answers out like this)

a) 8

9

2

3

4

3

3

2

4

3 , or if you prefer

8

11

b) 14

5

4

5

7

2

5

4

7

2

c) 14

25

7

5

2

5

5

7

2

5

5

21

2

12 , or if you prefer

24

111

To sum up fraction arithmetic – you need to know that there are different rules for different operations. A “common denominator” is ONLY necessary when you’re doing addition or subtraction. Do NOT expect to be able to do these quickly and make sure you show ALL your working!

2

1

5

2

1

2

5

3) Decimal Arithmetic You need to be able to add, subtract, multiply and divide decimals. Addition and Subtraction Just keep the decimal points in line! Remember to “carry” and “borrow” just as if the numbers were whole numbers (integers). Examples (set your work out in columns like this) 1) 2.32 + 0.784 2 . 3 2 0 (you can add this zero if you like!) + 0 . 7 8 4

3 . 1 0 4 2) 11.035 + 8.7 + 1.294 1 1 . 0 3 5 8 . 7 0 0 (again, you can add these zeros if you like!) + 1 . 2 9 4

2 1 . 0 2 9

3) 13.3 – 8.978

1 3 . 3 0 0 (when subtracting these zeros are important) – 8 . 9 7 8

4 . 3 2 2

You may be confident working these out in your head – please don’t be tempted to do so! Always

set your work out in columns!

Remember – the key for addition and subtraction is to KEEP THE DECIMAL POINTS IN LINE!!!

Multiplication

You need to be able to use a method of long multiplication – in the exam you can use any method

you like (I demonstrate two different methods below) but the ONLY method you’ll get “working

marks” for if you get the answer wrong is the “traditional” method (first example).

Start by working out the answer WITHOUT WORRYING about the decimal points at all. You then

need to think carefully about where the decimal point goes in your answer. Generally it does NOT

stay in line! There are two distinct methods for working out where to put it:

(i) ESTIMATION – work out roughly how big the answer should be by rounding the original numbers,

then put the decimal point in the “best place”

(ii) COUNTING DECIMAL DIGITS – count up the number of digits to the RIGHT of decimal points in the

question, then make sure that you have the same number of “decimal digits” in your answer.

1 1

1 1 1

1 1 1 2 2 9

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Examples – whatever method you use show your working clearly. Only the FIRST of these

techniques will gain method marks in the exam!

1) 1.2 × 3.4 (2 underlined digits are “decimal” digits)

Work out 12 × 34, then make sure that there are TWO digits to the right of the decimal point in your

answer (just like in the question)

1 2 × 3 4 3 6 0 (this line is 30 × 12) 4 8 (this line is 4 × 12) 4 0 8

So 12 × 34 = 408 giving our answer of 1.2 × 3.4 = 4.08 (TWO decimal digits)

Alternatively, to work out where the decimal point has to go, do a rough estimate...

1.2 x 3.4 1 x 3 = 3, so we put a decimal point in our answer to get something roughly equal to 3.

The best we can do is 4.08

2) “Diagonal method” (sometimes called Gelosian, sometimes called Napier’s Bones). You CAN use

this method, but it will NOT get you method marks if you get the wrong answer.

2.34 x 1.8 - work out 234 x 18, then make sure that there are THREE decimal digits in your answer

(since there are three underlined decimal digits).

So – 234 x 18 = 4212 giving our answer of 2.34 x 1.8 = 4.212 (three decimal digits)

Alternatively, to work out where the decimal point has to go, do a rough estimate...

2.34 x 1.8 2 x 2 = 4, so we put a decimal point in our answer to get something roughly equal to 4.

1

2 4 3

1

8

2

2

0

6 1 2

0 0

3

3

4

4

1

4 2 1 2 1

7

Division

In order to be able to divide decimals you need to be able to divide whole numbers. Basically, you

need a method of long division. There are several methods you could use but the one below is my

favourite.

Start by changing the question so that you are dividing by an integer (whole number). You do this by

moving the decimal point in BOTH parts of the question. You only need the SECOND number (the

one you are dividing by) to be an integer. Now do a long division and keep the decimal point of your

answer in line with the decimal point of the question.

