20

2 EXPRESSIONS AND SEQUENCES

In this chapter you will: distinguish the different roles played by

letter symbols in algebra and use the correct notation in deriving algebraic expressions

manipulate algebraic expressions by collecting like terms

use substitution to work out the value of an expression given the value of each letter in the expression

use the index laws applied to simple algebraic expressions

use the index laws applied to algebraic expressions with fractional or negative powers

generate terms of a sequence using term-to-term and position-to-term defi nitions of the sequence

derive and use the nth term of a sequence.

Neptune was the fi rst planet to be found by mathematical prediction. Scientists looked at the number patterns of the orbits of the other planets in the Solar System and correctly predicted Neptune’s position to within a degree. Using the predicted position, Johann Galle identifi ed Neptune almost immediately on 23 September 1846!

Objectives

You should be able to: simplify an expression where each term is in the same letter or letters use directed numbers in calculations use index laws with numbers.

Before you start

20

HI-RES STILL TO

BE SUPPLIED

21

2x, 3y and 2x 3y are called algebraic expressions Each part of an expression is called a term of the expression. 2x and 3y are terms of the expression 2x 3y. When adding or subtracting expressions, different letter symbols cannot be combined. For example 2x 3y

cannot be simplifi ed further.

Key Points

2.1 Letter symbols and notation

Work out the value of Simplfy1. 5 11 1. a a a a2. 3 7 2. b 2b 3b3. 1 3 9 3. 4c c 5c4. 5 (2 4) 4. 2xy 3xy xy5. 4 6 5. 3p2 5p2 4p2

6. 4 2

Get Ready

You can distinguish the different roles played by letter symbols in algebra and use the correct notation.

Objective

A company might use algebraic symbols and numbers as a shorthand method of describing the quantities of each item when doing a stock check.

Why do this?

1 Daniel has p marbles in a bag. He gives 8 marbles to Finlay.Write down an expression, in terms of p, for the number of marbles that Daniel has left in his bag.

Exercise 2A

Mary buys fi ve cups of tea at t pence per cup and three cups of coffee at c pence per cup.Write down an expression, in terms of t and c, for the total cost in pence.

The cost of fi ve cups of tea 5 t 5t pence.

The cost of three cups of coff ee 3 c 3c pence.

Total cost 5t 3c.

Example 1

This can be wri� en as t 5 or 5t.

Watch Out!

Do not try to combine the 5t and the 3c.

Watch Out!

Questions in this chapter are targeted at the grades indicated.

21algebraic expressions term expression letter symbol

2.1 Letter symbols and notation

Chapter 2 Expressions and sequences

22 collecting like terms

The sign of a term in an expression is always written before the term. For example, in the expression 4 2x 3y the ‘’ sign means add 2x and the ‘’ sign means subtract 3y.

The term x can be written as 1x. In algebra, BIDMAS describes the order of operations when collecting like terms

(see Section 1.5 for use of BIDMAS).

Key Points

2.2 Collecting like terms

You can manipulate algebraic expressions by collecting like terms.

Write down algebraic expressions for the following orders at Pete’s Chippy. (Look at the menu in Exercise 4A, question 5.)1. Two fi sh, one chips.2. Three sausages, two chips.3. Four meat pies, one sausage, one fi sh and four chips.

Get Ready

Objective Why do this?

2 Candles are sold in boxes. A small box holds 4 candles.Jo buys x small boxes of candles.a Write down, in terms of x, the total number of candles in these small boxes.A large box holds 10 candles.Jo buys 3 less of the large boxes of candles than the small boxes.b Write down, in terms of x, the number of large boxes she buys.c Find, in terms of x, the total number of candles in the large boxes that Jo buys.

3 Melissa made y cakes. Stuart made 3 more cakes than Melissa.Stuart put 5 sweets on the top of each cake.Write down an expression, in terms of y, for the number of sweets that Stuart used.

