Essential Mathematics Pb

Oxford

Essential Mathematics Book 2 David Rayner

Oxford University Press

Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan Oxford is a trade mark of Oxford University Press

© David Rayner All rights reserved. This publication may not be reproduced, stored or transmitted, in any forms or by any means, except in accordance with the terms of licences issued by the Copyright Licensing Agency, or except for fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988. Enquiries concerning reproduction outside those terms should be addressed to the Permissions Department, Oxford University Press. First published 1995 by Elmwood Press Reprinted 1996 by Oxford University Press ISBN 0 19 914662 4 Artwork by

Gopi Keilman Angela Lumley Eleanor Galvin Lisa Lee Emma Djonokusumo Susie Glass Kathryn Alcock Paulina Spencer Emily Lemoniati

Typeset and illustrated by Tech-Set, Gateshead, Tyne and Wear. Printed in Great Britain by

Butler & Tanner Ltd, Frome and London

PREFACE This is the second of a three book series written for pupils in the age range 11- 14 years. Many classrooms will contain children with a range of abilities in mathematics. This book is written to cater for this situation and also has an ample supply of questions and activities to stretch the most able. The author believes that children learn mathematics most effectively by doing mathematics. Interest and enthusiasm for the subject is engendered by working at questions which stretch the pupil forward in his or her knowledge and understanding. The author, who is a teacher, emphasises a thorough grounding in the fundamentals of number and algebra when working in the lower secondary classroom. It is hoped that the material in the books will stimulate young minds

and encourage logical thinking. Some exercises provide the necessary practice in the basic mathematical skills required for later study. There is no set path through the book and it is anticipated that most teachers will prefer to take sections in the order of their own choice. No text book will have the 'right' amount of material for every class and the author believes that it is better to have too much material rather than too little. Consequently teachers should judge for themselves which sections or exercises can be studied later. On a practical note, the author recommends the use of exercise books consisting of 7 rom squares. Many activities, investigations, games and puzzles are included to provide a healthy variety of learning experiences. The author is aware of the difficulties of teaching on 'Friday afternoons' or on the last few days of term, when both pupils and teachers are tired, and suitable activities are included. The author is indebted to the many students and colleagues who have assisted him in this work. He is particularly grateful to Christine Godfrey, Julie Anderson, David Moncur and Philip Cutts for their contributions to the text. The author would also like to thank Michelle Hawke for her work and sense of humour in editing and checking answers. David Rayner

CONTENTS Page

Part 1 1.1 Circles: circumference and area 1.2 Fractions, decimals and percentages 1.3

1.4 1.5

Part 2.1 2.2 2.3 2.4 2.5

2.6

Reflection Negative numbers Puzzles and investigations 1 Operator squares, Forestry problem.

13 18 24

2

Using a calculator Rotation Bearings Ratio Mixed problems Puzzles and investigations 2 Puzzles, Hidden words Perimeters and common edges.

Part 3.1 3.2 3.3 3.4 3.5

3

Part 4.1 4.2 4.3 4.4 4.5

4

Part 5.1 5.2 5.3

5

5.4

1 9

Pythagoras' theorem Sequences Formulas 3-D objects Mid-book review Multiple choice, Review exercises. Brackets and equations Volume Mental arithmetic Enlargement Puzzles and investigations 4 Crossnumbers, Round the class, Finding areas by counting.

Estimating Scatter graphs Averages Puzzles and investigations 5 Break the codes, Vending machine problem, Dominoes to hexominoes, Investigating tables.

27 30 37 45 50 56

62

68 72

79 82

89 101 108

112 118

123 127 132 140

Part 6.1 6.2 6.3 6.4 6.5

6

Speed Percentages Plotting graphs Probability Puzzles and investigations 6 Puzzles, Around and around, General Knowledge Quiz.

145 151 155 163 171

Part 7 7.1 Multiple choice tests 7.2 Revision exercises

174 178

Index

186

Part 1 c

1.1 Circles Circumference • The length of the perimeter of a circle is called the circumference. A radius is a line drawn from the centre of the circle to a point on the circle. A diameter is twice the radius.

• Find 8 circular objects (tins, plates, buckets, wheels etc.). For each object, measure the diameter and the circumference and write the results in a table. Use a flexible tape measure for the circumference or wrap a piece of string around the object and then measure the string with a ruler. For each pair of readings, work out the ratio (circumference 7 diameter). You should find that the number in the!:.. column is about the same each

Object

Circumference c

diameter d

Tin of tuna

28·6cm

8·8cm

d

time. Work out the mean value of the 8 numbers in the !:.. column. d

. .. ...

. . (circumference) • For any Circle, the exact value of the ratio - - . - - - - is a diameter number denoted by the Greek letter n.

Since~= n,

we can write

G9J. Learn this formula.

0

button, which will give the value of n Most calculators have a correct to at least 7 significant figures: 3·141593. • The number n has fascinated mathematicians for thousands of years. The Egyptians had a value of 3·16 in 1500BC. In about 250 B.C. the Greek mathematician Archimedes showed that n

c d

3·25

2

Part 1

was between 3~~ and 3~~- He considered regular polygons with many sides. As the number of sides in the polygon increased, so the polygon became nearer and nearer to a circle. Ludolph Van Ceulen (1540--1610) obtained a value of n correct to 35 significant figures . He was so proud of his work that he had the number engraved on his tombstone. In 1897 politicians in Indiana, USA displayed a complete lack of mathematical understanding when they passed a local law stating that the value of pi was to be taken as 4. The law was soon discarded! In more recent times a book was published, imaginatively titled 'pi' . The book consisted of the first one million decimal places of pi, calculated by computer. The book was not a best seller. (a) Calculate the circumference of the circle shown.

-

7 em

\ J

-

radius= diameter = circumference = =

7cm 14 em n x 14 44-0cm (3 s.f.)

(b) A circular tin of diameter 9 em rolls along the floor for a distance of 3m. How many times does it rotate completely? circumference = n x 9 = 28·274334cm 3m= 300cm

300 Number of rotations = - - - 28·274334 = 10·61 The tin makes 10 complete rotations.

Exercise 1

In Questions 1 to 8 calculate the circumference of the circle. Use the 'n' button on a calculator or take n = 3·142. Give the answers correct to 3 significant figures, unless told otherwise. 1.

2.

3

Circles

5.

9. A bicycle wheel of diameter 80 em makes 20 complete rotations as the bicycle moves forward in a straight line. Find the circumference of the wheel and work out how far the bicycle moves forward. Give your answers in metres. 10. In a coin rolling competition Gemma rolls a one ·pound coin on its edge a distance of 4· 2m. A one pound coin has diameter 2·2cm. How many times did the coin rotate completely? 11. A tennis ball of diameter 7 em and a golf ball of diameter 4· 25 em roll in a straight line so that each ball makes 20 complete revolutions. Which ball will go further and by how much? Give your answer to the nearest em. 12. A car tyre has a radius of 37 em. (a) How long is its circumference in em? (b) How many complete rotations will the tyre make if the car travels 2 km? 13. Which has the longer perimeter and by how much: an equilateral triangle of side 10 em or a circle of diameter 10cm? 14.

An airplane propeller has blades which are 92 em long. (a) How far will the tip of one blade travel in one revolution? (b) If the propeller makes 2 revolutions per second, how far will the tip of one blade travel in one minute? Give your answer to the nearest metre.

15. A newt walks around the edge of a circular pond at a speed of 2 cmjs. How long will it take to walk all the way round if the radius of the pond is 1· 3 m?

4

Part 1

16. A tin of tomatoes has diameter

7·5 em. The tin is wrapped in a paper cover which is long enough to allow 1em overlap for fixing. How long is the cover?

I

1 em overlap

17. A push chair has wheels of

diameter 66 em at the back and wheels of diameter 18 em at the front. The pushchair travels in a straight line and the rear wheels rotate completely 84 times. (a) How far in metres does the chair travel? (b) How many complete rotations do the front wheels make?

18. A trundle wheel can be used for measuring distances along roads or pavements. A wheel of circumference one metre is pushed along and distance is measured by counting the number of rotations of the wheel. Calculate the diameter of the wheel to the nearest mm. 19. The perimeter of a circular pond is 11· 7 m. Calculate the

diameter of the pond to the nearest em. 20. The tip of the minute hand of Big Ben is 4·6 m from the centre

of the clock face. Calculate the distance, in km, moved by the end of the minute hand in one year (365 days). 21. A cycling track is a circle of diameter 150m. The wheels of a

bicycle have diameter 82 em. How many times will the wheels of the bicycle rotate completely when the bicycle travels ten times around the track? 22. The tip of the second hand of an electric clock

moves at a speed of 1 cmjs. Calculate the distance x, from the tip of the second hand to the centre of the clock.

8

5

Circles

Perimeters Calculate the perimeter of the shape. The perimeter consists of a semi-circle and 3 straight lines. 1t X 10 . . l Lengt h of semt-ctrc e = - - -

I

2

=nx5cm

4cm

~

.·. Perimeter of shape= (n x 5) + 4 + 10 + 4 = 33·7 em (to 3 s.f.)

lOcm

Exercise 2

Calculate the perimeter of each shape. All arcs are either semi-circles or quarter circles. 1.

3.

2.

8.5

15 em

6.

t

t

3cm

2.5 em

~

7cm

+

9cm

8.

9. Scm

--r

t

4cm

~

4cm

__ L

7cm

'

'

:-.----- 8.5 em-------+-:

lOcm

~m

6

Part 1

10.

12.

Area of a circle (a) The circle below is divided into 12 equal sectors

(b) The sectors are cut and arranged to make a shape which is nearly a rectangle. (one sector is cut in half).

1

radius, r

l (c) The approximate area can be found as follows: length of rectangle ~ half circumference of circle n x 2r ~---

2 nr rectangle~ r rectangle ~ nr x r ~ nr 2 ~

width of .·. area of

If larger and larger numbers of sectors were used, this approximation would become more and more accurate. This is a demonstration of an important result.

~

Learn this formula.

7

Circles

Find the area of each shape.

-

(a) {

26cm

\

(b)

\ I ---3 .2cm-

radius = 13 em area = nr 2 2 = n(13 ) = 531 cm 2 (3 s.f.)

The shape is a quarter circle 2 n(3 ·2) area = ----'-------'4 = 8·04cm2 (3 s.f.)

On a calculator, press:

On a calculator, press:

@]0@]000

Cill0Cill00G0G

Exercise 3

In Questions 1 to 8 calculate the area of each circle correct to 3 s.f. 1.

2.

3.

4.

5.

6.

7.

8.

8

Part 1

In Questions 9 to 22 g1ve your answers correct to 3 s.f., where necessary. 9. When hunting for food, an eagle flies over a circular region of radius 3·5 km. What is the area of this region in km 2? 10. A carton of 'Verdone' weedkiller

contains enough weedkiller to treat an area of 100m2 . A circular lawn at Hampton Court has a radius of 16·5 m. How many cartons of weedkiller are needed to treat this lawn? In Questions 11 to 14 find the area of each shape. All arcs are either semi-circles or quarter circles and the units are em. 13.

11.

1

Scm

l

7 L...-------1

10

+

14.

D

-7------ 62 (b) 61 --> 63 (c) 61--> 64 (d) 61 --> 65 11. Draw axes with values from -7 to +7 and draw triangles with the following vertices: 61 : (3, 1) (7, 1) (7, 3) 62 : (1, 3) (1, 7) (3, 7) 63 : (7, -1) (3, -1) (3, -3) 64: (-1, -7) (-3, -7) (-3, -3) 65 : (-2, 2) (-6, 2) (-6, 0) 66 : (3, -4) (3, -6) (7, -6) Describe fully the following rotations or reflections: (a) 61-->62 (b) 61-->63 (c) 61 --> 64 (d) 61 --> 65 (e) 63--> 66

2.3 Bearings Bearings are used by navigators on ships and aircraft and by people travelling in open country. Bearings are measured from north in a clockwise direction. A bearing is always given as a three-figure number. A bearing of 090° is due east. If you are going southwest, you are on a bearing 225°. North

North

North Richard

310°

James is walking on a bearing of 035°.

Mary is walking on a bearing of 146°.

Richard is walking on a bearing of 31 oo.

38

Part 2

Exercise 7 1. Ten children on a treasure hunt start in the middle of a field and begin walking in the directions shown on the right. On what bearing is each child walking?

2. Ten pigeons are released and they fly in the directions shown below. On what bearing is each pigeon flying?

North

North

H

Janet S

outh

B

Wendy

3. Competitors in an orienteering event travel between 5 checkpoints A, B, C, D and E, which are joined by straight paths. B

E

South

39

Bearings

Measure the bearing for each part of the course below (a) (i) A to B (b) (i) C to A (c) (i) C to B (ii) B to C (ii) A to E (ii) B to D (iii) C to D (iii) E to D (iii) D to A (iv) D to E (iv) D to B (iv) A to C 4. Copy and complete (a) Due west is a bearing of (c) North-west is a bearing of

(b) South-east is a bearing of (d) Due south is a bearing of

Relative bearings The bearing of A from B is the bearing that we would take if we wanted to travel from B to A. It is helpful to draw arrows on diagrams, as below. North

(a) The bearing of B from A is 040°. (b) The bearing of A from B is 220°.

The position of an object, like a ship or an aircraft, can be fixed when its bearing is known from two different points. On the map, the ferry 'Micheline' is on a bearing of 030° from Dover and 121 o from Lowestoft. There is only one place it can be.

North

(a) The bearing of D from C is 120°. (b) The bearing of C from D is 300°.

North

1

40

Part 2

A reminder:

Parallel lines occur frequently in this topic. x + y = 180° in each diagram.

Exercise 8

1. (a) For each diagram, write down the bearing of (i) B from A, (ii) A from B. North

North

North

North

North

(b) What is the connection between the two answers in each case? (c) The bearing of a point C from a point D is 075°. Without sketching a diagram, write down the bearing of D from C.

2.

North

North Batford

North

3. North

Denfield

Find the bearing of (a) Batford from Ashby (b) Clamp ton from Ashby (c) Clampton from Denfield (d) Clampton from Batford (e) Denfield from Clampton (f) Batford from Clampton (g) Ashby from Denfield

Find the bearing of (a) R from P (b) T from Q (c) R from T (d) Q from R (e) P from R (f) Q from T (g) P from T

North

North

41

Bearings

North

4. Without using a protractor, decide which point is on a bearing of about (a) 010° from F, (b) 270° from D, (c) 160° from C, (d) 230° from B, (e) 320° from F, (f) 070° from A, (g) 340° from G.

1

c

e

eo

e E

e

eo

F

5. Use the diagram in question 4. From which point is: (a) F on a bearing of 140°? (c) E on a bearing of 270°?

(b) F on a bearing of 190°? (d) B on a bearing of 280°?

6. (a) On squared (or graph) paper mark points A and B, 8 units apart. Leave space above and below the line AB.

(b) Mark the point P which is (i) on a bearing 070° from (ii) on a bearing 320° from (d) Mark the point R which is (i) on a bearing 124° from (ii) on a bearing 180° ftom

A B. A B- ·

(c) Mark a point Q which is (i) on a bearing 162° from A (ii) on a bearing 225° from B (e) Measure the bearing of (i) P from Q (ii) R from Q

42

Part 2

7. The map shows several features on and around an island. Axes are drawn to identify positions. [eg The coordinates of the cave are (9, 3).] North

y

9f

+

I

+

r· · · · · · ··+

r········· ··+

+·····~~~··

+············+

8 7 ..

6 ..

3f

+

l-if--1-~""".-+·

2 .

...........f......

···············t······

2

X

3

4

5

6

7

8

9

10

11

Four commandos, Piers, Quintin, Razak and Smudger, are in hiding on the island. Find the coordinates of the commandos, using the following information. (a) The castle ruins are due south of Piers and the waterfall is due west of him. (b) From Quintin, the bearing of the satellite dish is 045° and the shipwreck is due south of him. (c) From Razak, the bearing of the waterfall is 315° and the bearing of the castle ruins is 045°. (d) From Smudger, the bearing of the cave is 135° and the bearing of the waterfall is 225°. (e) The leader of the commandos is hiding somewhere due north of the shipwreck in a hollow tree. From this tree, the castle ruins and the cliffs are both on the same bearing. Find the coordinates of this hollow tree.

Scale drawing Some problems involving bearings can be solved using a scale drawing. Since bearings are measured from north it is convenient to use squared paper. This enables you to place your protractor accurately in a vertical position. Begin questions by drawing a small sketch to get an idea of where the lines will go. Choose as large a scale as possible for greater accuracy.

43

Bearings

A ship sails 7 km on a bearing 040° and then a further 8 km on a bearing 300°. How far is the ship from its starting point? We will use a scale of 1 em to 1 km. (a) Mark a starting pointS and measure 40° from a vertical line through S and draw a line at this angle. (b) Mark a point A, 7 em from S, and draw a vertical line through A. (c) Draw a line at 60° to the vertical, as shown, (for the bearing 300°). (d) Mark a point B, 8 em along the line. (e) Measure the distance SB. Answer: The ship is 9·7 km from its starting point. [An answer between 9·6 km and 9·8 km would be acceptable].

Exercise 9

Use a scale of 1 em to represent 1 km, unless otherwise stated. 1. Point A is 8 km due east of point B and point C is due south of

point A. The bearing of point C from point B is 137°. Find the distance between B and C. 2. P is 6 km due north of Q and R is 8 km due west of Q. Find the bearing of R from P. 3. A ship sails 7 km on a bearing 330° and then a further 9 km on a bearing 074°. How far is the ship from its starting point? 4. A ship sails 8·5 km on a bearing 118° and then a further 10 km on a bearing 021 o . How far is the ship from its starting point? 5. An eagle leaves its nest and flies around its highland territory in three stages.

1st stage 2nd stage 3rd stage

Bearing

Distance

065° 154° 255°

6·5km 11 km 8km

(a) Make a scale drawing to show the journey. (b) On what bearing does the eagle have to fly to return to its nest?

44

Part 2

6. Point G is 9 km from F on a bearing of 130° from F. Point H is 10 km from F on a bearing of 212° from F. What is the bearing of G from H? 7. Point X is 7·2km from Yon a bearing 215° from Y. Point X is also 8·5 km from Z on a bearing 290° from Z. How far is Y from Z? 8. Use a scale of 1 em to represent 10 km. Mark the fixed points P and Q so that P is 70 km due north of Q. (a) At noon ship A is on a bearing 230° from P and 300° from Q. At 1500 ship A is on a bearing 162° from P and 096° from Q. (b) Mark the position of ship A at noon and at 1500. (c) On what bearing is ship A sailing? (d) At what speed is ship A sailing? 9. Two explorers, Tina and Karen, have a tent at camp A. Tina leaves the tent and walks on a bearing 064° for 12 km to reach point B. A little later Karen leaves camp A and walks on a bearing 111 o for 8 km, to arrive at point C . Karen hears strange noises and decides that she would be better off with Tina, who has a big gun. Draw a scale diagram to show Tina's route to point B and Karen's route to point C. How far, and on what bearing, must Karen walk to arrive at point B? 10. A boat left Dornoch to sail to the oil rig 0. After two hours it was at a point A, on a bearing of 252° from 0 and 329° from Banff. (a) Make a copy of the map on squared paper and find point A. [Use the squared background to locate Dornoch, Banff, John o'Groats and point 0 .] From A, the boat went on a bearing of 090° for 60 km to a new position B. (b) Draw this part of the voyage on your map and mark position B. The boat then began to sink and a rescue boat sailed, at a speed of 50 km/h, to point B from Banff.