To do this write down the first ten multiples of the number you are dividing by....then do some

mental arithmetic to work out the remainder at each stage.

As with quite a lot of what we have seen – do NOT expect to be able to do this quickly.

Example

8.058 ÷ 3.4

We don’t want to divide by 3.4 – change the question so that we divide by 34.

8.058 ÷ 3.4 becomes 80.58 ÷ 34

Now write out the first ten multiples of 34:

1 × 34 = 34 2 × 34 = 68 3 × 34 = 102 4 × 34 = 136 5 × 34 = 170 6 × 34 = 204 7 × 34 = 238 8 × 34 = 272 9 × 34 = 406 10 × 34 = 340 (There are short cuts for this and you won’t always need ALL the multiples) Now set the division out CLEARLY, making sure you’re careful with your remainders... Keep the decimal points in line!

3 4 8 0 . 5 8

Step 1 – how many 34s in 8? – too small so... Step 2 – how many 34s in 80? 2 with 12 left over Step 3 – how many 34s in 125? 3 with 23 left over Step 4 – how many 34s in 238? 7 exactly... So - 80.58 ÷ 34 = 2.37 and the answer to the original question is the same: 8.058 ÷ 3.4 = 2.37

2 . 3 7 12 23

8

4) Ratio You need to understand what we mean by two or more quantities being in a particular ratio. You

need to understand ratio notation (eg 2 : 3 or 3 : 6 : 7 etc). You need to be able to simplify a ratio

into its simplest form and also into the forms 1 : n and n : 1 where n is not necessarily an integer.

You need to understand how to change between writing the relationship between two quantities as

a ratio and writing one of the quantities as a fraction of the other (or of the whole).

You need to be able to solve problems involving ratio.

You need to be able to share a given quantity in a particular ratio.

Simplifying ratios

We can simplify a ratio in a very similar manner to simplifying a fraction – we divide (or sometimes

multiply) each “part” by the same number. We have finished when the different parts contain no

common factor. We are not allowed to use decimals or fractions in our final answer (although they

may appear in the question).

Examples

1) 4 : 12 simplifies to 1 : 3 (divide both numbers by 4) 2) 12 : 18 simplifies to 2 : 3 (divide both numbers by 6) 3) 1.5 : 2 simplifies to 3 : 4 (multiply both numbers by 2) 4) 1.5 : 4.5 simplifies to 1 : 3 (multiply both numbers by 2, then divide by 3) Writing ratios in the form 1 : n and n : 1 All you need to do here is divide by whatever number you need to change to 1. Examples

3 : 7 in the form n : 1 – divide by 3 and leave 7 ÷ 3 as a fraction – 1 : 7/3 (1 : 2.3 is OK)

5 : 8 in the form n : 1 - divide by 8 and leave 5 ÷ 8 as a fraction – 5/8 : 1 (0.625 : 1 is OK)

Sharing in a ratio Step 1 : add all the quantities in the given ratio Step 2 : divide the total amount by this sum Step 3 : multiply what you get by each “part” of the ratio Example Divide (or share) £72 in the ratio 1 : 3 : 4 Step 1: 1 + 3 + 4 = 8 Step 2: 72 ÷8 = 9 Step 3: 9 × 1 = 9, 9 × 3 = 27 and 9 × 4 = 36 So the three required quantities are £9, £27 and £36.

.

