Edexcel post 16 maths sample chapter

1

Numbers

Objectives Before you start this chapter, mark how confident you feel about each of the statements below: I can use and compare positive and negative numbers.

I can order positive and negative integers I can add, subtract, multiply and divide integers including negatives I can use negative numbers in context I can use index notation for squares and cubes I can find squares, cubes and roots with and without a calculator I can write a number as a product of its prime factors I can use prime factors to find HCF and LCM of two numbers ** I can find the HCF and LCM of two numbers using other methods I can use index notation for powers of 10, including negative powers.

I can use the law of indices.

I can covert large and small numbers into standard form I can add, subtract , multiply and divide number sin standard form I can interpret a calculator display using standard form.

Check in questions • Complete these questions to assess how much you remember about each topic.

Then mark your work using the answers at the back of the book.

• If you score well on all sections, you can go straight to the Revision Checklist and Exam-style Practice questions at the end of the chapter. If you don’t score well, go to the chapter section indicated and work though the examples and practice questions there.

1

Arrange these numbers in order of size, starting with the smallest:

a 3603 33 060 33 36 363 b 521 1250 2501 12 005 120 c 64 46 -640 -406 4060 6004 d 7340 -437 3047 -73 407

SEE 1.2

2

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2

Calculate the value of:

a 25 3

c 343

d 492

Write these numbers as products of their prime factors:

a 50 4

3

b 83

b 360

SEE 1.2 SEE 1.3

c 16

Decide whether these statements are true or false.

SEE 1.3

a The HCF of 20 and 40 is 4. b The LCM of 6 and 8 is 24. c The HCF of 84 and 360 is 12. d The LCM of 24 and 60 is 180.

5

Simplify the following, leaving your answers in index form.

a 6 ×6 3

6

b 12 ÷ 12

5

d 64

SEE 1.4

2 3

SEE 1.5

b 0.000 46

Work out the following calculations.

Leave in standard form.

a (3 × 10 ) × (4 × 10 ) 4

8

c (5 )

2 3

Write in standard form:

a 64 000 7

-3

10

SEE 1.5

b (6 × 10 ) ÷ (3 × 10 ) -5

6

-4

Work these out on a calculator:

a (4.6 × 1012) ÷ (3.2 × 10-6)

SEE 1.5

b (7.4 × 109)2 + (4.1 × 1011)

1.1 Positive and negative numbers Place Value Each digit in a number has a place value. The value of the digit depends on its place in the number.

The place value changes by a factor of 10 as you move from one column to the next.

Ten thousands

6

Thousands Hundreds

7

1

Tens Units

4

5

This number would be read as sixty seven thousand, one hundred and forty five.

Example Write these numbers in words.

001

a 538

b 2371

c 6 352 740

a five hundred and thirty-eight b two thousand, three hundred and seventy-one c six million, three hundred and fifty-two thousand, seven hundred and forty When ordering whole numbers: • put the numbers into groups with the same number of digits • for each group, arrange each number in order of size depending on the place value of the digits.

1.1 Positive and negative numbers

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Example Arrange these numbers in order of size. Put the smallest first.

002

26, 502, 794, 3297, 4209, 4351, 745 908, 5, 32, 85, 114, 54 321 This becomes: 5, 26, 32, 85, 114, 502, 794, 3297, 4209, 4351, 54 321, 745 908

Example Gill buys some premium bonds for £1050. Write this figure in words.

003

This number is one thousand and fifty pounds.

Directed Numbers Integers are whole numbers that can be positive or negative. Positive numbers are above zero.

Negative numbers are below zero.

Integers are sometimes known as directed numbers.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Example Put the correct symbol, > or <, between these numbers.

004

a -5, -2 b -4, -6 c 5, -2 a -5 < -2 b -4 > -6 c 5 > -2

Directed numbers are often seen on the weather forecast in winter.

On this weather map, Aberdeen is the coldest place at -8°C and London is 6 degrees warmer than Manchester.

Aberdeen (-8°)

Manchester (-4°)

London (2°)

4

1 Numbers

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Adding and subtracting directed numbers Look at the following: The temperature at 6am was -5°C. By 10am it had risen 8 degrees.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-5°+ 8°= 3° So the new temperature was 3°C. It is useful to draw a number line to help when answering questions of this type.

Example Find the value of -2 - 4

005

Start -6

-5

-4

-3

-2

Finish -1

0

1

2

3

4

5

6

When the number to be added (or subtracted) is negative, the normal direction of movement is reversed.