4 Angela packs pencils and pens into boxes.Each box can hold either 12 pencils or 10 pens.Angela packs some boxes with pencils and some more boxes with pens.Write down an expression, for the total number of pencils and pens Angela packs.

5 This is part of a menu at Pete’s Chippy.Pete takes an order of: two meat pies, three sausages, one fi sh and four portions of chips.a Write down an algebraic expression that Pete could use to write down this

order of food. Defi ne each letter that you use.b Mrs Smith orders x meat pies and y portions of chips. Write down an expression, in terms of x and y, for the total cost of Mrs Smith’s order.

MenuChipsMeat pieSausageFish

£1.20£1.80£0.90£2.00

DAO2

A03

A03

23like terms

2.2 Collecting like terms

Simplify these expressions.

a 3e 2f e 5fb g 3h 6g 7hc 4p 2q 1 3p 5q

a 3e 2f e 5f 3e e 2f 5f 4e 7f

b g 3h 6g 7h g 6g 3h 7h 7g 4h

c 4p 2q 1 3p 5q 4p 3p 2q 5q 1 p 3q 1

Alfi e is n years old. Bilal is 3 years older than Alfi e. Carla is twice as old as Alfi e.Write down an expression, in terms of n, for the total of their ages in years.Give your answer in its simplest form.

Alfi e n years Bilal (n 3) yearsCarla 2n years

Total n (n 3) 2n n n 3 2n n n 2n 3 4n 3 years

Example 2

Example 3

4e and 7f are not like terms.

3 7 4so 3h 7h 4h

2 5 3so 2q 5q 3q

4p 3p 1p which is wri� en as just p.

This can be wri� en as 3 n.

This is a correct un-simplifi ed expression.

Remove the brackets.

This is in its simplest form.

This can be wri� en as 2 n or n2 or 2n.

1 Simplifya 5x 2x 3y y b 3w 7w 4z 2zc 3p q p 4q d 4a 3b a 2be c 2d 5c 4d f 3m 7n m 4ng 5e 3f e 4f h 2x 8y 3 2y 5i 3p q 2 5p 4q 7 j 9 a 2b 5a 4 3b

Exercise 2B

Examiner’s Tip

Rewrite each expression with the like terms next to each other.

Chapter 2 Expressions and sequences

24 evaluate

If you are given the value for each letter in an expression then you can evaluate the expression.

Key Point

Work out the value of each of these expressions when a 5 and b 3.

a 4a 3b b a 2b 8 c 2a2 4b

a 4a 3b 4 5 3 (3) 20 9 11

b a 2b 8 5 2 (3) 8 5 6 8 3

c 2a2 4b 2 (5)2 4 (3) 2 25 12 50 12 38

Example 4

Positive negative negative.

Work out the multiplication fi rst (BIDMAS). Negative negative positive.

It is only the value of a (5) that is squared.

2.3 Using substitution

Given the value of each letter in an expression, you can work out the value of the expression by substitution.

Write expressions, in terms of x and y, for the perimeter of these rectangles.1. Length 2x 4, width y 2 2. Length 3y 3, width x 5 3. Length 4x 5, width y 2.

Objective

In your science lessons you need to be able to substitute into formulae when carrying out many calculations.

Why do this?

Get Ready

Examiner’s Tip

Replace each letter with its numerical value.

2 Georgina, Samantha and Mason collect football stickers. Georgina has x stickers in her collection. Samantha has 9 stickers less than Georgina. Mason has 3 times as many stickers as Georgina.Write down an expression, in terms of x, for the total number of these stickers. Give your answer in its simplest form.

3 The diagram shows a triangle.Write down an expression, in terms of x and y, for the perimeter of this triangle.Give your answer in its simplest form.

