Basic mathematics

basiс mathematiсs сonIenт

. make sense of maths in everyday situations . get to grips with the

essentials

. use numbers сonfident|y and aссurate|y

teaсd

yourself

basiс mathematiоs

alan graham

For over 60 Уears' more

than

50 million people have learnt over 750 subjeсts the teасfi youпself way, with impressive results.

be where you want to be with teaсh yourself

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undeвtanding the

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rcаsons to be оheedul about is this book really for

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mathematics

sсhoo|?

wаs at

sсhool?

сalсu|ator? сhild? Шe mаgiо number mасhine numbers, numbers еverywhеre saying hel|o to your сa|сulator introduсing the сounting numbers ordering numbеrs tens апd units hundrеds, thousands and beyond сhi|dren and numbers summary сomments on еxеrсises сalсulating with nшmbeв properties of numbers what arе the Тour ru|es'? knowing what sum to do doing sums on paper how d0 the four ru|es conneсt? summary аnswers to eхerсises for сhapter 08

3 4 5 6

wil| it he|p t0 use a

6

оan I he|p my

7 9 10 1 1

13 15 15 18

20 21

22 23 24 27 32 34 40 42 43

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05

frас.ti0пs

47

what is a fraсtion?

48

how t0 piсture a fraсtion

50

sсаttergraphs аnd |ine

fitting fraсtions into the number |ine

50

mis|eading grаphs

113

Whаt are equivalent fraсtions?

51

summary

116

adding and subtraсting fraсtions

52

answers to exercises for сhаpter 08

117

multip|yiпg aпd dividing fraсtions

54

summary

54

is aIgebna abstrасt and

praсtiсe exerсise

54

аlgebra as shorthand

121

answers t0 exerсises for сhapter 04

56

са|cu|ating with formu|ae

124

aпswers t0 prасtiсe exerсise

56

proving With a|gеbra

126

decimals

58

spreadsheets

129

dесima| fraсtions

59

summary

130

what is the poiпt

06

point?

104

stаtistlсаl grаphв barсhаrt and pieсharts

09

105

grщhs

109

119

шslng a tormшla

irrеlevаnt?

|

120

62

answers to еxerсises for сhapter 09

1з1

dividing fraоtions

65

puzlвs, games аnd diYeвions

134

aп overview of deсimals

66

1

number p|аte games

135

pnaоtiоe exerсise

68

2

pub сriсket

135

summary

68

136

answers t0 exerсises for сhapter 05

69

aпswers to praсtiсe exerсise

71

3 guess my number 4 thestoryof'12 5 fingеr tables

6 7 8 9

1З7 't38

of' the deсima|

guess the number

72

p0tt0пhgвs

73

whаt is a pe]centаge?

75

сhanging а frасtion to a percentаge

75

why bother with рerсеnhges? сa|сu|аtiпg perсentаge inсreases аnd

76

10

magiс squares magiс triang|e

1з6 't36

upsidе down

138

|ogiсa|ly speaking

138 't39

10 сalсu|ator invеrsions 11 four4s

141

78

12 the bells, the bells!

141

persistent probIems with perоenhges

82

13

return joumey

141

praсtiсe eхerсise

84

14 fiпd the numbers

'141

аnswers to exerсises for сhapter 06

85

.|5

answers to praсtiсe exerсise

86

mвasшdng

88

what do we measure?

89

why do we measure?

90

how do we meаsure?

91

how aссurate|y should we measure?

93

reduсtions

07

08

еxplore and exp|ain the pattem

summary

101

16 |arge аnd smal| sums .t089 17 аnd a]lthat 18 go|d pieсes 19 initia|ly spеаking 20 nim 21 guessthe number 22 са|сu|аtor snookвr 23 plасе invadeв

answers to exerоisеs for chapter 07

102

answers for сhapter 10

imperia| and mstriс

unib

95

141

142 142 143 143

144 144 145 145 147

11

dlagпostiо qшE

1t2

quiz

154

solutions to

pail two

fie

quiz

tij

157

detаilеd сomments on the so|utioпs

160

appвnd!сш. mathematiв in

169

aс.tioп

сalсu|atiпg а best buy

1т1

reading the 24-hour сloсk

174

bus aпd railwаy timehbles

178

сheсking the supermarket bi|l

182

uпderstanding a shop rесeipt

185

оheсking the VAт

187

оooking with figures

189

buyiпg a TV set

194

willitfit?

196

measures of alсohol

198

understaпdiпg barcodes

202

lunk mаiland frеe offeв

205

winning on the Nationа| Lottоry

207

sаfe trаvel

213

world population

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214

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211

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o Мany thanks to.lVеndy Austеn, Carriе Graham, Jamеs Griffin and Sally Kеnny for their hеlp in thе prеparation of this book.

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ln this chapter you will learn:

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.

whyyou can suсceed at

learning maths, even if you failed before why a calculator will help how to develop a positive attitude to maths.

ls this book really for me?

I wondеr what madе you dесidе to buy this book! No doubt eaсh individual hаs his or hеr own spесial rеasoп. As author, I сlearly сant mееt evеryonе's еxaсt neеds, so I triеd to writе this book with two pаrtiсular typеs of rеadеr in mind. I will сall thеm Marti аnd Меl.

undеrstanding of these things, you сould аlways skim-reаd these сhaptеrs, if only to boost your сonfidеnсe.

Marti, you havе an intеrest in helping somеonе elsе, say a сhild, thеn read thеsе three сhaptеrs rмith a tеaсher's hat on.

Thеy should provide you urith some ideas about how you сan help somеonе elsе to grasp thеse ideas of basiс arithmеtiс.

16 likе

MаЁi Iv1ar.tr

Chaptеrs 01 to 03, rмhiсh deal with basiс idеas of what a numbеr is (hundrеds' tеns and units), how to add, subtraс, multiply and divide and how to makе a start with a simplе four. funсtion сalсulator. Hourеver, evеn if you do havе a bаsiс

wаs nеithеr pariсularly good nor partiсulаrly Ьad at

mathеmаtiсs, Ьut was ablе to gеt by. Shе сouid usually work оut sеt by hеr tеaсher at sсhool, but nеvеr really undеrstоod Щ

why'uц' th9.y worked. Shе had a \Iagoe sensе thai there

lмas,

can I suссeed now if l fai|ed at school?

еxplorеd Ьut somеhow it nеvеr happenеd for hеr. Shе now has two сhildren agеd 10 and 6. Shе is awarе that hеr own attitudеs to

Thеre arе a numbеr of rеasons that sсhool mathematiсs may have bееn dull and hard to grasp. Hеrе are-some resourсеs that you may have now whiсh wеren't availablе to you when you

potentially, an еxсiting mеntal world of mathеmatiсal idеas to Ье

mathеmаtiсs arе being passеd on all thе time to hеr сhildren and shе rмants to bе aЬlе to enсouragе and help thеm morе еffeсivеly.

Thе сrunсh for Мarti сamе rмhеn her dauфеr askеd hеr about the differеnсе Ьеtwееn odd and еvеn numbers. Marti knеrм whiсh numЬеrs wеrе odd and whiсh wеrе еvеn' Ьut shе сouldn't rеally explain why. She miф hаvе bouф this book to undеrstand somЬ of thе bаsiс .whys' in mаthematiсs,

Mel Меl nеvеr got on with mathеmatiсs in sсhool. Hе lost сonfidеnсе rмith it at an.ear|У stagе and сonstantly had thе fееling of .if only thе tеaсher and thе othеr pupils knеw how littlе I knoщ thеy'd ЬЬ

shoсkеd'. As the years Wеnt on, hе lеarned to сovеr up his problеms and as a result hе always had a bad fееling whе.nеvеr mathеmatiсs сropped up * a сomЬination of fear of bеing сauф 9yt цd guilt at not propеrly faсing up to it. Hе has a good job Бut his fear оf mathеmatiсs rеgular1y сausеs him problеms. He miф have Ьouф this book in ordеr to lay to rёst thе ghost of Iris mathematiсal failurе onсе and for аll.

Whilе your namе is unlikеly to be Меl or Marti, maybe therе is an aspесt of thеir ехpеrienсеs of, and hopеs for, mathematiсs lеarning that yоu сan identify with.

Should l start at Chapter through the book?

0l аnd read right

Thе honеst answеr to this quеstion is, .It all dеpends

mathеmatiсs

is

rеasonaЬly sound, you might

. . .'. If

likе to

your Ъkip

were 15.

ReIevanсe Therе ц'еrе undouЬtedly many morе important things going on in most peoplе's lives at the age of 8, 12 or 15 than lеarning about adding fraсtions or solving equations. Mathematiсs just doesnt seеm to bе partiсularly important or rеlevant at sсhool.

But аs you get oldец you are Ьettеr ablе to аppreсiatе somе of the praсtiсal appliсations of mathеmatiсs and how thе various mаthеmatiсal ideas relate to the world of home and work. Еven a willingnеss to еntertain abstraсt idеas for thеir сuriosity value alonе may sеem more attraсtive out of a sсhool сontехt. one studеnt сomments:

Yеs, I think you сan assoсiatе more with it. !Иhen you 8еt to our agе, when you've a family and a homе, I think you

сan assoсiatе more

if you do put it to morе

things. You сan see it bеtter in your mind's eye.

praсtiсal

Gonfidence fu an adult lеarneц you сan takе a maturе аpproaсh to your study o{ mathеmatiсs and be honest in admitting whеn you don't understand. As onе adult lеarner said aftеr an unехpесedly exсiting and suссessful mathеmatiсs lesson сonduсеd in a small informal sеtting:

I fёlt I сould ask thе sort of question that I wouldnt havе

derеd ask at sсhooL like .!0hat is a dесimal?'

Мotivation A third faсtor in your favour now is motivation. Childrеn

аrе

required to attеnd sсhool and to tlrrn up to thеir mathеmatiсs lessons whеther thеy want to or not. In сontrast, you havе madе a сonsсious сhoiсe to study this book, and thе diffеrеnсе in motivation is сruсial. Perhaps you have сhosеn to read the book bесausе you neеd a bеttеr grasp of mаthеmatiсs in ordеr tо Ье more еffесtivе аt work. Or mayЬе you arе a rnathеphobiс parеnt who wants a bettеr outlook for your own сhild. or it is possiblе

thаt you havе аlways rеgrettеd that your mathеmatiсal

undеrstаnding got lost somеrлrhеrе, and it is simply timе to lay that ghost to rest.

Maturity аnd experience You arе a YerУ diffеrеnt pеrson to when you wеrе 8 or 15 years old. You now havе a riсhеr voсabulary and a widе еxpеrienсe of lifе, Ьoth of чrhiсh will hеlp you to grasp сonсеpts you nеvеr undеrstood Ьеforе.

Hаsn't mathemаtiсs сhаnged sinсe l was аt sсhoo]? Мathеmatiс s has changed a Ьit sinсе you wеre at sсhool but not by as muсh as you think. The сhanges havе oссurrеd more in the languagе of mаthеmаtiсs than in thе topiсs сovетеd. Basiс arithmеtiс is still сеntral to primary mathеmatiсs and today's 12.yеar-olds arе still having the same sort of proЬlems rмith dесimals and fraсtions as you did.

Will it help to use a сalсulator?

.four funсtion' сalсulator (i.e. one expеnsivе) than a simplе whiсh doеs the four funсions of add, subtraсt, multiply and dividе).

Thеre arе а numbеr of rеasons why this Ьook has Ьеen written assuming a сalсulator is to hand and somе of thesе are sрlt out in morе dеtail at thе start of the next сhaptеr. But two rеasons to think about herе are: Firstly, thе сalсulator is more' than simply a сalсulating dеviсe. As you rмork through this Ьook, you тvill sее how the сalсulator сan bе usеd as a mеans of leaгning and еxploring mathеmatiсs. As onе student сommеntеd: .

That чras somеthing that I еnioyеd; knouring that you сan divide and gеt larger numbеrs. I mean, that was a thing that, without thе сalсulatoц I would have been, . . . would never havе bеlievеd, but bесausе it was instant, it rrras something I сould sее straight arлray.

Sесondly, being сompеtеnt with mathеmatiсs isn't iust some aЬstraсt skill that dividеs thosе urhо сan from those who сan't. Мathеmatiсal skills should Ье usеful and insightful to you in your rеal тrorld and in the rеal world that most of us inhabit.

This сhaptеr ends with a sесtion on сhildrеn's mathematiсs. This thеmе will reсur throuфout the book and I hoр that, еvеn if you arе not yourself a parent' you йll find it interеsting. My exрrienсе of working with adults has bееn that lookiпg at

сhildrеn's еrrors and еxpеrienсеs in mathematiсs сan bе a .way in' for thе adults, giving a bettеr undеrstanding fasсinating of thе kеy idеas of mathеmatiсs.

can l help my сhild?

Мany рarents who werе unsuссessful at

mathematiсs

саr. onсе you'vе lеarnt whiсh kеy to go to' you do it and that's it. It's just a сasе of lеarning. No, it didn't frightсn

thеmsеlvеs havе the dеpтеssing eхperienсe of watсhing their сhildren follow in mother's or fathеi's footsteps. Being hopеlеss at mathеmatiсs sеems to run in families: or does it? T0hilе it сеrtainly seems likсly that сonfidenсe in mathernatiсs Passеs .mathematiсal ability' from parent to сhild, it is less сеrtain that is inheritеd in this way. So if you сonsidег yourself to be a duffеr at mathematiсs, thеre is no rеason to сondеmn your сhild to the samе fate. Indeed, thеrе is muсh that you сan do to build your сhild's сonfidenсе and stimulate his or hеr intеrest in the suЬiес.

You will nеed to gеt a саlсulator Ьеforе rеading thе following сhaptеrs. You won't nееd anрhing morе sophistiсatеd (or morе

might be in the kitсhеn, in a suрrmarkеt of on a long journey.

Yes. Мost adults have a сalсulator, but pеrhаps many may rarеly usе thеm. This Ьook should hеlp you ovеrсome аny anхiеty you may fееl. As onе studеnt еxplains: ItЪ only a thing with buшons, isnt it. All right, it takеs you a while to know what eaсh Ьutton is, Ьut it's likе driving a

mе.

Try to stimulatе your сhild's thinking and сuritrsity aЬout mathematiсal idеas. The idеal setting for thеse сonvеrsations

How, thеn, сan parеnts еnсouragе in thеir сhildrеn a сuriosity аnd ехсitemеnt about mathematiсal ideas? Below are a few

g.еnеral pointеrs to indiсаtе thе sort of things you might say and

do with your сhild to aсhiеvе this aim by сrеating what wе profеssionals hеlpfully сall a .mathеmatiсally s"timulating environmеnt'.

.

o

о

.

.!7hеre

possiЬlе try to rеspond to yoЕr сhild's quеstions positivеly. Еnсouragе the dеsirе to have a go at all answеr' еvеn if thе answer is wrong. wrong answеrs should bе оpportunitiеs for learning, not oссasions for punishment or huЫliation.

Try to think about what makеs сhildrеn tiсk and thе sort of things thеy might Ье сurious aЬout.