-4 - (-3) is the same as -4 + 3 = -1 When two = signs or two - signs are together, these rules are used: + (+) ( =+ - (-) ( =+

}

- (+) ( =-

Like signs give an addition

( -2 + (-3) = -2 - 3

+ (-) ( =-

-3 - (+5) = -3 - 5

= -5

}

Unlike signs give a subtraction

6 - (-4 ) = 6 + 4

= -8

( 5 + (-2) =5-2

= 10

=3

Multiplying and dividing directed numbers Multiply and divide directed numbers as normal and then find the sign for the answer using the following rules.

• Two like signs (both + or both -) give a positive answer • Two unlike signs (one + and the other -) give a negative answer.

(-) × (-) = +

Example Try these.

006

(+) × (+) = +

-20 -2

a -6 × 3

b -4 × (-2)

c

a -18

b 8

c 10

d

9 -3

(-) × (+) = -

d -3

(-) × (+) = -

Exam tips It is really important to remember the rules of multiplying and dividing by negative numbers, because they are also used in algebra.

1.1 Positive and negative numbers

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Practice questions 1 Complete each statement by using < or >. a 2 5 b 2 -5 c 41 14 d -3 -9 2 Find the value of: a -2 - 3 b -3 + -6 c 5 - -2 d -15 + 8 3 Which two of these have the same value? a -8 + 6 b -8 ÷ -4 c -4 × -1 d -6 - -4 e -28 ÷ -7 4 At midday in London, the temperature was 7°C. By midnight, it had fallen by 10°. What was the temperature at midnight?

5 At 6am when Jeff went to work, the temperature was -3°C. By 11am, it had risen to 3°C. How much had the temperature risen?

1.2 Square, cube and triangular numbers Square numbers Square numbers are whole numbers raised to the power of 2. For example 52 = 5 × 5 = 25 The first 12 square numbers are:

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) Square numbers can be illustrated by drawing squares: 1

4

9

16

25

…

You need to know the square numbers up to 15 × 15.

12 = 1 × 1

6

22 = 2 × 2

32 = 3 × 3

42 = 4 × 4

52 = 5 × 5

1 Numbers

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Cube numbers Cube numbers are whole numbers raised to the power 3. For example 53 = 5 × 5 × 5 = 125 (five cubed) Cube numbers include: 1 8 27 64 125 216 (1 × 1 × 1) (2 × 2 × 2) (3 × 3 × 3) (4 × 4 × 4) (5 × 5 × 5) (6 × 6 × 6)

... 1000 ... (10 × 10 × 10)

Cube numbers can be illustrated by drawing cubes: 1

8

27

64

125

13 = 1 × 1 × 1

23 = 2 × 2 × 2

33 = 3 × 3 × 3

43 = 4 × 4 × 4

53 = 5 × 5 × 5

…

Triangular numbers The sequence of triangular numbers is 1, 3, 6, 10, 15, ... Each time the difference between the preceding numbers goes up by 1.

1

3

+2

6

+3

10

+4

15

…

+5

Triangular numbers can be illustrated by drawing triangle patterns.

1

3

6

10

15

1

1+2=3

3+3=6

6 + 4 = 10

10 + 5 = 15

…

Square roots and cube roots

Is the square root sign. Taking the square root is the opposite of squaring.

Example Find 25

007

25 = 5 or -5 since (5)2 = 25 and (-5)2 = 25

When a number is square rooted it can have two square roots, one positive and one negative.

A surd is the square root of any number that is not a square number. It cannot be written exactly as a decimal.

For example 2, 3, 5, 6, 7,….. are all surds.

3

is the cube root sign. Taking the cube root is the opposite of cubing.

3

3

Example Find a 27 b -125

008

3

a 27 = 3 since 3 × 3 × 3 =27 3

b -125 = -5 since -5 × -5 × -5 = -125 1.2 Square, cube and triangular numbers

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Notice that when you calculate a square root you should have ± before it. When you calculate a cube root it will either be positive or negative (but not both).

Reciprocals The reciprocal of a number ax is ax.

Example Find a 47 b 4 c 2x

009

a The reciprocal of 47 is

7 4 1 4 2 x

b The reciprocal of 4 is since 4 can be written as 41) c The reciprocal of 2x is

To find the reciprocal of 1 12, first write it as an improper fraction = 32 and then take the reciprocal.

Hence the reciprocal of 1 12 =

2 3

See Chapter 2 for a definition of an improper fraction

Practice questions 1 Which of these are square numbers? a 48 b 16 c 1 d 81 e 400 2 Which of these are both a square number and a cube number? a 1 b 9 c 16 d 64 e 100 3 Find the value of: 3

a 81 b 64 c 13 d 1002 4 Find the next two numbers in this sequence.