4x � 2y

10y � x

2x � 5yA03

25

2.4 Using the index laws

1 Work out the value of each of these expressions when x 4 and y 1.a x 3y b x y c 2x 5y 3 d 4x 1 2y

2 Work out the value of each of these expressions when p 2, q 3 and r 5.a p q r b 2q 3r 5p c 2q r 3pd 6 q 2r p e 5p 3q2 f p2 2q2 r2

Exercise 7C

Example 5

You can use the laws of indices to simplify algebraic expressions. See Section 1.6 for the index laws.

Key Points

a Simplify c3 c4

b Simplify 5y3z5 2y2z

a c3 c4 c c c c c c c c7

b 5y 3z5 2y 2z 5 y 3 z 5 2 y 2 z

5 2 y 3 y 2 z 5 z1

10 y 32 z 51

10 y 5 z 6

10y 5z 6Using x p x q x p q

2.4 Using the index laws

Get Ready

Objective Why do this?

You understand and can use the index laws applied to simple algebraic expressions.

To write large numbers, like the speed of sound, indices are often used to shorten the way the value is written.

Simplify these expressions using the index laws.1. x3 x2 2. x6 x4 3. (x2)3

Watch Out!

Group like terms together before attempting to use the laws of indices.

Watch Out!

z is the same as z1

Note: 3 4 7.

Chapter 2 Expressions and sequences

26

1 Simplifya m m m m m b 2p 3p c q 4q 5q

2 Simplifya a4 a7 b n n3 c x5 x d y2 y3 y4

3 Simplifya 2p2 6p4 b 4a 3a4 c b7 5b2 d 3n2 6n

4 Simplifya 5t3u2 4t5u3 b 2xy3 3x5y4 c a2b5 7a3b

d 4cd5 2cd4 e 2mn2 3m3n2 4m2n

Exercise 2D

a Simplify d5 d2

b Simplify 10x2y5

______ 2xy3

a d5 d2 d5 __

d2 d d d d d _______________ d d

d3

b 10x2y5

_______ 2xy3 is the same as 10x2y5 2xy3

10x2y5 2xy3 (10 2) (x2 x) (y 5 y 3)

5 x21 y 53

5 x y 2

5xy2

Example 6

Using x p x q x p q

1 Simplifya a7 a4 b b5 b c c

8 __

c5 d d4 d3

2 Simplifya 6q5 3q3 b 12p7 4p2 c 8x6 2x5 d

20y8

____ 2y

3 Simplifya 15a5b6 3a3b2 b 30p3q4 6p2q c 8c4d7

_____ 2c2d3 d 6x3 2x4 ________ 4x2

e 5m2n 4mn2 ___________ 2mn2

Exercise 2E

Note: 5 2 3 Examiner’s Tip

Write fractions, such as p5

__ p3

as p5 p3.

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2.5 Fractional and negative powers

Simplify (2c3d)4

Method 1(2c3d)4 (2)4 (c3)4 (d)4

16 c3 4 d1 4

16 c12 d4

16c12d4

Method 2(2c3d)4 can be wri� en as 2c 3d 2c 3d 2c 3d 2c3d

2 2 2 2 c 3 c 3 c 3 c3 d d d d

16 c 3 3 3 3 d 4

16 c12 d 4

16c12d 4

Example 7

Using x p x q x p q

Using (x p)q x p q

1 Simplifya (a7)2 b (b3)5 c (c 3)3 d (d 2)8

2 Simplifya (2p3)2 b (3q2)4 c (5x 4)2 d (� m4

___ 2 ) 33 Simplify

a (2x 3y2)4 b (7e5f 3)2 c (5p 5q)3 d (� 2x4y2

_____ 3xy4 ) 3

Exercise 2F

2.5 Fractional and negative powers

Simplify these expressions.1. (a3)6 2. (3y5)3 3. (� 4a3b2

_____ 2a2b5 ) 2

You can use the index laws applied to algebraic expressions with fractional or negative powers.

To write very small numbers, like the radius of a molecule, negative powers of 10 are used.