(c) Measure the bearing to Banff from B.

45

Ratio

11. One day at noon, on the radar screen of the Golden Hinde, Sir Francis Drake sees two ships. One is a defenceless Spanish galleon, laden with treasure, and the other is a French pirate ship. Sir Francis

North

1 Spanish galleon

From the Golden Hinde, the Spanish galleon is 9 nautical miles away on a bearing 240° and the French pirate is 17 nautical miles away on a bearing 260°. The Spanish galleon sails due south at 8 knots. The French pirate sails south-east at 11 knots and Sir Francis sails at 14 knots on a bearing 211 o . At 1300 the captain of the Spanish galleon surrenders to the nearest ship. Who gets the treasure? [One knot is a speed of one nautical mile per hour]. 12. At 0400 a customs patrol boat is 20 km due south of a suspect cargo ship. The cargo ship is sailing at a steady speed on a fixed bearing of 070°. The patrol boat sails on a fixed course at a speed of 26 km/h and intercepts the cargo ship at 0530. (a) On what bearing did the patrol boat sail? (b) At what speed was the cargo ship sailing? [Use a scale of 1 em to represent 2km]

2.4 Ratio A ratio is a convenient way of writing a fraction. Suppose diluted orange drink consists of 1 part squash and 4 parts water. The ratio of squash to water is written with a colon, 1 : 4. There are 5 parts altogether so, of the mixture, is water and is squash.

4

t

A ratio can be simplified if all the numbers have a common factor. So the ratio 2: 6 can be simplified to the ratio 1 : 3. The ratio 3:6: 15 can be simplified to the ratio 1:2:5.

46

Part 2

A prize of £50 000 is shared between Ken, Len and Ben in the ratio 2: 3: 5. How much does each person receive? Add together 2, 3 and 5. There are 10 parts. Ken's share is ?o· He receives 120 of 50000 = £10000. Len's share is ?o· He receives 130 of 50000 = £15000. Similarly, Ben's share is to and he receives £25 000.

In a hall, the ratio of chairs to tables is 7: 2. If there are 10 tables, how many chairs are there? chairs : tables = 7 : 2 Multiply both parts by 5. chairs: tables= 35 : 10 So there are 35 chairs and 10 tables.

Exercise 10 1. Simplify the ratios

(a) 2:10

(b) 6:21

(c) 8: 12: 20

2. Share £60 in the ratio 1 : 3.

3. Share £63 between two sisters in the ratio 3 : 4. 4. Divide 6000 g of gold between three prospectors m the ratio 3:4:5. 5. A ballet dancer divides her 17

hour day into practice, performing and resting in the ratio 9 : 5 : 20. How many hours does she practice? 6. Lee, Mike and Neil formed a

syndicate to enter a giant lottery. They agreed to share their winnings in the ratio of their contributions. Lee paid £1, Mike paid 60p and Neil paid 25p. Together they won £1480000. How much did Neil get?

47

Ratio

7. Antifreeze is put into a car's radiator to prevent the water freezing. A mixture of antifreeze to water of 1:3 is a 25% mix. (a) What would a 33±% mix be? (b) How much antifreeze is needed to make 12 litres of the 25% mix? (c) How much antifreeze is needed to make 12 1itres of the 33±% mix? 8. The number of pages in a magazine was increased from 48 to 60. What will the new price be if the price, which was 36p, is increased in the same ratio? 9. A photo was enlarged in the ratio 2: 5. The enlarged photo was 6·5 em long. How long was the original? 10. The ratio of squash to water in a drink is 3:8. How much squash is used with 4 litres of water? 11. In 100 g of peanuts there is: 25 g protein; 10 g carbohydrate; 12 g fibre; 53 g fat. A tennis player's special high protein diet requires him to eat 300 g of protein per day. The player is sponsored by a peanut manufacturer and on certain days he is allowed to eat only peanuts! How many grams of peanuts must he eat so that he gets 300 g of protein? 12. Concrete for paths consists of cement, sand and gravel in the ratio 1 : 2: 4 by volume. What volume of sand is needed to make 3m 3 of concrete? Give your answer correct to 2 d.p. 13. Bread is made from flour and yeast in the ratio 30 to 1. (a) How much yeast is mixed with 960 g of flour? (b) How much flour is needed to mix with 400 g of yeast? 14. In the diagram, 1- of the circle is shaded and of the triangle is shaded. What is the ratio of the area of the circle to the area of the triangle?

t

48

Part 2

On a map of scale 1 : 3 000 000, Edinburgh and Newcastle appear 5 em apart. What is the actual distance between the towns? 1 em on map = 3 000 000 em on land. 5 em on map = 5 x 3 000 000 em on land. 15000000cm = 150000m = 150km Edinburgh is 150 km from Newcastle.

Exercise 11 1. On a map whose scale is 1 : 100 000, the distance between two villages is 7 em. What is the actual distance in kilometres between the two villages?

2. Two towns are 9 em apart on a map whose scale is 1 : 5 000 000. Find the actual distance between the two towns. 3. On a map whose scale is 1 : 200 000 the distance between two towns is 8·5 em. Find the actual distance between the towns. 4. The distance between two points is 30 km. How far apart will they be on a map of scale 1 : 50 000?

5. The length of a section of motorway is 15 km. How long will it be on a map of scale 1 : 100 000? 6. A world map is drawn to a scale of 1 : 80 000 000, while a map of Great Britain is drawn to a scale of 1 : 3 000 000. On the map of Great Britain, the distance from Land's End to John o'Groats is 36 em. How far apart are the two places on the world map? 7. Find the ratio (shaded area): (unshaded area) for each diagram. (a)

(b)

(c)

8. The scale used in a motoring atlas is '1 inch to 3 miles'. Write

this in the form 1 : n. [1 mile

= 1760 yards]

49

Ratio

9. The main ingredients in a recipe

for a cake to feed 5 people are: bananas 350 g corn oil 75 ml raisins 100 g flour 200 g almond essence teaspoon.

!

Work out the quantity of each ingredient for a cake to feed 8 people. 10. A pond contains carp and trout. If

t of the fish are carp, what is

the ratio of carp to trout? 11. The three angles of a triangle are in the ratio 1 : 3: 5. What is the

size of the largest angle? 12. A photocopier enlarges pictures in the ratio 3: 8. The height of a

cliff is 2-4cm on the original picture. How high is the cliff on the enlarged copy?

13. An alloy weighing 210 g consists of copper, zinc and iron. There is three times as much copper as zinc and one and a half times as much zinc as iron. How much zinc is there? 14. When a torch shines onto a wall

from a distance of 12cm, it makes a circle of diameter 7·8 em. (a) What will be the diameter of the circle when the torch is 16cm away? (b) How far from the wall is the torch when the diameter of the circle is 5·2cm? 15. The ratio of Lucy's age to Helen's age is 3: 4. How old is Helen

if she is 7 years older than Lucy? 16. The recipe for 10 cakes calls for

300 g sugar, 400 g butter and 600 g mixed fruit. How many of these cakes can a cook make if he has 2000 g sugar, 2400 g butter and 6000 g mixed fruit?

50

Part 2

17. In a Fibonacci sequence, each successive term is obtained by adding the two previous terms. A Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (a) Write down the next four numbers in the sequence. (b) The ratio of successive pairs of terms can be found: l = 1 l = 2· l = 1·5 etc I ' I ' 2 . Find the next ten ratios of successive pairs of terms. . 5 , 8 , 13 e t c] [te 3 5 8 What do you notice?

2.5 Mixed problems Exercise 12 1. Work out (a) -17 + 4 (d) 8- (-3)

(b) -7-5 (e) -12..;.. (3)

(c) -3 x (-2) (f) -8 + (-4)

2. A 375 g packet of Bran Flakes costs £1-14. Calculate the cost of a 500 g packet, if both packets represent the same value for money. 3. Kevin is in charge of catering for a political conference and has been asked to purchase 50 litres of whisky. There are three sizes to choose from: 70 cl for £11 -55, lOOcl for £16 -55, 150cl for £24·50, [100 cl = 1 litre] What is the cheapest way to buy 50 litres and what is the cost? 4. Find the number indicated by the arrow on the scales below.

w

~ 0.4

0

(d)

1.02

1.1

(e)

1.2

I

~

1.3

I

20

(f)

2.6

I I

16

2.65

I

I

0

0.1

I

I

Mixed problems

5. The number of pages in a newspaper was increased from 36 to 64. What will the new price be if the price, which was 27p, is increased in the same ratio? 6. The square ACDE is cut into seven pieces. Find the area, in square units, of (a) triangle ED I (b) square BJI G (c) parallelogram FGHE.

7. A man gives a total of £7 to his two children so that his daughter receives 60p more than his son. How much does his daughter receive? 8. A chef uses 200 ml of oil in 4 days. How many days will a 10 litre drum of oil last? 9. (a) Using a calculator, express the following as decimals correct to 6 decimal places: (i) 3t (ii) G+ ~- 989) (b) Which of (i) and (ii) above is the better approximation to n? 10. Work out 33 2 , 333 2 and 3333 2 . Use your answers to predict the values of 33333 2 and 333333 2 . Exercise 13

1. At a European Union conference, all the delegates can speak French or English or both. If 72% can speak English and 45% can speak French, what percentage of the delegates can speak both languages?

51

52

Part 2

2. (a) Use a calculator to work out

(i) 350--;-- 99 (ii) 350 -7- 999 (iii) 350 -7- 9999 (b) Use your answers to predict the answer to (i) 350--;-- 99999, correct to 9 decimal places. (c) Predict the answer to 350 -7- 999999, correct to 11 decimal places.

3. Neha went to Spain when the exchange rate was 218-40 pesetas to the pound. She bought a C.D. which would cost £10·99 in England. Neha paid 2250 pesetas for the C.D. How much more or less expensive was the C.D. when.bought in Spain? Give your answer to the nearest penny. 4. Greta has lots of 24p and 19p stamps and she wants to waste as

little money as possible when posting 3 packets. Which stamps should she use if the required postage is: (a) 60p (b) 120p (c) 89p? 5. Six touching circles of radius 5 em are shown. Calculate the area of the triangle shaded.

6. Packets of the same kind of flour are sold in two sizes. A 2 pound bag costs 69p and a 1500g bag costs £1 ·05. Given that 1 kg= 2·205 pounds, work out which bag represents the better value for money. 7. (a) Calculate the total surface area of the solid cuboid shown. (b) How many of these cuboids could be painted on all faces, using a tin containing enough paint to cover an area of 20m 2?

4cm 9cm

8. Three babies, Petra, Quentin and Rusty, are all weighed on April 1st. After that, Petra is weighed every second day, Quentin every third day and Rusty every fifth day. So, for example, Petra is next weighed on April 3, Quentin is next weighed on April 4 and Rusty is next weighed on April 6. What is the next date when all three babies will be weighed on

the same day?

53

Mixed problems

9. Canada has an area of 9 980 000 km 2 and a population of 28 200 000. Hong Kong has an area of 1030 km 2 and a population of 7 300000. How many times is Hong Kong more densely populated than Canada? 10. A computer is advertised at £800 plus V.A.T. (a) Work out the full price, including V.A.T. at 17!%. (b) What percentage of the full price is V.A. T.?

Exercise 14 1. At the Post Office,

stamps are printed on large sheets 38 em across by 60 em down. How many stamps are there on each sheet?

2cm

liJ l! liJ 17~ ~ ~ ~~

2. For a ten year old child, the recommended dose of 'Calpol' is two spoonfuls. Each spoonful is 5 ml. The whole bottle of 'Calpol' contains 0·25 litres. How many doses can you get from the bottle for a ten year old child? 3. Every year the Government spends about £8·8 billion paying teachers (who deserve every penny they get). A wad of fifty £10 notes is about 5 mm thick. As a publicity stunt, the Minister of Education decides to make a single pile of £10 notes of total value £8·8 billion. How high would the pile be? [1 billion= 1000 million] 4. The cost, £C of hiring a coach for a journey of m miles is given by the formula C=75+2-4m A weightlifting club intend to hire a coach for a trip of 155 miles. The club charges 32 adults £11 each and 14 children £5·50 each. Will they raise enough money to cover the cost of the coach? If not, how much more money do they need?

54

Part 2

5. A tool shop sells drills which go up in steps of

inch to t inch. (a) How many different drill sizes are there? (b) Which size is half way between inch and

±

1 16

inch from

t

t inch?

6. Write down the most appropriate metric unit for measuring:

(a) (b) (c) (d)

the the the the

capacity of a car's fuel tank, height of the Eiffel Tower, mass of a Jumbo Jet, area of a small farm.

7. 5, 8, 14, 37, 296, 323, 529.

From the list of numbers above, write down (a) three prime numbers (b) a square number (c) a cube number (d) a number obtained by multiplying together two other numbers in the list. 8. (a) In 1995 Ian was paid £368 per week and 18% of his pay was

deducted for tax. What was his 'take home pay' in 1995? (b) In 1996 Ian received a pay rise of 6% but, in the budget, taxes were raised so that 20% of his pay was deducted. What was his take home pay in 1996? 9. The Day Return train fare from

Hatfield to London is £9-40 and an annual season ticket costs £2250. What is the smallest possible number of return journeys a person needs to make in a year so that it is cheaper to buy the season ticket?

10. The pupils in a school were given a general knowledge quiz. A mark of 30 or more was a 'pass'. Some of the results are given in the table.

Passed

Failed

Total

Boys Girls

267 174

452

Total

441

821

(a) Copy and complete the table with the missing entries. (b) What percentage of the boys passed the test?

(c) Of those who passed, what percentage were girls?

55

Mixed problems

Exercise 15

Some questions in this exercise require a use of the formulas for the circumference and area of a circle. 1. On Gemma's watch, the tip of the

seconds hand is 13 mm from the centre of the watch. (a) How far does the tip of the seconds hand move in 1 hour? (b) Find the speed at which the tip of the seconds hand moves, in mmjs. (c) The minutes hand is the same length as the seconds hand. At what speed, in mmj s, does the tip of the minutes hand move? 2. In the U .K. the petrol consumption for a car is given in 'miles per gallon'. The metric equivalent is given in 'km per litre' . (a) Given that 1 gallon= 4·54 litres and 1 mile= 1·61 km, convert a consumption figure of 30 miles per gallon into km per litre. (b) In France motorists use a different unit: 'litres per 100 km' . Convert 30 mile per gallon into litres per 100 km. 3. The same size jar of Nescafe can be bought for £3·65 in Britain and for 26 francs in France. The exchange rate is 8·05 francs to £1. (a) In which country is the coffee cheaper and by how much? (b) What would the exchange rate be if the two prices were the same? 4. A supermarket has two special offers, one on Bran Flakes and one on Frosties. Work out the missing numbers.

(a)

(b)

Bran Flakes 20% extra free •

g for the price of 375g

[More difficult!]

Frosties 30% extra free 325g for the price o f .

5. 'Muirfield' grass seed is sown at a rate of 40 grams per square metre and a 2 kg box of seed costs £6·25. Mrs James wishes to sow a circular lawn and she has up to £50 to spend on seed. Find the radius of the largest circular lawn she can sow.

56

Part 2

6.

A short multiple choice test paper consists of three questions. For each question, +2 marks are given for the correct answer, -1 mark if the answer is incorrect and 0 marks if the question is not attempted. (a) Find the maximum and minimum total marks which are possible for the three questions. (b) How many total marks (if any), between the maximum and minimum marks, are impossible to achieve?

7.

The symbols *, \7, 0, 0 represent numbers. Use the clues in (a), (b) and (c) to answer part (d). (a) *+\7=0 (b) *=\7+0 (c) *+*+\7=0+0+0 (d) \7=Howmany O's?

8.

As part of an advertising campaign, the message 'Exercise is good for you' is taped individually right around 500 000 tennis balls, each of diameter 6·5 em. Find the total cost of the tape for the campaign, given that a 33m roll of tape costs 96p.

9.* At a steady speed of 70m.p.h. a Rover Sterling travels 21 miles per gallon of petrol. If petrol costs £2· 70 per gallon, work out the cost per minute of driving the car. 10. * One square metre of the paper used in this book weighs 80 g.

The card used for the cover weighs 240 g per square metre. Measure the size of this book and hence calculate its total weight. Check how accurate you were by weighing the book on an accurate set of scales.

2.6 Puzzles and investigations 2 Puzzles 1. The totals for the rows and columns are given. Find the values of the letters.

(h)

(a)

w

y

X

z

24

E

D

E

y

y

y

y

36

A

B

D

z

y

X

X

26

E

c

E

X

z

y

w

24

D

A

c

24

30

32

24

E

E

D

c c c c c

42

41

47

35

E

45

E

41

E

41

A

33

c

43

38

57

Puzzles and Investigations 2

(c) Find P, Q, R, Sand find the letter hidden by an ink blot.

(d) This one is more difficult

s

Q

R

s

42

A

B

B

A

38

Q

Q

36

A

A

B

B

38

Q

Q

s

44

A

B

A

B

38

s

Q

p

R

41

B

B

A

B

49

44

36

-

Q

41

42

27

49

38

49

Q

2. In a code A= 1, B = 2, C = 3 and so on. The score for the word 'ABLE' is 20 [1 + 2 + 12 + 5]. (a) Find the score for the word 'APPLE'. (b) Find a word with a score of 30. (c) Find a word with a score of 50. Ask a friend to check your words in (b) and (c). 3. Find a triangle in which all of the angles are square numbers (in degrees).

4. (a) The year 1990 is ten times a prime number. When is the next year which will be ten times a prime number? (b) Write down all the prime numbers less than ten thousand with digits adding up to 2. [Do not include '2'] 5. Fill up the square with the numbers 1, 2, 3, 4, 5 so that each number appears only once in every row, column and diagonal.

4

5

I

3

5

Note that you are now allowed to write the same digit twice on any diagonal. The solution on the right is not allowed because there are two 2's on the diagonal marked with a broken line.

4

5

. . .z 3

A'

. z . .

58

Part 2

6. The diagram shows two equilateral triangles. The sides of the large triangle are twice as long as the sides of the small triangle. (a) What is the ratio of the area of the large triangle to the area of the small triangle? (b) An equilateral triangle and a regular hexagon have equal perimeters. What is the ratio of the area of the hexagon to the area of the triangle?

D

7.* The two-digit number 37 equals (3 x 10) + 7. The three-digit number 425 equals (4 x 100) + (2 x 10) + 5 (a) Write the two-digit numbers 84 and xy in this way. (b) The digit 6 is written at the right of a certain two-digit number to make a three-digit number. The new number is 555 more than the original two-digit number. What was the original number?

Hidden words (a) (b) (c) (d) (e)

Start in the top left box. Work out the answer to the calculation in the box. Find the answer in the top corner of another box. Write down the letter in that box. Repeat steps (b), (c) and (d) until you arrive back at the top left box. What is the message?