Rеsist providing еasy adult answers to your сhild's quеstions Ьut rathеr try to draw fuгther questions and theoriеs from

hеr/Ъim.

If' a сhild givеs а wrong answеr it is probably for a good rеason. Try to disсover thе сausе of the problеm - it сould Ьe, for еxample,.that your quеstion was noi сlеarly еxpressеd or that the сhild is simply not rеady for thе сonсеpt.

Thеrе may bе spесifiс еduсational toys and apparatus whiсh аrе hеlpful to havе around. Having said that, a four-yеar-old will learn to сorrnt as effeсtively or inеffесtivеly with pеЬblеs

or bottlе tops as with .propеr'сounting briсks! .horм' and .why' questions are at a |!n-allg -rеmеmbеr that highеr lеvеl of сuriosity than .rмhat' quёstions аnd thеsе

should bе еnсouragеd. (Fоr еxamрle, ,шhу

doеs

2 x 3 = 3 x 2?, is a morе stimulating quеstion than .what is 2 x 3?')

The biggеst barriеr tо learning mathеmatiсS js feаri thе fеаr of Ьeing shown up as not uпdеrstanding somеthing whiсh sееms to bе. patently obyious to thе rest of humanity. Thе bеst way of hеlping your сhild is to start Ьy trying to ovеrсomе that fеir in y-оurself. It doеsn't mattеr if you dont know a simple fraсtion from a сompound fraсturе; your сompetеnсе аt maфеmatiсs is lеss important than your willingness to Ье honеst аbout rлrhat you dont undеrstand yoursеlf.

Just- as with сooking, сarpentry or playing the piano, lеarning mathеmatiсs doеsn't happеn |ust by rеading a Ъook aЬout it.

Although explanations сan lеad to undеrstanding, praсtiсе is nесеssary if you arе to aсhieve mastery.

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o In this chapter You will loarn:

с how to say .hello' to your

.

calculator

about numbelв and how they arэ ]эprэsonted о how сhildron first find out about numbers.

Numbers, numbers everytrrhere

Еvеrywhеrе you look, numЬеrs sееm to lеap out. Thеy liе hidden in rесipе books, arе stampеd onto соins and printеd onto stamps, flash up on supеrmarket сhесk-outs, providе us with brеakfast rеading on сornflakes paсkеts, arе displayеd on busеs, spin round on pеtrol pump dials . . . . It sееms that' urhatever task wе want to pеrform, numbеrs havе some rolе to play. Hеrе are few morе examples. о A farmеr will сhесk that all thе сows arе in by couпting them as thеy go through thе gatе. . At сhurсh, the viсar rеads out thе hymn numbеr. МеmЬеrs of thе сongrеgation arе ablе to find thе hymn in their hymn

Ьooks only

о

о

sеquenсe.

if thеy know how

numbers аre orgапizеd in

Thе rесipе book says .pour into a 9-inсh squarе tin аnd Ьakе at 180"с for 45-50 minutеs'. For this to makе sеnsе, wе nееd to know aЬout meаsurement of sizе in inсhеs, tеmperaturе in dеgreеs Cеlsius and timе in minutеs.

Compеtitive sport is tоtally based on numbers, usually in thе form of scoriпg - half-timе sсores' numbеr оf runs, highest break, winning times . . . .

You сan probaЬly think of lots morе еxamplеs of your own. It's hard to imaginе a rмorld .Whatwith no numbеrs. Horм would wе survivе without thеm? altеrnativе Ways might wе think up to organize human aсtiviф if wе сouldn't сount on numbеrs to сontrol thеm? Еxеrсise 2.1' is dеsigned to hеlp you to be morе awarе of thе rolе that numbеrs play in your lifе.

ExERсlsЕ 2.1 How long саn you last without numbers?

Take a waking pеriod оf, sаy, 30 minutеs of your lifе and sее how mаny situations requirе some usе and understanding of numbеrs. Then rеad on.

I triеd

the.!Иithin еxеrсise whilе writing this сhаpter on my just а fеw sесonds I found that I was сomputеr. adjusting thе linе шidth of the tехt to еxaсtly sixtеen and a .

half сеntimеtrеs.

Minutеs latеr thе 'phonе rang and I was aцrarе that I statеd mу tеlephone numbеr on piсking uP the rесеivеr.

The 'phone сall was to fix a meеting, so I had to сonfirm dаte ind time in my diary.

Shortly afterwards I цrent to makе a сup of сoffeе (I dont havе a very long attеntion span!). This involvеd putting milk and йatеr into a сup' plaсing it into thе miсrowavе and keying 222 (2 minuteЪ and22 sесonds) into tЬe titner.

Numbеrs are the building Ьriсks of mathеmatiсs. So, just.likе Ьеins aЬlе to rеad, therJarе сеrtain basiс mathematiсal skills that"you nееd iп oid.' to livе a normal lifе. For еxample, bеing ablе io: . rеad numЬers and сount

. tеll thе timе о handlе monеy whеn shopping . wеiф and measurе . undЪrstand timеtablеs and simplе graphs.

But as wеll as having a usеful, praсtiсal side, solving problems with mathеmatiсs сai also be сБallenging and fun. Anyone who

has donе drеssmaking or саrpеntгy knows that mathematiсs сan bе usеd in two ways.-onе is_in thе praсtiсal sеnsе of mеasuring

and using paше'nj and diagrams. The other is more abstraсt

.How сй. I сut out my pattern so as to use uP thе

-

lеast

.Can I usе thе symmеtry of thе garment to makе fwo matеrial?,, сuttings in onе?' Thе plеasurе you gеt from solving thеsе sorts of prJblеms is what Ьas kept matЬеmatiсians going for fouг thousand years! Мost of all, mathеmatiсs is a powerful tool for exprеssing and ideas. Sadly, too fеw pеoplе evеr gеt a sensе of .pourеr to ёxplain'that mаthеmatiсs offers. "o.-o''i.ating the Lеt us rеturn now to thеsе building bгiсks, thе numbers, and see what sеnsе your сalсulator makеs of thеm.

Saying hello to your сalсulator

Most basiс сalсulators look somеthing like this (sее oveг).

Еxamination boards have largely dеsфed thеir ехaminations on thе assumption that all сandidatеs havе aссеss to a

disрlay screen

саlсulator. thе

.off'key

the number pad

the 'on' and

.с|еar'key

OFF

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7

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4

5

6

1

2

3

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'multiply'key

.subtraсt'key

he'add'key

+

the

.еqua|s'key

Lеt us begin at the beginning; with thе сounting numbers 1, 2,

3,4,5, . . .

No doubt thеy look familiar еnough. But bеforе you rеaсh for pеnсil and papеr' lеt's sее how thеy look on your сalсulator.

E;XERCISE 2.2 Entering numbers Switсh on your сalсulator and key in the following sequеnсе

123456789

the numbеrs arе displayеd. .deсimal point' the

Bеforе wе taсklе thе hard stuft you should start by saying hеllo to your сalсulator. Usual еtiquеttе is as follows. о Рrеss thе .on'switсh (probaЬly markеd lЪшl or |бr'vсI) о Now kеy in thе numЬer 0.7734 о Turn your сalсulator upsidе down аnd read its rеsponse on thе sсrеen.

Good! Having now еxсhangеd .hЕLlos', this is сlеarly thе Ьasis for a good working relationship!

As was ехplainеd in thе prеvious сhaptеr, this Ьоok aims to tеaсh you Ьasiс mathеmatiсs tllith а саlсulаtor. Thеrе arе a numЬеr of good rеasons for prеsеnting thе bоok in this way, some of whiсh arе listеd bеlow.

с

lntroducing the сounting numbers

Look сarеfully at thе sсrеen.

Vritе doтrn what is rесorded and mаkе a notе of еxaсtly how

o

zero

Мost importantly' thе сalсulator providеs a powеrful aid to lеarning mathеmatiсs. Sеe if you are morе сonvinсed about this by thе timе you have finishеd this book.

the'divide'key

For somе typеs of problеm, сalсulators takе thе slog оut of thе arithmеtiс and allow you to foсus youf attеntion on undеrstanding whаt thе problem is about.

l

If you look сarеfully at еaсh number, you should sее that it is made up of a sеriеs of littlе dashеs. For еxamplе, thе thrее rеquirеs fivе dashеs and is writtеn as follows.

You might likе to сonsider in thе neхt еxеrсisе whiсh of thе numbers from 0 to 9 rеquirеs thе fеwest and whiсh the most dashеs to be displayed.

Еl(ЕRclsE 2.3 Displaying numbens

a

Cheсk how thе other numЬеrs are rеpresеnted and сount the numbеr of dashеs eaсh number nеeds. Thеn fill your answer into thе taЬle Ьеlow (one, thе 3, hаs already beеn donе for you).

NUМBЕR0 1 2 3 4 5 6 7 8

DAsHЕs

5

9

Ь

if you сan skеtсh any nеw соmЬinations of dashes that are not a|readу сovеrеd by thе numЬеrs. If you wеrе a сalсulator dеsignеr, сould you turn thеsе into anything Sеe

usеful?

Your taЬle should now enablе you to answer thе quеstion posеd thе fеwеst and earliеr, namely to spot whiсh numЬеr rеquirеs .!7as your еarliеr guеss vrhiсh thе most dashеs tо Ье displayed.

Ordering numbers Мost сalсulatofs сan Ье sеt up to produсе numbеr sеquenсes.

Hеre is a simplе onе to try.

Prеss

ГТl

1

Now prеss Г=l repеatеdly.

сorrесt? In faсt, the numЬеr 1 usеs lеast dashеs (two), whilе thе 8 usеs most (all sevеn).

You should

Thеrе are a few сomЬinations not сoverеd bу the numbеrs. For

Еithеr: Press Or: Prеss

еxample:

seе thе

сounting nlrmbеrs in sеquеnсе: 2, з,4, 5,6, . .

.

If this has not happеned, try thе follоwing.

1

Ггl Г+l

1-

Г+l 1 and thеn thе Г=l rеpеatеdly.

and thеn thе Г=] rеpeatedly.

!Иhat is gоing оn hеrе is that your сalсulator is doing a сonstant .add to perform .сonstant'

1'

Е

сalсulation. Thе сaрability

сalсulations like this is availаЬlе on most сalсulators in somе form. It is an еxtrеmеly useful featgte and еxaсtly how and when it сan bе usеd will Ье exрlainеd later.

Tens and units

and

L

Г

onе possiЬle appliсation of thesе shapеs is to spеll lеttеrs. For еxample, thе .Е' above сould bе usеd to rеfеr to oЕrrоr' if an impossiblе kеy sеquеnсе was prеssеd. Thе .C'сould pеrhaps be usеd to mеan .Constant opеrating' or mayЬe .Сareful, you'rе hitting my keys too hard!'

Lеt's have anothеr look at thе sеquеnсe of сounting numЬеrs.

Repeat thе instruсtion aЬove, using thе сonstant to add 1. As Ьеforе, rеpеatedly prеss Г=l until thе сalсulаtor displays 9. Thе display should look likе this.

rтl

Pausе for a momеnt and thеn press thе You should now sеe thе following.

ГEl

onсe morе.

Г_iб.|

Most pеoplе will reсognizе that this is simply a tеn, but notе

that it is quite diffеrеnt from thе prеvious numЬеrs. Foсus on thе faсt that thеre are now not one but two figurеs displayed hеrе. .!Иhat has happеnеd in thе movе frоm ninе to tеn is that thе ninе has сhangеd to a zero and a nеw figure, the 1, has appеarеd to thе lеft of thе zеro, thus:

a new figure, the 1, has appearеd in this сolumn

thе nin€ has сhanged to a zero

It is worth rеflесing on the faсt that numbers do not neеd to do

this. As you saw from Calсulatoг Еxerсise |.2, t|le сalсulator hаs а fеw more squiggles up its sleeve rr,hiсh -iфt b" used for eхtra numbеrs. For examplе, the numbеrs сould bе еxtеndеd to look likе this:

z0ro one

two

thrее four

r:l_l tL_l

seven

Г| I

eight

П

l_rll

l*l I

ninв

l-l

r_l

fivo

ten

eleven

twelve

l_

LI

l*l

I

tl

six

l:

l_l

So thе сolumns rеprrsent the tеns, and the onеs are thе units.

Thеrе arе thеrеfore twеnty thrее сoins herе: two tens and three

units.

If you fееl that you nееd praсtiсе at dеaling with tеns and units for slightly largеr numbеrs, havе a go now at Ехerсisе 2.4.

ЕxЕRсlsE 2.4 Praсtising with tens and units Switсh your сalсulator on and onсe аgain sеt up thе сonstant .add 1' by prеssing thе sеquenсе:

tЕ 1 tI

сalсulator.)

iJ

But, as it happеns, our numbеr systеm doesn't look likе this.

Probably for thе reason that most humans wеre born with tеn

r

(or by anоthеr method' depеnding on your partiсular

Now entеr the number 37 and prеss thе Г=l k.y four timеs.

Thе result should bе 41. [n other lмords,

37

fingers and tеn toеs' wе havе сomе to an agreemеnt that only ten uniquе сharaсtеrs arе nееdеd for сounting. Thеse ate0,L,2,3, 41 5r617r8 and,9.

..

.

сount on four morе

Thеsе tеn сharaсtеrs arе сallеd the numerаIs. Thе word numеral

rеfers to hoтr wе пrrite numbеrs, rather than bеing сonсеrnеd with hoчr many things thе numbеrs represent.

Aftеr сounting as far as 9, we simply $ouP numЬеrs in tens, and сount how many tens and how many lеft over. For еxamplе, herе is a sсattering of just somе of the onе pound сoins that I happеned to find urhen tidying uP my daughtеr's money bох.

Notе that prеssing any key at this point other than the numbеr kеys and Г=l is likеly to dеstroy thе соnstant setting. If, for ехаmplе, you havе alrеady prеssеd tlrе .сlеer' kе5 proЬably markеd Гб-.l or М, you impеtuous person, you will nееd to rеsеt the сonstant Ьy rе-kеying til 1 Е or an equivalеnt key sеquеnсе.

Onсе again, at thе risk of Ьeing Ьoring, you arе rеmindеd not to сlеar the sсrееn aftеr eaсh sеquenсе' as this will.dеstroy thе сonstant sЭtting. Епter Press

In ordеr to сount them, I

ЕA

miф

44 group them in tens' as follows.

**Ф

59 IJ

66

EEE ЕEtat=Е Etзt=t=EEEЕ l=ll=ll=ll=l

Ехpeаеd, E"эсpeаеd,

tens 4

6 8

7

шtlits Ansцleт

This idea of grouping in tens is thе very basis of сounting up as far as 99. Thе next seсtion goes bеyond tens аnd units, into thе world of hundrеds and thousands.

Prеss ГТl 100 аnd thеn thе Г=I rеpеatеdly.

Or: Press 100 ГТl ГТl and then thе Г=l repeatеdly. or: Press 100 Гтl 1'00 and thеn thе Г=1 repеatedly.