1, 3, 6, 10, 15, ….., …..

5 Give the reciprocal of each of these numbers.

1 c 4 d 3 a 5 b 5 15 3 9 20 e

1.3 Factors, multiples and primes Factors Factors are whole numbers that divide exactly into another number.

Example Find the factors of 12.

010

The factors of 12 are 1, 2, 3, 4, 6, and 12.

Multiples If one number is multiplied by another, the result is a multiple of the first number. The numbers in the multiplication tables are all multiples.

8

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Example Find the multiples of 5.

011

Multiples of 5 are 5, 10, 15, 20, 25,...

Since: 1 × 5 = 5 2 × 5 = 10 3 × 5 = 15 4 × 5 = 20 5 × 5 = 25

Prime numbers A prime number has only two factors, 1 and itself.

The prime numbers up to 20 are: 2, 3, 5, 7, 11, 13, 17, 19 Note that 1 is not a prime number.

Any positive integer can be written as a product of prime factors.

Prime factors Apart from prime numbers, any whole number greater than 1 can be written as a product of prime factors. This means the number is written using only prime numbers multiplied together.

A prime number has only two factors, 1 and itself. 1 is not a prime number.

The diagram below shows the prime factors of 60.

60 30 15 5

2

Divide 60 by its first prime factor, 2.

2

Divide 30 by its first prime factor, 2.

3

Divide 15 by its first prime factor, 3.

We can now stop because the number 5 is prime.

As a product of its prime factors, 60 may be written as: 60 = 2 × 2 × 3 × 5 or in index form: 60 = 22 × 3 × 5

Highest common factor (HCF) The highest factor that two numbers have in common is called the HCF.

Example Find the HCF of 60 and 96.

012

Write the numbers as products of their prime factors.

60 = 2 × 2

×3×5

96 = 2 × 2 × 2 × 2 × 2 × 3 Ring the factors that are common.

60 = 2 × 2

×3×5

96 = 2 × 2 × 2 × 2 × 2 × 3 These give the HCF = 2 × 2 × 3

= 12 1.3 Factors, multiples and primes

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Lowest (least) common multiple (LCM) The LCM is the lowest number that is a multiple of two numbers.

Example Find the LCM of 60 and 96.

013

Write the numbers as products of their prime factors.

60 = 2 × 2

×3×5

96 = 2 × 2 × 2 × 2 × 2 × 3 60 and 96 have a common factor of 2 × 2 × 3, so it is only counted once.

60 = 2 × 2

×3×5

96 = 2 × 2 × 2 × 2 × 2 × 3 The LCM of 60 and 96 is

=2×2×2×2×2×3×5

= 480

Example Buses to St Albans leave the bus station every 20 minutes. Buses to Hatfield leave the

014

bus station every 14 minutes.

A bus to St Albans and a bus to Hatfield both leave the bus station at 10 am.

When will buses to both St Albans and Hatfield next leave the bus station at the same time?

You need to find the LCM of 20 and 14.

20 = 2 × 2 × 5 14 = 2

×7

EV

PS

LCM = 2 × 2 × 5 × 7 [1] LCM = 140 Both buses will leave at the same time 140 minutes later, i.e. 2 hours and 20 minutes.

Time they leave together = 12.20 pm You could also list both the times of the buses from Hatfield and St Albans and find the time that is the same in both lists. This is a useful check.

Buses to St Albans : 20, 40, 60, 100, 120, 140, Buses to Hatfield : 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, Hence both lists have 140 minutes, i.e 2hours and 20 minutes.

Time they leave together = 12.20 pm

Exam tips In questions like the one above, always check that you have answered the question.

Notice that this question not only requires you to find the ‘140’ but then to change it back into the time they will meet.

Practice questions 1 Which of these are not a multiple of 3? a 18 b 25 c 32 d 304 e 6000 2 Which of these are prime numbers? a 1 b 2 c 9 d 17 e 47

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1 Numbers

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3 Write each of these as a product of prime factors. a 24 b 60 c 96 d 140 MR

4 Find the highest common factor of: a 48 and 60 b 96 and 140 5 Find the lowest common multiple of: a 24 and 140 b 60 and 96

1.4 Indices An index is sometimes called a power.

ab

The base

The index or power

Laws of indices The laws of indices can be used for numbers or algebra. The base has to be the same when the laws of indices are applied. See Chapter 5, where indices are applied to algebra.