1. Try not to use a calculator for this one.

612 6

9

+1·5 1·5

0·8

0·77

T

w

5% of 400

5 --2 0·1

32 B

50 000 -7- 200

~of 450

15

R

1·56

X

T

23

13

0·6

X

20% of 65 18

2·6

u

5- (7- 2)

180

0·9 2

-

0·1 2

0·15

E

R

E

s

~of 48

l- 0·1

(0·2) 2

6·4 -7- 0·2

0·27

250

E

0·6

0·04

0·62

v

10

c

0·2

20

0

s 18

X

48

A

0

D

N

10% of 2

lo (5·5 + 2·2)

0·3- 0·03

~ + 0 ·02

34 345

E ~ 4

of

~ 5

59

Puzzles and Investigations 2

2.

(-6)

5

X

N

E

12-;-(-1)3

!-0·6

17+(-25)

-30

10

X

R

c

8- (-4)

0 ·1- 0 ·3

-3 + 10 - 4

-6

-0·2

u (-2- 6)

X

(-2)

H

81-100

(7- 9) 2

10

(-60)

X (-

250

(-6)

X

!) 12

0

F

-8

5

(-3)

X

I

3

E

24

-0·1

36

s

N

0

s

L

30 (-5) 2

-4

- 19

-5

4

(-6)

(-2)2 + (-1)2

-12

16

-72

L

I

N

I

s

(-12)-;-3

-6-7 +8

-7+17

3-8+5

-2!- 3}

3. 1

-2

19

-2

(3- 9) 2

412 (8- 14)

-1

X

X

(-3)

0

(-!)

(-3)) + 20

(-0·1)

X

-1000

15 + (-16)

800 -;- (- 2) 3

47

0·05

(-0·1)

4

X (

-4! y

E

8- (-4)- (+6)

-i)

- 184

-100

T

s

L

E

27-270

X

2!2

M

0 ·4

((-9)

18

A

c

H

16-200

-80

(-10) 3 -243

16

M

s

I

-8!+4

( -20) -;- (- 8)

(-8) -;- (- 2) 2

36

6

-0·2

0 ·01

c

A

y

N

c

-35 - 45

8-12 + 20

-1!+6

!- 0 ·2

17 + (- 2)- (- 4)

60

Part 2

Perimeters and common edges In this work, one centimetre squares touch either along an edge or at a corner.

A

Shape A has 2 common edges, shown by the thick lines, and shape B has 3 common edges. The perimeter of shape A is 12 em and the perimeter of shape B is 10 em. B

• Using four squares, draw a shape with 1 common edge (c = 1). Write down the perimeter, pcm. Is there more than one shape with one common edge? Do all shapes with one common edge have the same perimeter? • Again using four squares, draw a shape with 4 common edges (c = 4). Write down the perimeter, p, of the shape. • Draw shapes with c = 2 , c = 3, c = 0 and each time write down the perimeter. • Try to find a connection between c and p. Either write the connection in words or as a formula, 'p = . . .'. Five squares or more

• Draw several shapes with 5 squares with c = 0, 1, 2 .. . . Find a formula connecting c and p. [Write p = . .. ]

• Now draw diagrams with 6 squares and again find a formula connecting c and p. • Look at your formulas for shapes with 4, 5 and 6 squares. Predict, without drawing any more shapes, what the formula might be for 7 squares: • Now draw several shapes with 7 squares, count c and p for each one, and check if the formula you predicted does work. • Now go further. Predict a formula for shapes with 10 squares or 100 squares. The most general case is the shape consisting of n squares, where n is any whole number (greater than 1). Try to write a formula connecting c, p and n.

61

Puzzles and Investigations 2

Extension

• Equilateral triangles can be drawn on isometric paper. For shape X, c = 2 and p = 8 . For shape Y, c = 1 and p = 10.

• X

•

y

• Find a formula connecting p and c for shapes with 4 triangles. • Work methodically and try to find a formula for shapes with any number of triangles. Further extension

• Without drawing any shapes, try to predict a formula, connecting p and c, for 3 hexagons. • Draw shapes with 3, 4, ... hexagons and check if your formula works.

• • •

•

Part 3 3.1 Pythagoras' theorem Below are three dissections which demonstrate a result called Pythagoras' theorem. Pythagoras was a famous Greek mathematician who proved the result in about 550 B.C. The first dissection works only for isosceles right angled triangles. The second works only when the sides of the triangle either side of the right angle are in the ratio 2 : I.

,, '

'' ''

'\

I

'

''

''

'

''

' \•'' ,,

" The most impressive dissection is the third, which is Perigal's dissection. It has been left for you to complete. For this method: (a) X is the centre of square (b) PQ is parallel to AB, the longest side of the triangle (c) RS is perpendicular to PQ.

CD

Your task is to rearrange the four together with pieces of square square @ to fit exactly into square

CD

G).

\

' '

''

''

''

''

'

63

Pythagoras' Theorem

All three dissections demonstrate Pythagoras' theorem ....

'In a right angled triangle, the square on the hypotenuse is equal to sum of the squares on the other two sides.' The 'hypotenuse' is the longest side in a right angled triangle.

The theorem can be used to calculate the third side of a right angled triangle when two sides are known.

Find the length x (a)

(b)

7

x2 + 32 = 62 x 2 + 9 = 36 x 2 = 27

X2=42+72 x 2 = 16 + 49 x

2

= 65

J65

X

x=vYf

= 8·06 (3 s.f.)

x

Remember:

x = 5·20 (3 s.f.)

The side on its own in the equation is the hypotenuse

Exercise 1 Give your answers correct to 3 s.f. where necessary. The units are em unless you are told otherwise. 1. Find x.

(a)

(b)

5

3~ 4

(c)

(d)

X

\\7

64

Part 3

2. Find y.

(a)

(b)

(d)

(c)

y

3. Find the side marked with a letter. (b)

(a) 7

4. A ladder of length 5 m rests against a vertical wall, with its foot 2m from the wall. How far up the wall does the ladder reach? 5. A ship sails 40 km due south and then a further 65 km due east. how far is the ship from its starting point?

6. A square has diagonals of length 24 em. Find the length of a side of the square to the nearest em. 7. What is the longest shot you could have to play on a snooker table measuring 12 feet by 6 feet?

8. Calculate the length of a side of the largest square which can be drawn inside a circle of radius lOcm.

9. The square and the rectangle have the same perimeter. Which has the longer diagonal and by how much? Scm

65

Pythagoras' Theorem

10. Calculate the height of the isosceles triangle shown.

11. Calculate the vertical height and hence the area of an equilateral triangle of side 14 em. 12. A car driver can take one of two routes from A to C: Route l . From A to B and then B to C. This route is along main roads and he travels at 50 mph. The angle between AB and BC is 90°. Route 2. From A straight to C along a short cut. This is a minor road and he drives at only 30mph. (a) Which route takes the shorter length of time? (b) By how much in hours is it shorter?

-

50 mph

_!?J I

12 miles

c

t

50 mph

A

Harder questions (a) Calculate x.

(b) Find x andy

A

(i)

~·

(ii)

8

(i) x 2 = 32

In L.ABD, 2 AB + 82 = 11 2 AB 2 =57 In L.ABC, x 2 = AB 2 +5 2 x 2 =57+ 25 x = 9·06 (3 s.f.)

+ 42

x=5 This is a '3, 4, 5 triangle'. It is an important right angled triangle which can be 'spotted' to quickly solve other triangles. (ii) The sides 6 and 8 are both twice as long as the sides of triangle (i). ... y = 10.

66

Part 3

Exercise 2

Give answers correct to 3 s.f. where necessary. The units are em unless you are told otherwise. Look for 3, 4, 5 triangles to shorten the working. 1. Find the length x.

(a)

(b)

5

X

5

(d)

(e)

4

(f)

8

2. The diagram shows a rectangular

box (a cuboid). Calculate the length of (a) AB (b) AC (c) AD. [Draw triangle ACD].

3. Calculate the length of the longest diagonal of a cube of side lOcm. 4. The inside dimensions of a

removal lorry are 2·5 m by 3m by 4m. A pole vaulter's pole is 5·7m long. Will it fit inside the lorry? 5. A farmer is shooting at a target which is 18m above the ground. The target is 120m north and 60 m east of the farmer . How far is it, in a straight line, from the farmer's feet to the target?

D

67

Pythagoras' Theorem

6. The diagram shows water lying in a semi-circular channel. Calculate the maximum depth d of water in the channel.

7. Calculate x andy.

(a)

""10 X~

(b)

\J,,

2x

8. In the rectangle calculate: (a) BC (b) area of triangle ABC (c) x.

Pythagoras' theorem in circles A circle is drawn through the corners of a square of side 8 em. Find the shaded area. Let the length of the diameter be x em. By Pythagoras', x 2 = 82 + 82 X= 11 ·313708 radius = 5·6568542 area of circle= n x 5·6568542 2 = 100·53096cm2 area of square = 64 cm 2 shaded area = 36·5 cm 2 (3 s.f.) Notice that we have approximated to 3 s.f. only at the very end of the calculation.

Exercise 3

In Questions 1 to 3 find the shaded area. Lengths are in em. All arcs are either semi-circles or quarter circles. You do not always have to use Pythagoras' theorem. 2.

3.t3 ---6-

68

Part 3

In Questions 4 and 5 find the total area.

4.

5.

semicircles

quarter circle with centre 0

\

'semi circle of diameter 10 em 6.

The area inside the running track shown is 10200m2 . Find the length x.

7. This diagram contains two semicircles. Calculate the shaded area, given that the diameter of the larger semi-circle is 12 em.

8. * This diagram has one quarter circle and two semi-circles. Calculate the shaded area.

---IOcm--

3.2 Sequences Differences Different numbers of lines are drawn below and the maxtmum number of crossovers for each is shown. lines 2 3 4 I crossover

3 crossovers

6 crossovers

5

crossovers

1 3

6 10

69

Sequences

One method for predicting further results is to look at the differences between the numbers in the 'crossovers' column. The differences form an easy pattern so that we can predict that there will be 15 crossovers when 6 lines are drawn.

lines

crossovers

differences

2 3 4 5 6

1 3 6 10

2 3 4

(3)

(ill

/

ipredictions

In some sequences in the next exercise you may need to find the 'differences of the differences', called the second differences. [In the table above the second differences are all 1.] • Consider the sequence below.

• The first, second and third differences are shown below.

2 10

30

2 10 30 68 130

68 130

First difference

Second difference

Third difference

8 20 38 62

12 18 24

6 6

An obvious pattern is seen. Exercise 4

1. Below is a sequence of rectangles where each new diagram is obtained by drawing around the outside of the previous diagram, leaving a space of 1 unit.

diagram 1 3 squares

diagram 3 35 squares

diagram 2 15 squares

(a) Draw diagram 4 and count the number of squares it contains. Enter the number in a table and use differences to predict the number of squares in diagram 5. (b) Now draw diagram 5 to check if your prediction was correct.

diagram

squares

differences

1 2 3 4

3 15 35

12 20

70

Part 3

2. Square grids are drawn by a computer which takes 1 second to draw a line of length 1 em.

diagram 1 will take 4 seconds

diagram 2 will take 12 seconds

Find how long it will take to draw diagrams 3 and 4 and enter the results in a table. Use the sequence of times to work out how long diagram 8 would take to draw.

diagram 3

diagram

time (sec)

differences

1 2 3 4

4 12

8

3. Below are the first three members of a sequence of patterns of hexagons made with sticks. diagram I

diagram 2

diagram 3

/~

/~

/~

/~/~

/~/~

I~/~/ I I

I I I /~/~/~ I~/~/~/ I I I

I~/I 6 sticks

I

I

I

Draw diagram 4 and count the number of sticks it contains. Write your results in a table and then predict the number of sticks needed to make diagram 6. 4. Below are three sequences. Use differences to predict the next two numbers in each sequence.

(a) 1 6 13

(b) 3 6 13

(c) 11 14

22

24

33

39

35 53

(j)

(J)

(J)

(J)

(J)

(J)

22

I

Formulas

71

5. The numbers 1, 2, 3, .. . 96 are written in a spiral which starts in the centre.

There are, in fact, many sequences in the pattern. (a) In the row marked A, the numbers are 5 22 51 92. Predict the next number in this sequence. (b) Predict the next number in the rows marked B and C.

6. Use first, second and third differences to predict the next number in each of the sequences below.

(a) 2

5 13 28 52 87

(b)

3 11 31 69 131

(c) 1

(j)

(j)

4 13 34 73

(j) 7. A 1 x 1 square contains just one square but a 2 x 2 square contains 5 squares. [Four 1 x 1's and one 2 x 2] A 3 x 3 square contains 14 squares (check this).

Draw a 4 x 4 square and 5 x 5 square and count the number of squares in each. Predict the number of squares in an 8 x 8 square.

D

EE

72

Part 3

8. Playing cards can be used to build (rather unstable!) 'houses'. Houses with 1 storey, 2 storeys and 3 storeys are shown below, together with the number of cards required to make each one.

/\ /\ 1 storey 2 cards

1\ 1\1\ --1\1\ 1\1\/\ 2 storeys 7 cards

3 storeys 15 cards

(a) Draw a house with 4 storeys and count the number of cards required. Write down the sequence of the numbers of cards needed for 1, 2, 3 and 4 storeys. (b) Ceri has a very steady hand and decides to build a house with as many storeys as possible. How many storeys will it be possible to build if she has 5 packs of 52 cards?

3.3 Formulas Finding a formula The sequence of diagrams below shows rectangles made from sticks.

Suppose we wanted a formula connecting the number of sticks with the number of rectangles. • Let the number of sticks be s. Let the number of rectangles be r. • Count the number of sticks and the number of rectangles m several diagrams.

s=6 r= 1

s = 10 r=2

s = 14

r=3

s = 18 r=4

73

Formulas

• We see that the number of sticks is '(the number of rectangles) x 4 plus 2'. The formula can be written s = (r x 4) + 2 The expression 'r x 4' is usually written '4r' so the formula becomes 1

s = 4r+ 2

1

Notice that we can now calculate the number of sticks for any number of rectangles. For example when there are 100 rectangles, r = 100. Using the formula, s = 4 x 100 + 2 s = 402. The diagram with 100 rectangles has 402 sticks. The method for finding a formula has three stages: • Introduce symbols (like r and s in the above example). • Collect data (often this involves drawing diagrams). • Look at the data to 'spot' a connection. Sometimes it helps to write the data in a table. Exercise 5 1. Squares can be made from matches as shown. Call the number of matches m and the number of squares s.

m=4 s= 1

m =7 s=2

m = 10 s=3

(a) How many matches are there in the diagram which has 10 squares? (b) How many matches are there in the diagram which has 50 squares? (c) Write down a formula connecting m and s. Write it in the form 'm = ' 2. Triangles can also be made from matches. Call the number of matches m and the number of triangles t.

m=3 t= 1

m=5 t=2

m

t

=7

=3

(a) How many matches are there in the diagram which has 20 triangles? (b) How many matches are there with 100 triangles? (c) Write down a formula connecting m and t. Write it as 'm=

74

Part 3

3. In these diagrams black squares are surrounded on three sides by white squares. Let the number of black squares be b and let the number of white squares be w.

=1

b=2

b=3

w=5

w=6

w=7

b

Find a formula connecting b and w, writing it in the form 'w= 4. In the diagrams below rectangles are joined together and dots are drawn around the outside with 2 dots on a long side and one dot on a short side.

• • •L-1----~1· • •

• • • • • • • •• • • L-1_-~...l_ ___jl· •I I I I• • • • • ~.--~.~~.~-.~~.~-.~

Call the number of dots d and the number of rectangles r. Find a formula connecting d and r. 5. Find the new formula connecting d and r when the rectangles are joined along their longer sides.

• •

• • •

:[]: :[ill: • • • •• 6. Numbers are written in a 'wave' pattern. The top numbers are indicated by arrows.

4 3

1st top number

2nd top number

3rd top number

~

~

~

5

9

13

6

8 7

10

12 11

17 14

16

15

(a) What is the lOth top number? (b) What is the nth top number? [n stands for any whole number].

75

Formulas

7. Here is another 'wave pattern'. 1st top number

2nd top number

3rd top number

•

•

15

3

2 1

•

9

4

8 5

7 6

10

14

11

16

13

17

12

(a) What is the 100th top number? (b) What is the nth top number? 8. In these diagrams a shaded letter 'L' is surrounded by white squares.

Find a formula connecting the number of shaded squares and the number of white squares.

Substituting into a formula (a) The cost, in pounds, of printing a book is given by the formula cost= (number of pages) x 0·2 + 12·5 Find the cost of printing a book with 210 pages. Using the formula, cost= (210 x 0·2) + 12·5 = 42 + 12·5 cost of printing book = £54·50 (b) A formula used to calculate velocities is v = u +at. Find the value of v when u = 200, a= -10, t = 6. v = u +at V = 200 + (-1 0) X 6 v = 200-60 v = 140

19 18

76

Part 3

Exercise 6 1. Here is a formula c = 7t- 3. Find the value of c when (a) t = 2 (b) t = 10

(c) t =-!-

2. Using the formula p = 70 - 4x, find the value of p when (b) X= 10 (c) X= 20 (a) X= 1 3. Below are several different formulae for z in terms of x. Find the value of z in each case. (a) z = 15x - 60, x = 4 (b) z = 2 (3x + 5), x = -1 10- X (c) z = - - , x = 5

2

4. In the formulae below t is given in terms of n and a. Find the value of t in each case. (a) t = 3a- lOn; a = 5, n = 1 (b) t = 20a + 7n- 4; a= 1, n = 2 (c) t=an+11; a=4,n=3 (d) t = 5 (3a- 8n); a= 0, n = 1 5. Find the value of c, using the formulae and values given. (a) c = mx + 9; m = -2, x = -3 (b) c = l3t- t 2 ; t = 3 (c) c = 3pq + p 2 ; p = 5, q = 0 (d) c=(2a+b) 2 ; a=1,b=3 6. If T = 3y + 4i, find the values ofT when (b) y = 3 (c) y = -1 (a) y = 1 [Remember 4i = 4 Ci)J 7. If A = d 2 (a) d = 2

-

3d+ 5, find the values of A when (b) d = 5 (c) d = -2

8. The velocity, v, of an accelerating dragster is given by the formula v = u +at. Find v when u = 0, a= 22·5 and t = 3.

9. The charge, £C, made by a chef to cook for a group of p people is given by the formula C=7p+65. (a) What is the charge for a group of 20 people? (b) How many people are in the group if the charge is £282?

77

Formulas

10. From a point h metres above the

sea, the distance to the horizon is dkm. The formula connecting d and his d= 3·S8Vh (a) A sailor in the crow's nest of a ship is 20m above the sea. How far can he see to the horizon? (b) How tall is a lighthouse from which the horizon is 19 km away? 11. The weight w of the brain of a

Stegosaurus is connected to its age, A, and its intelligence quotient, I, by the formula 2 A +I/ A W=--S000 Find w, when A = 20 and I= 2.

Expressions An expression does not have an equals sign. For example: 3x- 7; x 2 + 7y; ab - c2 . These are all expressions. Below are three expressions involving a, b, c and d. Find the value of each expression given that a= S b = -2

c=3 d= -1. 3a- c (ii) =3xS-3 = 1S- 3 = 12 Notice that the working goes down

(i)

2b + d = 2(-2) + (-1)

(iii)

=-4+(-1)

3c- 4b =3(3)-4(-2) =9+8

= -S

=

17

the page, not across. This helps to avoid errors.