Hundreds, thousands and beyond

As you do this exerсisе, try to devеlop a sensе of what to еxpесt at thе nехt press of ГТl, partiсularly whеn thе numbеr in the hundrеds соlumn is a 9.

E]xЕRGlsE 2.5 Handling hundreds Srмitсh your сalсulator on and onсe again sеt up thе сonstant .add 1' by a method suitaЬle for your partiсular maсhinе.

To еnd this seсtiоn, it is usеful to bе ablе to say numbеrs as well as to writе and intеrpret thеm.

Now еntеr thе numЬer 96 and press the

Sеt your сalсulator сonstant t,o add 1'423.

r

k"y four timеs.

Thе result should bе 100. [n othеr тrords,

9

6

сount on four morе

ЕxERclsE 2.6 Saying numbers out loud This is donе by prеssing

B

...

100

14231.

гт1 742з (or 1423

tE t!

or 1423

Е k.ь try to say out loud thе numbеr you sее on thе sсrееn. Inсidentаlly, it hеlps if yоu say the numbеrs vЕRY LoI.]DLY INDЕЕD. Don't worry if оthеr mеmbеrs of Now, eaсh timе you prеss thе

your housеhold or your dog think you arе сrossing thе final frontier. This is normal Ьеhaviоur for mathеmatiсal geniusеs.

.!Иhеn

You should not рrеss any kеy othеr than thе number kеys and |E. As bеfore, if you have dеstroyed thе сonstаnt' you will nееd to rеsеt it by re-keying Е 1 tа or аn equivalent kеy sеquenсе.

Rеmеmbеr also not to сlеar thе sсrеen after eaсh sequеnсr.

Press Eхpeсted Ехpected Ехpeсted hшndrеds tens units Aпsшеr 1,98 Еt=t= 2 0 1 201 399ГаEЕ]вE 4 4 404 0 696 Еt=Elt=EЕ7 0 2 702 897 r=1sixtееntimеs 9 7 3 913

Eпter

Thе largest numbеr you сan produсе with thгeе figurеs is ninе

hundred and ninеty-ninе. If you want to gеt any larger, you nеed to rеgroup and сrеatе a new сatеgory of .tеn hundrеds', whiсh чre сall onе thousand.

Let's еxplore thousands by sешing the сalсulator to сount up in muсh larger steps; this time in intervals one hundred at a time.

You may bе aЬlе to work out for yoursеlf how to do this. If not, try one of thе following sеquеnсes.

you have сompletеd this ехеrсisе, you сan сhесk your

answеrs rмith thosе givеn at the еnd of thе сhaptеr.

Thе number systеm rеprеsents a сеntral themе in thе rмork of tеaсhеrs of сhildren in thе еarly yеars of sсhooling. You may havе сhildrеn of your own or may Ьe intеrеstеd in how thеsе idеas arе taсklеd with young сhildrеn. Thе final sесtion of this сhaptеr invitеs you into thе infant сlassroom to sее what things go on and what sorts of notions and diffiсulties young сhildrеn havе with number.

D(ERCISE 2.7 Ordering your figures

Using thе figurеs 3, g,4 and 6 onсe only, writе dorмn:

a Ь

thе largest possiblе four figurе numbеr thе smallеst possiblе four figure wholе numbеr.

Bеyond thousаnds and tеns. of thousands liе two additional words that you should knorлr: millions and Ьillions.

A million is a thousand thousand or 1 000 000.

A billion is a thousand million or 1 000 000 000.

Inсidеntally, notiсе how, withлery |arge numЬеrs likе thеsе, thе digits are oftеn groupеd in thrееs simply to make them easier to rеad.

To еnd this seсtion, thе tеrm plаce ualue neеds to bе еxplainеd.

Plaсе valuе is rеally anоthеr way of desсribing thе wholе idеa of hundrеds, tеns and units. It rеfеrs to thе kеy idеa of how our number system works, namеly that the .plaсе, of a digit (in othеr words, its position in thе number) is what dеtеrminеs its value.

For example, thе two.digit numbеr 37 is written аs 3, followеd on thе right Ьy 7. By сonvention' wе agreе that the first of thеse digits, thе threе, rеfеrs to thrее tеns whilе thе sеvеn rеfers to

sеvеn units. To undеrstand this point prinсiplе of .plaсе valuе,.

is to undеrstand

the

Children and numbers For most сhildren, numbеr words first сomе into their world

through songs (.Fivе little spесklеd frogs sitting on a log,, .Onе, two, threе, four, fivе, onсе I сaught a fish alivе', and so on). A fеaturе of many suсh songs is that фе numbers arе sung in sequеnсе - sometimеs thе numЬers go up and somеtimes thеy go dorмn. This property of thе \ivay that numЬеrs follow on from еaсh othеr in sequеnсе "l.,2,3,4, 5... is the ordinаl property of numbеrs (ordinal as in thе .order').

TЬe nцtпber line shown bеlow is a helpful way of еnabling сhildrеn to form a mеntal piсturе of thе sеquеnсe of numЬers.

Howеver, therе is quite a stеp for сhildrеn from understanding numbers in a numbеr linе, to bесoming aware of what thе thrеe. ness of threе rеally mеans in tеrms of thе number of oЬjeФs - 3 toys, 3 Ьгiсks, 3 swееts, 3 сalсulators, and so on. [t is a majoг

Ьгеakthrough for a сhild when shе lеarns that .thrее' is a dеsсription that сan Ье appliеd to a цrholе variety of diffеrеnt сollесions (or sеts) of things. Shе will nееd many oссasions in whiсh she physiсally grasps thrеe objeсts beforе thrее-nеss bесomеs a сonсеPt that shе сan grasp mentally. This idea of a numЬer desсriЬing houl manу thiпgs thеre are is known as thе саrdinаl property of numbеrs. Мany sсhool aсtivitiеs in thе early years deal with this сonсеpt Ьy mеans of play involving сounting out various objeсts and so on.

-

briсks, bottlе tops, matсh boxes,

Finally, iust in сasе you thought that hеlping сhildrеn to

undегstand numЬer Was a straightforward еxеrсisе, try to piсk your vlay through the following exсhange Ьеtwееn a four-yеarold and his tеaсher, rесordеd in thе Ьook Vаllу's Stories Ьу

Vivian Palеy.

.Wе havе threе 12s in this room', .!Иally said onе day. .A round 12, a |ong |2, and a short 1'2. TЬe round 12 is the boss of thе сloсk, thе long 12 is thе rulеr, and thе short 12 is on a сalеndar.' .!Иhy is tЬe L2 on thе сalеndar a short Lz?,,I askеd.

.Mе and Еddiе mеasurеd it. [tЪ rеally a fivе. It сomеs out

five on thе rulеr.'

.Right. It's fivе.'\Раlly starеd thoughtfully at thе сloсk. .I'm likе the Ьoss of Marсh Ьесausе my birthday is Мarсh \2.The 12 is on thе top of the сloсk.'

Summary In this сhaptеr you rlrrrе introduсеd to your сalсulаtor and shown how to usе its сonstant faсility. You wеre thеn shown how to usе thе саlculаtor сonstant to сount in onеs оr in any

interval, and you Wеrе thеn askеd to сount your way through our wholе number systеm' Ьasеd on tens, hundrеds and thousаnds. By сlоsely еxamining how numbеrs arе rеprеsented on thе сalсulator displag we lookеd at the nutl'еrаls, i.e. how numbеrs arе wriffеn. But numЬеrs have other featurеs worth exploring. Firstly, they fоrm a natural sеquenсе. This is known as thеir ordinal propеrry and is niсely rеpresentеd on a number .how Iine. Theу arе also important as a way of dеsсriЬing many', whiсh is thеir саrdiпаl pfopеrty.

Gomments on exercises Еxerсises 2.| to 2.5 2.6 written on the sсreeп

no additional сommеnts Saying numbеrs out loud Sаid vr,nу LIUDLY INDЕЕD

1423 onе thousand four hundrеd and twentythrеe

2846 two thousаnd еight hundrеd and forty.

slx

4269 four thousand two hundrеd and sixtyninе

5692 fivе thousand six hundrеd and ninеtvtwo

71|5

sеvеn thousand one hundred and fiftееn

1|384

elеvеn thousand thrее hundred aпd eiфty-

8538

eight thousand fivе hundred and thirty. еight 996,l' nine thousand nine hundrеd aпd sixty. onе

four

12807 trлrelvе thousand еight hundred and sеven

the laгgеst possiblе four fфrе urholе numbеr is 9643 b thе smallеst possiblе four figure wholе number is 3469.

2.7 a

o g s) o o s) 1+ a э

:

з

сг

Е :

-

Пl

э

GT \

t fl

Цl

J

ln this chapter you wil! learn: . some important number words - like prime, square, odd and even r what сa|culations you need in different situations о about the .four ru]es': add, subtraсt' multiply and divide r how negative (i.9. minus) numbers fit in.

Nof* ln this and all subsеquеnt сhaptеrs, сomments on exеrсisеs arе given at thе еnd of the сhapter.

Properties of numbers Bеforе taсkling the sorts of сalсulation that we normally do with

numbers, it is worth looking at some of thе moге useful propеrties of numbеrs. You will find out what is mеant Ьу odd, euen, prime, reсtangulаr and squаre numbers. Thеse tеrms are bеst undеrstood by sееing thе numЬers arrangеd in pattеrns on thе taЬlе in front of you. So, Ьefore reading on' try to get hold of tеn or twеlvе small idеntiсal objeсts (сoins, Ьuffons, or papеr

сlips, for examplе).

Еven or odd

ooooО

Choosе a sеlесtion of your objeсts (any numbеr Ьetwееn 1 and 72 will do) and try to arrange them into two rows, like this:

ooooo

If' like me' you сhosе a numbеr of сoins that produсed two

еqual rows, thеn your selесtion was an еvеn numЬer of сoins. In this сase, сhoosing tеn сoins prоduсed two еvеn rows of fivе еaсh. So ten is an evеn number.

Howеvеr, perhaps your selесtion didn't work out likе this and, whеn thе two еqual rows wеrе formеd, thеrе was an odd one ovеr _ likе this:

ooooo ooooo

Whеn any sеlесtion of things that are laid out into two rоws produсе an odd onе ovеr' then thеrе must havе Ьееn an odd numЬеr of thеm. In this сasе, сhoosing еlevеn сoins produсеd

two еvеn rows of fivе eaсh plus an odd onе ovеr. So еlеvеn is an odd numЬеr. Еxеrсisе 3.1 will give you Prасtiсе at dесiding whethеr a numbеr is evеn or odd.

ЕxЕRсlsE 3.1 Even or odd?

For eасh numbеr, mark thе box сorrеsponding to whеthеr it is еvеn or odd. Thе first onе has bееn donе for you.

Numbеr

7

2L286310

Еvеn П П t] П t] П odd El П ППППППП 11

f] Пt]

Prime, reсtangular and square Now сhoosе siх of the сoins. Notiсе that thеy сan bе arranged in a rесtangle as two rows of thrее, like this:

ooo r-\r-)r-) \_,/ \_/

\-./

or as thrее rows of two. likе this:

r-\^ \_/ \-/ tltt

\_/

\_-/

ltt,

\_-/ \_-/

Еithеr way, six is a reсtangulаr number bесausе it caп Ьe arrangеd in thе form of a rесtanglе. Similarly, the number 18 is rесtangular beсаusе 18 сoins сould bе arrangеd likе this.

oooooo oooooo oooooo Now try thе samе task with sеvеn сoins. You will soon find that this is impossiЬlе. No mattеr how you move them around, the сoins will not form a гесtangle; they сan only Ье plaсed on a linе. likе this.

ooooooo

EXERсISE 3.2 Prime, Пectangular or squаre?

For еaсh numЬеr, mаrk thе boxеs сorresponding to whеthеr it is prime, rесtangular or square. Thе first onе has bееn done for you. Notе that 9 is Ьoth rесtangular and squarе, so two boxеs havе bееn markеd.

Numbеr

Primе Reсtangular Squafe

9721285 ПППППП

tЕПt-]Пt_]П EПf]ППП

3417

ПП t] t_]П П t]П t]

I{l/}|вEвflEц *ie5 oьв.PRln€sufr,

So sеven is not a rесtangular number.

Any numbеr whiсh сant bе aгrangеd in thе shapе of a reсtanglе is сallеd prime, so 7 is a prime number.

You havе already beеnjntroduсеd to the idea of a square. You

Pausе for a fеw momеnts now and think about what othеr

should also know that thе opposite of a squarе is сallеd a squаre root, and it is writtеn as \Г. A few ехamplеs should illustratе what square root means.

Now сhoosе

Thе squаrе of 5, writtеn as 5,, is 25.

Thе square root of 25, writtеn as {25 is 5.

primе numbеrs thеre аrе.

oo o oo n oo a)

9 сoins and arrangе thеm into threе rows and thrее

сolumns, as shown bеloцr.

Notiсе thаt the сoins have formеd a squarе shapе. Thе numbеr 9 is a square numЬеr bесausе it саn Ьe arrangеd in thе form of .!7hat othеr square numЬеrs сan you think of} a squarе.

Thе nеxt exеrсise will give you thе opportunity to prасtise your understanding of these threе tеrms' prime, rесtangular and squarе numЬers.

The squarе of 9, writtеn as 9,, is 8]',.

Thе squarе root of 81, written as "/81 is 9.

Thе square of 10, writtеn as 10,, is 1,00.

Thе squаre root of 100, wriшen as "/100 is 10.

Whаt are the .four rules'?

The four most basiс things you сan do to any two numЬеrs is add (+), subtraсt (-), multiply (x) or dividе (+) them. Тhеsе are .opeгations'). known as thе four rules (somеtimеs сalled thе four I've sеt out Ьelow somе simplе examplеs explaining what еaсh of these four rules mеans. Kеep your сoins hаndy and, if you think it helpful, aсt out thе various сalсulations using thе сoins. Еven thouф you may fееl a bit daft doing this, you will find that this will help you to rеmemЬer it later. (Having said that, I have found that when I havе beеn a bit daft, other pеople sееm

to rememЬеr it later.)

ЕxЕRсlsЕ 3.4 AppIe

Adding

a

Нerе is a simplе example of an adding situаtion.

I start with 2 gold

rings

v^

... and my truе lovе givеs mе 3

b

O

с d

morе O

How many gоld rings do I havе altogеthеr

?o o

O

OO

fol This сalсulation wоuld bе entеrеd in your сalсulatorlr as follolлrs. о First, switсh on.

о

Thеnprеss2Е3ta

and thе rеsult, 5, appеars on thе display.

If you fееl you nееd somе morе еxеrсisеs on thе rings, use your сalсulаtor to do Еxerсisе 3.3 norм.

ExERсlsE 3.3 Give me a ring

a

You havе

].3 gold rings and your truе lovе givеs

fo!!y!

You have 13 applеs and you givе away How many do you havе left?

You Ьave 42 apples and you give away How many do you havе lеft?

You havе 89 applеs and you gеnеrously How many do you have left?

You have 277 apples аnd you foolishly Нow many do you have left?

17.

give awаy 53. givе away 218.