1

a n × am = an + m

a0 = 1

am = a

a n ÷ am = an - m

a1 = a

(an)m = an × m

a m = ( a)

a-n =

1 an

m

n

m

n

Example Simplify the following, leaving your answers in index notation.

015

a 52 × 53

b 8-5 × 812

c (23)4

a 52 × 53 = 52 + 3 = 55

b 8- 5 × 812 = 8- 5 + 12 = 87

c (23)4 = 23×4 = 212

Example Evaluate: a 42 b 50 c 3-2 d 3612 e 623

016

a 42 = 4 × 4 = 16 1

d 36 2 = 36 = 6

Hint: Evaluate means to work out.

b 50 = 1 2

c 3-2 = 2

e 6 3 = ( 8) = 22 = 4 3

1 32

=

1 9

Example Simplify the following, leaving your answers in index form.

017

a 72 × 75

b 69 ÷ 62

c

a 72 × 75 = 77

b 69 ÷ 62 = 67

c 1

37 × 32 310 7 3 × 32 = 39 310 310

d 79 ÷ 7-10 = 3-1

d 79 ÷ 7-10 = 719

Example Evaluate: a 33 b 70 c 6413 d 81 2 e 5-2 f ( 49)

018

a 33 = 3 × 3 × 3 = 27 1 2

d 81 = 81 = 9

-2

1

b 70 = 1 e 5-2 =

1 52

=

1 25

3

c 64 3 = 64 = 4 f

( 49) = ( 94) -2

2

=

81 16

1 = 5 16

Exam tips ET Questions involving negative powers are tricky. Remember to make the negative power into a positive power first by taking the reciprocal.

1.4 Indices

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Practice questions 1 Write each expression in power notation. a 3 × 3 × 3 × 3 × 3 = b 10 × 10 × 10 = c 2 × 2 × 2 × 2 × 2 × 2 = 2

73 × 75 = c 217 × 23 = d 39 ÷ 33 = a 52 × 52 = b

3

7 = 820 ÷ 815 = c a 65 ÷ 64 = b 74 6

4 Given that 34 = x2 work out the value of x.

5 Evaluate: 1

1

a 120 b 64 2 c 343 3 d 8-1 e 10-3

1.5 Standard index form Standard index form (standard form) is useful for writing very large or very small numbers in a simpler way.

When written in standard form a number will be written as:

a × 10n A number between 1 and 10 1 < a < 10 The value of n is the number of places the digits have to be moved to return the number to its original value.

If the number is 10 or more, n is positive.

If the number is less than 1, n is negative.

If the number is 1 or more but less than 10, n is zero.

Example 1 Write 2 730 000 in standard form.

019

●●

2.73 is the number between 1 and 10 (1 < 2.73 < 10)

●●

Count how many spaces the digits have to move to restore the original number.

The digits have moved 6 places to the left because it has been multiplied by 106 2.73 2730000.

So, 2 730 000 = 2.73 × 106

Example 2 Write 0.000 046 in standard form.

020

●●

Put the decimal point between the 4 and 6, so the number lies between 1 and 10.

●●

Move the digits five places to the right to restore the original number.

●●

The value of n is negative.

So, 0.000 046 = 4.6 × 10-5

Standard form on a calculator To put a number written in standard form into your calculator, you use the following keys:

12

x10 x EXP

or

EE

1 Numbers

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For example, (2 × 103) × (6 × 107) = 1.2 × 1011 would be keyed in as: 2 x10 x 3 × 6 x10 x 7 = or 2 EXP 3 × 6 EXP 7 =

Doing calculations in standard form Work out the following using a calculator. Check that you get the answers given here.

Example (6.7 × 107)3

021

Example

022

Example

023

(6.7 × 107)3 = 3.0 × 1023 (2 s.f.)

(4 × 109) (3 × 104)2 (4 × 109) (3 × 104)2

= 4.4

(5.2 × 106) × (3 × 107) (4.2 × 105)2 (5.2 × 106) × (3 × 107) (4.2 × 105)2

= 884.4 (1 d.p.)

On a non-calculator paper you can use indices to help work out your answers.

Example (2 × 103) × (6 × 107)

024

(2 × 103) × (6 × 107) = (2 × 6) × (103 × 107)

= 12 × 103+7

= 12 × 1010

= 1.2 × 101 × 1010

= 1.2 × 1011

Example (6 × 104) ÷ (3 × 10-2)

025

(6 × 104) ÷ (3 × 10-2) = (6 ÷ 3) × (104 ÷ 10-2)

= 2 × 104-(-2)

= 2 × 106

Example (3 × 104)2

026

(3 × 104)2 = (3 × 104) × (3 × 104)

= (3 × 3) × (104 × 104)

= 9 × 108

You also need to be able to work out more complex calculations.