Exercise 7

Find the value of the expressions given that

1. 5. 9. 13. 17. 21.

Sa-c 4b + c

b- c a 2 + b2

ab + c bd + c2

2. 6. 10. 14. 18. 22.

2b +a 2d- a 7- 2a ac + b

Sd- 2c 2(a- c)

a= S b= -2 c=3 d= -1 3. 7. 11. 15. 19. 23.

a+d Sb+10 2S + Sb 6- 2c b2 + cd 3 (a+ d)

4. 8. 12. 16. 20. 24.

3c- b a+ b + c 3a- 4d d2 + 4 Sa+ b + d a(c +b)

78

Part 3

25. Given that x = 3 and y = -4, find the value of each of the following expressions. (c) 3x- y (a) 4x + y (b) x 2 + / (f) 5(x-y) (d) / - x (e) xy + 12 26. Given that p = -2 and q = 5, find the value of each of the following expressions. (a) 2p + q (b) p- q (c) 3 (p + q) (d) / - 2q (e) pq + 12 (f) p(2q + 1)

27. Given that m = 6 and n = -1, find the value of each of the following expressions. (c) m (5- n) (a) m 2 + n2 (b) 3mn + 20 (d) m+n+1 (e) n(m 2 -n 2 ) (f) mn (5m + 2n)

Race game START

w-3 1 - 3x 2(a -3)

1- y

t 3n- 9 a-2 3x

-

-t

X

•

2(3- x) 4-p

a+2

c+1

4-x

Players take turns to roll a dice. The number rolled gives the value of the letter in the expression on each square The value of the expression determines how many squares the player moves (forward for a positive number, backwards for a negative number). For example, if you are on the square 'x - 3' and you throw a 5 you move forward 2 places. The winner is the first player to move around the circuit. [You can also play 'first player to make 3 circuits' or any other number.]

X

+ 5 p+3 3(2- x)

(4- x?

t

+ 1 2x- 7 6- m

-

2n n

-

2y -p + 5 b+5 3z 11 -3t

2(a + 1)

-8 + c

79

Formulas

3.4 3-D Objects A drawing of a solid is a 2-D representation of a 3-D object. Below are two pictures of the same object. (a) On squared paper.

(b) On isometric dot paper.

The dimensions of the object cannot be taken from the first picture but they can be taken from the second. Isometric paper can be used either as dots (as above) or as a grid of equilateral triangles. Either way, the paper must be the right way round (as shown here). N.B. Most of the questions in this section are easier, and more fun to do, when you have an ample supply of 'unifix' or 'multilink' cubes.

Views With complicated shapes, it is easier to 'see' the shape if some of the faces are shaded. In (b) all the faces which would be seen from the left are shaded. (b) (a)

Notice that in the object above we cannot tell whether there are six cubes or seven, with one hidden at the back. To eliminate any doubt we can draw 3 'views': (a) from above, called the plan view; (b) from the left. (c) from the right.

n plan

from left

from right

~

80

Part 3

Exercise 8 1. Using four unifix cubes, you can make several different shapes.

A and B are different shapes but C is the same as A.

A.!

B.!

C)(

Make as many different shapes as possible, using four cubes, and draw them all (including shapes A and B above) on isometric paper. 2. In the box are four different shapes A, B, C and D. Underneath the box there are 12 shapes. Make shape A and decide which of the shapes below are the same. Repeat for shapes B, C and D. One shape from the lower 12 is neither A, B, C nor D. Which shape is that?

A

c

B

D

~ 1.

©

2.

3.

6.

7~8.~

9.~10.~

11.

12.

81

3-D Objects

3. You need 27 small cubes for this question. Make the four shapes below and arrange them into a 3 x 3 x 3 cube by adding a fifth shape, which you have to find . Draw the fifth shape on isometric paper. [The number next to each shape indicates the number of small cubes in that shape]. (a)

(b)

(c)

4. Make the objects below, using cubes. On squared paper, draw the plan view, the view from the left and the view from the right of each object.

(a)

(b) plan

/~

left~ n2:ht

This object consists of 5 cubes

This object consists of 7 cubes.

82

Part 3

5. Make each of the objects whose views are given below. Draw an isometric picture of each one.

(a)

left

plan

(b)

right

left

plan

right

tj=J ~ (c)

plan

left

JJ

right

Eb dJ

tj=J

.

3.5 Mid-book rev1ew Multiple choice test 1. Change ~6 to a percentage

2. Work out (- 3) + (-7)

A 48% B 148% c 67±% D 27%

5. On a map of scale 1 : 50 000, the length of a road is 6 em. Find the actual length of the road on the land.

A -4 B +4 c 10 D -10

6. Find x , correct to 2 d.p.

5~

A 3km B 30km

C 300m D 30000cm

A 17·50 B 74·00 c 8·60 D 4·90

7

3. Work out, correct to 3 significant figures, 4·9

1·92

-

0·72

4. The bearing of A from B is about:

~

B

~North

A 0·637 B 1·70 c 2·89 D 1-42

A 120° B 300° c 60° D 240°

Use the diagram below for questions 7 to 10

83

Mid-book Review

7. The image of 63, after reflection in the y axis, is

A 61 B 62

c

64 D 65

8. 64 is the image of 65 after reflection in the line:

A x=2 B x=1

9. The centre of the rotation which moves 61 onto 65 is

A (2, 5) B (-5, 0)

10. The image of 62 after rotation through 180° about the point (0, 0) is

11. Find the next number in the sequence 5, 7, 11, 17, 25,

16. Which is the odd one out?

c y=1 D y=2

A 64 B 61

c

66 D none of the above

A 33 B 35

c 36 D 37 A £1 B £55

c

£30·25 D £28·50

13. Solve the equation 2x-1=11

A 6 B 6l

c

5

A 045° B 225°

18. Which point does not lie on the line y = - x?

A (-3, -3) B (2, -2)

19. Find the shaded area, in cm 2 .

A 16- 2n B 8- 2n

20. Which is the largest number?

A 0·2 B o.2i

21. Find the length x.

A 13 B v'13

D·~ 2

~

c 12 D none of the above

c 22% 21 D TOO c J53 DVS

A 2 B -2

24. Find the ratio (shaded area): (unshaded area)

A 3:4 B 4: 1

c

A -2 B 2

c 0·858 D 9·72

23. Find the value of 2x 2 , when x = -1

9 D -9 15. When minus 5 is subtracted from 7, the answer IS

c (0, 0) D ( -4, 4)

A 1 B 11

l

A -11 B 11

c 135° D 315°

22. If z = 3a - n, find the value of z when a= -2 and n = 5

D 5 14. Work out 4- (7 X 2)- 1

2

TOO

17. A ship sails south-west. On what bearing does it sail?

4

12. 55% of £55 is

c

D 5l

c

(0, 0) D none of the above

A 20% B 0·2

EEJ

c -11 D none of the above c 4 D -4 c 3: 1 11_ D 31_. 2. 2

84

Part 3

Review exercise 1 Where necessary, give answers correct to 3 significant figures. 1. Work out (a) -6 + 9 (d) 127(-3)

(b) (-3) X (-5) (e) (-3)-(-4)

(c) 8- (-3) (f) -8 + 15

2. Work out. Give your answer to the nearest penny where necessary.

(b) 6-2% of £12·60 (d) of 17% of £8 ·88

(a) 16% of £28 (c) of £26

n

i

3. Solve the equations (a) Sx- 3 = -2 (c) 4x- 1 = 2x + 9

(b) 8 + 7x = 6

4. Work out, correct to 3 significant figures

(a) ~ + _22_ 4·1 11 ·7

(b)

(d) (2·3% of 111)

2

10·99 8·2-1-47

(e) 4·2 +

++

(c) 8·6- 1-42

(f) 8·63 2

1 17

-

5·592

5. Find the next two numbers in each sequence.

(a) 33, 42, 51 , .. . (b) 3, 6, 12, 24, (c) 2, 5, 10, 17

6. The Ace, King and Queen of diamonds are removed from a pack of cards. Find the probability of selecting (a) another Ace (b) another diamond (c) the ace of spades. 7. Work out the angles marked with letters. (a)

(b)

(c)

Y ---+~il----

X

8. Find the perimeter and area of the shape, which consists of a semi-circle and a rectangle.

t

6cm

~

-IOcm-

9. A bicycle wheel has a radius of 35 em. On a journey the wheel turns 650 times. How far has the bicycle travelled in metres?

85

Mid-book Review

10. The outer diameter of a bicycle wheel is 95 em. How many complete turns does the wheel make when the bicycle travels a distance of 350m? 11. A circle is drawn to touch the

four sides of a square of side 8cm. Calculate the shaded area.

Scm

12.

A square is drawn inside a circle of radius 6 em. Calculate the shaded area.

13. Calculate the area of each shape. All the arcs are semi-circles (a)

(b)

----t0cm-4cm

Review exercise 2 1. Use differences to find the next number in each of the following

sequences.

(a) 3

15 23

(b) 2 4 12 26 46

(J)

(J)

5 9

(c)

7 9

17 37 75 137

(J) 2. Here is a sequence of squares surrounded by dots.

• ·D· •

•• ••• •I I I I• ·OJ· • • • ••

(a) How many dots will surround the diagram with a row of 10 squares? (b) How many squares are in the diagram which has 104 dots?

86

Part 3

3. The diagrams show a sequence of shapes made from cubes of side 1 em. The volume of the first member of the sequence is 1 cm 3 and its surface area is 6cm 2 .

(a) Find the volume and the surface area for the second and third shapes in the sequence. (b) Calculate the volume of the shape which has a surface area of (i) 22cm2 (ii) 82 cm 2 .

4. A solid brass ingot weighing 1·24kg is melted down and cast into small models of the Colosseum each weighing 250 mg. How many models can be made?

5. (a) Change to decimals

t

(i) (ii) 34% (b) Change to percentages

(i) 0· 22

(ii) ~1

(iii)

i

(iii)

1 50

6. In rectangle KLMN point X is the mid point of LM and KXN = 20°. Draw a diagram and work out MNx. 7. Work out

(a) (-8) 7 (- 2) (e) 6x(-2)

(b) ( -3) 2 (f) -5 + 5

(c) -7 + 10 (g) 9- 14

(d) -2-(-3) (h) 127(-3)

8. Port A is 60 km due north of port B. A ship sails 60 km due east

from port B to arrive at point C. Draw a diagram to show the positions of A, B and C. What is the bearing of C from A?

9. Reflect shape P in line 1 and label the image P'. Now reflect P' in line 2 and label the image P". Describe fully the single transformation from Ponto P".

line 2

87

Mid-book Review

10. Reflect shape A in line 1 and label

the image A'. Reflect A' in line 2 and label the image A". Describe fully the single transformation which maps A onto A".

11. Draw axes with values from -6 to +6 and draw 61 with vertices at (1, 4), (1, 6), (4, 6). (a) Reflect 61 in the line y = x onto 62 (b) Reflect 62 in the x axis onto 63 (c) Reflect 63 in the line x = 3 onto 64 (d) Reflect 64 in the line y = x onto 65 (e) Describe fully the single transformation which: (i) maps 61 onto 63 (ii) maps 65 onto 61. 12. The two different objects shown

each consist of 5 cubes. Make each object using 'unifix' cubes and draw the object from a different angle.

•

Review exercise 3 Give answers correct to 3 significant figures where necessary. 1. Draw points X and Y in the middle of a piece of squared paper.

(a) Mark the point A which is on a bearing of 050° from X and on a bearing of 330° from Y. (b) Mark the point B which is on a bearing of 180° from X and on a bearing of 240° from Y. (c) Measure the bearing of A from B. 2. On a new diagram draw points X and Y as in Question 1. (a) Mark the point C which is on a bearing 025° from X and on a bearing 310° from Y. (b) Mark the point D which is on a bearing 130° from X and on a bearing 180° from Y. (c) Measure the bearing of C from D.

•

•

88

Part 3

3. Find the length of the side marked x. (a)

7

(b)

(c)

{7 4. A ship sails 15 km due East and then a further 25 km due South. Calculate the distance of the ship from its starting point. [Do not use a scale drawing]. 5. Ben is 35m North-East of Alan and Chris is 80m South-East of Ben. Draw a diagram showing the positions of Alan, Ben and Chris. Calculate the distance from Alan to Chris.

D.....------.

6. X is the midpoint of the side AD of the rectangle ABCD. Write down the ratio area of 6BDX : area of 6 ABX

7. Make an accurate drawing and measure the length or angle x . (a)

n

(b)

lOy 7cm

om

Scm

8. (a) Look at the sequence of numbers Al , A2, A3 , . .. Work out (i) AS (ii) AlOO (b) Look at the sequence of numbers Bl , B2, B3, . .. Work out (i) B6 (ii) B20

81

82

8

16

5

9

4

10

3

11

6

7

12

Al

17

83

18

26

15

19

25

14

20

24

13

21

22

A2

23

A3

Part 4 4.1 Brackets and equations 4

Brackets The area of the whole rectangle shown can be found by multiplying its length by its width. Area = 4 (x + 2) Alternatively the area can be found by adding together the areas of the two smaller rectangles. Area = 4x + 4 x 2 We see that 4 (x

+ 2) =

4x (x

1 X

+ 2)

4 X2

2

....___ _ _ ___, +

4x + 4 x 2

In general a number or symbol outside a pair of brackets multiplies each of the numbers or symbols inside the brackets. 5(x+2)=5x+10 3 (x - 2) = 3x - 6 -4(2x- 1) = -8x + 4 -(3x + y- 4) = -3x- y

+4

a(x+b)=ax+ab b (y - t) = by - bt -(x- 2) = -x + 2 n (a-b-c)= na- nb- nc

A common error occurs when no number is written before the brackets as in ' -(x- 2) '. To avoid difficulty you can write the expression as '-1(x- 2)' and then multiply the terms inside the brackets by ( -1). Remove the brackets and simplify. (a)

3(a- 2)- 2(a- 4) = 3a- 6- 2a + 8 =a+2

(b)

2(x+7)-(x-2) = 2(x + 7)- 1(x- 2) = 2x + 14- x + 2 = x+ 16

Exercise 1 In questions 1 to 15 remove the brackets. 1. 4. 7. 10. 13.

2(3x + 4) -2(2x + 3) 5(x + 2y - 2) -(4y + 3) -(1- x)

2. 5. 8. 11. 14.

5(5x- 2) 3(2x + 3y - 1) 2(6x + 1) 9(3- x) -2(3x + 10)

3. 6. 9. 12. 15.

10(2x- 1) -4(x- 1) -(3x- 1) 5(7- 3x) -q(1 -a+ 2b)

90

Part 4

In questions 16 to 35 remove the brackets and simplify. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34.

2(4x + 3) + 4(3x- 4) 5(x - 1) - 3(x - 2) 4(3x + 1) + (2x- 1) 3(2x- 1)- (x + 1) 7(2x- 1) ~ 4x 6x + 3(x + 1) 3(x- 1)- (x- 3) 2(a + 1) +(a- 5) 3(x- y) - 2(x - y) 2(x + 2y)- (2x + y)

17. 19. 21. 23. 25. 27. 29. 31. 33. 35.

3(4x + 5)- 2(x + 5) 6(2x- 1) + 3(1 + 2x) 2(4+x)+(5x-2) 5(4x- 2) - (2x- 1) 5(3- 2x) +lOx 4x + 5(2 - x) 4(1 - 2x) - (1 - 2x) 5(2a- 1) + 3(1 - 2a) 4(2x + y) + (x + 3y) 6(x- 3y)- 3(x + 2y)

Using letters for numbers Many problems in mathematics are easier to solve when algebra is used. When letters like x, y , a or n are used in the working of a question, it is important to remember that the letters stand for numbers. (a) Suppose I start with the number 7, add 6 and then divide the result by 100. The number . 7 +-6, wh"1ch 1s . - 13 . I h ave at t h e en d 1s 100 100 (b) Suppose I start with the number x, add 5 and then divide the result by z. The number I . x+5 h ave at t h e end 1s - - . z 5 Remember: x is a number, z is a number and x + is a number. z

(c) Suppose one pistol weighs 600 grams. Then five pistols weigh (600 x 5) grams and x pistols weigh 600x grams [Remember 600x means '600 x x'.] (d) Suppose a prize of £n is shared equally between y people. Each person receives £ !!... y

Exercise 2

In Questions 1 to 12 write down what you are left with after following the instructions. If any of your answers contain brackets, do not remove them. 1. Start with x , double it and then add 5. 2. Start with y , subtract 7 and then multiply the result by 3. 3. Start with y, treble it and then subtract 8. 4. Start with n, double it, subtract 7, and then multiply the result

by 4.

Brackets and Equations

91

5. Start with x, multiply by 4 and then take the result away from 12.

6. Start with a, multiply by 7, take the result away from 10 and then multiply the result by 3. 7. Start with h, subtract t and then multiply the result by 5. 8. Start with p, add 5 and then multiply the result by d.

9. Start with a, double it and then add A. 10. Start with n, multiply by k, subtract u and then add b.

11. Start with e, subtract u, add h and then multiply the result by k. 12. Start with x, subtract t, add y and then divide the result by u. 13. A model of Stonehenge consists of n stones. How many stones are there

in three such models? How many stones are there in x models?

14. A small bag of peanuts contains y nuts, and a large bag contains 3 times as many. If a boy buys a large bag and then eats 10 nuts,

how many are left in his bag?

15. A basketball player scored x points in each of his first three games and then he scored y points in each of his next two games. How many points did he score altogether? 16. Bill used to earn £n per week. He

then had a rise of £r per week. How much will he now earn in w weeks? 17. How many drinks costing z pence each can be bought for

(a) £3

(b) £n?

18. In a shop: tapes cost £t each;

C.D.s cost £c each; records cost £r each.

Write down the total cost of (a) 3 tapes and 1 C.D. (b) 1 record, 5 tapes and 2 C.D.s (d) x C.D.s andy tapes (e) p records and 5 tapes

(c) m tapes and 3 C.D.s (f) 200 C.D.s and z tapes.

92

Part 4

19. In a butcher's shop: chickens weigh m kg each;

ducks weigh x kg each; turkeys weigh z kg each. Find the total weight of (a) n chickens, y ducks and t turkeys. (b) v ducks, 8 turkeys and p chickens. 20. The height of a balloon increases at a steady rate of x metres in t hours. How

far will the balloon rise in n hours?

21. Unleaded petrol costs z pence per litre, which is x pence per litre less than 4 star. How much do I pay for n litres of 4 star? 22. 'Think of a number'. Ask someone to follow these instructions:

(a) Think of a number. (b) Add 3 to the number. (c) Multiply the answer by 5. (d) Subtract 7 from the new number. (e) Double the answer. (f) Subtract 6 from the last number. (g) Read out the final answer. You can now work out the original number as follows: 'Subtract 10 and divide by 10' [E.g. if the final answer is 370, the original number was (370- 10) ...;- 10. It was 36.] Try this a few times and then explain why it works by using algebra.

Solving equations The main rule when solving equations is

~me thin~o You You You You

can can can can

both

sid~

add the same thing to both sides. subtract the same thing from both sides. multiply both sides by the same number. divide both sides by the same number.

