Mathematics THREE (EXTENSION) FOR COMMON ENTRANCE
Serena Alexander
I n t r o d u c t i o n
● Notes on features in this book Words printed in blue and bold are keywords. All keywords are defined in the Glossary at the end of the book.
Example Worked examples are given throughout to aid understanding of each part of a topic.
Exercise Exercises are provided to give pupils plenty of opportunities to practise what they have learned.
Extension Exercise Some exercises contain questions that are more challenging. These extension exercises are designed tow stretch more able pupils.
Summary Exercise Each chapter ends with a summary exercise, containing questions on all the topics in the chapter. Useful rules and reminders are scattered throughout the book.
Activity The National Curriculum for Mathematics reflects the importance of spoken language in pupils’ development across the whole curriculum – cognitively, socially and linguistically. Activities to develop these skills are interspersed between the chapters. These are essential for developing pupils’ mathematical vocabulary and presenting a mathematical justification, argument or proof.
ix
Working with numbers
1
You already know a great deal about numbers. You should also know that there are many different ways of working with them, particularly when they are very large or very small. The number system continues infinitely, without end, in both the positive and negative directions. In this chapter you will look again at some basic information that will help you with later chapters.
● Natural numbers (also called cardinal numbers) Natural numbers are the numbers you use for counting: one, two, three, four, …. You can add, subtract, multiply and divide them. Although you can use a calculator to complete calculations with natural numbers, remember that you can save a great deal of time if you can do simple arithmetic in your head. Try this exercise.
Exercise 1.1 Write the numbers from 1 to 60 down your page before you start. Make a note of the time as you start and check how long it takes you to complete the exercise. For this exercise only, just write down the answers. It should take you no more than 10 minutes! 1 7 × 5
6 94 + 17
11 318 + 122
16 319 − 123
2 8 × 9
7 12 × 8
12 3 × 15
17 191 + 325
3 36 + 123
8 124 + 74
13 72 ÷ 12
18 424 ÷ 8
4 42 ÷ 7
9 126 ÷ 9
14 368 − 143
19 8 × 15
5 82 − 51
10 45 − 17
15 4 × 19
20 308 ÷ 11
21 25 × 7
27 331 − 117
33 305 − 199
39 35 × 6
22 302 + 99
28 504 ÷ 3
34 417 + 392
40 399 + 417
23 125 ÷ 5
29 680 − 156
35 45 × 7
41 225 − 186
24 425 − 106
30 19 × 3
36 720 ÷ 36
42 225 ÷ 45
25 13 × 3
31 127 + 523
37 237 + 283
43 225 ÷ 15
26 199 + 249
32 144 ÷ 8
38 200 − 137
44 19 × 6 1
45 19 × 9
49 128 + 512
53 13 × 13
57 125 × 4
46 360 − 179
50 207 + 153
54 15 × 15
58 20 × 500
47 95 × 5
51 305 − 125
55 75 × 3
59 3600 ÷ 90
48 99 × 8
52 288 ÷ 12
56 25 × 9
60 100 000 ÷ 80
● Long multiplication and long division You should have been able to use simple mental arithmetic to answer all of the questions in that exercise. When calculations become more difficult, you may think you need to use a calculator. However, you may not have a calculator with you, or you may need to use a pencil and paper to do the calculations. You should be able to add, subtract and carry out short division and multiplication quite easily. Make sure that you can complete long multiplication and long division as easily. Remember that for long multiplication you break the smaller number into a multiple of ten and a unit.
Example Multiply: 5264 × 63 The estimate tells you how many columns you need.
Estimate: 5000 × 60 = 300 000
63 is 60 + 3 Then: 5264 × 63 = 5264 × (3 + 60)
Next to each row of the calculation, write down what you are multiplying by. HTh TTh Th
5
+
H
T
U
2
6
4
×
6
3
1
5
71
91
2
×3
3
11
53
82
4
0
× 60
3
3
1
6
3
2
The carried numbers from the multiplications are in these rows.