Beforе wе lеavе adding and subtraсting (to movе on to multiplying and dividing), havе a look at thе diagram Ьеlow, whiсh illustratеs a num.ber liпe. This is a usеful way of represеnting numbеrs and thе opеrations of addition and suЬtraсtion. Thе numbеr line itsеlf is nothing more than a straight linе with thе numЬеrs plaсеd in sеquenсe on it. Thе diagram bеlow has beеn drawn to show the сalсulation 8 - 3.

Thе proсеdurе is to start at the valuе 8 (on the right hand arroЧ/' labеПed

.Start'), takе thrеe steps to thе lеft, and thе answеr is thе

point on the numbеr linе whеrе you stop. In this examplе, this is at thе valuе 5.

So8-3=5.

Stop

you 13 morе.

How many do you have altogеthеr?

You hаvе 9 gold rings and your truе lovе givеs you 45 morе.

How many do you havе altogеthеr? с You have 29 gold rings and your truе lovе givеs you 83 morе.

How many do you have altogеthеr? d You have 3|6 gold rings and your true lovе gives уou 477 morе. How many do you havе altogеthеr?

4.

b

a пumber line showing the сalсu|ation 8 _ 3

Subtraоting Hеrе is a simple eхamplе of a suЬtraсting situation. СС.С..СС You start with 8 applеs and gеnеrously givе 5 away

rlll

aaa--+aaaaa

How many applеs do you havе lеft? . С.

This сalсulation would Ье еntеrеd in your сalсulatоr as follows. о First, switсh on. о Then press 8 Г:-l 5 Г=l Thе rеsult, 3, appеars on thе display.

If you fеel you nееd somе morе praсtiсе at giving away apples, rrsе your сalсulator to do Еxerсisе 3.4 now.

In gеneral, сalсulations on thе numЬer linе are organizеd as follows. о Additions involve taking stеPs to thе right. о Subtraсtions involve taking steps to thе lеft.

You сould spend a fеrм minutеs now сhесking that you сan rеpresent simplе сalсulations likе this one on a numbеr linе.

Еsther's 30 еggs

0000000000 0000000000 0000000000

Еaсh egg-box holds six eggs

|0оTl

Multiplying An egg-box сontains six еggs.

Р"т.#iъ{,;;.,;Тj1;.'

r r ш

In this еggs-samplе you found threе lots of siх things. This is еntеred into the сalсulаtor as follows:

t0001

How many еgg-Ьoхеs цrill shе nееd?

o_о.0l |оOTl l0о-оl

Ю00l |000| l000l

Юml ЮоTl 10001 10001 This is entered into thе сalсulator as follows: Prеss 30

Е

6

Е

and the result,5, appears on thе display.

6 Г=l and thе rеsult, 18, appеars on thе display. this .divide' sign meапs this

,times'

.shared

.how

by' (,vofe: |t mаy be showп on a сomputer аs /)

sign meаns mаny lots of, (Note: it may bе sh0wn on a сomputer as *)

If you fееl you nееd to сraсk а fеw morе еggs' use сalсulator to do Еxеrсisе 3.5 now.

ЕxERсlsE 3.5 A run on eggs

yorrr

to do Еxerсise 3.6 now.

ЕxERсlsE 3.6 A run on eggs

(1)

At yоur loсal supermarkеt somronе tеlls you thаt thе еggs are

going сhеap. Naturally, you rush over to Ьuy somе. a You have just bought 4 boxеs of eggs. Нow many eggs havе you bought? b Your friеnd buys 18 boxеs оf еggs. How many еggs has she bought? с A Ьakеry Ьuys 1.1"4 boxеs of еggs. How many еggs havе thеy bought? d That day, thе supermarkеt sеlls 1673 Ьoхеs of еggs, аll going сhеap. How many еggs havе thеy sold?

Dividing Division is the fouпh and most trouЬlеsomе of the four rulеs.

Hеrе is anothrr typiсal evеryday sсеnario rлrhеrе division might сrop up. Еsthеr kееps сhiсkens in hеr baсk gardеn.

If you fееl you neеd to Ьox a fеw more еggs' use your сalсulаtor

on

a Мonda5 shе

сolleсts 30 еggs. How many Ьoхеs of еggs will shе nееd?

a

(2}

On a Мonday, shе сollесts 48 еggs. How many egg-boxеs will

she nееd?

Tuesday, shе сollесts 42 eggs. How many еgg-Ьохеs shе nееd?

Over a weеk, shе сolleсts 258 еggs. Hoпr many еgg-boxes shе neеd?

Ь on

will

с

will

d An

egg paсking station сolleсts |5 732 еggs. If thеy paсk thеm into Ьoxes of 18, how many egg-boхеs will they nеed?

If you have just taсklеd Еxеrсise 3.6, you may hevе notiсed that the numbеr of еggs were delibеratеly fixеd so that thе division workеd out еxaсtly. In othеr words, in every сasе' thеre wеrе thе

сorreсt number of еggs to fill еaсh box сomplеtely, with no sParе eggs lеft ovеr. In thе real world, of сoursе, сalсulations rarely work out so nеatly. !Иhat to do when division doеsn't work out exaсtly is disсussеd in Chaptеrs 04 and 05.

Knowing what sum to do Whаt сausеs somе сonfusion whеn using the four rulеs is that diffеrеnt pеoplе usе diffеrеnt words to dеsсriЬе thеm. Most of thеsе terms arе listed in Еxerсisе 3.7. Seе how many you

EXAП/IPLE l Ca|сulate5x4+7

Solution

rесоgnizе.

Тhe сalсu|ator sequenсe is

ExЕRсlsЕ 3.7 The terms used to desсribe the four rules

EXAMPLЕ 2

Tiсk whiсh tеrm rеfеrs to whiсh rulе.

Term

+

divide

from

goes into how many more how many less less minus multiply plus

produсt share sum subtraсt take away times

!7ith a сalсulator to hand, aсtually doing a сalсulation is usuаlly straightforward. Thе rеal skills arе knowing what sum to do and knorдr how to intеrprеt your answеr. Hеrе is an illustration.

giving the

answer,27.

I have four bottles of milk delivered eaсh weekday and seven at the weekend. How many bottles are delivered in a week?

x

add and differenсe

5 Гi-l 4 ГТl 7 r=1

Solution

Тhe сa|culation is made up of two pаrts.

Five /ots of 4 bott|es (one for eaоh of the five weekdays) is a mu|tip|iсation: 5 x 4

and seven at the weekend is an addition: + 7

So the so|ution, as before, is 5 Г x l ц ГT1 т Г=1 Giving the answer, 27 bottles.

Hеrе arе somе for you to try yoursеlf.

Е)(ЕRсlsE 3.8 Galсulаtions in context a A bus sеts out from the dеpot with 27 peop|e.

Calсulatе the numЬеr of pеople on thе Ьus, if: (i) aftеr the first stop, 18 pеoplе gеt on and 9 gеt off. (ii) after thе sесond stop, 12 pеoplе gеt on and2l get off. (iii) after thе third stop, 5 pеoplе gеt on and 16 gеt off.

b Ovеr eight minutes' thе ovеn

с

tеmpеraturе rosе from 21 dеgrееs Celsius to 205 dеgrееs Cеlsius. (i) !Иhat was thе tеmpеrature risе? (ii) Vhat was thе avеrage tеmPеraturе risе pеr minutе?

Dеnisе drinks 15 glassеs of watеr pеr day. How many will shе havе drunk:

(i)

in a wееk? (ii) during thе month of July? (iii) in a yеar? d A bottlе of winе is said to providе enough for about 7 glassеs.

If a glass holds about 10 сl, how muсh does a bottlе hold?

Doing sums on paper

ЕxA]t,lPLЕ 2

In this sесtion you arе asked to put your сalсulator to onе sidе and look at how сalсulations wеrе donе bеforе сalсulatoгs wеrе inventеd. No doubt you spent many hours at sсhool adding, subtraсting, multiplying and dividing' Hеre is a summary of the mеthods as thеy arе сurrеntly taught in most sсhools. Thе sоlutions havе bеen brokеn down into stagеs to hеlp you follow what is going on.

Stаge

Adding with penсiland paper

their сoIumns

ЕXAмPLЕ

1

Ca|сu|ate 27

+

Solution Stage'l

Stage 2

Stage 3

Set out the sum like this, with units under units and tens under tens

Add the units

Write the 5 in the units сo|umn ...

27 +68

+E 15

orlten5units Stаge 5

... and write the 1 in the tens сo|umn

Add the tens

+

6r_8

5

269

1

Set out the sum like this, with uпits (U)' tens (t) and hundreds (H) in

2+6+'l =9

and write the 9 in the tens оo|umn

27

+ 6'8

95

27 +68 5

Stage 3

Stаge 2 Add the units

3+9=12

17з +269 2

Write the 2 in the

3

+9

units сolumn .'.

12

1 ten 2 units

stаge 4

Stage 5

Stage 6

... and write the 1 in

Add the tens

Write thе 4 in the

the tens оo|umn

Stаge 4

27

+

Solution

HтU 17з +269

68

tens units

Ca|сu|ate 173

7+6+'l =14 14 tens is

173

tens сo|umn ..'

1

hundred and 4 tens

173

173

+26'9

+26'9

2

2

42

Stage 7

Stage 8

Stage 9

... and write the 1 in

Add the hundreds

Write the 4 in the

+2619

ths hundreds сo|umn

1+2+1=4

hundreds сolumn

+2,69 42

+2'69 42

+

.|7з

1тз

173

2'61

442

ЕxЕRсlsE 3.9 Prаctising adding with penсil and pаper Try thеsе addition sums now. You сan сheсk your answers using your сalсulator.

a 43+67 b 162+77 c 437 +239 d The

sum of 259 and952

Subtrасting with penсil аnd pаper

EXAмPLЕ

Sudden|y, a сustomer Walks in - с|ear|y a big spender - and asks for 27 tins of beans. Starting with the seven tins, you have no

1

Ca|сulate 98 - 45

Solution

Stage I

Stаge 2

stаge 3

Set out the sum like this, with units under units and tens under tens

subtraсt the units

Write the 3 in the

tens units

-

98

I

-5

units сolumn ...

98

-45

8-5=3

3

4_5

сhoiоe but to open up one of the cases. You пow have 10 + 3 = 13 |oose tins, and you count out 7 of them into the оustomerЪ tro|ley. Then you hand over two сases (two lots of tens)' making 27 in a||. Whаt you have Ieft аre three unopened сases and six loose tins. Your stoсk is now 36 tins. .dесomThis approaсh to subtraсtion is сalled thе method of .brеaking .dесomposing' up' or position', Ьесause it involves lots of tens (or lots of hundreds, thousands, and so on).

Rеmembеr, deсomposition is only nесеssary whеn you arе trying to takе bigger from smallеr and you hаve to go and opеn uР a bigger сasе.

Stage 4

stаge 5

ЕH]tiPLE 3

Subtraсt the tens

Write the 5 in the

Ъking 48 сoins from 95 сoins Set the сoiпs out Stаgэ 1

98

-45

3

9-4=5

tens сolumn

-

98 4_5

Ninety five сoins

53

This first ехamplе was сarеfully сhosеn so that' in both thе tеns and in thе units сolumn' you rмеre suЬtraсting а smallеr numЬrr from a biggеr nlrmЬеl Thе nеxt examplе shows whаt to dо if, at anУ point in thе саlсulаtion, you havе tо takе biggеr from smallеr.

ExAпnPLЕ 2

in сo|umns, with ten сoins in eaсh

сolumn, thus:

r39333t33 {** Ф and 5 units

9 tens

Stage

2

Тake away 8 сoins

UnfortunateIy there are only five loose оoins, so we cannot take 8 away direсtly. We need to break up one of the columns of ten into |oose сoins, making a total of 8 teпs and 15 units thus:

Subtraсt 27 from 63

This time, you are asked to taсk|e the problem from your imagiпation, rather than by fo|lowing a set of rules. lmagine that you are a shop-keeper selling tins of beans. Тhe tins сome paсked in сases, ten tins to a саse. on the floor behind the сounter are six unopened сases. A seventh оase is open on the сountel with just three tins in it. So your сurent stoсk of beans is 63 tins (six lots of tens and three singles).

3333З333

*--s€ъand 15 units

You сan now take away the 8 units.

15-8=7 Тhis leaves 8 tens and 7 units. thus:

33r33333 #d"_.-*itъ @6n ФФ 8 tens

Stage

3

and 7 units

Hеrе is holм thе dесоmposition proсеss would lоok on papеr.

15-8=7

Write the answer

Write the аnswer into the units сolumn

оolumn

8-4=4

into the tens

815

815

* 48

-48

7

47

Ca|сu|ate on paper 95

Еxaсtly thе samе approaсh appliеs whеn you arе subtraсting numbеrs of hundrеds, thousands and so on' as thе nехt еxample shows.

випnPLЕ 5 Ca|сu|ate on

paper

442

- 48

- 17з

Stage I

Stage 2

Set out the sum like this,

Subtrасt thе units

with units (U)' tens (Г)

and hundreds (H)

HтU 442 -17З

Е)(AIiPLЕ 4

2

--з 2-3=?

Stage 3 Deсompose one of the 4 tens to get

з tens and 12 units

4

з12

_17з

Gan't take bigger from smaller

Stage I

Stаge 2

Stage 3

Set out the sum like this, with tens (I) and units (U)

Subtraсt the units

Deсompose one of the 9 tens to get I tens and 15 units

-

Subtraсt the tens

Solution

So the answer is 47: i.e. 4 lots of ten and 7 units.

95

Subtraоt the units

Тake away the 40 оoins

33t3 ъ{.-..i> 3313

тU

Stage 5

Тhis сonfirms the answer we got before, 47.

You now take away 4 lots of ten from the remaining 8 lots of ten, leaving 4 lots of ten.

Solution

Stage 4

5

-8

5-8=?

tl_g

сan't take bigger from smaller

Stagr 4

Stage 5

Subtraоt the units

Subtraсt the tens

'12-3=9

Write the answer in the units сolumn

815

-48

4

312

.17з

I

4 з'2

-173

I

3-7 =?

Again' сan'l take bigger from smaller

Stage 6 Deсomрose one of the 4 hundreds to get 3 hundreds and 13 tens

3'з12

-17З

I

Stage 7

Stage 8

Stagв 9

Subtraсt the tens

Subtraсt the hundrвds

Write the 2 in the

13-7=6

Write the answ€r

in the

tens сo|umn

з,з,2

=173 69

-173

-17з 269

=2

ЕxERсIsЕ 3.10 Practis.ng subtraсting with penсil аnd pаper Try these suЬtraсtion sums now. You сan сhесk your answers using your сalсulator.