Solve the equations (a) 3x- 1 = 5 3x = 5 + 1 [Add 1 to both sides] 3x = 6 x = = 2 [Divide both sides by 3]

t

(b)

3x + 1 = 2x + 9 3x- 2x = 9- 1 x=8 Remember: 'x terms on one side, numbers on the other side'.

93

Brackets and Equations

Exercise 3 Solve the equations. 1. 3x- 2 = 13 4. 5 + 2x = 6 7. 5 = 3x- 1 10. 4 = 6x+ 5

4x + 1 = 25 7 + 3x = 22 7 = 15- 2x 7x- 1 = -8

2. 5. 8. 11.

3. 6. 9. 12.

7x- 2 = -1 3 = 4x + 1 10 = 12- 3x 3-x= 10

In Questions 13 to 24, begin by putting the x terms on one side of the equation. -

13. 4x + 3 = 2x - 5 16. 6x + 1 = 2 - 3x 19. 2x- 8 = llx + 12 22. 16x + 9 = 12x - 3

14. 17. 20. 23.

7x- 5 = 2x + 8 7x- 2 = 1 - 3x 3x - 9 = 4x + 4 1 - 1Ox = 6 - 5x

15. 18. 21. 24.

3x + 7 = 8x + 2 5 - x = 2x - 7 2 + 8x = 5 - x 4 - 5x = 4 + 7x

Equations with fractions (a) 2x = 5

~=-2

(b)

3

X

2x = 15

[Multiply by 3]

x -- l 2l

4= -2x 4

[Divide by 2]

-2 -2=x

X -

(c)

X

-+ 3 =

[Divide by -2]

-=X

-712

[Multiply by x]

4 (d) - - 1 = 14

7

2

X

~=4

4 - = 15

[Subtract 3 from both sides.]

2

[Add 1 to both sides.]

X

;t x ~ =

4x2

[Multiply both sides by 2.]

4

;(- = 15x ,X

x=8

[Multiply both sides by x.]

4 = 15x 4 15-X

Exercise 4 Solve the equations. X

X

1. - = 4 3

2. -=2

X

5

X

4

6. 2x = 1

5.- = -5 5 6 9. - = 7

3 4

8 13. 3 = -

11.- = 1

X

X

14.

3. = 3

X

_!_Q

7

7. 3x = 2 4 2

10.- = 9

X

X

4.- = -2

3. 5 = -

8. 5x = 2 2

12.

X

8

4

16. -2 = 100

15.- = -11

X

~ =.!.

X

X

Questions 17 to 32 are more difficult. 17.

X

-+ 1 = 3

5

X

18.-- 1 = 8 2

19.

X

-+ 9 = 5

8

X

20. 6 +- = 10 3

94

Part 4

1

1

21.-x+9=20

=

22. -x- 6

2

3

1 25. x- 3 = -x

11

X

2

4

29. --2 = 4

30.

X

4

-+ 2 =

1= 0

27. 3 -~=~ 4 2

28. - - - = 1

31.

1

4

24.

3

26. 7-- = 1

3

2

23. -X+ 8 = 10

7-2 = 5

X

32.

X

-X-

5 9

1

X

2

3

-+ 1 = 2x

4

Equations with brackets Many of the more difficult problems which appear later in this section involve forming equations with brackets. Once the brackets have been removed the method of solution is similar to that for the equations dealt with earlier.

(a) 3(2x- 1) = 2(5- x) 6x- 3 = 10- 2x 6x+ 2x = 10 + 3 8x = 13 X=

(b) 2(3x - 1) - (x - 2) = 5 6x- 2-x+2 = 5

5x X

lt

=5

=1

Exercise 5 Solve the equations.

1. 4. 7. 10. 13. 16.

3(x + 4) = 2(x + 5) 3(x + 5) = 2(4- x) 8(x- 3) = 2x 5(x - 1)- (x + 2) = 0 6 - 2x = 5(1 - x) 3x + 2(2x + 1) = 4(3 + x)

2. 7(x + 2) = 4(x + 6)

5. 8. 11. 14. 17.

4(1 - 3x) = 9(3 + x) 2(x + 1) + x = 7 2(3x- 1)- 3(x + 1) = 0 8 + 3(2x + 1) = 9 6x-2(3x-1)=4x

3. 6(x- 4) = 2(x- 1)

6. 9. 12. 15. 18.

Problem solving with equations Many mathematical problems are easier to solve when an equation is formed. In general it is a good idea to start by introducing a letter like 'x' or 'h' to stand for the unknown quantity. Steven is thinking of a number. When he doubles the number, adds 4 and then multiplies the result by 3, the answer is 13. What number is he thinking of? Let the number he is thinking of be x. He doubles it, adds 4, multiplies the result by 3. We have, 3(2x + 4) = 13 6x + 12 = 13 6x = 1 _I

X-6

Steven is thinking of the number

1;.

7(2x + 1) = 2(5 + 4x) 7(x - 2) - 3 = 2(1 - x) 4(x+1)+2(1-x)=x 3(1 - x)- (3 + x) = 0 (5- x)- (x- 10) = 15

Brackets and Equations

Exercise 6 In each question, I am thinking of a number. Use the information to form an equation and then solve it to find the number.

1. If I subtract 2 from the number and then multiply the result by 5, the answer is 11. 2. If I double the number and then subtract 7, the answer is 4. 3. If I multiply the number by 4, add 3 and then double the result, the answer is -2. 4. If I treble the number, add 2 and then double the result, the answer is 9.

5. If I add 4 to the number and then multiply the result by 7, I get the same answer as when I subtract 1 from the number and then double the result. 6. If I multiply the number by 7 and subtract 10, I get the same answer as when I add 2 to the number and then double the result. 7. If I multiply the number by 5, subtract 2, and then multiply the result by 4, the answer I get is the same as when I double the number and then subtract 3. 8. If I double the number, add 3 and then multiply the result by 5, I get the same answer as when I double the number and then add 21. 9. If I subtract 2 from the number and then multiply the result by 9, I get the same answer as when I take the number away from 3 and then double the result. 10. If I subtract the number from 2 and then multiply the result by 4, I get the same answer as when I add 1 to the number and then multiply the result by 5. 11. If I treble the number, add 5 and then double the result, I get the same answer as when I double the number and then subtract from 11. 12. If I double the number, add 4 and then divide the result by 3, I get the same answer as when I subtract the number from 10 and then double the result.

95

96

Part 4

Ebony had saved £54 and Hayley had saved £14. After both girls have been babysitting, for which they each receive the same amount of money, Ebony has three times as much as Hayley. How much did they each receive for babysitting? Let the amount they each received be £x. To help 'see' the problem, draw a table to show their money. Ebony Hayley @ @ before: after:

(£54+x)

@4B

Ebony has three times as much as Hayley. 54+ x = 3(14 + x) 54+ x = 42 + 3x 12 = 2x 6=x They each received £6 for babysitting.

Exercise 7

Answer these questions by forming an equation and then solving it. 1. (a) Find x if perimeter is 18 em.

(b) Find x if the area is 6cm2 .

xem

(x

+ 4) em

(x- 2) em

2. Sally has 5 times as many sweets as her brother Paul, but, as she

is feeling generous, she gives him 10 of hers so that they now each have the same number. How many did Paul have originally? 3. The diagram shows two angles in an isosceles triangle. Find the angles in the triangle.

4. In the quadrilateral, AB = x em,

BC is 2 em less than AB and CD is twice as long as BC. AD is 1 em longer than CD. If the perimeter of the quadrilateral is 33 em, find the length of AB.

97

Brackets and Equations

5. Sam has £71 and Tim has £30. They are both paid the same

money for a paper round. Now Sam has twice as much as Tim . . How much were they each paid? 6. The diagram shows a road from A to E.

A to B is 5km more than D to E. C to D is twice the distance from A to B. Cis midway between Band D . If the total distance from A to E is 91 km, find the distance from D to E. A

7. The sum of four consecutive whole numbers is 98. Let the first number be x and write down the other three numbers in terms of x. Find the four numbers. 8. The sum of four consecutive odd numbers is 216. Find the numbers. 9. The triangle and the rectangle have the same area. Find x.

~ ,D -12-

(x+ 3)

10. There were x people on the ElOl bus when it left Northolt. It

then stopped only at Greenford, Hanwell and Baling. At Greenford, 5 people got on and nobody got off. After leaving Hanwell, there were three times as many people on it as when it arrived there. There were 42 people on the bus when it arrived in Baling. Form an equation in x and solve it to find the number of people on the bus when it left Northolt. 11. Lily pours 5 barrels of water into

an empty tank and her sister Shelley pours in a further 12 litres through a hose. Later on, their father fills 2 barrels from the tank. There are 45 litres of water still in the tank. Use x for the number of litres in a barrel. Make an equation and solve it to find x .

In:

5~---------L:JL:J L:JL:JL:J

TANK

+ 12 litres through hose 2 barrels

98

Part 4

12. (a) The area of rectangle P is five times the area of rectangle Q. Find x. (b) The value of x is changed and the areas of the unshaded rectangles become equal. Find the new value of x.

9

l

p

5

j 2

Q

-x-

• 13. In a full season, Aston Villa played 40 league games. They won x games and lost 7 games. They got 3 points for each win, 1 point for each draw and 0 points for each defeat. Altogether they got 75 points. How many games did they win?

14. My daughter asked how old I am.

I answered 'In 20 years, I'll be twice as old as I was 12 years ago.' How old am I? 15. You have three consecutive even numbers so that the sum of

twice the smallest number plus three times the middle number is four times the largest number. Find the three numbers.

z

16. The diagram shows a rectangular

pond ABCD surrounded by a uniform path of width 2m. AB is three times as long as BC and WX is twice as long as XY. Find the dimensions of the pond.

y

D..-------;c Pond

B

A

w 17. Angle A is 30° more than the sum of angles B and C.

X A

Find the three angles of the triangle.

c B

99

Brackets and Equations

'L' puzzles (b) Here is another.

(a) This is an 'L' puzzle. 3 + 9 = 12 3+1=4

A

(c) Find all the numbers in this puzzle given that the number in box B is twice the number in box A.

B~ b2j

Let the number in box A be x. Then the number in box B is 2x. Write ? for the corner number.

A

B

Across the top row: ? + x = 12 SO

? = 12-

F 7

X

Using this value in the left hand column, 12- x + 2x = 17 x=5 The puzzle can now be completed:

Exercise 8 1. Copy and complete the following puzzles. You do not need an equation.

(a)

(b)

j+~ +

(c)

-

+

~+g=a

llij

In question 2 onwards use the method shown in the above example to form an equation and then solve it to find the missing numbers. 2. The number in box B is twice the number in box A. Start by letting x be the number in box A.

A

B~ ~

3. The number in box B is four times the number in box A.

A

B~

100

Part 4

4. The number in box B

is twice the number in box A. [Notice that box B is different to Qu. 3].

6. The number in box B

is four times the number in box A.

8. The numbers in boxes A and B add up to 20.

A

BEF A

·~ A

·~

5. The number in box B

A

is three times the number in box A. B

7. The numbers in boxes A and B add up to 20. Let the number in box A be x.

9. The number in box B is 5 less than the num-

ber in box A.

18

s 8

A

·~ 28

A

.EF

10. The diagram has changed here but the principle is similar to the above. 9 + 5 = 14; 9 + 7 = 16; 14 + 4 = 18; 16 + 2 = 18.

Find all the missing numbers given that: (a) the number in box B is four times the number in box A. (b) the number in box C is nine times the number in box A.

B

Aoo B

11. Find all the missing numbers given that:

(a) the number in box B is one less than the number in box A. (b) the number in box C is one third of the number in box A.

C

A

00

C

4

12. This one is more difficult. Find all the missing numbers given that: (a) the number in box A is three times the number in box C. (b) the number in box B is two more than the number in box A. (c) the number in box D is one more than twice the number in

box B.

B

A~C

D~

101

Volume

4.2 Volume Blocks A and B are each made from eight cubes, measuring 1 em x 1 em x 1 em. They each have a volume of 8 cubic em, which is written 8 cm3 .

Rectangular blocks like these are called cuboids. A cube like block B, in which all dimensions are equal, is a special kind of cuboid. The volume of a cuboid is given by the formula ,

~(length) x (width) x (height))

(a) Find the volume of the cuboid

(b) Find the height of the cuboid, given that its volume is 25 cm 3 .

lcm

Volume= 2 x 6·5 x 1 = 13cm3

5

X

4

X

h = 25 li h- 20 h = 1·25cm

Exercise 9 1. Find the volume.

2. Find the length x . (a)

~4cm

~m volume = 70 cm 3

(b)~

Sc~x volume = 90 cm 3

102

Part 4

The objects in questions 3 to 8 are made from centimetre cubes. Find the volume of each object.

3.

4.

5.

6.

7.

8.

9. Use isometric paper to draw the following objects: (a) a cuboid with volume 45 cm3 (b) aT-shaped object with volume 15cm3 (c) an L-shaped object with volume 20cm 3 (d) any object with a volume of 23 cm 3 . 10. Some of the shapes below are nets for closed boxes. Decide which nets will form closed boxes and work out the volume of the box in each case, giving your answer in cubic units.

(c)

(b)

(a)

I

I

I

:

i ...._ .___ -2-

(d)

-4-

11. *(a) Measure the dimensions of this book. (b) Work out the internal dimensions of a cardboard box which would take 50 books. Assume the books are in two adjacent piles of 25. (c) Allow for the thickness of the cardboard box by adding 1 em to the length, width and height to give the external dimensions. About how many books could be carried in a lorry with internal measurements 3m x 4 m x 2m?

103

Volume

12. *The diagram shows a section of a wall. The dimensions of a brick, allowing for cement, are as shown . .L:?J75mm

l.J.....-22's nun

112.5

mm

(a) Find the number of bricks in a section of a similar wall 90cm long. (b) One brick costs 22p. Find the cost of the bricks required for a wall of length 3 km, built around the perimeter of the grounds of a stately home.

Prisms The volume of the object shown can be found by dividing the object into layers, indicated by the thick lines. Each layer contains 6 cubes and there are 4 layers. The volume of the object is 24cm3 . An object which can be cut into identical layers like this is called a prism. A prism has the same cross section throughout its length. Volume of a prism= (Area of cross section) x (length) Any cuboid is a prism since it has the same cross section throughout its length. Find the volume of the prism shown. All the angles are right angles and the dimensions are in em.

4

Area of cross section = 4 x (3 + 3 + 3) + (3 x 2) = 42cm2 . Volume of prism = 42 x 8 = 336cm3

104

Part 4

Liquids The volume of a liquid is usually given in litres or millilitres (ml) 1000 ml = 1 litre and l ml is the same as I cm 3 . The diagram shows a cubic metre of water. 1m3 = 100 x 100 x 100cm3 3 = 1000 000 cm 3 So 1 m = 1 000 000 ml = 1000 litres

lOOcm

lOOcm

Exercise 10

In Questions 1 to 6 find the volume of each prism. All the angles are right angles and the dimensions are in centimetres.

3.

2.

1.

3 8

4.

6.

5.

7. A uniform metal rod of length 5m has a volume of 3750cm 3 . Find the area of the cross-section of the rod. 8. A vertical tower of height 32m has a square cross-section. Find

the length of a side of the square if the volume of the tower is 4608 m 3 . 9. Find the volume, in litres, of

the water trough shown.

t

20cm

•

2

105

Volume

10. Find the capacity, in litres, of a rectangular tank with internal dimensions 60 em by 20 em by l m. 11. Some steps are to be made in concrete. (a) Calculate, in cubic metres, the volume of concrete needed. (b) The concrete is mixed, by volume, from cement, sand and stones, in the ratio 2 : 3 : 6. What volume of sand is required for the steps? 12. The cross section of a plaster moulding for ceilings is a quarter circle cut from a square. 1 cm 3 of the plaster weighs 1· 2 g. Calculate the weight of a 4 m length of this moulding. 13. The diagram shows the cross section of a swimming pool. Water is pumped into the pool at a rate of 20 litresjsec. How long, in hours and minutes, will it take to fill the pool?

lm

14.*When water freezes it expands in volume by 10%. How many ice cubes, measuring 3 em x 3 em x 3 em, can be made from 20 litres of water?

Cylinders A cylinder is a prism because it has the same cross section throughout its length.

Volume= (area of cross section) x (length)

(volume=~

\

1 -h-

106

Part 4

(a) Find the capacity, in litres, of the oil drum shown

(b) A cylinder of radius 3 em has a volume of 200cm 3 . Find the length of the cylinder.

1

IOOcm

l 56 em

The oil drum is a cylinder. Volume of oil drum = n x 28 2 x 100 cm 3 3 = 246 000 cm (to 3 s.f.) Capacity of oil drum = 246 litres (to 3 s.f.)

V = n r 2h 200 = n X 32

X

h

200 = h nx 9 length of cylinder= 7·07 em (3 s.f.)

Exercise 11

Give answers correct to 3 significant figures , where necessary. 1. Find the volume of each cylinder.

(a)

(c)

(b)

4 em

t Scm

U+

!

3.2 em

E-~~0.9cm --7cm-

2. Cylinders are cut along the axis of symmetry. Find the volume of each object. (a)

8 em

(b)

cv 3. Find the volume in litres of a cylindrical tank of radius 40 em and height 35 em.

4. The lead in an unsharpened pencil is in the shape of a cylinder of diameter 2 mm and length 16 em. Find the volume of the 3 lead in cm .

107

Volume

5. A mine shaft 200m long is dug with

the cross-section shown. Calculate the volume of earth which must be removed to make way for the shaft.

3m

3m

6. Water is poured from the full cylinder

~m

A into the empty tank B. Will all the water go in?

Scm

+

15 em

7. An empty cylindrical tank of height 70 em and diameter 1 metre

is to be filled from a tap which delivers water at the rate of 150 ml per second . How long will it take to fill the tank? Give your answer to the nearest minute. 8. How many times can the cylindrical

glass be filled from the large drum which is full of milk?

3cm

drum

9. A solid sculpture weighing 5·7 kg

is made of metal and 1 cm 3 of the metal weighs 7·8 grams. The sculpture is to be melted down into solid cylinders of diameter and height 4 em . How many complete cylinders can be made?

10. The cross-section of a metal pipe is shown below. Calculate the

volume of metal used to make a pipe of length 10m.

108

Part 4

11. 'Vache qui rit' cheese portions are packed in circular boxes. Each portion is 1-4 em thick and has a top surface which is a sector of a circle of radius 5 em.

(a) Calculate the volume of the cheese in a full box and hence the volume of one portion. (b) 100cm3 of 'Vache qui rit' weighs 113 grams. Find the weight of one portion of cheese. 12. A solid rectangular block has dimensions 35 em x 28 em x 18 em. Calculate the volume of the largest cylinder which can be cut from this block.

4.3 Mental arithmetic Different people use different methods to perform calculations in their heads. Here are two ways of working out 59+ 58:

59+ 58: 60+60 1+2 120-3

120 3 117

59+ 58: 50+ 50 9+8 100 + 17

100 17 117

Here are two more examples:

Find 5°lo of £88. 10°lo of £88 = £8·80 5°lo of £88 = £4-40

£10- £1-85 £10 - £2 = £8 £8 + 15 p = £8 ·15

The questions in the tests below should be read out by a teacher and you should not be looking at them. Each question should be repeated once and then the answer, and only the answer, should be written down.