The carried numbers from the s 1 1 1 r addition are in this row. e b 5264 × 63 = 331 632 m u n h The method for long division is similar to that for short division t i w but you show all the working. This is so that you can calculate the g remainders clearly and accurately. n i k r o W
1 2
L o n g m u l t i p l i c a t i o n a n d l o n g d i v i s i o n
Examples (i) Divide: 8921 ÷ 37 Estimate: 9000 ÷ 40 ≈ 220 Th
3
7
H
T
U
2
4
1
8
9
2
1
7
4
1
5
2
1
4
8
r4
3
7
×
2
7
4
1
4
1
3
7
-
4
1
3
7
×
4
4
8
2
8921 ÷ 37 = 241 r 4
(ii) Divide: 861 ÷ 35 Estimate: 900 ÷ 40 ≈ 22 H
3
5
T
U
t
2
4
6
8
6
1
0
7
0
1
6
1
1
4
0
2
1
0
2
1
0
-
-
-
Remember that when you have remainders you can write extra zeros after the decimal point and carry on dividing.
861 ÷ 35 = 24.6
3
Exercise 1.2 Multiply or divide, showing all your working clearly. 1 27 × 36
6 3648 × 91
11 213 × 69
16 1809 × 55
2 147 × 29
7 5624 × 72
12 1161 ÷ 43
17 1008 ÷ 45
3 389 × 47
8 6293 ÷ 29
13 2713 × 28
18 999 ÷ 037
4 391 ÷ 17
9 7746 ÷ 36
14 13 500 ÷ 36
19 3005 × 17
5 989 ÷ 23
10 9900 ÷ 48
15 2145 × 42
20 1155 ÷ 42
● Problem solving When solving problems, you need to consider the question carefully and see if you need to add, subtract, multiply or divide. Sometimes you may need to do more than one calculation to work out the answer. Write every calculation out neatly so that you can check it through. Do remember to estimate first so you can check your answer looks sensible.
Remainders There are different ways of dealing with remainders, depending on the problem. 1 Leave it as a remainder. 2 Round up to the nearest whole number. 3 Round down, ignoring the remainder. 4 Write the remainder as a fraction.
s r e b m u n h t i w g n i k r o W 1
5 Continue dividing and writing extra zeros after the decimal point. Note that you can always reach a definite answer if the number that you are dividing by ends with a 2, 4, 5 or 8 and sometimes if it ends in 3, 6, 7 or 9
Exercise 1.3 1 There are 365 days in a year and 24 hours in a day. How many hours are there in a year? 2 How many minutes are there in a week? How many seconds are there in a week? 3 In the imperial system of weight there are 16 ounces in a pound and 14 pounds in a stone. How many ounces are there in 7 stone?
4
I n t e g e r s
4 Julius Caesar said, with a smile: ‘There are one, seven, six, oh yards in a mile.’ How many yards are there in five and a half miles? 5 A fast food shop sells on average 176 beefburgers each hour. How many beefburgers does the shop sell in a 17-hour day? 6 There are 22 pupils in my class and we all have milk at break-time. Milk comes in cartons containing one-third of a pint. A school term lasts 12 weeks and there are five days in each week. How many pints of milk does my class drink that term? 7 Rough books come in cartons of 500. How many classes of 22 does one carton supply, assuming every pupil needs one rough book? 8 My teacher is photocopying our mathematics exam. The exam is on four sheets of paper and there are 92 pupils taking the exam. (a) How many sheets of paper will she use? (b) The photocopier copies 24 sheets in a minute. How long does it take my teacher to photocopy the exam for all the pupils? 9 We have sold 245 DVDs of the school play at £9.50 each. (a) How much money have we taken? (b) It cost £450 to film the school play and then £4.20 per DVD. How much profit or loss did we make? 10 The school chef buys 36 litres of milk at £2.40 for 4 litres, 18 dozen eggs at £1.95 per dozen and 11 kg of flour at £1.40 a kg. From that he makes a batter and then cooks 850 pancakes. What is the cost per pancake, correct to the nearest penny?