Get Ready

Objective Why do this?

Examiner’s Tip

You must apply the power to number terms as well as the algebraic terms.

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Chapter 2 Expressions and sequences

28

The laws of indices used so far can be used to develop two further laws.x4 x4 x44 x0

Alsox4 x4 1 since any term divided by itself is equal to 1Therefore x0 1

In generalx0 1

The laws of indices can be used further to solve problems with fractional indices.The square root of x is written √

__ x, and you know that:

√__

x √__

x x

Using xp xq xp q

x 1 _ 2 x

1 _ 2 x 1 _ 2

1 _ 2 x1 xand so, x

1 _ 2 √__

x

Also, x 1 _ 3 x

1 _ 3 x 1 _ 3 x, showing that x

1 _ 3 3 √__

x

In general x

1 _ n n √__

x

Key Points

x3 x4

x x x ____________ x x x x 1 __

x

Also, using xp xq xpq

x3 x4 x34 x1

Therefore x1 1 __ x

In generalxm 1 ___

xm

Simplify (3x4y)2

(3x4y)2 1 _______ (3x 4y)2

1 _____ 9x8y2

Example 8

Using xm 1 ___ xm

Using (xp)q xp q

1 Simplifya a1 b (b2)1 c c2 d (d 3)1

2 Simplifya (e 3)2 b (f 2)4 c (x1)2 d (y1)1

3 Simplifya (x2y7)0 b (2x4y5)0 c (5p2q4)1 d (3c3d)3

e (� 2p3q ____ 3r2 ) 2

Exercise 2G

Examiner’s Tip

Remember that a negative power just means ‘one over’ or ‘the reciprocal of’.

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29sequence rule terms of the sequence

2.6 Term-to-term and position-to-term defi nitions

Simplify (8x 6y 4 ) 1 _ 3

(8x 6y 4 ) 1 __ 3 8

1 __ 3 (x 6 ) 1 __ 3 (y 4 )

1 __ 3

3 √__

8 x 6 1 __ 3 y 4 1 __ 3

2 x2 y 4

_ 3

2x 2 y 4

_ 3

Example 9

Using x 1 __ n n √

__ x

Using (x p)q x p q

1 Simplifya (9a4 )

1 _ 2 b (16c2 ) 1 _ 4 c (27e3f 9 )

1 _ 3 d (100x3y5 ) 1 _ 2

2 Simplifya (a4 ) 1 _ 2 b (8c3 ) 1 _ 3 c (32x9y5 ) 1 _ 5 d (x2y6 ) 1 _ 4

Exercise 2H

A sequence is a pattern of shapes or numbers which are connected by a rule (or defi nition of the sequence). The relationship between consecutive terms describes the rule which enables you to fi nd subsequent terms of the sequence.Here is a sequence of 4 square patterns made up of squares:

Pattern 1 Pattern 2 Pattern 3 Pattern 4

Key Points

2.6 Term-to-term and position-to-term de� nitions

Continue these number patterns.1. 2, 4, 6, 8, 10, … 2. 4, 9, 14, 19, 24, 29, … 3. 1, 3, 5, 7, 9, …

You can generate terms of a sequence using term-to-term and position-to-term defi nitions of the sequence.

Objective

To recognise trends in specifi c illnesses in a country or the world, patterns linking data are often used.

Why do this?

Get Ready

Examiner’s Tip

Remember that the denominator of the index is the root.

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Chapter 2 Expressions and sequences

30 term-to-term position-to-term

Each pattern above is a term of the sequence;

is the 1st term in the sequence,

is the 2nd term in the sequence, etc.

The number of squares in each term form a sequence of numbers, 1, 4, 9, 16, … You can continue a sequence if you know how the terms are related: the term-to-term rule. You can continue a sequence if you know how the position of a term is related to the defi nition of the

sequence: the position-to-term rule.