67

d

Thе diffеrеnсе bеtweеn 259 and 952

-43

ь L62_77 c 437 -239

add? t]

6Г=12E 2Е2Ег---1 6|т12 Е] Г_l

3'3'2

69

a

4Гт:]3ЕГ-----,

hundrеds сoIumn

3'3'2

3-1

3Гтl3tЕ3tЕ3tзГ'--l

Нow many lots of 3 did you

Negative numbers You may havе notiсеd that the suЬtraсtion sums rлrhiсh you havе bееn askеd to do so far havе beеn aшifiсiаlly .set up' sо that you havе bееn taking smaller from largеr. Basiсall5 subtraсtion is normally seеn vеry muсh in terms of .taking аway' objесts, and so, if you start off with thrее oЬjесts, you сan't takе morе than

thrеe аway. Howеvеr, subtraсtion doеsn't always involve moving oЬjесts around. Lоok at these two examplеs: . Althou8h I had only {3 in my bank aссount' I wrote a

.

сhеquе for {5.

Thе tеmpеraturе was 3oС and it droppеd a furthеr five degrееs оvеrnight.

How do the four rules сonneсt?

The four rulеs are, of сoursе, сlosеly сonnесtеd to еaсh othеr. Ехеrсisе 3.11 is designеd to lеt you .disсovеr' some of thеsе сonnесtions for yoursеlf. It is best done with the hеlp of a сalсulatoц though this is not еssеntial.

ЕxЕRс|sЕ 3.11 connecting thэ four rulэs Fill in the blanks in the sеquеnсеs bеlour.

(Thosе blanks marked in а long rесtanglе, thus, Г_---l, refеr to a numbеr that rеsults from a сalсulation. Blanks markеd Г-l .rulе, likе + or refer to a -.) thе Ьlanks, jot down on thе right.hand side a Aftеr сompleting undеr .Commеnt'what you think your result suggеsts aЬout thе four rulеs.

Keу

sequеnсe

Е] Г-----l E 2 tз Е] Г-_-_l П 3 tз г---з з в12 Е] Г-_-l Г] 2 t= г-q 4гfl 3Ег--lt]3Е]Г--l 20 Гтl5 E] Г.--l t] 5 Е г---zq 3 rт12 4 Гтl 3

Thе banking systеm hasnt ground to a halt or thе thеrmometеr .lVе simply solvе the problеm еxplodеd as a rеsult of thеsе еvents. Ьy inventing a new sеt of numbеrs less than 0. In bank statements thеse have thе lешеrs o/D (standing for ovеrdrаwn) bеsidе them.

Usually, howеver, wе just сall them minus, or negаtiue numbers.

So rеally thе numbеr linе should be еxtеndеd to thе lеft to look likе this.

-б4J-2_1012з4 number Iiпe showing negative numbers

SuЬtraсting follows:

5 from 3 сan bе shown on thе numbеr line

as

Stop

Сo?nn'ent -2-,1

012з4

Starting at thе numbеr 3 (thе right-hand arrow), take five stеps .minus two'. to the lеft, taking us to thе answer, _2, said as

You сan сhесk this rеsult by prеssing thе сorrеsponding key sеquеnсе on youf сalсulatоr. 3

tf

5

Е

Now havе

a go at thе praсtiсе exеrсisе Ьelow, whiсh should help to сonsolidatе somе of thе key points of the сhapter.

ЕxERсIsE 3.12 Prаctice exerсise 1 Sеt yourself a fеw simplе .sums' using

thе four rulеs of +, -, +. Cheсk yorrr answеrs using a сalсulator. Do the involving + and - again Ьy drawing a numbеr linе and moving, respeсtivеly, right or left. Chесk that you gеt the samе answеr as with the сalсulator.

The plaсe valuе of the 6 in the numbеr 365 is ten.

TИhat is the plaсe value of the digit 6 in the following numbеrs?

х and .sums'

2

Number 365 614 496 Plасе ualue ten

|6

042

109З 46|

7"C is thrее degreеs lеss than 10.C. Find the temperaturе whiсh is thrеe dеgrеes less than thе

following:

Tеrпperаture"С 10 Three dеgrеes lеss 7

4 21 -6 -10 0 3

-3

Doеs it matter in what ordеr you add, subtraсt, multiply and divide numbers?

For еxample, doеs 23 Гxl 15 |=l give thе samе answer as

15

Гxl 2з

т+-з,

Usе your сalсulator to еxplorе.

(i)

Does thе squarе of an even numЬеr always give an even numbеr? (ii) Does thе square of an odd numbеr always give an odd numbеr? (iii) Are all odd numЬers pгimе? (iv) .lVгitе Are all prime numbеrs odd? (v) down thе numbеrs from 1 to 20, and indiсate whethеr еaсh numbеr is primе, reсtangular' odd, еvеn or square.

Summary This сhaptеr startеd by eхamining some of thе Properties of numЬеrs _ whеthеr they arе evеn or odd, primеi reсtangular or squarе' for examplе. Nеxt, thе so-сalled .fouг rules' of add (+),

subtraсt (_), multiply (x) and dividе (+) wеre еxplainеd, both on papеr (in thе Answеrs to Еxеrсise 3.7) and with thе hеlp of a сalсulator. You wеre shown horм to usе a .numbеr line' to rеprеsеnt numЬеrs and simplе сalсulations. Thе languagе of

arithmеtiс is an arеa whiсh сausеs сonfusion and words |ike surn and produсt werе dеfined. Thе four rules arе, of сoursе, сlosеly сonnесtеd to eaсh othеr and thеsе interсonneсtions wеrе еxplored rryith thе help of a сalсulator. Finally, you wеrе introduсеd tо nеgativе numЬеrs; these сan Ье thought of as thе numЬеrs that appеar to thе left of zero on thе number line.

Answers to exercises for Ghapter 03 3.1 Numbert1,721,2 86310 Еvеn t-] П ГХ] t! Et!ППt! odd t! E t_l tl Пt-]EEt-] з.2

NumЬеr97212 Primе ПtIEП Rесtangular Еt_]ПE] Squarе Et-]ПП 3.3

a b c d

3.4

a b с d

3.5

a b c d

3.6 a b с

d

1.3 +

13 = 26 rings

9+45.= S4rings 29 + 83 = 112 rings 31.6 + 477 = 793 rings

.l'3_4= 9applеs

42.- L7 = 25 applеs 89 - 53 = 36 applеs 277 _218 = 59 applеs

6x4=

24eggs

6x78 = 108 еggs 6x114= 684еggs 6 х 1,673 = 10 038 еggs 48+6= 8Ьoхеs 42+6 = 7boxеs 258+6= 43boxеs 15 732 + 18 = 874 boxеs

8534 LL f] t! в t-] E tЕППt! П ППГ]E П

3.7

3.10

Term

+

add and differenсe

a

x

a

3.11 a

from

a

goes into

a

how many more how many less

a

a a

less

minus multiply plus

a

3.8 a

(i)

с

a a

3.9 a

b

a a

+ 78 _ 9 = 36 pеoplе left + 12 _ 2L = 27 pеoplе lеft ... + 5 - 16 = 16 peoplе left

27

(i) 205 -21' = 184 dеgrees (ii) Avеragе tеmpеraturе rise pеr minutе = #= 23 dеgrееs. (i) 7 х 15 = 105 glassеs of Watеr in a wеek (ii) 31 x 15 = 465 glassеs of wаtеr during thе month of July (iii) 365 x 1"5 = 5475 glassеs of watеr in a yеar.

с d

43

+

67

=1.1.0

162+77 =239

437 +239 =676

259+952=1271.

Сomtпent

Ггl

|т12 Г=l Г----з

3.12

1 No сommеnts 2 NumЬer

a

7xl0=70cl

Е Г-З E 2 tз Г--з 3 ta Г---l tз Г-l Г-l 2 Е] Г---l tE 3 tз Г----А tЕ 5 E Г-20 3Г+l 3tE3E3Г=1 r*тa 6

a

(ii) ...

(iii)

Ь

Kеу sеquеnсе з Г-112 E] 4 Гтl 3 Гт-1 3 Гт12 tf, 4 Гil 3 E] Г--та 20 Г+-l 5 Г=l Г----"

6|':.2L]2E2EГ_-l

a

subtraсt take away times

a

How many lots of 3 did you add?

4.v1 3 Г=l г---тa

a

produсt share sum

1.62-77=85 437*2з9=|98 952-259 = 693

b с d

a

divide

67-43=24

1

365 61'4 496 1'6042 1'09З461 valuе ten hundrеd unit thоusаnd tеn 1"0 42r*6 -10 0 3 -3 3 Ъmpеraturе "C 118*9 *13-3 0-6 Thrее dеgrееs less 7 Plaсе

4- Thе ordеr doеsnt mattеr when adding and multiplying but it doеs fоr suЬtraсting and dividing. For exаmplе:

2+3=3+2=5 Adding Мultiplying 2xЗ=3х2=6 But

2-3 SuЬtraсting -

_1, whеrеas 3_2=1' = 2 + 3 =3,whеrеas 3 1 2 = 1Ь Dividing 5 (i) Thе square of an еvеn numbеr always givеs an

еvеn

numbеr.

(ii) Thе squarе of an odd number always givеs an оdd numbеr. (iii)Not all odd numЬеrs arе primе (for еxamplе, 9, whiсh is

3x3).

(iv)All primе numbеrs arе odd with onе еxсеption, namеly thе numbеr 2.

40

(v)

Number

pnme

rcctangular

oсld

$ *

t

o q)

2

a

3

a

4 a

7

a

11

1з 14 15 16 17 18 19

20

a a a

a

a a

a a a

a

a

a a

a

a

a

a

a

a a

a

a

a

a

a

12

a

a

a

9 10

square

a a

5 6

I

even

a

1

о| rl

a

a

a I

a

I

I I I

a a

No/e: By thе wa5 thе numbеr 1 has not bееn markеd as еither { prime number or a rесtangulaг numbеr. It doеsn't сomfortably fit into еithеr сategor5 and-сant bе сlassifiеd in this way.

{r д)

o r+ :r o a э

ln this chapter you will leаrn: о how to picture a fraсtion

. .

about equivalent fractions how to оalcu|ate with

fraсtions'

Deсimals, noщ. and frасtions. I dont know, I just didn't sееm to grasp them ... I just found them boring. I

сouldnt

сonсеntratе on them at all. They werеn't interesting еnough for me. It wasn't affraсtivе enough.

Sheilа, a friепd

Not thе most promising statr to a сhapter on fraсtions perhaps! [t сertаinly. sеems to Ьe the сasе that, while most peoplе knЬw roughly whatЪ going on whеn thе fouг rulеs arЪ appliеd to чhole numbers, sums тrith frасtions сan Ьring thе Ъhutters down. The first thing you should rca\ize is ihat fraсtions, deсimals and perсentages arе all vеry similar. As yоu rлrill sеe ovеr the next threе сhaptеrs, фy,'. all slightly diiferent wаys of^desсribing the. samе thilg. Ttrhat we сall frЪсtions - things ПЁe |, } and so on - should rеally Ьe ca||ed comnaon fractions. in faсt thesе sorts of fraсtions are not quite as .сommon' as they used to bе. Inсrеasingly the morе awliчrard сommon fraсtions, like f, and J'' for eхamplе, are bеing геplaсed Ьу decimаI fraitions.

Dесimal fraсtions look likе 0.3, 0.725, and'so on and arе dealt with in Chapter 05. But first of all, lеt's find out what a fraсtion is and where it сomеs from.

What is a fraction?

Sabine and Sam arе four. They havе nеver hеard of a fraсtion. I produсеd thrее squares of сhoсolate and said that they wеre to be sharеd bеtwеen thеm. Thеy took onе squafе еaсh. Iriow what about that third squarе? \trell, you сan be sure that thеy wont givе ig to mе' or thеir favouritе сharity. Sabinе and Sаm may not havе hеard of a fraсtion, but thеy are quitе сapаblе of invеnting

This сan Ье writtеn as follows 7I3

=2 remainder 1,.

But, as with thе square of сhoсolatе, we dont always want to lеavе thе remaindеr .unsharеd'. If thе rеmaindеr of 1' is аIso sharеd out amongst thе 3 (pеoplе) thеy eaсh gеt an extrа one

third, as shown Ьеlolп.

./ ''фф

e

ффффффФ.*mшс *фф

t

three lots of.two аnd one third'

So, the morе сomplеtе answеr to this division sum is: 7

l 3 = 2li.e.

7 dividеd by 3 givеs 2 and a third.

It is important to understand why fraсtions arе wriцen as thеy arе. Thе fraсion l rеally is anothеr way of writing 1 dividеd Ьy 3' or Ll\. So thе top numbеr in a fraсtion (сallеd the numеrаtor) is thе numbеr of things to Ье shared out. The bottom numbеr (the dеnominаroz) tеlls you how many sharеs therе will Ьe. the top number is the numerаtor

one whеn thе oссasion arises. As will bе Ъxplained belo{

fraсtions саn Ьe thouфt of as the .broken bits' tЬat fiе bеtwееn

thе rмholе numbеrs.

Fraсtions oссur quite naturally in division (i.e. sharing) whеn thе sum doеsnt dividе eхaсtly. For ехamplе, sharing sevеn doughnuts amongst 3.

ффффффф;Фф 7 doughnuts

sharвd...

Еxеrсisе 4.1. will hеlp you grasp this important idea.

,.'фф *фф ... into thrse lots of 2 ...

ф аnd 1 |eft ovвr (the remainder)

EXERCISE 4.1 Sharing cheese A boх of proсеssеd сhееsе has siх sеgmеnts. Sharе two

boxеs

еquallyaпrongthrееpеoplе.1 | (i) How many sеgmеnts / \-,fi.)

;l'":"'::#:Т'Т,;-

\

,.--Г-хt l | \ \ / \-,\

ИTv \-Т'

will еaсh pеrson rесеivе?

\-]--l

\-1--l

How to piсture a fraсtion Thе сommon fraсtions likе a half, thгее quarters and two thirds arе partof еveryday lаnguage. You should find it helpful to have a mеntal piсture of a fraсtion. Thе piсturе whiсh is in my mind

(and whiсh is usеd

in most sсhools whеn fraсtions aie first

introduсеd) is to imagine a wholе as a сompletе сakе. This сan Ье сut into sliсеs reprеsеnting various fraсtiЬns, likе this:

Until frасtions arе introduсed, numЬеrs сan be thouфt of as a set of points еqually spaсеd on thr linе (i.е. thе rлrholйumbеrs).

But as your piсturе of numbеrs еxpands, you сan see that thеre arе lots of other points Ьеtwееn 1, and 2, Ьеtwееn 2 and 3, and so on. Нow many are thеrе? Are there any gaps at all on thе numbеr linе whеn thе fraсtions are addеd? Thеse arеn't quеstions with еasy answеrs but you might сarе to think aЬout

thеm.

Finally, herе is a remindеr of how the сakе diagram and thе number linе сan hеlp your mеntal piсture of fraсtions.

Cakе diagram.-+ fraсtions arе .bits'оf a wholе Numbеr linе --+ fraсtions fill in the gaps .betwееn wholе

whole

half

quarter

1

t

L

three quadeв 4

Thеse sorts of piсturе arе hеlpful as a lмay of undеrstanding what a fraсtion is. But if you nеed to сompare fraсtions or d6 сalсulations with them, then you neеd morЪ than piсturеs. For еxamplе:

l

Is 3 of а сakе biggеr than

of it?