● Integers An integer is a whole number, so natural numbers are all integers. Integers, however, also include 0 (zero) and negative numbers such as 3, 2 and 1 −
−
−
Negative integers are the set of integers (whole numbers) that are less than zero. Negative numbers can be added, subtracted, multiplied and divided in a similar fashion to natural (positive whole) numbers. Remember what happens when you are calculating with negative numbers. For example: 4 − 4 = 0 but 4 − ( 4) = 8 −
5
It can help to draw a number line to see what is happening.
Examples (i) 2 + 3 +3
–10 – 9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
2
3
4
5
2 + 3 = 5 (ii) 2 + ( 3) −
+ (–3) –10 – 9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2 + ( 3) = 1 −
So
−
2 + 3
=
5 but
( 2) + 3 = 1 −
Similarly 5 − 3 = 2 but 2 × 3
=
2 + ( 3) = 2 − 3 −
3 − 5 = 2 and 2
=
−
6 and
( 2) × ( 3) = 6
6 ÷ ( 3)
=
−
2 and
( 6) ÷ ( 3) = 2
−
−
1
−
−
3
×
=
5 − ( 3) = 5 + 3 = 8
−
6 but
6 ÷ 3 = 2 but
and −
−
−
−
When calculating with negative numbers you should write down the question, then your working and finally the answer.
Example −
−
It is not good practice to write an operator (+, −, ×, ÷) next to a raised negative sign, so always use brackets.
3 − ( 7) −
3 − ( 7) = 3 + 7 −
−
4
=
s r e b Exercise 1.4 m u Add, subtract, multiply or divide, showing all your working clearly. n h t 1 3 − 4 6 8 + 5 11 3 × ( 4) i w g 2 5 − 8 7 13 − 9 12 12 − ( 5) n i k 3 3 + ( 6) 8 ( 3) × 7 13 14 − 8 r o W 4 ( 4) × ( 4) 9 12 ÷ ( 3) 14 4 − 8 −
−
−
−
−
−
1
−
−
5 3 × ( 3) −
6
−
−
−
10 ( 24) ÷ ( 8) −
−
15
−
4 − ( 8) −
16 5 × ( 2)
21 4 − 9
26 ( 5) + ( 2)
17 16 ÷ ( 4)
22 7 − 3
27 8 − 3
18 7 − ( 5)
23 4 × ( 3)
28
−
8 × ( 2)
19
24 12 ÷ ( 6)
29
−
24 ÷ 8
25 5 + ( 6)
30 100 ÷ ( 10)
−
−
−
25 ÷ 5
−
20 ( 5) × ( 5) −
−
−
−
−
−
W o r k i n g w i t h n u m b e r s
−
−
−
● Working with numbers Factors and multiples Any natural (whole) number can be written as the product of other numbers. 24 can be written as 1 × 24, or 2 × 12, or 3 × 8 or 4 × 6 1, 2, 3, 4, 6, 8, 12 and 24 are all factors of 24
●
24 is a multiple of 1, 2, 3, 4, 6, 8, 12 and 24
●
If you had to write the multiples of 5 that were less than 30 you would write: 5, 10, 15, 20, 25
Prime numbers and composite numbers A prime number is a number that has exactly two factors: itself and 1 The set of prime numbers starts: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... A composite number is any positive integer greater than one that is not a prime number.
Prime factors
Note that the number 1 only has one factor (itself) and so does not qualify as a prime number. It is a very special number with other properties that mean that it cannot be prime.
A composite number can be broken down, by successive division, into factors that are prime numbers. These prime numbers are called the prime factors of the number.
7
Example Find the prime factors of 210 2
2
1
0
3
1
0
5
3
5
5 7
7 1
210 = 2 × 3 × 5 × 7
Sums and products A sum is the result of an addition. The sum of 5 and 6 is 11 A product is the result of a multiplication. If you are asked for the product of 5 and 6 the answer is 30 When you write 210 as:
210 = 2 × 3 × 5 × 7
you have written it as the product of its prime factors.