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5) Rules for positive, integer indices Important: “integers” are whole numbers and “indices” is another word for “powers” Let’s start by making sure you know what we’re talking about.

xn (x to the power n) means “x multiplied by itself n times” 23 (2 to the power 3 or 2 cubed) means 2 × 2 × 2 (equals 8) 52 (5 to the power 2 or 5 squared) means 5 × 5 (equals 25) 65 (6 to the power 5) means 6 × 6 × 6 × 6 × 6 (equals 7776) The three rules you need to know are:

xa × xb = x(a + b)

xa ÷ xb = x(a - b)

(xa) = x(ab)

The first two rules only work for powers of the SAME number. If you want to simplify expressions involving powers of DIFFERENT numbers, this will only be possible if you can write the different numbers as powers of the same number. For example: 43 × 82 = (22) × (23) = 26 x 29 = 215 Examples 24 x 27 = 211 312 ÷ 34 = 38

(52) = 512

8

5

13

5

94

5

3322

5

32

33

3

3

33

3

33

3

279

)()(

b

3 3

6

10

6) Rounding (decimal places and significant figures) Basic aim – to take ANY starting number (normally positive, but the rules will work if it’s negative too!) and ROUND it (give a rough value) to a given level of accuracy. Decimal Places If a number has lots of decimal digits then we may be asked to round to a given number of decimal places. The method for this is ALWAYS the same..... Step 1 – always start at the decimal point Step 2 – count the designated number of places Step 3 – decide whether the digit you’ve arrived at stays the same (if it is followed by 0, 1, 2, 3 or 4) or “rounds up” (if it’s followed by 5, 6, 7, 8 or 9). The only time this can get a little bit complicated is if a “9” rounds up, basically “pushing” an extra 1 into the previous column and leaving a zero in the starting column. ALWAYS CHECK that your answer has the same number of “decimal places” as the question asks for – this may mean leaving zeros in the final column(s) Examples Round 1.58297 to 1dp, 2dp, 3dp and 4dp 1.58297 = 1.6 (1dp) 1.58297 = 1.58 (2dp) 1.58297 = 1.583 (3dp) 1.58297 = 1.5830 (4dp)

Round 28.0279 to 1dp, 2dp and 3dp

28.0279 = 28.0 (1dp) 28.0279 = 28.03 (2dp) 28.0279 = 28.028 (3dp)

This is the first

“decimal digit” – it

rounds up to a 6

because it is followed

by an 8

This is the second

“decimal digit” – it

stays the same

because it is followed

by a 2

This is the third

“decimal digit” – it

rounds up to a 3

because it is followed

by a 9

This is the fourth “decimal

digit” – it rounds up because

it is followed by a 7. Because

it is a 9 and should round up

to 10, we add one to the

previous column and fill this

column with a zero

This is the first

“decimal digit” – it

stays the same

because it is followed

by a 2

This is the second

“decimal digit” – it

round up to a 3

because it is followed

by a 7

This is the third

“decimal digit” – it

rounds up to an 8

because it is followed

by a 9

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Significant figures We may be asked to round ANY number at all to a given number of significant figures. The method for this is ALWAYS the same..... Step 1 – start at the first NON ZERO digit Step 2 – count the designated number of places Step 3 – decide whether the digit you’ve arrived at stays the same (if it is followed by 0, 1, 2, 3 or 4) or “rounds up” (if it’s followed by 5, 6, 7, 8 or 9). The only time this can get a little bit complicated is if a “9” rounds up, basically “pushing” an extra 1 into the previous column and leaving a zero in the starting column. ALWAYS CHECK that your answer is of the same “order of magnitude” as the starting number (we are giving an approximate value so it has to be the right size) – sometimes we need to fill in columns of zeros to make sure our answer is correct! Examples Round 27603 to 1sf, 2sf, 3sf and 4sf 27603 = 30000 (1sf) 27603= 28000 (2sf) 27603 = 27600 (3sf) 27603 = 27600 (4sf)

Notice that the final two answers are the same (!). Rounding to significant figures can be a little bit more complicated than rounding to decimal places. Round 0.018275 to 1sf, 2sf, 3sf and 4sf 0.018275 = 0.02 (1sf) 0.018275 = 0.018 (2sf) 0.018275 = 0.0183 (3sf) 0.018275 = 0.01828 (4sf)

This is the first

“significant figure” – it

rounds up to a 3

because it is followed

by a 7. We need to fill

in 4 zeros.

This is the second

“significant figure” – it

rounds up to an 8

because it is followed

by a 6. We need to fill

in 3 zeros.

This is the third

“significant figure” – it

stays the same

because it is followed

by a 0. We need to fill

in 2 zeros.