Find a the next term, and b the 12th term of the sequence of numbers: 1, 4, 9, 16, …

1st term 2nd term 3rd term 4th term 5th term1 4 9 16

3 5 7

a The diff erence between the 4th and the 5th term is 9 and so the 5th term is 16 9 25.

b The 6th term 62 36, the 7th term 72 49, etc.

The 12th term 122 144.

Example 10

Find a the term-to-term rule,b the next two terms, and

c the 10th term for each of the following number sequences.

1 2 5 8 11

2 4 2 8 14

3 19 12 5 2

4 1 3 6 10

5 0 2 6 12

Exercise 2I

The diff erence between consecutive terms increases by 2.This is the term-to-term rule which enables you to fi nd subsequent terms of the sequence.

The numbers 1 (12), 4 ( 22), 9 ( 32), 16 ( 42) and 25 ( 52) are the fi rst fi ve square numbers.

In this way a term of the sequence can be found by the position of the term in the sequence.

31arithmetic sequence difference zero term

2.7 The nth term of an arithmetic sequence

An arithmetic sequence is a sequence of numbers where the rule is simply to add a fi xed number.For example, 2, 5, 8, 11, 14, … is an arithmetic sequence with the rule ‘add 3’.In this example the fi xed number is 3.

This is sometimes called the difference between consecutive terms. You can fi nd the nth term using the result nth term n difference zero term. You can use the nth term to generate the terms of a sequence. You can use the terms of a sequence to fi nd out whether or not a given number is part of a sequence, and

explain why.

Key Points

2.7 The nth term of an arithmetic sequence

Find 1. the rule, 2. the next two terms, 3. the 10th term for each of the following number sequences.a 1, 4, 7, 10, … b 4, 1, 2, 5, 8, … c 124, 118, 112, 106, 100, …

You can use linear expressions to describe the nth term of a sequence.

You can use the nth term of a sequence to generate terms of the sequence.

Objectives

To be able to predict how many people might catch fl u, epidemiologists need to develop a general rule.

Why do this?

Get Ready

Here are the fi rst fi ve terms of an arithmetic sequence: 2, 5, 8, 11, 14, …

a Write down, in terms of n, an expression for the nth term of the arithmetic sequence.b Use your answer to part a to fi nd the 20th term.

zero term 1st term 2nd term 3rd term 4th term 5th term–1 2 5 8 11 14

3 3 3 3 3

difference a The zero term is the term before the fi rst term.

Work out the zero term by using the diff erence of 3.Zero term 2 3 1

The nth term n diff erence zero term nth term n 3 1 3n 1

b For the 20th term, n 20When n 20, 3n – 1 3 20 1 60 1 59 So the 20th term is 59.

Example 11

Inverse of 3.Examiner’s Tip

Always check your answer by substituting values of n into your nth term.For example,1st term, when n 1, 3n 1 3 1 1 2 ✓2nd term, when n 2, 3n 1 3 2 1 5 ✓ 3rd term, when n 3, 3n 1 3 3 1 8 ✓ etc.

Chapter 2 Expressions and sequences

32

1 Write down a the difference between consecutive terms b the zero term for each of the following arithmetic sequences.

0, 2, 4, 6, 8, …

7, 3, 1, 5, 9, …

14, 9, 4, 1, 6, …

2 Here are the fi rst fi ve terms of an arithmetic sequence: 1, 7, 13, 20, 26, …a Write down, in terms of n, an expression for the nth term of this arithmetic sequence.b Use your answer to part a to work out the i 12th term, ii 50th term.

3 Here are the fi rst four terms of an arithmetic sequence: 7, 11, 15, 19, …a Write down, in terms of n, an expression for the nth term of this arithmetic sequence.b Use your answer to part a to work out the i 15th term, ii 100th term.

4 Here are the fi rst fi ve terms of an arithmetic sequence: 32, 27, 22, 17, 12, …a Write down, in terms of n, an expression for the nth term of this arithmetic sequence.b Use your answer to part a to work out the i 20th term, ii 200th term.