Unique factorisation property Every number that is not zero or one can be written as a product of prime factors, but every number has a different set of prime factors. This means that every number is unique. This is known as the unique factorisation property. Consider the number 1533 You can write is as the product of its prime factors: 3
7 × 73
×
Then 3, 7 and 73 are the prime factors of 1533
s r e b m u n h t i w g n i k r o W
No other number can have the same product of prime factors.
Index numbers When the product of prime factors includes repeats of prime factors you can use index notation to record them. 16 = 2 × 2 × 2 × 2 So 16 = 24 4 is the index number that indicates there are four 2s in the product.
1 8
W o r k i n g w i t h n u m b e r s
Exercise 1.5 1 Which of these numbers are prime numbers? 5 17 25 27 32 37 48 2 Which of these numbers are factors of 36? 1 2 3 4 5 6 7 9 10 36 72 360 3 Which of these numbers are multiples of 12? 1 2 3 4 5 6 8 12 24 36 4 (a) Which of these numbers are prime factors of 42? 1 2 3 6 7 14 21 42 (b) Which of these numbers are prime factors of 20? 1 2 4 5 10 20 5 Which of these numbers are prime factors of 17? 1
2 5
17 34
6 (a) What is the product of 12 and 7? (b) What is the sum of 13 and 3? (c) What is the product of 6 and 7? 7 Continue the list of prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... as far as 50 8 List all the factors of: (a) 65
(b) 101
(c) 19
(d) 72
9 Write each of these numbers as a product of its prime factors. (a) 504
(b) 136
(c) 1000
(d) 945
10 (a) List all the factors of 24 and all the factors of 42 (b) Which factors do they have in common? (c) Which is the highest? This is the highest common factor (HCF) of 24 and 42 11 Find the highest common factor of each pair of numbers. (a) 8 and 10
(b) 20 and 30
(c) 100 and 360
12 List the first 10 multiples of 4 13 List the first 10 multiples of 6 14 List the first 10 multiples of 10
9
15 Look at the lists in questions 12–14. What is the lowest number that is a multiple of: (a) 4 and 6
(b) 4 and 10
(c) 6 and 10?
These numbers are called lowest common multiples (LCM) of each pair of numbers. 16 Find the lowest common multiple of each pair of numbers. (a) 8 and 10
(b) 20 and 30
(c) 100 and 360
● Highest common factors and lowest common multiples
In the previous exercise highest common factors (HCF) and lowest common multiples (LCM) were quite easy to find by inspecting the various factors and multiples. For larger numbers it is not always as simple. Consider the question: What is the highest common factor of 210 and 375? To find out, you need to write each number as the product of its prime factors.
Example What is the HCF of 210 and 375? 2
2
1
0
3
3
7
5
3
1
0
5
5
1
2
5
3
5
5
2
5
7
5
5 7
1
210 = 2 × 3 × 5 × 7
5 1
375 = 3 × 5 × 5 × 5 3 = 3 × 5
The common factors are 3 and 5 and so the HCF is 3 × 5 = 15
s r e b m u n h t i w g n i k r o W
Suppose you had been asked to find the lowest common multiple of 210 and 375 For smaller numbers you could look at the first few multiples. For larger numbers you need to look at the prime factors.
1 10
First you need to find all the prime factors.
To find the HCF you need to work out the product of all the factors that are common to 210 and 375: 3 and 5
The lowest common multiple is the product of all the prime factors of one number and any extra prime factors in the other(s).