Fitting fraсtions into the number Iine

Thе nеxt stеp is to-.."understand how fraсtions fit into thе sequеnсе of nцmЬdrs that you lookсd at in Chaptеr 02. For examplе:

are.

What are equivalent fraсtions? .What

is thе differеnсe bеtweеn sharing two сakеs аmong four pеoplе or sharing one сake bеtwееn two peoplе} W.еll, sinсе evеrybody еnds up with half a сakе, thеrе is no diffеrеnсе in thе

i

sharе that eaсh pеrson gеts. Thе first lot of pеoplе aсtually gеt of a сakе but that would sеem to be thе samе as }. So iandL are fraсtions whiсh arе thе sаme; yet thеy aге diffеrent - they havе thе samе valuе but havе differеnt numеrаls top and bottom. Thе rлrord used to.dеsсribе this is equiuаIeпce.

.!7e

2* is onе third of thе way Ьеtwееn

2 and

3# is sеven tеnths of thе way betweеn

3

would sаy that landL are equiuаIent frасtioпs. Еxеrсisе 4.2 (ii) will give you a сhanсе to try to spot somе more equivalеnt fraсtions. Тry it now.

3 and 4

Thе diagram bеlow shows how thesе fraсtions fit on thе numЬеr line.

lr tl

2t

3fr

I

, *

a пumber |ine showing frасtions

в(ERсlsЕ 4.2 FINDING EQU|VALENт FRAстIoNS (i) Мark with an arrow еасh of thеsе numbers on thе numbеr linе Ьеlow:

2tr,',4h,1'ь

tl lt 012

numbеrs' And now you're rеady to add and subtraс fraсions. !Иеlt neaгly ...

Bеfore that it rмould bе usеful to know what equiualenf fraсtions

,

*,

(ii) Find thrее fraсtions equivalеnt to еaсh of thе following (thе first sеt has Ьеen donе for vou): Fraсtion з

Еquivalеnt fraсtions

6212

8, 12,

41

L.

This timе the sliсеs of сake aren't thе samе sizе, so wе сant just add thеm together. Thе way out of this problеm is to сut both fraсtions until all thе sliсеs are thе samе sizе - likе this:

1,6

B

9 10

I s24

1!

20

Now, with all the sliсеs еqual to ё, they саn be аddеd. Thе сalсulation looks likе this:

Adding and subtraсting fraсtions

J+2 2 Both fraсtions are сhangеd to еquivalеnt .sixths'

!Иhеn adding fraсtions, it is hеlpful to think of thе sliсеs of сakе.

Fоr ехamplе: *+

= 2+

=I

i=?

=14

q-ф= Herе the sliсes arе all thе samе sizе togеther, like this:

(*

eaсh) so wе just add thеm

*+E=f

It is usual to write thе

ansrлrer

in thе form of thе

simplest

еquivalеnt fraсtiоn, so the answrr' 8, сan be writtеn as

}.

Howеvеr, whаt happеns whеn you havе to add fraсtions like ttrе

following?

1=*.=?

But why did I сhoosе to suЬdivide eaсh fraсtion into sliсes of t?

The rеason is that * is thе еasiеst fraсtion tЬat a Ьa|f and a third will Ьrеak up into. I сould havе usеd sliсеs of * or * but that would havе bееn unneсessarily сompliсatеd. By thе way, this proсеss of Ьrеaking fraсtions up into smallеr sliсеs so that they сan be added or subtraсtеd is сallеd .finding thе smallеst сommon denominator'.

To summarizе, finding thе lowеst сommon dеnominator means finding thе smallеst numbеr whiсh both thе denominators will divide into. (RеmemЬеr that thе dеnominator is thе Ьottom numЬеr in thе fraсtion.) This number thеn bесomеs thе new denominаtor. Thus in the еxаmplе abovе, thе loцrеst numbеr that 3 and 2 both dividе into is 6, so 6 is the nеw dеnominator.

Now try Еxеrсisе 4.3 (thе first onе has been donе for you).

ЕxЕRсlsЕ 4.3 Adding and subtraоting frасtions Complеte thе tablе bеlow: Саlсulаtioп Equiuаlent frаctions

1+* Ь+1

l+t

t-t

*+*

*-1

-El2a

-L 12

Ansшer

цt2

Мultip|ying and dividing fraсtions

2

How oftеn in your lifе have you had to multiply or dividе two

fraсtions outsidе a sсhool mathеmatiсs lesson? I suspесt that thе answer is, fоr most pеoplе, nеvеr. I therеforе dont intend to dеvotе muсh spaсе to this diffiсult and rathеr pointlеss exеrсisе.

Howеvеr, it ls useful to knorм a fеrлr basiс faсts - for еxamplе, a

half of a ha|f is a quartеr, and a tеnth of a tеnth is onе

hundrеdth. Thеrе arе also a fеw praсtiсal situations (like sсaling

thе ingrеdiеnts of a rесipe, for example, whеn you Want to produсe a smаllеr or largеr сakе than thе one in thе rесipе) whеre multipliсation and division of vеry simplе fraсtions may Ье helpful. This is probably еasiеr to understand by looking at dесimal fraсtions, so wе shall return to this topiс in Chaptег 05.

. . . .

Fraсtions сan bе thought of as bits of wholе numbеrs.

A usеful way of rеpresenting fraсtions is as sliсеs of a сakе. Еquivalеnt fraсtions, likе } and i havе thе samе value and сorrеspond to thе samе sizе of sliсе of thе сakе.

Adding and subtraсting fraсions usually involves rewriting the fraсtiоns as еquivalеnt fraсtions. This mеans finding a сommon dеnominator (thе bottom numbеr in thе fraсtion), and adding thе numеrators (the top numbеrs in the nеrм fraсtions).

Мultiplyrng and dividing fraсtions is eаsiеst to undеrstand when thе fraсtions аrе writtеn as deсimal fraсtions (sее Chaptеr 5).

Prасtiсe exetЕise

1

Shаrе thе following equally. (Thе first onе has bеen done for

you.)

a

1t сakеs amongst 4 peoplе. Еaсh

а2 b4

d12 e3t

t7t

с10

!Иritе thе appropriatе fraсtions onto thе sliсеs of the сloсk.

Now add thе fraсtions togеthеr.

Cheсk that they add to

1.

Thе fraсtion * сan bе wriшеn more simply as *. \Гritе the following fгaсtions in thеir simplest form. (onе has been donе for you.)

Summary о

Thе number 1 сonsists of 3 thirds. Hou, many thirds are thеrе in thе following numbеrs?

gets

17 сakеs amongst 5 pеoplе. Еaсh gеts 5 сakеs аmongst 6 pеoplе. Еaсh gеts

20 сakеs amongst 3 pеoplе. Еaсh gets

сakеs

П."k", П."k.. П."k.,

Fraсtionst*t*8+*.&#?, Simplеst

(i)

form

з

Changе all thе fraсtions bеloцr to twelfths. (Thе first one has bееn doпе for you.)

(ii) Now raпk thеm in order of size, putting a rank of 1 against thе largеst fraсtion and 6 against thе smallest.

(Again, onе has bеen done for you.)

|12Ьe' Fraсtions Frасtions as twelfths l,, Rank 6 A sum of

3

{600 000 was left

to Ье

shared among thrеe

сharitiеs, as follows: Charity A was to rесeivе onе quaffеr.

Chaгity B was to rесeivе two thirds.

Chaгity C тyas to rесеivе thе rеst.

Calсulate: a thе fraсtion of thе sum that wеnt to Charф b the aщount of monеy duе to еaсh сharity.

C

Answers to exercises for Ghapter 04 (i) Еaсhpеrsongеtsff =!

4.1

=4sеgmеnts. (ii) Еxprеssеd as a fraсtion, еасh pеrson gets * or in othеr

words,3ofabox.

4.2 (i)

Ь

llll llll

lЬ

z1

ц?'

с

'1

-^6

д-Ц

10=Т

d 12=Ч е 3i=8 f.

гraсЕlons

Equiualent frасtions

a

6 81

?5-12 41 10r 18 j15 20t 50

+

210

z28шa 60;

I

6s

1 4t

.L 24

zo

Еquiuаlеnt

&+* +1 1*3

t** l+tr 2*2

llr_L1 +-)

20-20

Charity B was to rесеivе Charity С was to rесеivе

e00

1000 300

24

11

frаctions

*_* *+*

18

ь

2_ 22

-L 11

3

х

Ьx

f,600 000 = {400 000

f600 000 = {50 000 (As a quiсk сhесk, thеsе amounts of monеy should add to {600 000. {150 000 + d400 000 + {50 000 = {600 000.)

24

18t

1.212 2t

ш

11

48

16 48

Rank326415 6 a Charity C was to rесеivе 1 - (* * &) = L _# = Ь b Charity A was to rесеive trx 1600 000 = {150 000

16

1 8r

=#.=1.

84.5.t264д9 10 6 10 18 9

Simplestform t t ь 3 3 * ь Fraсtions 1 1 2 +,z ь Fraсtionsastwеlfths * t + Ь * *

ansrдrеrs hеrе.

I 12э

71=Ч

The fraсtions arе *' Ь'# andL. Thеsе сan bе rецrriffеn in twеlfths and addеd. as follours.

Frасtion

*.ь l+*

EI."k.,

*+h*Ь*t=Ч6

(ii) Therе afе many possiЬlе

Ь

d 20 сakes amongst 3 pеoplе. Еaсh gеts

b

012345

Саlсulаtioп

|Тl."k",

2a

IIIl llrr

4.3

с 5 саkеs amongst 6 pеoplе. Еaсh gеts

Ansшеr #

1=

L*

z=1'ь

* *

20

Answers to praсtiсe exerсise I a

17 сakеs аmongst 4 pеoplе. Еaсh gets

l2Jl.д.,

b

17 сakеs аmongst 5 pеoplе. Еaсh gеts

ГJТl.,k.,

Five fingers on eaсh hand (well, four fingers and onе thumb) sееms to Ье a rеasonable numbеr to possess. Any fewer and we wouldnt Ье aЬlе to play thе .Мoonlight Sonata'with thе same

panaсhе; any morе and rме'd havе a bit of a struggle putting on a pair of glovеs. It may not surprisе you that the link bеtwееn thе numЬer of our tеntac|es and dесimals is, wеll, morе than

tenoo\s..!7'hаt I'm really saying, thеn, is that thе rеason our numbеr systеm is basеd on thе numbеr tеn is beсausе humans

have сountеd on their ten fingеrs for thousands of years. .Dесimals' (from thе Latin deсi mеaning tеn) is rea||у a way of desсribing the tеn-nеss of our сounting systеm. Howеvеr, it usually rеfеrs to dесimal fraсtions. And thеrе is no shortage of thosе around us. Just listen to sports сommеntators, for examplе:

... thе winning time of 10.84 sесonds

smаshes thе world

rесord Ьy two hundrеdths of a sесond. a long jump of 8 mеtrеs, point2l.

... ... thе winning sсorеs for thе pairs iсе skating arе

a.

follows: 5.9, S.8, 5.9 ...

o o

as

Dесimal points appear whether rме afе talking about monеy ({8.14) or measurement (2.3L mеtres) and will appеar on a сalсulator display at thе touсh of a button. Now find yoursеlf a сalсulator and you сan usе it to invеstigatе exaсtly what a dесimal fraсtion is.

tt

Decimal fractions

з

s) :

What is a decimalfrаetion?

A deсimal fraсtion is simply anothеr way of urriting a сommon

frасtion. [n this seсtion you сan usе your сalсulator to disсovеr how fraсtions and dесimals arе сonnесtеd.

сл

Еl(ERсlsЕ ln this сhapter you will learn:

r

about the'ten-ness' of numbers . why we use a decimal point and where we put it с about the сonneсtion between fraсtjons and

.

decimals

how to сa|cu]ate with

deсimals.

5.1

Deriving deсimals from fraсtions

For еaсh boх of quеstions Ьelow: a writе dorмn thе answеrs in fraсtions b usе vour сalсulator to find.Conсlusion' thе answеrs in dесimals box. .o-pl"t. thе blank in thе

"

Thе first one has bеen started for vou.

Keу

sequenсe

г=1

1,rт1 2

2Гт1+г=l

Frасtion

Dесirnal Conсlusion

ь

0.5

Г.i 10Гn *

5 50 Г+-l ].00Г=l

1Гтl

Гтl

100Г=l

3 ГТ-I4 Г=l 6 Гт-l 8 75 rт1 100 Г=l

Е]

1Е10 10

йth

Thе dесimal for

}

is

The dесimal for

*

is П

the сorrеsponding deсimal fraсtions addеd.

IбЗ|

4 г=l

2Г+l 8Е] 5 Гтl20 E]

25

fraсions. Hеre are thе .сakеs' from Chapter 04, but this timе

Гтl

Thе dесimal for

t=]

i is

Thе dесimal for -1is

100 Г=l

who|e 1

quarter

hа|f

three quаrters l, or 0.75

l. or 0.25

l, or 0.5

Lеt's nour turn to the uray wе rеpresеnt deсimal fraсtions on a numЬеr linе. Again, sinсе fraсtions and dесimals arе rеally vеry similar, it is not surprising that they сan both bе rеpresentеd in thе samе way. Fоr examplе, thе fraсtion l and the deсimal 0.75 sharе thе samе position on the number linе. Thus:

П Гl

i,

As сan be disсovеrеd from the kеy sеquеnсеs above, сonvеrting

or 0.75

from fraсtions to dесimal fraсtions is very straightforward using E сan bе сonvеrtеd to a

a сalсulator. For еxample, thе fraсtion

deсimal fraсtion by dividing following kеy sеquеnсе.

5 by 8, i.е. by

prеssing the

Е]

5 Гт-1 s This produсеs thе rеsult 0.625. [n othеr words, the fraсtion

E

has thе sаmе valuе as thе dесimаl

fraсtion 0.625. Еxеrсisе 5.2 will give you praсtiсе at сonvеrting from fraсtions to deсimal fraсtions.

ЕxЕRclsЕ 5.2 GonveЁing from frаctions to deсimal

fraсtions

Now usе your сalсulator to find the deсimal values of fraсtions in thе tablе below.

Tablе

thе

1

_.1131_L2-L2-L1J. 2 4 4 10 5 5 10 10 20 8 rraсflon Dесimal

3

Picturing deсimal fraсtions You may rеmеmbеr from thе prеvious сhaptеr that fraсtions сould bе hеlpfully rеprеsentеd using sliсеs of a сakе. Sinсе therе is suсh a сlose link Ьеtчrееn fraсtions and dесimal frасtions, it

follows that thе same hеlpful piсturеs apply to dесimal

Having madе a сonneсtion bеtwеen fraсtions and dесimals, thе nехt eхеrсisе (сallеd .Guеss and press') gets you working iust with deсimals. Thе idеa is to writе down your gшеss as to what thе аnswеr will bе for eaсh сalсulation. Thеn you press the kеy sеquеnсе on the сalсulator and sее if you are riф. Thе aim of this еxеrсisе is to help you sеe thе сonnесtion Ьеtrлreеn dесimals аnd wholе numЬеrs.