This is the fourth

“significant figure” – it

stays the same because

it is followed by a 3. We

need to fill one

ADDITIONAL zero.

This is the first

“significant figure” – it

rounds up to a 2

because it is followed

by a 8.

This is the second

“significant figure” – it

stays the same

because it is followed

by a 2.

This is the third

“significant figure” – it

rounds up to a 3

because it is followed

by a 7.

This is the fourth

“significant figure” – it

rounds up to an 8

because it is followed

by a 5.

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7) Estimation When a question asks you to “Estimate” an answer, it’s looking for a specific method. Step 1: round EVERY value in the question to 1 significant figure (look at the previous section if you’re unsure) – make sure you show this step CLEARLY Step 2: showing any intermediate steps, work out the answer Examples (set your answers out like this – write the question, then write an “is approximately equal to” sign, then show how you round, then work out the answer) Estimate the following: 1) 54 × 87

54 × 87 50 × 90 = 4500 2) 716 × 82.65

716 × 82.65 700 × 80 = 56000

3) 872

9852319

.

..

403

120

3

620

872

9852319

.

..

Square roots You can get a rough idea of the value of a square root by comparing the value in the square root with the nearest square number. You are expected to know all the square numbers up to 152 = 225. Examples

Estimate the value of 62

The nearest square number to 62 is 82 = 64.

So 62 is a little bit less than 8.

So a rough estimate for 62 is 7.9 (7.8 would be fine too)

Estimate the value of 126

The nearest square number to 126 is 112 = 121.

So 126 is a little bit more than 11.

So a rough estimate for 126 is 11.2 (11.1 or 11.3 would be fine too)

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8) Factors, Primes and Multiples You need to (i) Know what these are (don’t get factors and multiples muddled up!) (ii) Make sure you know your tables, otherwise this is a bit tricky FACTORS go exactly into a number It is often better to think of them as “hanging around in pairs” Examples All factors of 24: 1,24 2,12 3,8 4,6 All factors of 40: 1,40 2,20 4,10 5,8 All factors of 100: 1,100 2,50 4,25 5,20 10 (notice that 100 has a “lone” factor – because it is a square number) A PRIME NUMBERS has exactly 2 factors – itself and one. You should be able to recognise prime numbers under 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 and for larger numbers you may be asked to work out whether they are prime. MULTIPLES of a number are the numbers that it goes exactly into (ie its times table) Examples First 5 multiples of 6: 6, 12, 18, 24, 30 First 5 multiples of 11: 11, 22, 33, 44, 55 You can check if a number is a multiple of 2, 3, 4, 5, 6, 9 or 10 by using a divisibility test A number is a multiple of 2 if..............it ends in 0, 2, 4, 6, 8 (ie it is even) A number is a multiple of 3 if..............its digit sum is 3, 6 or 9 A number is a multiple of 4 if..............its last two digits are a multiple of 4 A number is a multiple of 5 if..............its last digit is 0 or 5 A number is a multiple of 6 if..............it is a multiple of 2 AND 3 A number is a multiple of 9 if..............its digit sum is 9 A number is a multiple of 10 if............its last digit is 0 There are other tests for divisibility that you may be familiar with (eg 7, 11) but you should be OK with all of the above.