109

Mental Arithmetic

Test 1 1. What 1s a half of two thousand three

hundred? 2. Add together £7·65 and 40 pence.

3. If I have 55 pence change from a ten pound note, how much have I spent? 4. Find the mean of 27 and 31. 5. Work out 5% of £68. 6. 30% of a class prefer BBC1 and 34% prefer lTV. What percentage prefer the other channels? 7. How many millimetres are there in 70 em? 8. Between 6.00 p.m and midnight the temperature falls by l5°C. The temperature at 6.00 p.m is 6°C. What 1s the temperature at midnight? 9. How many days are there in 40 weeks? 10. Write down ten million millimetres m kilometres. 11. What number is three times as big as sixty-one? 12. Jim is paid £98 a week. About how much is that in a year? 13. How much less than 270 is 16? 14. From nine times seven take away twelve. 15. My watch reads five past six. It is 15 minutes fast. What is the correct time? 16. How many minutes are there in 6 hours? 17. Seven oranges costing eleven pence each are bought with a £5 note. What is the change? 18. What four coins make 62 pence? Give two possible answers. 19. Add together £3·90 and 65 pence. 20. A man died in 1991 aged 75. In what year was he born? 21. How many 2 pence coins are needed to make £4? 22. A piece of string 70 em long is cut into four equal parts. How long is each part? 23. Add up the first three prime numbers. 24. True or false: 10% of £5 ·50 is SSp? 25. What is a half of five hundred and ten? 26. How many weeks are there in three years? 27. A T.V. programme starts at 8·52 p.m. and finishes at 9·15 p.m. How long is the programme? 28. Two angles of a triangle are 75° and 10°. What is the third angle?

29. A car travels at 40 miles per hour for 21

hours. How far does it go? 30. Work out one squared plus two squared

plus three squared.

Test 2 1. Add together 15, 25 and 60. 2. How many millimetres are there in a

kilometre? 3. Find the length of the perimeter of a

regular hexagon of side 15 em. 4. Find the change from £10 when you buy

two magazines for 95p each. 5. Give a rough estimate for the square root of 110. 6. Find the cost of 60 eggs at £1 per dozen. 7. A car is travelling at a steady speed of 80 m.p.h. How far does it go in 15 minutes? 8. Find the difference between 111 and 20. 9. Work out I+ 22 + 33 . 10. Through what angle does the minute hand of a clock move between 8·50 and 9·30? 11. Work out roughly the area of a semicircle of radius I 0 em. 12. Is it true or false that a metric tonne, which is 1000 kg, is about the same as the imperial ton? 13. A boat is sailing due West. On what bearing is it sailing? 14. At noon the temperature is -4°C. By midnight the temperature has fallen by 12°. What is the temperature at midnight? 15. A large brick weighs 2 kg. Roughly what does it weigh in pounds? 16. Work out 1% of £115. 17. A plant grows 5 em every day. How many days will it take to grow 70 em? 18. A charity collection is made into a pile of 500 20p coins. How much was collected? 19. Add together 98 and 57. 20. True or false: At a steady speed of 60m.p.h you go 1 mile every minute. 21. James has one of each of the coins from 1p to 1 pound. What is their total value? 22. Three angles of a quadrilateral are 80°, 120° and 100°. What is the fourth angle?

110

Part 4

23. A tree was planted in Kew Gardens m 1880. How many years ago was that? 24. What is 80% as a fraction? 25. How many items costing £25 each can you buy with £1000? 26. What five coins make 68p? 27. Calculate the length of the perimeter of a rectangular field measuring 120m by 50m. 28. Work out 0·23 multiplied by 100000. 29. Increase a price of £500 by 1%. 30. Answer true or false: is greater than

(tf

t·

Test 3 1. What is the number which is 400 less than 50000? 2. Find the change from a £5 note after buying 3 pounds of apples at 25p per pound. 3. A girl faces south and turns clockwise through right angles. In which direction is she now facing? 4. A film, lasting hours, starts at 8·50. When does it finish? 5. Work out 100- 5-4. 6. Name the date which is 5 months before the 1st of March. 7. What is the maximum number of obtuse angles you can have in a quadrilateral? 8. Write 210 as a percentage. 9. Of the people in a room, a quarter were American, ten per cent were Russian and the rest were British. What percentage were British? 10. Which of these fractions is the larger: ~ or

lt

lt

.l? 10.

11. True or false: a weight of 7 stones is less than 70kg. 12. Work out 1% of £30. 13. Write in words the answer to 10 x 100 x 1000. 14. Add together 2, 3, 4 and 5. 15. A car travels 40 miles in 30 minutes. How far will it travel at this speed in ~ hour? 16. Jim spends 30% of his money on a book and 60% of his money on clothes. If he

had £2 left, how much did he have at first?

t

17. Write as a decimal: plus 1 ~00 • 18. A bucket contains 2 litres of milk. How much is left, in ml, after 200 ml is removed? 19. How many hours and minutes is it from 7-42 a.m until noon? 20. A steel bolt weighs 250 g. How many bolts weigh 50 kg? 21. If 20 magazines cost £8-40, find the cost of 5. 22. An ice cream costing 43p was paid for with a £1 coin. Which three coins were given as change? 23. In January, Steve weighs 90 kg. By July his weight is reduced by 10%. What does he weigh in July? 24. Find the total surface area of a cube of side 1 em. 25. Work out (7 x 103 ) + 10. 26. Write 5·5 recurring correct to one decimal place. 27. A 1Op coin is 1· 7 mm thick. What is the height, in em, of a pile of 100 coins? 28. Estimate the length of a side of a square of area 150cm2 . 29. Work out of £69 30. True or false: 10 em is about 4 inches?

t

Test 4 1. How many 20 pence coins are needed to make £10? 2. What number is mid-way between 3·1 and 3·18? 3. Work out 5% of £290. 4. True or false: one foot is approximately 15cm? 5. Add together 3·1 and and give the answer as a decimal. 6. One sector of a pie chart represents 10% of the whole chart. What is the angle of the sector? 7. Find the approximate area of a circle of diameter 20 em. 8. I pay for a cake costing £6·20 with a £20 note. What change do I receive? 9. Who is taller: Jim who is 6 feet tall or

t

Dave who is 1 metre 40 tall?

Mental Arithmetic

10. What five coins make 66p? 11. A rectangle measures 1·2m by lOcm.

What is its area in square metres? 12. A rope of length 5 feet 2 inches is cut in

half. How long is each piece? 13. A film started at 8·15 and finished at

10·50. How long was the film in hours and minutes? 14. Which has the longer perimeter: a square of side 10 em or a circle of diameter lOcm? 15. What fraction is equivalent to 12t%? 16. Find the cost of 6 litres of wine at £3·50 per litre. 17. How many 24p stamps.can be bought for £2? 18. Add together 45 and 135. 19. How long will it take to travel 60 miles at a speed of 30 m.p.h? 20. Work out 3 x 30 x 300. 21. What is the angle between the hands of a clock at 4 o'clock? 22. Find the cost of buying a newspaper for six days if each paper costs 45p. 23. Work out two fifths of £75. 24. How many prime numbers are there between 30 and 40? 25. I am thinking of a number. If I double it, add one and then square the result the answer is 49. What number am I thinking of? 26. Work out% plus! and give the answer as a decimal. 27. Divide one million by 20. 28. Two angles of a triangle are 64° and 74°. What is the third angle? 29. In a quiz, Ricardo got 18 out of 24. What percentage is that? 30. Increase a price of £500 by 10%.

Test 5 1. At a fair, the charge for playing a game

was 15p. If 30 people played, how much money was raised? 2. My train leaves at 1410. How many minutes do I have to wait if I arrive at the station at 1339?

Ill

3. The area of a triangle is 75 cm2 . Its base measures 25 em. What is the height of the triangle? 4. One eighth of the children in a class walk to school. What percentage of the class is this? 5. A man was born in 1956. How old will he be in 2004? 6. In an isosceles triangle the odd angle is 70°. How large are each of the two equal angles? 7. Tins of dog food are packed in cases of 6. If I want 50 tins of dog food, how many cases must I buy? 8. The time in New York is 5 hours earlier than the time in England. If I want to telephone New York at 1430 their time, what time will it be here? 9. I think of a number, multiply it by 7 and subtract 3. The result is 18. What number am I thinking of? 10. A plank of wood measures It metres by 10 em. What is the area of the plank in square metres? 11. Which is largest: 0·71, 0·7 or 0·077? 12. A bar of chocolate costs 16p. I buy as many as I can for £1. How much change will I receive? 13. There are 30 houses in my street and 10 of them are for sale. What percentage is this? 14. Add together 19, 51 and 32. 15. By how much does a half of 82 exceed 35? 16. A coat which costs £29·95 goes on sale for £24. What reduction is this? 17. Work out 5% of £4·80. 18. Two angles in a quadrilateral are each 95° and a third angle is 1ooo. What is the fourth angle? 19. Give an estimate for 83·7 x 9·6. 20. What number is a quarter of 130? 21. How many days will there be in 1998? 22. Rosie is going on a 2 week holiday. She leaves on the 27th of July. On what date will she return? 23. What is 5% as a simplified fraction? 24. What is the number exactly half way between 2·7 and 2·71?

112

Part 4

4.4 Enlargement When a photographer makes an enlargement of a picture, he chooses the size of the new picture. If his original picture is 8 em by 5 em and he wants the new picture to be 16 em by 10 em, he needs an enlargement with a scale factor of 2. The scale factor of an enlargement can be found by dividing corresponding lengths on two pictures. The scale factor for the enlargement of this picture is 1·2 [i.e.

m

Check that :~ is also equal to 1· 2. 48

30

36

For an enlargement, the object and image must be exactly the same shape. Rectangle A is an enlargement of rectangle B but triangle C is not an enlargement of triangle D.

B

D

c

A

Exercise 12

In questions 1 to 6 compare the two diagrams and state whether or not one is an enlargement of the other. Where it is an enlargement, state the scale factor. 1.

2.

3.

4.

5.

6.

113

Enlargement

7. This picture is to be enlarged to fit the frame. Find the height of the frame. 60mm

ht?

Frame

40mm

50mm

8. Picture B is either enlarged or reduced to fit the frames A, C and D. Find the lengths x, y and z. All lengths are in mm. D

c B A

60 y

20

X

30

33

Centre of enlargement In everyday use, the word 'enlarge' means 'increase' or 'expand'. Mathematicians use enlargement to describe a transformation, in the same way that reflection and rotation are transformations. A mathematical enlargement can either increase or decrease the size of an object, depending on the scale factor. If the scale factor is a number less than 1 (and greater than 0), the image is smaller than the object but the process is still called enlargement. A mathematical enlargement always has a centre of enlargement as well as a scale factor. The centre of enlargement is found by drawing lines through corresponding points on the object and image and finding where they intersect. For greater accuracy it is better to count squares between points because it is difficult to draw construction lines accurately over a long distance.

z

114

Part 4

In the second diagram, A'B'C' is an enlargement of ABC with scale factor 2 and centre 0. Observe that OA' = 2 x OA OB' = 2 x OB OC' = 2 X oc Always measure distances from the centre of enlargement.

To fully describe an enlargement we need two things: the scale factor and the centre of enlargement.

(a) Draw an enlargement of L'd with scale factor 3 and centre 0.

Notice that OA' = 3 x OA.

(b) Draw an enlargement of shape P with scale factor± and centre 0 .

Notice that OB' =

t x OB.

In both diagrams, just one point on the image has been found by using a construction line or by counting squares. When one point is known the rest of the diagram can easily be drawn, since the size and shape of the image is known.

Exercise 13

In questions 1, 2, 3 copy the diagrams and then find the centre of enlargement. 1.

2.

3.

115

Enlargement

In questions 4 to 10 copy the diagram and then draw an enlargement using the scale factor and centre of enlargement given. Leave room for the enlargement! 4.

7.

6.

9.

y

11. Copy the diagram.

(a) Draw the image of 6.ABC after an enlargement scale factor 2, centre (0, 0). Label it A'B'C'. (b) Draw the image of 6.DEF after an enlargement scale factor 2, centre (-5, 4). Label it D'E'F'. (c) Draw the image of 6.GHI after an enlargement scale factor 3, centre (-5, -2) . -+-+---+-+---+-+----'~-+--+---+----x Label it G'H'I'. 5 (d) Write down the coordinates of A', D' and . . . .;. i G'. -2

+-I ·· -3

+ ·············!

+

!············· + -4 1

......; ................ ; ........... ; ................;................; .. -5 1

12. (a) Draw x andy axes with values from 0 to 12 and draw 6.1

(b) (c) (d) (e)

with vertices at (6, 4), (6, 6), (5, 6). Enlarge 6.1 onto 6.2 with scale factor 2, centre (7, 6). Enlarge 6.1 onto 6.3 with scale factor 3, centre (6, 3). Enlarge 6.1 onto 6.4 with scale factor 2, centre (3, 8). Write down the coordinates of the right angled vertex of 6.2, 6.3 and 6.4.

+

!············· +

+············+·······

······!···············!················!··· ······+············+··········•

116

Part 4

13. (a) Draw axes with values from 0 to 12 and draw 61 with vertices at (5, 4), (3, 4), (3, 6).

(b) Draw 62, the image of 61 under enlargement with scale factor 2, centre (5, 2). (c) Draw 63, the image of 61 under enlargement with scale factor 3, centre (2, 3). (d) Draw 64, the image of 63 (not 61!) under enlargement with scale factor centre (11, 0). (e) Draw 65, the image of 63 under enlargement with scale factor %, centre (11, 12). (f) Write down the coordinates of the right angled vertex in 62, 63, 64 and 65.

t.

14. Copy the diagram. (You will need

it because you have to draw several construction lines). Find the scale factor and centre for each of the following enlargements: (a) 6A _. 6F (b) 6B---> 6C (c) 6D---> 6C (d) 6B---> 6E (e) 6B _. 6D.

15. Draw axes with values from 0 to 14.

Draw quadrilateral ABCD at A(2, 12), B(4, 10), C(2, 8), D(3, 10). (a) A'B'C'D' is an enlargement of ABCD with A' at (4, 14) and C' at (4, 6). Complete the quadrilateral A'B'C'D'. (b) A *B*C*D* is an enlargement of ABCD with A* at (8, 12) and B* at (14, 6). Complete the quadrilateral A*B*C*D*. (c) Draw A 0 B0 COD 0 , which is an enlargement of A*B*C*D* with scale factor and centre of enlargement (2, 6).

i

0

(d) Write down the coordinates of A

•

Enlargement

16. Copy the diagram. Describe fully each of the following enlargements (a) square 1 ---+ square 3 (b) square 2 ---+ square 5 (c) square 2 ---+ square 7 (d) square 3 ---+ square 5 (e) square 7 ---+ square 5

17. Copy shape A inside the rectangle shown. Shape A is enlarged so that it just fits inside the rectangle. Draw the enlargement of shape A and mark the centre of enlargement.

18. (a) Draw axes with values from 0 to 14. (b) Draw the following shapes: (i) Rectangle A at (9, 11), (11, 11), (11, 12), (9, 12) (ii) Triangle B at (2, 10), (2, 12), (1, 12) (iii) Square C at (3, 6), (4, 6), (4, 7), (3, 7) (c) Enlarge A onto A' with scale factor 2, centre (10, 14) (d) Enlarge B onto B' with scale factor 3, centre (0, 11) (e) Enlarge C onto C' with scale factor 4, centre (3, 8) (f) Write down the areas of shapes A, A', B, B', C, C'. . (area of A') ; (area of B') ; (area of C') . (g) Work out the ratios: area of A area of B area of C (h) Write down any connection you observe between the ratios above and the scale factor for each enlargement.

117

118

Part 4

4.5 Puzzles and investigations 4 Cross numbers Make three copies of the pattern below and complete the puzzles using the clues given. To avoid confusion it is better not to write the small reference numbers 1, 2, .. . 19 on your patterns. Write any decimal points on the lines between squares.

I

3

2

4

5 7

6

9

8

10

II

13

12

14

15

16

17 19

18

Part A Across

15% of 23 Next prime number after 23 One-third of 2409 X Solve the equation - = 3·8 5 6. Area of a circle of diameter 30 em (to 3 s.f.) 7. (71·6)2- (t+ io) 9. 245 2 - (3 3 X 2 2 ) 2 3 13. 7 + 7 + 7 15. + 3 X 13 17. Last 3 digits of (567 x 7) 18. 50 m written in em 19. The value of 2x 2 , when 1. 2. 4. 5.

t

X--

-2.!2

Down 1. Volume of a cube of side 15 em 2. One minute to midnight on the 24 hour clock 5·2 17 3. - - + - - t o 3 s.f. 0·21 0 ·31 2 5. (lli) to the nearest whole number 8. 12- 1~0 10. Prime number 11.2 5 -3 12. of 3675 13. North-west as a bearing 14. of 11% of 12000 16. Number of minutes between 1313 and 1745.

t t

Check: There should be 5 decimal points in the puzzle.

119

Puzzles and Investigations 4

Part B Across

Down

1. (0·5 7 t) X 123 2. 1001 7 77 4. Cost, in pounds, of 3 ·2 kg of powder at 6p per gram. 5. [number of letters in the word 'squared' ]2 6. 33i% of 2802 7. 8·14 - (1 ·96 x 0 ·011) to 4 s.f. 9. 7391 X 11 13. 1 1 + 2 2 + 3 3 + 4 4 15. 104 - (2 X 20 2 + 9 X 7) 17. Number of minutes between 0340 and 1310. 18. (5 across) 2 x 2 ·7 - {0 19. 19 + +

io t

(1 across) x (2 across) t + i + t+t to 3 d.p. 20 2 - J4 42 ·2 - (8·1 X 0·13) to 3 s.f. Product of the 4th, 5th and 6th prime numbers. 10. Number of minutes taken to travel 12 km at 60 km/h. 11. (2 3 X 3 2 ) + 2 2 + 2 12. Number of hours in a leap year. 13. 13% of £22 ·80, to the nearest penny 14. Next in the sequence 31 ·7, 95·3 , 286, ... 16. Length, to the nearest mm, of a diagonal of a square of side 24cm.

1. 2. 3. 5. 8.

Check: There should be 6 decimal points in the puzzle.

Part C Across

Down

1. South-west as a bearing. 2. Inches in a yard 4. Last three digits of (112 + 22)2 5. 4 score plus ten 2 6. (26t) , to the nearest whole number 7. 24·3 + 357 + 87·04, 1 ·9 24 3 ·7 correct to 2 d.p. 9. (2 across) 2 + across)(middle digit of 5 down) 13. 7 stone 8 pounds in pounds 15. last 3 digits of (407 x 21) x 11 17. ~+t+t+~ 18. The length, correct to 2 d.p., of the diagonal of a rectangle measuring 31 em by 41 em. 19. Next in the sequence 5, 9, 17, 33, 65, .. .

v(l

555 1. Solve the equation-- = 2 X

2. 11% of £323·11, to the nearest penny 1·23 3. , correct to 2 d.p. 1·4 - 0 ·271 5. Next in the sequence 114, 229, 459, ... 8. Area, in cm 2 , of a rectangle measuring 1 ·2 m by 11 em 10. (A square number) - 1 11. Change 144km/h into m/s 12. J (4 across) x (13 across)+ (10 down) 13. Angle in degrees between the hands of a clock at 2 ·30 14. A quarter share of a third share of a half share of £152 ·16 16. (inches in a foot) x (ounces in a pound).