ЕxЕRcIsE 5.3 Guess аnd p]ess

CаIcalаtion

ta 0.5 гil 2 a 0.25 Г;l + г;t 0.5 Гi..l 10 Е 4 гтl 10 |=-l 0.1 гтl 0.1 tз 0.1 гxl 10 Е 0.5

гтl

0.5

Guеss

1

Prcss 1

What is the point of the deсima| point?

If thе тrorld сontained only wholе numbеrs, wе would nеvеr nеed a deсimal point. Howеvеr, it is hеlpful to bе aware that dесimal numbers arе simply an ехtеnsion of the wholе number systеm. If you think of a wholе number, thе rules of plaсе vаlue tеll us what eaсh of these digits rеprеsents. Thus, thе last digit of a wholе numЬеr shows how many units it сontains, йе sесond last digit gives the numbеr of tens and so on. For еxamplе, thе numЬеr twеnty-four is цrrittеn as:

Howеvеr, whеn wе start to usе numbеrs rмhiсh inсludе bits of a whole (i.е. with dесimals) somе othеr.plaсеs'arе neеdеd. Thesе rеpresеnt thе tеnths, hundredths, thе thоusandths, and so on.

The dесimаl point is sitnplу а rпаrker tо show wherе thе units (whole numЬers) end and the tеnths Ьеgin. You'll gеt a Ьettef idеa of this by disсovеring whеn thе dесimal point appears on

your сalсulator. As you do Ехеrсise 5.4, watсh out for the dесimal point ...

Blank сhecks

400

25

10

E П

Е1

Е] 10E1 П E tТ-]10 Е]t-] Е tЕ] 100 t= t-]

1 4 Е]100 Е i:] 7 H1000Еf]

E П Е1 10 Г= П til 10E] П tn 10 Е t-] Е 10 tз П E 10 E]Г] t!

10

point

(1)

.

(0.1) (0.01)

(0.001)

Using the four rules With deсimаIs If you aren't sufe how to usе the four rulеs of

+, -, x and + with dесimal numbers, why don't you еxpеrimеnt with your сalсulator? You will quiсkly disсovеr that thе four rulеs work in exaсtly the samе way for dесimals as for wholе numbеrs, and for that rеason addition and subtraсtion of dесimals arе'not

ЕxЕRсlsE 5.5 Multiptying with decima! frасtions The first сolumn in this tablе givеs you thrее multipliсation сalсulations involving fraсtions. For еaсh сalсulation: a сhangе thе fraсtions to dесimals (Column 2) b usе yоur сalсulator to multiply thе dесimals (Column 3) с сhangе thе dесimal answеr baсk to a frасtion (Column 4) Calculation in fractions

ЬxЬ

Сomplеte thе blanks and thеn сhесk with your саlсulator.

100[i]

(10)

Baсk in Chaptеr 04, I suggestеd that you сould learn аЬout multiplying and dividing fraсtions by еxpеrimеnting with your сalсulator. Еxеrсisе 5.5 is dеsignеd to hеlp you do just that.

-дtчjЭJ

5.4

(100)

spеlt out hеrе as a sеpаratе topiс.

2

ExERсlsЕ

Huпdreds Tens Units Dесimаl Tenths Haпdredths Thousаndths

10Е1 10 10

П

П E t] tз

You may have got a piсturе of thе dесimal point jumping one plaсe to the lеft еvery timе you dividе by 10. Aсtually, this is a

slightly misleading piсturе. Мost сalсulators rмork on a prinсiplе of a .floating deсimal point'. This mеans that thе deсimal movеs aсross thе sсrееn to kеep its position Ьеtwеen thе units and thе tеnths digit.

The key point to rеmеmЬеr is that thе deсimal point is nothing morе thаn a mark separating the units from thе tеnths. Bеlow I'vе written out the morе сomplеtе sеt of .plасе valuеs' еxtеnding beyond hundreds, tеns and units into dесimals.

Calculation in Decimal answer decimal form (use a са|сulator) 0.5 Х 0'5

0.25

Fraction answer -1

lx* txЬ Now lopk at сolumns (1 ) and (4) and sеe if you сan spot the rulе fоr multiplying fraсtions. Think about this for a vrhilе bеforе rеading on.

Rule for mu|tiplying frасtions Тo multip|y two fraсtions, say 8 and 1z, muItiply the numerators (3 x 1' giving 3)' then mu|tiply the denominаtors (5 x 2' giving 10). Тhe answer in this сase is *: i.e. 8 x * = *i* = *

Е)(A]IiPLE

Dividing fraсtions

1

Dividing fraсtions is a morе painful and lеss usеful skill than multiplying thеm аnd I dont proposе to wastе muсh timе on it

Look at the fo||owing muItipliоation:

ixi

As before' these tv,o frаctions сan be сonverted into deсimal form' so the сalculation саn be rэwritten as fol|ows.

0.75 x 0.4

Pressing 0.75

ГП

0.4 Г=l on the сalсuIator gives an answer of

0.3, or -i.

By way of a сheсk, we сou|d apply the ru|e on the previous page.

We mu|tiply the two numeratoБ and then the two denominators' as follows.

3u2

4^

fu2 э

5= 4хs =

20

Тhis can be simplified to

*

(remember from Chapter 04 that and'or аre equiva|ent fractions).

*

So, it doesn't matter whether multiplication is done in fraction form or deоimal form; the resu|t is the same either way.

here. This topiс is one of sеvеral .сasualtiеs' of thе сalсulator agе whiсh is no longеr relеvant and whiсh, in my view, is simply nоt

worth learning. tf you аre in а situation whеrе you nеed to dividе fraсtions, a good strategy is to сonvert the fraсtions to dесimаls and pеrform thе division on youf сalсulator. Hеrе are two еxamplеs.

DGlt,|PLЕ

э'z 4т5

l

Rewriting as deсimaI fraсtioпs, this gives: 0.75 + 0.4

Using the сalсulatol this gives an answer of 1.875.

A

ru|e of thumb whiсh used to be tаught for dividing fraсtions is to turn the fraсtion you are dividing by upside down and proсeed as for mu|tipliсation. So' in this сase we get:

1+l=Ixi=f If you сheсk on your сalсu|ator (by pressing 15 ГT-l 8 Г=l), you wil| see that the fraction Т has the same value as the earlier answer of 1.875.

Е)(AiNPLE 2 Look at this multip|iсation:

2Ix 4t Again' these two fractions сan be оonverted into deсima| form, so the сa]сu|ation сan be rewritten as fol|ows.

ЕXAП,|PLE 2 Look at this division:

.

2.25 x 4.6

Pressing 2.25

r{

4.6

|=

l on the сalculator gives an answer of

10.35.

Again, these two fractions сan be сonverted into decimal form, so the calсu|ation.сan be rewritten as fo|lows: 4.2 + 1.5

As before, We сan сheсk this agaiпst the method of multiplying fraсtions desсribed in the box. However, the number 2i must be rewritten as i аnd 43 must be rэwritten as ?.

9'

4

-н-gJ'& 5 - 4х5 ^

20 --Фz

Finally' just to cheоk that the two methods produce the same rвsult' this fraction оan be сonverted to deсimal form by dМding the numerator by the denominatoБ thus: Pressing 2o7 |т.ia 20 answer of 10.35.

гn

on the саlculator сonfirms fhe previous

4+11

Pressing 4.2

гт1

1.5

Г=l on the сalсuIator gives an answer of

2.8.

As before, We сan сheсk this against the method of dividing fraсtions desсribed above. Howeve1 the number 4i must be rewritten as ? and 11 must be rewritten as i.

t+2=T

xl=ffi-f

whiсh сan bе simplified to €.

Fina|ly' just to оheсk that the two methods produсe the same resu|t, this fraction сan be сonverted to deсimal form by dividing

the numerator by the denominator, giving the same answer as before.

Ехеrсisе 5"6 сontains some сalсulator aсtivitiеs whiсh should help you to beсome more сonfident with dесimals.

ЕxЕRсlsE

decimals

largе numbеr into thе сalсulator display and thеn rеpеatеdly prеss Г=l. At first, watсh what happens. Later, try to prеdiсt

what will happеn.

An overuiew of decimals

Set thе сonstant to multiply by 10. Thеn еntеr a small dесimal fraсtion and rеpеatеdly prеss ta. Try to makе sеnsе of what is going on and thеn try to prеdiсt what will happеn nеxt.

Sеt thе сonstant to add 0.1 and rеpеatеdly prеss Г=l. !7ithout pressing the .Clеаr' kе5 entеr a |arge dесimal

As your сonfidenсе чrith dесimals grows, you urill сomе to

appreсiate how dесimal numbers are a natrrral еxtеnsion of our wholе numbеr system. What this mеans in praсtiсе is being aЬle to undеrstand plaсе valuе. So just as you add up to 10 units and thеn swap thеm for onе tеn' so you add up to tеn hundredths

numЬеr and kееp prеssing Г=l.

Set thе сonstant to add 0.01 and rеpеat what you havе just donе in part с.

Rеpеat parts с and d but with the сonstant set for subtraсtion

and swаp them for onе tenth. This is illustratеd in the two addition sums Ьеlow: 6 3

10

ten units arе

written as 1 in the tеns сolumn

Using the сonstаnt to investigаte

Sеt your сalсulator's сonstant to dividе by 10.,Nеxt enter a

14+5=2.8.

1

5.6

0.01 0.06 0.03 0.10 ten hundredths arе writtеn as

in thе tеnths сolumn

f

in еaсh сasе.

With a friеnd, play thе gamе .Guеss thе numbеr', the rules of whiсh arе explainеd at thе еnd of thе сhaptеr.

Praсtiсal situations involving dесimals abound, thе most ].

(Notе that this rеsult, 0.10, will be shown simply as 0.1 on the сalсulator display)

obvious examplе Ьеing monеy. Thus {3.46 represеnts 3 wholе pounds, 4 tеnths of a pound (i.е. 4 tеn-pеnсеs) and 6 hundrеdths of a pound (i.е. 6 pеnсе).

13.46

one oЬvious ProPеrty of wholе numbеrs is that thе morе digits a numЬеr has, thе biggеr it is. Unfortunately this is zof true for

deсimal numЬers. For еxamplе, the numbеr 5.831659 is

aсtually smallеr than, say, 7.2. Don't bе unduly imprеssed by a long string of digits. what maшers is thе position of the deсimal point. You nееd to sеe beyond this string of digits and gеt a sеnse of how big thе numbеr асtually is. For ехample, it is morе useful to know that 5.831659 is betweеn 5 and 6 (or |ust less than 6) than to quotе it to siх dесimal plaсеs.

Sеnsibly used, сalсulators arе an eхсеllent means of sееing beyond the digits of a numbеr. For ехamplе, еarlier in thе сhapter, in Еxеrсisе 5.4, you rrrerе askеd to pегform reрated division by 10 and thеn observе urhat happеnеd to thе deсimal point of thе answer. This is an еxеrсisе whiсh you сan do with any statring numЬеr of your own сhoiсе, and thе rеpeatеd

division by 10 сaп bе more effiсiently done by using thе

сalсulator's сonstant faсilitv.

Howevеr, thе monеy rеpresentation

of deсimals сan

bе

сonfusing. We sау {'3.46 as.three pounds foщ,-siх', rathеr than .thrее point four six', whiсh is thе more сorrесt deсimal form.

This lаttеr vеrsion emphasizеs thе dесimal plaсe value of еaсh

digit. otherwisе you ian gеt into troublе when dеaling with sums of monеy likе one pound and ninе pеnсе, whiсh is often mistakеnly written as {I.9, ratllеr than f,L.09.

.!Ие

grow up with deсimals and mеtriс units like mеtrеs' сenti-

metrеs, kilograms, millilitres, and so on all around us. Howеvеr, we still havе fееt and inсhes, pounds and ounсеs, and thеsе

units, known as impеrial units, arе thе onеs that many adults still fееl h,рpy with. Thеsе units arе ехplainеd in somе dеtаil in Chaptеr 07.

Thе main advantage of metriс units is that thеy are basеd

еntirely on tens' hundrеds and thousands; for example, thеrе аrе 100 сеntimеtrеs in a mеtrе' 1000 metrеs in a kilomеtrе, 1000

millilitres in a litre, and so on. Contrast this with the oldfashionеd 14 pounds in a stone, 12 inсhеs in a fоot, 7760 уafls in a milе, and so on - rеаlly a сomplеtе shambles!

Prасtiсe exerсise

1 a Мark

the numbеrs 0.35 and 0.4 on the numbеr linе below.

If you want to eщ>lorе numbеrs, the сalсulator is an еxсellеnt A variety of сalсulator aсtivities wеre suggеstеd whiсh should all сontributе to your undеrstanding of, and сonfidеnсе with, dесimals. plaсе to start.

Finally, hеrе is a сheсklist of thе sort of things you should aim to know аЬout deсimals. Nofe: You will havе thе opportunity of using dесimals again rмhеn wе look at units of mеasurе in Chaptеr 07.

Deсimals сheсklist You should have the aЬility to:

0.5

,

b Vhiсh

о know that the

of the two numbеrs, 0.35 or 0.4, is bigger?

. . . .

[n thе number 0.6, thе 6 stands for 6 3 Ring the numbеr nеarеst in sizе to 0.78 0.7 17010.8 t80 I .08 t7 4 Мultiply by 10: 5.49 ..+ ) Add onе tеnth: 4.9 --+

6 Thе numbеr Ьеlow is

о

.

aЬut

.

4 in

the numbеr 6.L43 refers to four

hundrеdths mark dесimal numЬеrs on thе number linе arrangе deсimal numbеrs in ordеr from smallеst to biggеst multiply and dividе dесimal numЬеrs Ьy 1.0, 100 and 1000 knoцr that 3.45 is half way Ьetwеen 3.4 and 3.5 know that 0.25 mеans 1and 3.75 means 3l handlс units (metrеs, pounds (f,), kilograms) in praсtiсal situations ,

know roughtly what answеr to expесt in a сalсulation involving dесimals.

Answers to exerсises for Ghapter 05 21

22

7 How many diffеrеnt numbers сan you write down Ьеtwеen

0.26 and 0.27?

8 Whiсh of these numЬеrs is largец 24.9|257 or 83?

Summary This сhapter should have helped you to makе the link bеtweеn frасtions and dесimals. I hopе that, aftеr геаding it, you now have a сlearer sеnsе of how deсimal fraсtions (i.е. numbeгs likе 0.56' 45.03 and so on) fit into the way thе number system is

organizеd. !Иhile digits to the lеft of the dесimal point reprеsent thе numbеr of units, tеns, hundreds, and so on' thе digits to thе right arе thе tеnths, hundrеdths, thousandths, and so on.

Your сalсulator should havе providеd you with most of thе

answrrs tо these еxеrсisеs. Horrеveц hеrе arе somе of thе main points.

5.L Keу sequence Frасtion Deсirпаl Сonclusioп 1

гт-l2 г=l

2 Гтs 4 =.