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9) Prime Factorisation, Highest Common Factor (HCF) and Lowest Common Multiple (LCM) If you know your prime numbers, and if you can divide, then you can use a FACTOR TREE to work out the prime factors of a number and hence write ANY positive integer as a product of its prime factors, in INDEX FORM if required. Start by finding a factor pair of the starting number. If one of these factors is prime, circle it and stop that part of the tree there. For any non-primes continue writing as a factor pair until you’re left JUST with circled primes. Example: 1540 You’ve got choices for how to start. You could start 2 × 770, but I think it’s good to get the numbers small quickly so I’ll start with 10 × 154 The completed factor tree gives us 1540 = 2 × 2 × 5 × 7 × 11 (try to write primes in ascending order) We can write this “in index form” (using powers) as 1540 = 22 × 5 × 7 × 11 While there are quite a few ways of drawing out the tree (you could have started with 2 and 770) all the different ways will give the same set of prime factors. Highest Common Factor of two (or more) numbers The HCF of two (or more) numbers is the LARGEST number that is a FACTOR of both (all) of them. For small numbers it’s usually easiest to work this out by writing out a list of all factors of each number. Examples (i) Find the HCF of 36 and 48. Factors of 36: 1, 36 2, 18 3, 12 4, 9 6 Factors of 48: 1, 48 2, 24 3, 16 4, 12 6, 8 Comparing lists gives 12 as the HCF. (ii) Find the HCF of 24, 32 and 56. Factors of 24: 1, 24 2, 12 3, 8 4, 6 Factors of 32: 1, 32 2, 16 4, 8 Factors of 56: 1, 56 2, 28 4, 14 8, 7 Comparing lists gives as the HCF. Lowest Common Multiple of two (or more) numbers The LCM of two (or more) numbers is the SMALLEST number that is a MULTIPLE of both (all) of them. Again, for small numbers it’s usually easiest to work this out by writing out a list of multiples of each number.

1540

10 154

2 5 2 77

7 11

15

Examples (i) Find the LCM of 12 and 18. Multiples of 12: 12, 24, 36, ...... Multiples of 18: 18, 36, ... Comparing lists gives 36 as the LCM. (ii) Find the HCF of 8, 18 and 24. Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, ..... Multiples of 18: 18, 36, 54, 72, ...... Multiples of 24: 24, 48, 72, ...... Comparing lists gives 72 as the LCM. As the numbers get bigger, or when there are more numbers, or if the numbers are just a bit nasty you’re often better off using the prime factorisation of each number and a VENN DIAGRAM. Example Find the HCF and LCM of 56 and 84 Step 1: work out prime factorisation of 56 and 84

56 = 23 × 7, 84 = 22 × 3 × 7 Step 2: put these factors into a Venn Diagram

Step 3: the HCF is the product of all the primes in the INTERSECTION (middle) In this case 2 × 2 × 7 = 28 Step 4: the LCM is the product of all the primes in the whole diagram In this case 2 × 2 × 2 × 7 × 3 = 168 (This number can often be quite big so the question might ask you to leave it as a product of prime numbers in index form)

56 84

2

7

3 2 2

16

10) Fraction/Decimal Equivalence You need to know that fractions and decimals are different ways of representing numbers, and you

need to be able to change between the two forms (easy example: 1/2 = 0.5)

Just like your tables, some of these equivalences you should just know. If you're unfamiliar with any of these below please try to learn them. Terminating decimals (ie decimals that stop)

1/2 = 0.5

1/4 = 0.25

1/5 = 0.2

1/8 = 0.125

1/10 = 0.1

3/4 = 0.75 2/5 = 0.4 3/8 = 0.375 3/10 = 0.3

3

/5 = 0.6 5/8 = 0.625

7/10 = 0.7

4/5 = 0.8

7/8 = 0.875

9/10 = 0.9

Recurring decimals (ie decimals that go on repeating forever - the digit under the dot repeats, or in the case of the final column, both the digit repeat) 1/3 = 0. 3 1/6 = 0. 61 1/9 = 10 . 1/11 = 900 . 2/3 = 0. 6 5/6 = 0. 38 2/9 = 20 . 2/11 = 810 . 4/9 = 40 . 3/11 = 720 . 5/9 = 50 . 4/11 = 630 . 7/9 = 70 . 5/11 = 540 .

8/9 = 80 .

6/11 = 450 .

7/11 = 360 .

8/11 = 270 .

9/11 = 180 .

10

/11 = 090 .