Check: There should be 7 decimal points in the puzzle.

120

Part 4

Round the class: a game for the whole class

You' re next 1'f you . 11 have thlS-

• Fifty cards like those above are handed out around the class. • One person has a START card. He/she reads the number in the ring at the top of the card (11) and then reads the statement 'You're next if you have this +8' . • Someone in the class will have the card shown in the middle. That person says '19' and then reads 'You're next if you have this x 2'.

You' re next if you have this X 2

• This process continues around the room until card 50 with the word 'END' is reached. The object of the game is to complete the 50 cards as quickly as possible. • The class can be split into two teams. Each team completes the 50 cards against the stopwatch and the winner is the team with the quicker time. • Teacher's note: The cards appear in the Answer Book from which they can be photocopied and cut up. A master sheet is also printed for the teacher to keep track of what is going on.

Finding areas by counting dots • Shapes can be drawn on dotty paper, with vertices on dots.

v:o D

E

• •

•

• • F

• •

•

0: G

• •

•

121

Puzzles and Investigations 4

Triangle D has 6 dots on its perimeter. For short, we will write p=6. Square E has 8 dots on its perimeter and 1 dot inside. We will write p = 8 , i = 1. Write down the values of p and i for shapes F and G. • Is there a connection between the values for p , i and the area A for each shape? Here are some of the areas: For shape D: p = 6, i = 0, A= 2. (area is in square units) For shape E: p = 8 , i = 1, A = 4. For shape F: p = 8, i = 2, A = 5. For shape G: p = 4, i = 1, A = 2. It is certainly not easy to see a connection or a formula that works for all the shapes. • Make the problem simpler. When you have a complicated problem it is often helpful to make it simpler. We will start by drawing shapes with no dots inside (i = 0].

CJr::r I:::I

•

•

•

Record the values for p and A in a table:

•

P

A

6 8 10 3 5

2 3

Try to find a connection between p and A. Write it in the form 'A = .. .'.

•

• Now draw shapes with one or more dots inside. Record the values for p, i and A in a new table. Try to find a connection between p , i and A. Write it in the form 'A= . . .'.

A

p

8 10 4 5

1

2 1 3

4 6

DD ~

~ •

•

•

122

• Predictions If you think you have a formula for A, in terms of p and i, use it to predict the area of each of the shapes below.

Check by calculating the areas by 'traditional' methods.

Part 4

Part 5 5.1 Estimating In many real life situations, it may not be necessary or even possible to obtain the exact answer to a problem. An estimate, which gives a rough answer, may be more realistic. The chef in charge of ordering food for meals in a large restaurant can never be sure how many people he needs to cater for each day. He could estimate the likely number of customers and make his order accordingly. A farmer may not know the exact area of all his fields when he orders fertilizer. He could estimate the amount required and if he ran out he could order some more. (a) Estimate the reading on each scale (i)

1

(ii)

3 Answer: about 3·2 or 3·3.

4

1

~--------------------~

20

10 Answer: about 16.

(b) Estimate the weight, in kg, of the following: (i) A standard loaf of bread. (ii) The President of France. Answers: (i) About kg----> kg. Use an ordinary bag of sugar, which is 1 kg, as a reference. (ii) About 70 ----> 90 kg.

1

i

Exercise 1

1. Estimate the reading given by each pointer. (a)

(b)

I 5

I 6

(d)

0

2

4

(e)

(f)

I 100

110

8

9

2

2.1

124

Part 5

2. Estimate what percentage of each diagram is shaded.

(a)

(b)

(c)

3. Write the decimal point in the correct place. (a) length of a football pitch (b) weight of an 'average' new born baby (c) width of this book (d) capacity of a wine glass (e) weight of a one pound coin (t) capacity of a household fridge (g) area of one face of a lOp coin

9572 3124 1831 1260 904 656 4524

L . . ._IL...-___________.

m

kg mm

ml g

cubic feet mm2

4. A quick way of adding lots of figures on a shopping bill is to

round every number to the nearest pound. So £2-43 becomes £2, £0·91 becomes £1, £0·24 becomes £0 and so on. (a) Use this method to estimate the totals below: 0.85 (i) WSKAS COCKTAIL (ii) PLN BAGUETTE H/EATING MINCE HAWAIIAN CRN PAIN AU CHOC PAIN AU CHOC PAIN AU CHOC BUTTER BUTTER EGGS PORK/CHICK/PIE MED.MAT.CHDR. HOT PIES POT. WAFFLES WHOLE BRIE MUFFINS BACON RASHERS BEETROOT LOOSE CHEESE

3.95 1.85 0.54 0.54 0.54 0.89 0.89 0. 78 2.03 1.21 1.47 1. 39 1.01 0.49 1.65 0.99 0.99

FOIL LETTUCE ROUND JW TUNA MAYO SOYA MILK SOYA MILK ORNGE MRMLDE YOGHURT SPGHTI/HOOPS CHEESE WHISKAS WHISKAS VINEGAR KING EDWARDS. UHT H/FAT MILK • APPLES WHISKAS PEACHES FROM. FRAIS

(b) Use a calculator to work out the exact total for part (i). Compare the answer with your estimate above. 5. On squared paper draw x andy axes with values from 0 to 10. Use as large a scale as possible. Draw a straight line through the points (0·5, 2) and (10, 7·5). Estimate the following, for points on the line. (b) the y value when x = 6 (a) the y value when x = 2 (d) the x value when y = 5 (c) they value when x = 4·5

(e) the x value when y = 6·5

(f) the x value when y = 2·5

0.49 0.65 0.24 0.75 0.47 0.47 0.74 0.99 0.26 1. 34 0.45 0.45 0.68 1. 99 0.26 1. 89 0.45 0.24 0.72

Estimating

125

Estimate the answers to the following: (a) 59·7 x 97 ·01 ~ 60 x 100. About 6000 (b) ,J23 ·97 x 0·198 ~ 5 x 0·2. About 1 (c) 42·86 ...;- 119·3 ~ 40 ...;- 100. About 0-4 (d) 4 ·75% of 384·3 ~ 5% of 400. About 20 Notice that, in the working, all figures are rounded off to just one significant figure . [E.g. 59·7 ~ 60, 97 ·01 ~ 100]. A common error occurs when small numbers, like 0·0 18, are incorrectly rounded to zero . (e) 3106 x 0·018 ~ 3000 x 0·02. About 60 (f) 0·00974 x 0·039 ~ 0·01 x 0·04. About 0·0004

Exercise 2

Do not use a calculator. Decide, by estimating, which of the three answers is closest to the exact value. Write the letter A, B or C for each part. Calculation 1.

A

B

c

109 30 200 50 2·5 million 0·1 8·8 2·5 0·13 1·3

99 70 400 5 250000 2 65 5 0·5 0-4

1100 700 4000 0·5 25000 10 108 25 1·3 4·1

0·7

3·1

10.

96·2 X 11 ·9 6·79 X 9·82 18·9 x 21·1 511 X 0·103 24 783 X 10·72 203-4 ...;- 19·87 82·78 ...;- 1·274 5·07 ...;- 0·197 1211 ...;- 979·6 193-4 ...;- 48 ·74

11.

lQ

12.

[ (!/ +C60 )

13.

649 312 1·35- 0-421

100

500

1000

14.

98 ·63 X 11 ·71 10·7 X 0 ·985

10

60

100

15.

,j98·73

16

31

2. 3. 4.

5. 6. 7. 8. 9.

7r2

X

7 2 ]

x 10

+

X

VIQ.f

,.j3.93 X 0 ·202 17. 9·83% of (2567·3 + 1·92) 18. ! of (20·27 x 20·11 ) 16.

7

212

4

3·5 0·04

0-4

0·001

1

250

400

2500

16

80

160

19.

-?9 of 1% of 6060

30

60

300

20.

~of 24·8% of! of 2403 ·62

20

200

2000

126

Part 5

Exercise 3

In Questions 1 and 2 there are nine calculations and nine answers. Write down each calculation and insert the correct answer from the list given. Do not use a calculator. 1. (a) 1·8 x 10-4

(b) (d) 4·02 X 1·9 (e) (g) 36·96 7 4 (h) Answers: 63·99, 18·72, 48·248

9·8 X 9·1 3·8 x 8·2 9·6 7 5 31·16, 4·08, 1·92,

(c) 7·9X8·1 (f) 8·15x5·92 (i) 0·11 + 3·97 9·24, 7·638, 89·18,

2. (a) 3·9 7 2 (b) 36·52 7 4 (c) 101-4 7 5 (d) 7·2 X 1·9 (e) 4·3 x 6·8 (f) 3·57 7 3 (g) 7·76 7 2 (h) 5% of 700 (i) 43·56 7 6 Answers: 13·68, 7·26, 29·24, 3·88, 1·19, 20·28, 9·13, 1·95, 35·0.

3. A boxer earned a fee of $8 million for a fight which lasted 1 minute 35 seconds. Estimate the money he earned per second of the fight. 4. A car does an average of 34·3 miles per gallon of petrol and petrol costs £2-45 per gallon. Mr Davis estimates that his total petrol bill for a year in which he drove 9875 miles was £350. Is this a reasonable estimate? If not, give a better estimate.

5. At a fun fair, customers pay 95p for a ride on a giant spinning wheel. The operator sells 2483 tickets during the weekend and his costs for electricity and rent were £114. Estimate his profit over the weekend. 6. Estimate, giving your answer correct to one significant figure:

(a) the number of times your heart beats in one day (24 h), (b) the thickness of one page in this book. 7. (a) In 1989 thousands of people formed a human chain right across the U.S.A., a distance of about 4300 km. Estimate the number of people in the chain. (b) Estimate the number of people needed to form a chain right around the equator. (Assume you have enough people volunteering to float for a while in the sea.) The radius of

the earth is about 6400 km.

127

Scatter Graphs

5.2 Scatter graphs John and Ruth are having an argument about test results. John thinks that children who read well at the age of 7 tend to do better at GCSE English when they are 16. Ruth thinks that there is little connection between these results and that other factors, like the type of secondary school attended, are more important. One way to help settle the argument is to take a group of children and to look at their reading test results at the age of 7 and their subsequent English GCSE results. Here are some results. A high score is 10 and a low score is 0.

Child

A

B

c

D

E

F

G

H

I

J

K

L

M

reading test at 7

5

7

8

9

7

3

4

5

10

9

8

9

4

English GCSE at 16

7

6

7

7

8

2

2

3

8

7

8

9

4

It is not easy to draw conclusions by

looking at the above results. A much clearer picture develops when the results are plotted on a scatter graph. The test results at age 7 are plotted across the page and GCSE results at age 16 are plotted up the page. In this group of children it is fair to say that, even though the points are scattered, there is a connection. The children who did well at 7 tended to do well at 16. Those who scored a low result at 7 tended to score a low result at 16. We say there is a correlation between the two sets of results.

10 9 -+·· ........, ... . . , 8

-+

: . . . . . . . : ..., . . . . . . .

7

6

2

2

3

4

5

6

Reading test at 7

7

8

9

10

128

Part 5

The correlation between two sets of data can be positive or negative and it can be strong or weak as indicated by the scatter graphs below. X

X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X X

X

X

X

X

X

X X

X X

strong positive correlation

weak positive correlation

strong negative correlation

When the correlation is positive the points are around a line which slopes upwards to the right. When the correlation is negative the 'line' slopes downwards to the right. When the correlation is strong the points are bunched close to a line through their midst. When the correlation 1s weak the points are more scattered. It is important to realise that often there is no correlation between two sets of data. If, for example, we take a group of students and plot their maths test results against their time to run 800 m, the graph might look like the one on the right. A common mistake in this topic is to 'see' a correlation on a scatter graph where none exists.

Time to run 800 m X X

X X X

X

X

X

X

X

X

Maths results

Exercise 4 1. Make the following measurements for everyone in your class: height (nearest em) armspan (nearest em) head circumference (nearest em) hand span (nearest em) pulse rate (beats/minute) For greater consistency of measuring, one person (or perhaps 2 people) should do all the measurements of one kind (except on themselves!)

Enter all the measurements in a table, either on the board or on a sheet of paper.

Name

Height

Arms pan

Head

Roger Liz Gill

161 150

165 148

56 em 49cm

129

Scatter Graphs

(a) Draw the scatter graphs shown below (i) arm span (ii)

hand span

height

pulse

(b) Describe the correlation, if any, in the scatter graphs you drew in part (a). (c) (i) Draw a scatter graph of two measurements where you think there might be positive correlation. (ii) Was there indeed a positive correlation? 2. Plot the points given on a scatter graph, with s across the page and pup the page. Draw axes with values from 0 to 20. Describe the correlation, if any, between the values of s and p. [i.e. 'strong negative', 'weak positive' etc.] (a)

(b)

(c)

s

7

16

4

12

18

6

20

4

10

13

p

8

15

6

12

17

9

18

7

10

14

s

3

8

12

15

16

5

6

17

9

p

4

2

10

17

5

10

17

11

15

s

11

1

16

7

2

19

8

4

13

18

p

5

12

7

14

17

1

11

8

11

5

3. Suppose scatter graphs were drawn, with the quantities below on the two axes. What sort of correlation, if any, would you expect to see in each case? (a) height of a man; height of the man's father (b) a person's pulse rate; a person's reaction time (c) outside temperature; consumption of energy for heating a home (d) value of a car; mileage of the car [for the same kind of car] (e) price of goods in U.K; price of similar goods in Germany (f) number of ice creams sold; outside temperature 4. An experiment: estimating angles (a) You state a list of 'target' angles like 25°, n o, 110°, etc and ask your partner to draw the angles, using just a pencil and ruler. (b) You now measure the angles drawn and, for each one, record the error. [The 'error' is always a positive number. For example, both 28° and 22° give an error of 3° from 25°]. (c) Draw a scatter graph with 'error' across the page and 'target angle' up the page. (d) Is there any correlation? Compare your results with those of other people.

130

Part 5

Line of best fit When a scatter graph shows either positive or negative correlation, a line of best fit can be drawn. The sums of the distances to points on either side of the line are equal and there should be an equal number of points on each side of the line. The line is easier to draw when a transparent ruler is used. Here are the marks obtained in two tests by 9 students. Student

A

B

c

D

E

F

G

H

I

Maths mark

28

22

9

40

37

35

30

23

?

Physics mark

48

45

34

57

50

55

53

45

52

A line of best fit can be drawn as

there is strong positive correlation between the two sets of marks. The line of best fit can be used to estimate the maths result of student I, who missed the maths test but scored 52 in the physics test. We can estimate that student I would have scored about 33 in the maths test. It is not possible to be very accurate using scatter graphs. It is reasonable to state that student I 'might have scored between 30 and 36' in the maths test.

50

40

30

lO

20 Maths mark

Here is a scatter graph in which the heights of boys of different ages is recorded. A line of best fit is drawn. (a) We can estimate that the height of an 8 year old boy might be about 123 em [say between 120 and 126 em]. (b) We can only predict a height within the range of values plotted. We could not extend the line of best fit and use it to predict the height of a 30 year old! Why not?

30

40

131

Scatter Graphs

Exercise 5 1. For each scatter graph state which line, p or q, is the line of best fit.

(a)

(b)

p

2. The following data gives the marks of 11 students in a French test and in a German test. French

15

36

36

22

23

27

43

22

43

40

26

German

6

28

35

18

28

28

37

9

41

45

17

(a) Plot this data on a scatter graph, with the French marks on the horizontal axis. (b) Draw the line of best fit. (c) Estimate the German mark of a student who got 30 m French. (d) Estimate the French mark of a student who got 45 m German. 3. The data below gives the petrol consumption figures of cars, with the same size engine, when driven at different speeds. Speed (m.p.h.)

30

62

40

80

70

55

75

Petrol consumption (m.p.g)

38

25

35

20

26

34

22

(a) Plot a scatter graph and draw a line of best fit. (b) Estimate the petrol consumption of a car travelling at 45m.p.h. (c) Estimate the speed of a car whose petrol consumption is 27m.p.g. 4. The table below show details of the number of rooms and the number of occupants of 11 houses in a street. Number of rooms

2

3

7

11

7

5

5

11

5

6

4

Number of occupants

2

8

5

2

6

2

7

7

4

0

1

(a) Draw a scatter graph (b) Can you estimate the likely number of people living in a house with 9 rooms? If so, what is the number? Explain your answer.

132

Part 5

5. Look at the scatter graph of height against armspan which you drew for question 1 of the last exercise.

(a) Draw a line of best fit on your graph. (b) A person of height 161 em enters the room. Estimate the armspan of that person.

5.3 Averages An average is a single number, chosen for a set of data, which can be used to represent that data. In a sense the average is a representative value, typical of the data from which it comes. The average can be found in different ways.

0

The mean

0

The median

0

The mode

0

All the data is added and the total is divided by the number of items. In everyday language the word 'average' usually stands for the mean.

When the data is arranged in order of size, the median is the one in the middle. If there are two 'middle' numbers, the median is in the middle of these two numbers [i.e. the mean of the two numbers].

The mode is the number or quality (like a colour) which occurs most often. Sometimes a set of data will have no mode, two modes or even more and this is a problem which we cannot avoid. Range

The range is not an average but is the difference between the largest value and the smallest value in a set of data. It is useful in comparing sets of data when the spread of the data

is important.

133

Averages

The mean weight of a team of 8 rugby players is 85·2 kg. Find the new mean weight when two new players arrive weighing 73 kg and 81·4 kg. Total weight of original 8 players = 85·2 X 8 = 681·6kg. Total weight of 10 players = 681·6 + 73 + 81-4 = 836kg :. Mean weight of 10 players = 836...;... 10 = 83·6kg

The marks achieved by 10 pupils in a test were as follows: 5, 4, 7, 7, 6, 8, 9, 6, 6, 8. For these marks, find (a) the mean,

(b) the median

(c) the mode

(a) Mean mark = 5 + 4 + 7 + 7 + 6 + 8 + 9 + 6 + 6 + 8 = 6·6 10 (b) Arrange the marks in order: 4 5 6 6 6 7 7 8 8 9

l The median is here. Median=

6

+7 = 2

6·5

(c) Mode= 6, since there are more sixes than any other number. (d) Range=9-4=5.

Exercise 7

1. The weights, in kg, of 10 refugees from a war were as follows: 52, 49, 47, 51, 46, 42, 44, 61, 55, 56. Find the mean weight of these people. 2. In several different garages the cost of one litre of petrol is 55p, 52·8p, 56-4p, 53-lp, 59p, 53·8p and 57p. What is the mean cost of one litre of petrol?

(d) the range.