5 Here are the fi rst four terms of an arithmetic sequence: 18, 25, 32, 39, …Explain why the number 103 cannot be a term of this sequence.

6 Here are the fi rst fi ve terms of an arithmetic sequence:7 11 15 19 23Write down, in terms of n, an expression for the nth term of this sequence.Pat says that 453 is a term in this sequence. Pat is wrong.Explain why. Nov 2005

Exercise 2J

2x, 3y and 2x 3y are called algebraic expressions.

Each part of an expression is called a term of the expression.

When adding or subtracting expressions, different letters cannot be combined.

The sign of a term in an expression is always written before the term.

The term x can be written as 1x.

In algebra, BIDMAS describes the order of operations.

If you are given the value for each letter in an expression then you can evaluate the expression.

You can use the laws of indices to simplify algebraic expressions.

The basic index laws can be used to develop further laws:

xp xq xp q xp xq xp q (xp)q xp q

and x0 1, for all values of x

and xm 1 ___ xm and x

1 _ n n √__

x where m and n are integers.

A sequence is a pattern of shapes or numbers which are connected by a rule (or defi nition of the sequence).

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Chapter review

33

Chapter review

The relationship between consecutive terms describes the rule which enables you to fi nd subsequent terms of the sequence.

You can continue a sequence if you know how the terms are related: the term-to-term rule. You can continue a sequence if you know how the position of a term is related to the defi nition of the sequence:

the position-to-term rule. An arithmetic sequence is a sequence of numbers where the rule is simply to add a fi xed number. This is

called the difference between consecutive terms. The nth term of an arithmetic sequence can be found using the result

nth term n difference zero term. You can fi nd the nth term of an arithmetic sequence using the result nth term n difference zero term. You can use the nth term of an arithmetic sequence to generate the terms of a sequence. You can use the terms of a sequence to fi nd out whether or not a given number is part of a sequence, and

explain why.

1 A cup of tea costs x pence. A cup of coffee costs y pence.Viv buys 3 cups of tea and 5 cups of coffee.Write down an expression, in terms of x and y, for the total cost.

2 Samantha is absent from school on n days. Georgina is absent from school on 3 more days than Samantha. Melissa is absent from school on 5 days less than Georgina. Write down, in terms of n, the number of days that Melissa is absent from school.

3 Simplify a 3x 4y 2x y b m 7n 5m 3n

4 Helen and Stuart collect stamps.Helen has 240 British stamps and 114 Australian stamps. a Write down an algebraic expression that could be used to represent Helen’s British and Australian stamps. Defi ne the letters used.Stuart has 135 British stamps and 98 Australian stamps.b Using the same letters, write down an algebraic expression that could be used to represent the total of Helen’s and Stuart’s British and Australian stamps.

5 To calculate the cost of printing leafl ets for a school fair, the printer uses the formula: C 40 0.05n

where C is the cost in pounds and n is the number of leafl ets printed.a How much would it cost to print 200 leafl ets?

b Can you suggest what 40 and 0.05 represent?

6 Work out the value of each of these expressions when x 2, y 3 and z 7a 3x y b x 2y c x 3y 2z d 5xy e x2 y2 z2

7 The formula used to convert temperatures in Fahrenheit, F, into Celsius, C, is given by:

C 5(F – 32) _______ 9

a Find C when F 77.b Use the formula to fi nd the freezing point of water in Fahrenheit.A newspaper headline read ‘Phew, what a scorcher! Temperature soars into the 100s.’c What temperature unit are they using? What is its equivalent in the other unit?