(i) I have left out equivalent fractions (ie 2/4 is the same as 1/2, 3/9 is the same as 1/3 etc) (ii) Hopefully you can see some patterns in the ninths and elevenths - they aren't that difficult to learn! Changing terminating decimals to fractions This will always work, but it may be tricky to cancel the fraction fully. Remember that you are expected just to “know” some of these (listed above) Step 1: count the number of decimal places in the starting decimal - for the denominator, use the appropriate power of 10, with as many zeros as you had decimal places to start with Step 2: use the “decimal digits” as the numerator of your fraction Step 3: cancel down as far as you can – this can be tricky so take your time

17

Examples 1) 0.78 Step 1: 2 decimal places, so we use 100 as the denominator Step 2: choose 78 as the numerator Step 3: cancel down by dividing top and bottom by 2

0.78 = 50

39

100

78

2) 0.8725 Step 1: 4 decimal places, so we use 10000 as the denominator Step 2: choose 8725 as the numerator Step 3: cancel down by dividing top and bottom by 25

0.8725 = 400

349

10000

8725

Changing a “nice” fraction to a decimal (if it isn’t one you know – see list above) Whenever you are asked to change a fraction to a decimal, start by making sure it’s in its simplest form (ie cancel down if necessary). Once you’ve done this you may well end up with a “nice” denominator – by this I mean a number that you can change fairly easily into a power of 10 (10, 100, 1000 etc) Then it’s just a question of writing an equivalent fraction with a power of 10 as the denominator, then changing it to a decimal. Examples

640100

64

25

16. (multiply top and bottom by 4 to get 100 on the bottom)

450100

45

20

9

40

18. (cancel down, multiply top and bottom by 5 to get 100 on the bottom)

6010

6

30

18. (cancel down and you get 10 on the bottom)

Once you’ve cancelled down, some examples of “nice” denominators are 2, 4, 5, 8, 10, 20, 25, 40, 50 because it’s quite easy to change them to 10, 100, 1000.

18

Changing any fraction to a decimal (if you can’t do it quickly) To be able to do this you need to be able to do division, and you divide the numerator by the denominator. If the denominator is “nasty” you will need to be able to do long division. Examples

1) Change 7

3 to a decimal.

Divide 3 by 7. In order to get the "decimal places" keep adding zeros to the 3 ie make it 3.00000... Then do your "normal" division. 0 . 4 2 8 5 7 1 4 ... 7 3 . 0 0 0 0 0 0 0 ... Step 1: how many 7s in 3? 0 remainder 3 Step 2: how many 7s in 30? 4 remainder 2 Step 3: how many 7s in 20? 2 remainder 6 Step 4: how many 7s in 60? 8 remainder 4 Step 5: how many 7s in 40? 5 remainder 5 Step 6: how many 7s in 50? 7 remainder 1 Step 7: how many 7s in 10? 1 remainder 3 Step 8: how many 7s in 30? 4 remainder 2 As soon as we get a remainder repeated (in this case the 3) we know that the decimal will recur.

So we have 71582407

3 .

2) Change 24

7 to a decimal.

This is going to be a bit trickier as I don't know my 24 times table. Just like with "normal" long division, write out the first 10 multiples of 24:

1 24 = 24 2 24 = 48 3 24 = 72 4 24 = 96 5 24 = 120

6 24 = 144 7 24 = 168 8 24 = 192 9 24 = 216 10 24 = 240 Now follow the same procedure as above 0 . 2 9 1 6 6 ... 24 7 . 0 0 0 0 0 ... Step 1: how many 24s in 7? 0 remainder 7 Step 2: how many 24s in 70? 2 remainder 22 Step 3: how many 24s in 220? 9 remainder 4 Step 4: how many 24s in 40? 1 remainder 16 Step 5: how many 24s in 160? 6 remainder 16 Step 6: how many 24s in 160? 6 remainder 16......repeated remainder so decimal recurs now

So, 6291024

7 .

3 2 6 4 5 1 3

7 22 4 16 16

19

QUESTIONS A) Four rules for positive and negative integers Write each question out then give its answer. Use the rules carefully! 1) 8 – 17 2) (-3) + 9 3) (-4) + (-2) 4) 8 – (-4) 5) (-13) – (-11) 6) (-5) x 8 7) 4 x (-9) 8) 12 x (-9) 9) 24 ÷ (-3) 10) (-42) ÷ (-7) B) Fraction Arithmetic Write out each question and show your working. If your answer is greater than one you may leave it as a mixed number OR as a top-heavy fraction. You answer MUST be simplified.