134

Part 5

3. Six girls have heights of l-48 m, 1·51 m, 1-47m, 1·55 m, 1-40m and 1·59m. (a) Find the mean height of the six girls. (b) Find the mean height of the remaining five girls when the tallest girl leaves. 4. The temperature outside a house was measured at midnight

every day for a week. The readings were: -20, -1 0, 30, -40, -1 0, oo, -20. (a) Find the mean temperature for the whole week. (b) Find the mean temperature for the last four days of the week. 5. Mrs Green gave birth to five babies (two girls and three boys)

which weighed 1·3 kg, 1·2 kg, 1-45 kg, 1·35 kg and 1·3 kg. What was the median weight of the babies? 6. Sally throws a dice eight times and wins 20p if the median score

is more than 3. The dice shows 6, 1, 2, 6, 4, 1, 3, 6. Find the median score. Does she win 20p? 7. The temperature was recorded at 0400 in seven towns across the U.K. The readings were 0°, 1°, -4°, 1°, -2°, -5°, -4°. What was the median temperature? 8. The number of occupants in the 33 houses in a street is as follows: 2 4 3 4 1 4 2 4 1 5 2 3 0 5 3 4 3 6 7 3 3 6 4 1 4 2 0 1 4 3 2 5 0 What is the modal number of occupants in the houses? 9. The test results for a class of 30 pupils were as follows:

Mark

3

4

5

6

7

8

Frequency

2

5

4

7

6

6

What was the modal mark? 10. Write down ten numbers so that:

the mode is 5; the median is 7; the mean is 9·2; the range is 15. Ask a friend to check your answer. 11. Make a list of 15 numbers (not all the same!) so that the mode,

the median and the mean are all the same value. Ask a friend to check your list.

12. The pie chart illustrates the number of different kinds of tree in a wood. What is the modal type of tree?

135

Averages

13. The mean of the numbers 3, 4, 4, 5, 6, 7, 7, 8 is 5·5. Write down, without any further working the mean of the numbers 303, 304, 304, 305, 306, 307, 307, 308. 14. The range for nine numbers on a card is 60. One number is covered by a piece of blu-tac. What could that number be?

10

61

44

•

15. The mean of the eight numbers 6, 6, 9, 3, 8, 11, 5, 2 is 6,\-. When a number xis added, the new mean of the nine numbers is 7. Find x. 16.*(a) For the first four months of the year Mr Lee is paid £900 per month. After a pay rise, he receives £1155 per month for the rest of the year. Calculate his mean pay per month over the whole year. (b) Mrs Lee receives £x per month for the first four months and then £y per month for the rest of the year. Calculate her mean pay per month in terms of X andy.

17.*The average height of n people is hem. One person of height x em leaves the group and one person of height y em joins the group. Find an expression for the new average height.

What do you understand by 'average'? (a) The mode is the easiest average to find . If a shopkeeper wanted to stock just one size of a certain product, he would choose to stock the most popular size which is the mode. The mode can be misleading, if used in the wrong application, because it takes no account of any numbers apart from the most common. The frequency charts show the marks for two classes in the same test. The mode for class A is only 3 marks, whereas the mode for class B is 8. With further calculation we find that the mean mark for class A is 5-4, which is slightly higher than the mean mark for class B.

frequency 7

frequency

....---

7

6

6

-

5 4

5

-

-

;--

r-

4

3

3

2

2

-

1--

-Jy 3

4

5

6

mark

7

8

-~

3

4

5

6

mark

7

8

136

Part 5

(b) The median is also easy to find and it gives the middle number in a set of data, which is clearly understood. The median takes no account of very large or very small numbers away from the median. For example, the two sets of data, P and Q below, have the same median but very different values for the mean and range: 3, 4, 4·5, 5, 5·2, 5·7, 110·2 Set P Set Q 3·2, 4·1, 4·8, 5, 5·3, 5·3, 5-4

i median (c) The mean takes longer to calculate but it does include all the data and is the most widely quoted average. One or two extreme values can, however, distort the mean so that it is not representative of the data. For example, the mean of the numbers in set P above is approximately 19·7. None of the numbers in set P is close to this 'average' value. Knowing which average to choose is a matter of judgement and the choice often depends on the data. The salaries of 9 employees of a firm are, in pounds, 8500, 8700, 8900, 9000, 9200, 9300, 9700, 13000, 155000. Which 'average' salary best describes this group? (a) (b) (c) (d)

There is no modal salary The median salary is £9200. The mean salary is £25 700 For this set of data, the median is a far more typical value than the mean. The mean value is distorted by the one very high salary of £155 000.

Exercise 8 1. In a maths exam, Stephen's result was 53%. The marks obtained

by the pupils in his group were 37, 39, 39, 40, 40, 41, 42, 42, 53, 90, 91, 95, 97, 99, 100 [All percentages]. When he tells his parents his mark, he also tells them what the group's 'average mark' was. Which average do you think he chose to give them? 2. The annual salaries of the workers

in a small company were as follows: 25 trainees £4800 each 15 on scale A £7800 each 10 on scale B £9500 each 2 on scale C £14 200 each

1 manager £45 000.

.. ·---·-·- --.. 0 0-------····-· 9 !illl

••

li - - - -

II .U

'

I

137

Averages

The boss wants to give the impression that the average salary is fairly high. The workers' representative, on the other hand, wants to quote an average salary which is very low. Which average should each person quote? 3. In a maths exam, Joe got 54%. For the whole class the mean mark was 51% and the median mark was 57%. Which 'average' tells him whether he is in the 'top' half or the 'bottom' half of the class. 4. In an experiment, several toy cars were run until their batteries ran out of power. The length of time for which they worked was recorded. The results were car

1

hours of working

12

2

3

4

1312 11

2

l

5

6

7

10 13 14

8 1012

(a) Find the mean and the median length of time for which the cars ran. (b) Which figure represents the better 'average' value? 5. Which is the best average for the set of data below? 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 80, 110. Give reasons for your choice. 6. In a test the marks for the boys were 6, 9, 8, 6, 5, 8 and the marks for the girls were 8, 7, 4, 7, 5, 6, 2, 9. (a) Find the mean mark for the boys. (b) Find the mean mark for the girls. (c) Find the mean mark for the whole class. 7. The mean height of 5 basketball players is 1·98 m and the mean height of 3 jockeys is 1·61 m. Find (a) the total height of the basketball players. (b) the total height of the jockeys. (c) the mean height of the group of 8 basketball players and jockeys. 8. The mean weight of 7 cars on a transporter is 724 kg. An eighth car of weight 908 kg is driven onto the transporter. Find the mean weight of the 8 cars.

138

Part 5

9. (a) Find the mean of the numbers 5, 7, 5, 6, 3, 9, 10, 7.

(b) When a number x is added to the list, the new mean is 8. Find x. 10. Find the mean of (a) 1, 2

(b) 1, 2, 3 (c) 1, 2, 3, 4

(d) 1, 2, 3, 4, 5 (e) 1, 2, 3, 4, ... .. . n

Frequency tables When a set of data consists of many numbers it is convenient to record the information in a frequency table. It is possible to find the mean, median and mode directly from the table as shown in the example below.

The frequency table shows the number of goals scored in 15 football matches. number of goals

0

1

2

3

4

5 or more

frequency

2

5

4

3

1

0

(a) We could find the mean as follows: ~+0+1+1+1+1+1+2+2+2+2+3+3+3+~

mean=-'--------------------------'-15 A better method is to multiply the number of goals by the respective frequencies. (0 X 2)+(1 X 5)+(2 X 4)+(3 X 3)+(4 X 1) mean = --'-----'------'-----'----'------------15 mean= 1·73 goals (correct to 2 d.p.) (b) The median is the 8th number in the list, when the numbers are arranged in order. The median is, therefore, 2 goals. (c) The modal number of goals is 1, since more games had 1 goal than any other number.

Exercise 9 1. The frequency table shows the

weights of 30 eggs laid by the hens on a free range farm. weight frequency

44g

48g

52g

56g

60g

5

6

7

9

3

Find the mean weight of the eggs.

139

Averages

2. The marks, out of 10, achieved by 25 teachers in a spelling test were as follows:

mark

5

6

7

8

9

10

frequency

8

7

4

2

3

1

Find (a) the mean mark (b) the median mark (c) the modal mark. 3. A golfer played the same hole 30 times with the following results.

score

3

4

5

6

7

8

frequency

2

11

5

5

3

4

(a) Find his mean score, median score and modal score on the hole. (b) Which average best represents the data? Explain why. 4. The graph shows the results of a survey of a number of families in which the number of children was counted. (a) What is the mode? (b) What is the median? (c) Which of the above is the more meaningful average for the data? Explain why.

11

Number of families

-

10 9

8 7

6 5 4

1--

r--

-

3 2

0

5. The table shows the weights of 50 coms. Find the weight x if the mean weight of the 50 coins is 1 gram more than the median weight.

2 3 4 5 6 Number of children in family

weight

frequency

2g 4g 5g

4 7 16 10 13

X

9g

-

1--

n 7

140

Part 5

6. The number of bedrooms in the houses in a street is shown in the table. (a) If the mean number of bedrooms is 3·6, find x . (b) If the median number of bedrooms is 3, find the largest possible value of x. (c) If the modal number of bedrooms is 3, find the largest possible value of x . 7. The weights of 20 ear-rings were measured. Copy and complete the table below so that the mean weight is 6·3 g, the median weight is 6 g and the modal weight is 7 g.

number of bedrooms

frequency

2 3 4

5 12 X

weight of ear-ring

5g

frequency

5

6g

0

5.4 Puzzles and investigations 5 Break the codes 1. The ten symbols below each stand for one of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 but not in that order. 8 V'D*i? 8 tt~I

Use the clues below to work out what number each symbol stands for. (a) + =? (b) v x v = e (c) ? +? +? = 8

* *

(d) D-

*

= 8 (e) tt +I= (f) 1t X f = f (g) ~-I= I

e

2. The ten symbols used in part 1 are used again but with different values. The clues are more difficult to work out. (a)

e + ~=I

(b) \7 + D =I

(c) 8 x D = 8

(d) 8 X ?=? (e) 1t 7 ~ = 8 (f) D + D + D + D = 8 (g) 1t- ~ = \7

(h)

* X* =j

7g

0

8g 2

141

Puzzles and Investigations 5

Vending machine problem • You have lots of 1 Op coins and 20p coins to put into a vending machine. In how many different ways can you put 40p in the machine? • This is not difficult. Make a systematic list: 10, 10, 10, 10 1 way 20, 20 1 way 20, 10,10} 10, 20, 10 3 ways. These count as different because the order 10, 10,20 of the coins is different. So there are 5 ways of putting in 40p. • How many ways are there of putting in 120p? This is much harder because there are so many ways of doing it. If you start to make a list you soon realise how difficult it will be. In many investigations it is a good idea to attempt an easier problem: • (a) Find the number of ways you can put in lOp, 20p, 30p, 40p etc. Put the results in a table and look for a sequence.

Amount put in

Number of ways

lOp 20p etc

1 2 etc

(b) Find the number of ways for SOp. Can you see a sequence in your results? If so use the sequence to predict the number of ways of putting in 60p. Now check your prediction by listing the ways of putting in 60p. • Hopefully you now have some confidence in your sequence. Use the sequence to predict the number of ways of putting in 120p.

Dominoes to hexominoes • A domino consists of 2 squares joined along an edge. There is only one shape of domino:

OJ

• Triominoes consist of 3 squares joined along an edge. There are two different triominoes:

142

Part 5

• A tetromino consists of 4 squares joined along an edge. Draw the four possible shapes for a tetromino. • A pentomino consists of 5 squares and there are 12 possible shapes. Find them. • Hexominoes, which consist of 6 squares, present a much greater challenge! There are 35 different shapes to find and you will have to be very methodical in your approach. (a) Start by drawing all possible shapes with six squares in a line. That is easy because there is only one shape. (b) Now draw all possible shapes with five squares in a line. Here are 2 of them:

D

D

(c) Continue drawing and make sure you do not repeat shapes. Use tracing paper to check for rotations or reflections of previously drawn shapes. Good luck!

Investigating tables Addition table • Draw your own addition table like the one shown on the right.

+12345678910 2 3 4 5 6 7 8 9 10 II 3 3

4

5 6

4 4

5

6 7

5 5

6

7

6 6

7

8

7

7

8 8

8 9 10 11 I2 8 9 10 II I2 I3

7

8 9 IO II 12 13 I4 9 10 II 12 13 14 15

8 9 10 II I2 13 I4 I5 16 9 IO II 12 13 I4 15 I6 17

9 9 IO II I2 13 14 15 I6 17 18

10 10 11 12 13 I4 15 16 17 18 I9 II II 12 I3 14 15 16 17 18 I9 20

• Here is a 2 x 2 square taken from the table:

Find the sum of the four numbers: 5

+ 6 + 6 + 7.

rrrrt EEl

Repeat for other 2 x 2 squares in the table. Write down what you notice. • A 3 x 3 square contains 9 numbers. Find the sum of the 9 numbers for several 3 x 3 squares. What do you notice?

IO II 12 II I2 I3 12 13 14

143

Puzzles and Investigations 5

• Predictions Before you find the sum of the 16 numbers in this 4 try to predict the answer. Now check to see if you were right.

X

12 13 14 15

4 square

13 14 15 16 14 15 16 17 15 16 17 18

Predict the sum of the numbers in a 5 x 5 square and a 6 x 6 square. Write down a general result for the sum of the numbers in any size square. • Questions (a) The highest number in a 3 x 3 square is 19. Without looking at the addition table, fill in the other 8 numbers in the 3 x 3 square. (b) The sum of the numbers in a 4 x 4 square is 272. Find the numbers in the square. (c) Let x stand for the number in the top right corner of a 2 x 2 square. Write down the other 3 numbers in the square, in terms of x, and work out their sum. Repeat this procedure for a 3 x 3 square. What do your results show?

Multiplication table • Draw your own copy of the multiplication table shown.

X 1

2

3

4

5

6

7

8

9 10

1 2

3

4

5

6

7

8

9 10

8 10 12 14 16 18 20

2 2

4

6

3 3

6

9 12 15 18 21 24 27 30

4 4

8 12 16 20 24 28 32 36 40

5 5

10 15 20 25 30 35 40 45 50

6 6

12 18 24 30 36 42 48 54 60

7

7 14 21 28 35 42 49 56 63 70

8 8

16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100

• Totals Find the sum of the 10 numbers in the first row: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. Find the sum of the numbers in the second row, the sum of the numbers in the third row and the sum of the numbers in the fourth row. What do you notice about your answers? Use your results to predict (without adding together all 100 numbers!) the sum of all the numbers in the whole square.

144

Part 5

• Opposite corners (a) Multiply the numbers in opposite corners of any 2 x 2 square. In the square shown, 6 x 12 = 72 9 X 8 = 72. Repeat for other 2 x 2 squares. What do you notice?

:Gnl

m

(h) Multiply the numbers in opposite corners of a 3 x 3 square:

8

8 X 30 24 X 10. Repeat for other 3 x 3 squares and larger squares. What do you notice? (c) What about rectangles? 2

3

4

4

6

8 10 12 14

5

6

9 10

16 18 20 24 27 30

7

30 35 36 42 42 49

Write down a general result for squares or rectangles. • Side and corner numbers (a) In the 3 x 3 square

2

® 4

0

6

6

®

®

12

48 56

add the side numbers (circled): 3 + 4 + 9 + 8 = 24. add the corner numbers: 2 + 6 + 12 + 4 = 24.

Repeat for other 3 x 3 squares. (b) Find the sum of the side numbers (circled) and the sum of the corner numbers in some 4 x 4 squares and in some 5 x 5 squares. What do you notice?

®

12 3@ 8 12 @

8@ @ @ 116 @ 15 18 21 @

IQ.o 24 @ 6@

@ @

0 0

10 15

• Predictions Predict the relationship between the corner numbers and the side numbers in a 6 x 6 square. Test your prediction on any 6 x 6 square from the table. • Extension (more difficult) In this 3 X 3 square @ +@ = 18 2

®

4

4

6

8

@

9

@

@X 6

@+@=54 x6=54. Take a 3 x 3 square from the top three lines of the table and then other 3 x 3 squares from further down the table, moving down one line at a time. What do you notice?

Find out as much as you can.

25 30 35

@ @

24 30 1136 42 48

= 18

Here is another 3 x 3 square from the top 3 lines of the table.

®

20 24 28

8

®

10

16 18 20

@

27

®

Part 6 6.1 Speed When a cyclist moves at a constant speed of 30 metres per second, it means that he moves a distance of 30 metres in 1 second. In 2 seconds he moves 60 metres. In 3 seconds he moves 90 metres and so on. We see that the distance moved is equal to the speed multiplied by the time taken.

~iinej ... CD We obtain two other formulas from CD: Remember:

~tance =

speed

... Q)

Divide both sides by time:

Divide both sides by speed:

These three important formulas can be remembered using a triangle as shown. [D is at the top] D To find S: cover S, and you have T

D To find T: coverT, and you have-

s

To find D: cover D, and you have S x T Note: The above formulas can only be used for objects moving at a constant speed. The units used for speed, distance and time in a question must be compatible. • If the speed is in miles per hour, the distance must be in miles and the time must be in hours. • If the speed is in metres per second, the distance must be in metres and the time must be in seconds.

146

Part 6

(a) A speedway rider is travelling at a steady speed of 25 m/s. (i) How far does the rider travel in 3·2 s? (ii) How long does it take him to travel a distance of 11 m? (i) distance travelled = speed x time = 25 X 3·2 =80m . k distance (ii) tlme ta en = - - speed = ~~ = 0-44s (b) A bird flies at a speed of 8 m/s for 10 minutes. How far does it fly? Change 10 minutes into 600 seconds. distance = speed x time =8x600 The bird flies 4800 m.

(c) A train travels 15000m in 20 minutes. Find the speed of the train in km/h. Change 15 000 m into 15 km. Change 20 minutes into ! hour. . distance . -speed of tram = - tlme

=lf 3

= 45km/h

Exercise 1 1. A tram travels a distance of 200m

at a speed of 25 m/s. How long does it take? 2. A man runs at a speed of 7·5m/s. How far will he run in 4 seconds?

3. An arctic tern flies a distance of 245 km in 9 hours. how fast does it fly? 4. A steamroller takes 180 seconds to travel 60m. What is its speed, in mjs? 5. How long does it take a train to travel 270 km at a constant speed of 90 km/h? 6. A partridge flies 3 miles in 15 minutes. What is its speed in

m.p.h.?

147

Speed

7. The people in a giant wheel at a fun fair move at a speed of 26 mjs. How far do they travel in 11- minutes? 8. A cyclist takes 21- h to travel 68 km. What is her average speed? 9. An airliner flies at a speed of 570 m.p.h. How far does it fly in 40 minutes? 10. A T.G.V. travels 567 km from Bordeaux to Paris at an average

speed of 252 km/h. Find the arrival time in Paris, if it leaves Bordeaux at 1410. 11. A boat sails at a speed of 13 knots for 2 days. How far does it

travel? [1 knot= 1 nautical mile per hour]. 12. The graph shows a car's return journey from Bath to Taunton. (a) Find the car's speed between 0700 and 0745. (b) Find the highest speed of the car during the JOurney. (c) Find the average speed for the whole journey, including stops.

100

60

40 20

0700

0800

0900

1000

13. In a grand prix, the winning car passed the chequered flag 0·3 seconds ahead of the next car. Both cars were travelling at 84 m/s. What was the distance between the two cars? 14. (a) Sam drives from Liverpool to York at an

average speed of 30 m.p.h. How long will it take in hours and minutes? (b) Mike takes 2-!h to drive from Hull to Newcastle. What was his average speed? (c) Nikki drives from Preston to Newcastle at an average speed of 42 m.p.h. and the journey takes 2 h 40 min. What is the distance from Preston to Newcastle?

'b-~ 126 ~ ~CJ