Example The mass of Saturn is 5.7 × 1026 tonnes. The mass of the Earth is 6.1 × 1021 tonnes.

027

How many times heavier is Saturn than the Earth? Give your answer in standard form.

Correct to 2 significant figures.

5.7 × 1026 6.1 × 1021

= 93 442.6

Now rewrite your answer in standard form.

Saturn is 9.3 × 104 times heavier than the Earth.

1.5 Standard index form

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Practice questions Exam tips Make sure that you check that your final answer is written in standard form.

1 A single cold virus is 0.000 000 02 m long.

Express this in standard index form.

2

1.2 × 105 = c 1.23 × 10−5 = d 1.234 × 10−1 = a 1 × 104 = b

3 One atom of gold has a mass of 0.000 000 000 000 000 000 000 000 33 g.

Express this in standard index form.

4 Work out each of these without using a calculator. Give all your answers in standard index form.

a (6 × 103) + (2 × 102) = b (6 × 103) − (2 × 102) = c (6 × 103) × (2 × 102) = d (6 × 103) ÷ (2 × 102) = 5 You probably have about 2 × 1013 red corpuscles in your bloodstream. Each red corpuscle weighs about 0.000 000 000 1 g. Work out the total mass of your red corpuscles, in kilograms. Give your answer in standard index form

REVISION CHECKLIST ●● Each digit in a number has a place value. The value of the digit depends on its place in the number. ●● Integers are whole numbers that can be positive or negative. ●● When multiplying or dividing positive and negative numbers: –– Two like signs (both + or both -) give a positive answer –– Two unlike signs (one + and the other -) give a negative answer. ●● Any whole number greater than 1 can be written as a product of its prime factors, apart from prime numbers themselves (1 is not prime). ●● The highest factor that two numbers have in common is called the highest common factor (HCF). ●● The lowest number that is a multiple of two numbers is called the lowest (least) common multiple (LCM). ●● Make sure you know and can use all the laws of indices. ●● A negative power is the reciprocal of the positive power. ●● Fractional indices mean roots. ●● Numbers in standard form will be written as a × 10n.

–– 1 < a < 10 –– n is positive when the original number is 10 or more.

–– n is negative when the original number is less than 1.

–– n is zero when the original number is 1 or more but less than 10.

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1 Numbers

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Exam-style Questions

8 2 5

1

Sam uses these cards to make 3-digit numbers.

Write down a number she could make which is less than 300.

2

Which of these has a value which is equal to the value of -13 + 8 ?

Tick all the boxes which are equal to this value.

5 - 10

-11 + 6

-8 - -3

10 - 5

3

Simplify 5 × 5 × 5 × 5. Give your answer in index form.

4

Circle the square numbers.

5

Write down the value of (-7)-3

6

What is the reciprocal of 3 18?

7

Jess simplifies 84 × 83 and gives the answer as 812. Is she correct?

Give a reason for your answer.

8

Find the highest common factor of 28 and 60.

9

Write 116 as a product of prime factors. Give your answer in index notation.

-4

13

120

9

10

Find the lowest common multiple of 12 and 30

11

Write down the value of 125 3

12

What is the value of 160. Circle your answer.

13

Which of these are surds? Tick a box or boxes to show your answer.

1

25

14

1

5 18

0

1

1 3

-4

4

16

3

48

2

Which of these is not a number in standard form? Circle your answer.

10 × 10-5

8 × 10-6

6.2 × 1013

4.9 × 103

1.01 × 1014

15

The light on Tom’s smoke alarm flashes every 50 seconds. The light on Tom’s house alarm flashes every two minutes. They flash together at 2 am. What is the next time they flash together?

16

Written as a product of prime factors, 70 = 2 × 5 × 7 Find the value of a if 700 = 2a × 5a × 7

17

Work out (8 × 103) + (6 × 105). Give your answer in standard form.

18

Find the value of (9 × 105) ÷ (3 × 103). Give your answer as an ordinary number.

19

The population of Iceland is 3 × 105. The population of Scotland is 5.4 × 106. How many times larger is the population of Scotland than the population of Iceland?

20

Write down the value of ( 23) .

-2

Now go back to the list of objectives at the start of this chapter.

How confident do you now feel about each of them?

Exam-style Questions

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