1) 7

1

3

2 2)

8

5

6

5 3)

8

12

7

31 4)

5

3

6

5 5)

7

51

2

13

6) 4

3

5

2 7)

5

2

3

21 8)

2

11

7

22 9)

9

7

8

5 10)

4

31

3

13

C) Decimal Arithmetic Show careful working for each question 1) 2.34 + 5.7 2) 12.3 – 5.67 3) 12.34 + 1.752 – 3.9 4) 2.6 × 15.3 5) 22.678 ÷ 4.6 D) Ratio 1) Simplify each of the following ratios. Write the question, then "=", then give your answer. a) 3 : 6 b) 12 : 2 c) 1.5 : 4 d) 1.2 : 5 e) 14 : 35 : 77 2) a) Write the ratio 4 : 7 in the ratio 1 : n where n is a fraction or a decimal b) Write the ratio 3 : 8 in the ratio n : 1 where n is a fraction or a decimal 3) The ratio of boys to girls on a school trip is 2 : 3 a) Write the number of boys on the trip as a FRACTION of the number of girls on the trip b) Write the number of boys on the trip as a FRACTION of the number of children on the trip c) If there are 45 children on the trip, how many are boys and how many are girls? 4) Showing your working carefully, share £63 in the ratio 3 : 4 5) Three bags contain a mixture of red and green balls. In bag A there are 30 balls and the ratio of red to green balls is 2 : 1 In bag B there are 24 balls and the ratio of red to green balls is 3 : 5 In bag C there are 28 balls and the ratio of red to green balls is 4 : 3 What fraction of ALL the balls are red? Give your answer in its simplest form. E) Rules for positive, integer indices For these questions, write the question then "=", then give your answer. 1) Simplify the following a) 23 × 24 b) 68 ÷ 62 c) (52) 2) Write as a single power of 2

a) 45 b) 163 c) 3

24

4

8x2

3

20

F) Rounding to a given number of decimal places or significant figures Round the following to the stated degree of accuracy (dp = decimal places, sf = significant figures). No need to explain your answer/show any working – but please write down the question and details of how you’ve rounded. For example: 4.527 = 4.5 (1dp) 1) 34.65 (1dp) 2) 1.8672 (2dp) 3) 12.8091 (3dp) 4) 342.898 (2dp) 5) 1.9765 (1dp) 6) 1875 (1sf) 7) 0.02876 (1sf) 8) 86026 (2sf) 9) 0.0030698 (2sf) 10) 8.976 (2sf) G) Estimation Estimate the following. Start by writing the question then show your working clearly.

1) 68 × 17 2) 893

851123

.

.. 3)

1870

87215

.

.. 4) 23 5) 170

H) Factors, Primes and Multiples 1)Write out this list:

2, 3, 11, 14, 17, 18, 21, 27, 29, 31, 36, 39, 40

From the list of numbers above, list.... a) all multiples of 7 b) all factors of 36 c) all primes 2) With reasons, decide whether 2, 3, 4, 5, 6, 9 and 10 are factors of number 4620. I) Prime Factorisation, HCF and LCM For all these questions, show your working carefully. 1) Write the number 1092 as a product of prime numbers in index form. 2) Work out a) the HCF of 45 and 60 b) the LCM of 24 and 30 2) Work out the HCF of 234 and 198 3) Work out the LCM of 80 and 52, writing your answer as a product of prime numbers in index form. J) Fraction/ Decimal equivalence 1) Write out each of these fractions, then give its decimal equivalence (no working required)

a) 5

3 b)

8

5 c)

3

2 d)

6

5 e)

9

8

2) Showing careful working, change these decimals to fractions in their simplest form: a) 0.24 b) 0.85 c) 0.256 3) Showing careful working, change these fractions to decimals, clearly showing recurring digits if necessary

a) 25

3 b)

40

13 c)

12

1 d)

15

7