H i g h e s t c o m m o n f a c t o r s a n d l o w e s t c o m m o n m u l t i p l e s
Example Find the lowest common multiple of 210 and 375 210 = 2 × 3 × 5 × 7
375 = 3 × 5 × 5 × 5
LCM = 2 × 3 × 5 × 5 × 5 × 7
2, 3, 5 and 7 are prime factors of 210, the extra prime factors are 5 and 5
5250
=
Exercise 1.6 1 Find the HCF of the numbers in each pair. (a) 504 and 945
(c) 945 and 1000
(b) 136 and 504
(d) 136 and 945
2 Find the HCF of 330 and 175 3 Find the HCF of 132 and 165 4 Find the HCF of 812 and 638 5 What is the largest number that divides exactly into both 1000 and 3600? 6 Peter picks 480 apples and Piper picks 600 apples. They pack their apples into identical boxes, with no apples left over. What is the largest number of apples that each box could take? 7 Find the LCM of the numbers in each pair. Use your calculator if you need to. (a) 504 and 945
(c) 945 and 1000
(b) 136 and 504
(d) 136 and 945
8 Find the LCM of 330 and 175 9 Find the LCM of 132 and 165 10 Find the LCM of 812 and 638 11 What is the smallest number that is a multiple of both 20 and of 36? 12 (a) Ollie and Millie are counting out loud, starting at 1. Millie calls out every fourth number and Ollie every seventh number. What are the first four numbers that they both call out? (b) Tom, Dick and Harry play the same game. Tom calls out every 20th number, Dick every 15th number and Harry every 25th number. What is the first number that they all call out?
11
● Order of operations When you have mixed calculations, you need to do them in the correct order. Multiplication and division must be done before addition and subtraction. Hence 5 + 7 × 3 = 26 not 36
You must do the multiplication first.
Examples (i) Work out: 5 + 7 × 3
(ii) Work out: (5 + 7) × 3
5 + 7 × 3 = 5 + (7 × 3)
(5 + 7) × 3 = 12 × 3
5 + 21
36
=
=
26
=
Can you see how the brackets make a difference to the calculation? You can use brackets to indicate that you must carry out the addition first. Sometimes a calculation may have numbers with indices. You must do those before addition and subtraction too.
Example Multiply: 3 × 53 3 × 53 = 3 × 125
Note that this is not (3 × 5)3 = 153
375
=
Think of BIDMAS to remember the rule. Brackets Index numbers Divide Multiply Add Subtract
s r Exercise 1.7 e b m Calculate the answers. u n h 1 14 − 3 × 2 t i w 2 (14 − 3) × 2 g n i 92 − 8 k 3 r 6 ×7 o W 4 5 × (28 − 4) ÷ 6 1
5 3 × 43 − 36 ÷ 32
12
6 5 × (11 − 4)2 ÷ 7 7 5 × 34 − 25 ÷ 8 8
3 4 − 24 5
9 (4 × 3)2 − (27 ÷ 9)3 10 4 × (62 − 18) ÷ 9
● Extension: Using prime factors to calculate However good your mental arithmetic is and however quick you are with pencil and paper calculations, it is always worth checking if there is an efficient way to carry out a calculation without writing it out in full. Consider 722 ÷ 27 You may spot that 27 has some common factors with 72. To be sure of calculating accurately you need to write the calculation as a product of prime factors. 722 ÷ 27 = (2 × 2 × 2 × 3 × 3)2 ÷ (3 × 3 × 3)
●
1
2×2×2×3× 3 ×2×2×2 31 × 31 × 31
=
=
2 × 2 × 2 × 2 × 2 × 2 × 3
=
64 × 3
=
192
×
3
1
×
3
1
It is often easiest to write a division as a fraction and cancel common factors.
You can also use prime factors to find roots. Under the rules of BIDMAS, roots are considered as index numbers.
Example Calculate: 3 3
3375
3
3375
÷
225
3375 = 3 × 3 × 3 × 5 × 5 × 5
225 = 3 × 3 × 5 × 5
3375 = 3 × 5 = 15
225 = 3 × 5 = 15
÷
225 = 1
You may need to combine roots and index number calculations.