Review exercise

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Chapter 2 Expressions and sequences

34

8 Simplifya y y y b x 3x c z3 z5 d p p6 e 2a2 8a5

9 Simplifya a6 a3 b b9 b4 c 21p4 3p d 24x5

____ 3x2 e 16a6b3 2a5b3

10 Find a the rule b the next two terms c the 12th term for this number sequence. 102 99 96 93 90

11 Write down a the difference between consecutive terms, b the zero term of this arithmetic sequence. 3 2 7 12 17

12 Here are the fi rst fi ve terms of an arithmetic sequence: 0, 4, 8, 12, 16, ... a Write down, in terms of n, an expression for the nth term of this arithmetic sequence.b Use your answer to part a to work out the i 20th term, ii 1000th term.

13 Here are the fi rst four terms of an arithmetic sequence: 204, 192, 180, 168, …a Write down, in terms of n, an expression for the nth term of this arithmetic sequence.b Use your answer to part a to work out the i 13th term ii 99th term.

14 Here are the fi rst fi ve terms of an arithmetic sequence: 3, 7, 11, 15, 19, ... Prove that the number 5381 is not a term in this sequence.

15 The nth term of a sequence is n2 4Alex says “The nth term of the sequence is always a prime number when n is an odd number.”Is Alex correct? You must give a reason for your answer. Nov 2008, Paper 2 Adapted

16 Here are the fi rst 5 terms of a sequence. 1 1 2 3 5

The rule for the sequence is “The fi rst two terms are 1 and 1. To get the next term add the two previous terms”Explain why, after the fi rst two terms, the other terms of the sequence are alternately even and odd.

17 Here are the fi rst four terms of an arithmetic sequence.5 8 11 14Is 140 a term in the sequence? You must give a reason for your answer.

18 Neal is asked to produce an advertising stand for a new variety of soup. He stacks the cans according to the pattern shown.

The stack is 4 cans high and consists of 10 cans.a How many cans will there be in a stack 10 cans high?b Verify that the total number of cans (N) can be calculated by the formula N h(h 1) _______ 2 when

n number of cans high.c If he has 200 cans, how high can he make his stack?

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Chapter review

19 Naismith, an early Scottish mountain climber, devised a formula that is still used today to calculate how long it will take mountaineers to climb a mountain. The metric version states: Allow one hour for every 5 km you walk forward and add on 1 _ 2 hour for every 300 m of ascent.a How long should it take to walk 20 km with 900 m of ascent?A mountain walker’s guide contains the following information for a particular walk.

Helvellyn HorseshoeGlenridding to Helvellyn via the edges (circular walk)Length: 8.5 kmTotal ascent: 800 mTime: 4 hour round trip

b Calculate how long this walk should take according to Naismith’s formula. Give your answer to the nearest minute.

c Suggest reasons why this time is different to the one in the guidebook.

20 Simplifya (a5)4 b (3b4)2 c (3e5f)3

21 The n th even number is 2n Show algebraically that the sum of three consecutive numbers is always a multiple of 6.

Nov 2008, Paper 4 Adapted

22 The expression 6x2 y

_____ 4y3 can never take a negative value. Explain why

23 Simplifya (x5) 1 b (y3)6 c (xy)0 d (2m2n) 4

24 Simplifya (16g6 )

1 _ 2 b (64xy6 ) 1 _ 3 c (y8)

1 _ 2 d (a4b10) 1 _ 4

25 a Simplify (� 9p4 ___ 4y2 )

1 _ 2

b Simplify (�2q3 ) 2

c Simplify (� 12xy3

_____ 3x5 y ) 1 _ 2

26 A 4 by 4 by 4 cube is placed into a tin of yellow paint.

When it has dried, the 64 individual cubes are examined.How many are covered in yellow paint on 0 sides, 1 side, 2 sides ...? Extension 1: Repeat the question for an n by n by n cube, and show that your expressions add up to n3.Extension 2: Repeat the question for a cuboid of dimensions l by m by n.

?? Jim has £x. Bill has £3 more than than Jim. Beccy has twice as much as Bill. Write down an expression, in terms of x, for the total amount in pounds they have altogether.

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