Example Calculate: (35 ×
225
441 )2
÷
225 = 3 × 3 × 5 × 5 225
15
=
441 = 3 × 3 × 7 × 7 441
(35 ×
21
=
225
÷
5 ×
441 )2 = =
252
=
625
× 3 × 5 2 3×7
7
13
E x t e n s i o n : U s i n g p r i m e f a c t o r s t o c a l c u l a t e
Extension Exercise 1.8 Use products of prime factors to calculate the answers to these. 1 128 ÷ 16
6 1252 ÷ 53
2 225 ÷ 15
7
4 452 ÷ 15 5
15
9
45
×
×
12 ( 65 × 26 × 10
45
16 ÷
52 )2
17
13 (2160 ÷ 63)2 1764
81
÷
×
63
35
×
3
÷
45
×
75
10 1602 ÷ (64 × 25)
27
11 72 × 362 ÷ 123
14
729
8 4902 ÷ 1225
11025
3
3
1225 4
40180
18 ( 4 1296
( 3 9261
÷
×
19
256 )
15 (98 × 56 ÷ 143)2
3
÷
3375 3
÷
3
÷
3
÷
144
343
64 )
91 × 21 × 39
20 ( 2025
×
÷
2 3
63
×
147
3375 )3
Summary Exercise 1.9 Answer questions 1–10 mentally. 1 52 + 33
6 324 − 138
2 165 − 41
7 108 × 4
3 12 × 8
8 116 ÷ 4
4 144 ÷ 12
9 148 × 3
5 334 + 159
10 138 ÷ 6
11 Answer these without using a calculator. Write down the question and any necessary working. (a) 7 + ( 4)
(f) 3 + ( 8)
(b) 4 − ( 7)
(g) 15 × ( 5)
−
−
−
−
−
s r e (c) ( 3) × ( 4) (h) 6 − 9 b m (d) 14 ÷ ( 2) (i) 3 + ( 9) u n (e) 4 + 12 (j) ( 18) ÷ ( 6) h t i w 12 Which of these numbers are prime numbers? g n i 1 4 5 9 13 24 31 99 k r o 13 Write down all the factors of 28 W −
−
−
−
−
1
−
14 Which of these are multiples of 6? 1 14
2
3
4
6
12
15
30
−
−
E x t e n s i o n : U s i n g p r i m e f a c t o r s t o c a l c u l a t e
15 Write down the HCF of: (a) 16 and 24
(b) 252 and 714
16 Write down the LCM of: (a) 8 and 10
(b) 168 and 462
Write down your working clearly for questions 17–18. Do not use a calculator. 17 On an intergalactic expedition I have packed 25 space pods weighing 12 kg each, 38 space suits weighing 28 kg each and 9 extra-terrestrial repellent missiles weighing 135 kg each. Have I packed more than the 2500 kg limit? 18 (a) The mass of one astronaut plus his kit is 84 kg. The total mass allowance on the space station is 4000 kg. If there are 23 astronauts how much mass allowance is there left? (b) One quarter of that remaining mass allowance is for food. Each astronaut is allowed 1 kg of food per day. How many days can the astronauts survive on their food allowance?
Activity: Largest and smallest sums and differences Take four digits: 6, 7, 8, 9 The largest number you can make with them is 9876 and the smallest is 6789 Now make two 2-digit numbers and find the difference and the sum. For example, you can make the numbers 78 and 96 The difference is 18 and the sum 174 Can you make two 2-digit numbers that give you a smaller difference or a larger sum? Find the smallest and largest sums and differences that you can make from two 2-digit numbers, using the digits 6, 7, 8 and 9 Now find the smallest and largest sums and differences that you can make from two 3-digit numbers, using the digits 4, 5, 6, 7, 8 and 9 Next find the smallest and largest sums and differences that you can make from two 4-digit numbers using the digits 2, 3, 4, 5, 6, 7, 8 and 9 Finally, find the smallest and largest sums and differences that you can make from two 5-digit numbers using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 Put your results in a table. Digits
Smallest sum
6, 7, 8, 9
Largest sum
Smallest difference
Largest difference
7
4, 5, 6, 7, 8, 9 2, 3, 4, 5, 6, 7, 8, 9
6047
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
What do you notice? Try to explain your answer.
15