5 Г+l 10 Г=l

\

0.5

Z 4

0.5

-lt0

0.5

100 50 Г5l 100 Г=l lo

1Г+l 4

Е]

2

гтl

5

ГТl 20Г=l

8 Г=l

1 8

.L

20

0.5

0.25

0.2s 0.2s

Thе dесimal for

iь

|Т.sl

25Г}-.1100Е ffi

3Г5l4t= 6г=-l s Е] 75rта 100 Е

0.25

1

0.75

8

0.75

;й

0.75

Г5l 10 г=l lo10 Гтl 100 г=l # S.2Fraсtion ь \ t Deсimal 0.5 0.25 0.75

Calculаtion

0.5 Г+l 0.5 Г=l 0.5 Гx l 2Г5l 0.25 ГП 0.5 ГХl

Answers to praсtiсe exerсise la

The dесimal for

iis

0.35 0.4

[о:ъ-l

0.1

1

5.3

Thе deсimal foг *is lбЭ5l

Thе dесimal for*'is |T.Л

0.1

+

+

з

*й * +

+

0.7 0.2 0.4 0.3 0.9 0.05 0.125 0.33

Press 1.

1.

+г=l 1.

10Гэl 5.

4 гтr 10EI 0.4 0.1 Г+l 0.1 Г=l 0.2 0.1 Г;l 10Е1 1'.

5.4 100 Е 10 Е] 10 Е1 10 Е1 1 tт-1 10 Г= 0.1 Е 10 Е 0.01 400 E 10 tз 40 Е 10 tз 4 tI] 10 Е 0.4 tn 10 E]0.04 25 E 10 Е]2.5 Е 10 Е 0.25 Е] 10 Е 0.025 tт-] 10 Е

0.5

b

0.4 is biggеr than 0.35 In thе number 0.6, thе 6 stands for 6 tеnths.

3 Ring thе nuцrЬеr nеarеst in sizе to 0.78 0.7 t 70 /Q9/ 80 | .08 t 7 4 Мultiply by 10: 5.49 -.+ 54.9 5 Add one tеnth: 4.9 --+ 5.0 1

6

0.0025 1 Е] 100 Г=] 0.01 4 Е] 100 0.04

7

E

Е

1000

EI

0.007

5.5 Calсulation Calсulation in Dесimal answer Fraсtion in fraсtions deсimal form (use a сalсulator) аnswer

Ьx*

*xl

lxl

5.6 No сomments

0.5 х 0.5 0.5 x 0.2 0.6 x 0.5

0.2s

*

0.1

a10

0.3

10

21

22

This numbеr is aЬout 21.85 Thеrе arе infinitely many numbеrs bеfwееn 0.26 and 0.27' For еxamplе, I сould writе out thе thousandths: 0.261, 0.262, 0.263, and so on up to 0.269. Thеrе arе ninе of thеse. But bеtwеen, say,0,262 and 0.263I сould write out ninе furthеr numbеrs, еaсh еxpressеd as a tеn thousаndth; 0.262,l', 0.2622,0,2623, and so on. Then [ сan write out numЬеrs in hundrеd thousandths, millionths, and so on. This proсеss сan сontinuе indеfinitely or until I fall ovеr rмith exhaustion' Although 24.9L257 сontаins more digits than 83, its valuе is only about 25, so 83 is larger.

Guess the number A game for two playеrs, basеd on the сalсulator сonstаnt.

Plауer А sесrеtly сhooses a numЬеr bеtweеn 1 and 20 - say 72

and prеsses 1 iE 12 Г=10. Thе final 0 is pressed in ordЁr to сlеar thе display. ([f your сalсulator has a .douЬle prеss'сonstant' thеn prеss 12

-

tn ttr

0 instеad.)

Plауer B has to guess whiсh numbеr A has сhosеn to hide in thе сalсulator сonstant by trying diffеrеnt numЬеrs and prеssing Г;l. Thе аim is for B to guess A's numЬеr in the fеwеst possible

guеsses.

Samplе play: B's attеmpts to guеss thе hiddеn numЬеr 1.2 are as

follows.

B presses

Display

Comments

16l'=l 15г=l

1.3333333 1.25

16 is too big 15 is too big 9 is too small '12 is the hidden number

9 Г=l

12 Г=1

o.75 1.

o o o

т'

-

э

1+

s)

GT

o a

ln this chаpter you will tearn: о why percentages are important . about the оonneоtion between percentages, fraсtions and deоimals о how to do percentage calсuIаtions с about сommon difficulties that people experience with perсentages.

Therе are roughly 20 million tеlephonеs in Russia and only about one million in lrеland. So it would seеm that people in Russia arе bettеr off in rеspeсt of aссеss to tеlеphonеs than thе Irish.

Thе faсts arе right but thе сonсlusion is wrong when you rea|ize

that the population of Russia is aЬout eighty timеs that of Irеland. In faсt about 75 per сеnt of housеholds in Ireland (that is, 75 out of еvеry 100 housеholds) havе a telеphonе, whereas only about 15 per сent of Russian housеholds havе onе.

Failing to сomparе likе with like сan rеsult in quitе inсorrесt сonсlusions, as this exаmplе has shown. Perсеntages arе a usеful

dеviсе for making fair сomparisons. Unfortunately, many pеople find pеrсеntagеs diffiсult. Govеrnmеnt rеports and eduсational resеarсhеrs havе сonfirmеd that amongst adults thеrе is a widеsprеad inаbilф to undегstand perсеntagеs. Yet this is dеspitе thе faсt that you сan't piсk up a newsPapеr or watсh TV without сoming aсross thе word Pеrсentage ovеr and ovеr. I opеnеd a dai|у newspapеr at random and quiсkly piсked out thе following two еxamplеs.

Rеad thе сuttings nour аnd try to makе sensе of how thе word perсentagе is bеing usеd. You will gеt anothеr сhanсe to read them at the end of the сhaptеr.

Labour attaсks mortgage insuranсe p|an u.du.l slmrmn.' сommunlty A'h|]. сo0т!.poпdant

l abour lаunсhеd аn assault lзn thе Govеrпmеnt's mort-

gagэ insuгаrrсе sсhеmе yester.

day, aосtsing miлЬtеrs of bеlг4,ins homеownеrъ with

mislеаdiпg promisеs. Aссord. ing to lяbour rcseагсtr, houу ing сosts will soon b.e 25 рr сmt highеr thaп they werе а

yeаr ago.

Gordon Brown'

slrаdow

фалcellor. told a nеws сonfеr. еnсe thаt thе сombiпаtion of highrr intefest rаtes' сuts itl mortgздр tшк rеliеf шtd the

"tt'!eat"

Italy's long dolсe vita REsЕARсtIЕRs are фing to ninooint the rcasons for thе Ьnl'er,itу among residеnts in

in oсtobеr. t4.000mаrt ',rar

сhildren don,t really undеrstand

rлrhat a perсеntagе is.

What is a percentage?

Thе first thing you should rea|ize аbout a perсеntage is that it is very similar to a dесimal and a fraсtion. Likе thеm, a Pеrсеntаgе is usеd to dеsсribе a .bit' of a numЬеr. But really it is nothing morе than a partiсular sort of fraсtion. Think baсk to Сhaptеr 04 on fraсtions, whiсh eхplained hоw two of morе fraсtions сould Ьe equiuаlent. Hеre arе four fraсtions whiсh arе еquivalеnt:

Lzl-so zt 4; 10r

100

Thеsе fraсtions arе equivalеnt Ьесause thеy sharе thе samе valuе of a half.

Now look at the last of thеsе four fraсtiоns, #. You might read it as .fifty oцt of a hundrеd'. A shoшhand way of saying this usеs the Latin word per cеntutt' mеaning .out of evеry hundred'.

So,

*

is thе samе thing as .fifty pеr сеnt'.

...<.д-.\

LTучч:у !Иеll, if a half (i,е. #) is thе samе as 50 per сеnt, what do you think а quarter Ьесomеs as a perсеntagеГ Thе аnsrмеr is 25 Ьесausе

*=

,',', or 25 per сent.

Campodimеlе' a mountaintop villаge аЬout 70 miles southеаst of Romе. Thе shrdiеs srцsest thе hаrrrlеt is a nеаr.ideal

Thе symЬol for .per сеnt'is

hеalthy buсoliс living. Мore thаn l0 pеr сeпt of thе йllagе's 9(Ю rеsidеnts аrе bеtrreеn 75 and 99 yеаrs old, the авэ of фe oldest геsidеnt. not "Cапtoodimеlе uniquе, Ъut t ЬеШеr'e ifs about аs сlose as you сan get to фe perЫ еnvironmеnt for a long

Ghanging a fraсtion to a perсentage By now you might havе цrorkеd out for yoursеlf how to сhange a frасtion to a pеrсеntage. If not' you сan геad thе mеthod цrhiсh ['ve summarized in two simple stеps bеlow. Lеt us takе thе еxample of сonverting the fraсtion 1 to a Perсеntage.

said.

Step

йnion of the сlаssiс Mediьrrдreaп diet and

is

life.'' Dr Alessаndro Mепoffi

of mort8аge

insrrr. arrсe mеапt that tпйсаl hous. iцs сosts' wltiф werе t3'ДЮ a lаst April' would рss thе

The reason for the сonfusion that most peoplе havе with perсentages is, I think, quite simplе. Many adults and most

Souгсе:

cшrdiаn2/2/95

"/o.

So 25% is really anothеr waу of writing iй or 25 per сеnt.

1

Change thе fraсtion into its dесimal foгm f - 0.8 ([f you find this hard to do in your hеad, prеss 4 .5 [E on your сalсulator.)

til

Step

2

sliсеs of the wholе .сake' (quartеrs and tenths' rеspесtively) arе not thе samе sizе. In order to makе a propеr сomparison, thе fraсtions neеd to Ье brokеn down to thе same sizе of sliсe' and hundrеdths are very соnvеnient. So hеrе goes ...

Еxprеss thе аnswеr in hundrеdths.

0.8 is the samе as 0.80, or 80 hundredths.

So, $ сonvеrts to 80%.

Yоu will probaЬly Want to praсtisе this, so try Еxеrсisе 6.1 now.

ExERсlsE 6.1 Changing fraсtions to perсentages Fill in thе Ьlanks in thе taЬlе bеlow. Thе first onе has Ьееn donе for vou.

Fraction

Decimal fractian

I 2

0.5

Percentaoe 5oo/o

137L

Frаction

Hundrеdths

4 -z_ 10

Perсentаgе

75% 70%

100

70 100

Clеarly 75"Ь is bigger rЬan70Y", so wе сan now сonсlude that

i is

biggеr than

ro..

L

If you look at a praсtica| еxamplе you will get a Ьеtter idеa of hoцr useful pеrсеntagеs are.

1

!Иhiсh of thе folloцring would rеprеsent the biggеr priсе risе?

10

-L 20 5

Likе fraсtions and dесimals, perсеntages сan Ье геprеsentеd оn thе numbеr linе. In the pеrсеntagе numbеr linе Ьеloщ 100% сorrеsponds to thе number 1,200Y" to 2 and so on.

a Brеad to go uP by 6p per loaf b A refrigerator to go up by {5

In onе sense thе answer сould be b, beсausе {5 is morе than 6p.

But, sinсe most people buy many morе loavеs of brеad than thеy do fridgеs, wе would pгoЬably bе morе сonсеrnеd if bread wеnt up by 6p pеr loaf. Thе only fait waу to сomparе thеsе priсе rises is to aсknowlеdge that 6p is a lot сomParеd with thе priсe of a |oaf. of bгеad, wherеas d.5 may not be so muсh сompared with thе priсe of a rеfrigerator. Using pеrсеntages allows us to makе сomparisons, taking aссount of thе priсes of еaсh itеm.

So, if wе сonvert thesе priсe rises to perсеntagе priсe risеs, a vetУ diffеrеnt piсture emergеs. In Еxеrсise 6.2 уol arе asked to havе a 8o at сalсulating thеsе tцro pеrсеntagе inсrеases. Dont тrorry if you сant do it straight^waУ, as thе method is еxplained bеlow.

ЕxЕRсlsE6.2 perсentage number line

GаlcuIatingpercentаgeinоlэаses

Complеtе thе table Ьelow. (['ve taken thе original priсe of Ьrеad to Ьe 60p per loaf and that of the refrфrаtor to bе {100.)

({)

In praсtiсе, pеrсеntagеs arе rarеly reprеsеnted in the form of а numbеr linе but I've inсludеd it to strеss thr similariw with fraсtions and dесimals.

originаl ({,) Brеad 0.60 Fridge 100.00

Why bother with percentages?

Solution The сalсulation of thе perсentage priсе inсrеasеs is illustratеd in

The main advantagе оf pеrсеntages is that thеy arе muсh еasiеr to сompare thаn fraсtions. For еxample, rvhiсh do you think is .!7riшen biggеr, l or *l likе this you сant really say, bесаusе thе

price

thе table over.

Price risе

Pеrсentаge priсe risе

0.06 5.00

Оriginлl

priсe ({)

({) 0.06 .5.00

Priсе risе

Brеad 0.60 Fridgе 100.00

Perсentаge p,riсe 8ж

x 100

ise

1

Еlиi,|PLЕ

5oo/o

= 1'0Yo

soGкs

,йx100=5Yo

In summary then, pеrсentagе priсе inсreasеs (оr dесreasеs) arе сalсulatеd as follows.

Improve your soсks |ife!

{4Ч$*+*

уs6

Pеrсеntagе priсе inсrеas. = Х 100. \-,rlgrnаI prlсе

is that we tеnd to buy Ьrеad еvery wееk, so this pйе risе is фесting our shоpping Ьill evеry wееk. Fridgеs, on thе othеr hаnd., arе a lpry rarе purсhasе and еvеn a i.5 priсe risе will simply not affесt most peоple most of thе timе.

What is the sa|e priсe of a pair of these soсks?

Solution

Since 50% is

*, ttrere is a reduction of half of 92.50. Тhis reduсtion is = Е1.25.

So the new priсe is Е2.50. Е1.25 = Е1.25.

LеtЪ nour look in morе detail at how to сalсulatе perсentage inстеasеs and rеduсtions.

Е)(AмPLE

Galсulаting percentage inсreases аnd reduсtions wherе thе pеrсеnиges сonvеrt'to very simple frасtions (е.g. 100 per сеnt' 50 pеr cent,25 pеr сent or 10 per сеnt)' it should bе possiЬle to do thе сalсulation in your hеad. Howeveц for anything more сompliсatеd, I would always usе a сalсulator to сafсulаtе pеrсentagе сhangеs. Hеrе, first, arе two examples whiсh сould probably Ье donе in your hеad.

a pair

Now 50o/o oft!!

Perhaps you werе aЬle.to сonfirm my сalсulation that brеаd wеnt up in priсе Ьy 10 per сеnt rмhеrеas the fridgе went up Ьy only 5 pеr сеnt in priсе.

T!r9rе is, of-сoursе, another reason that this inсгеаse in the priсe of bread vrill саuse more сonсеrn than that of thе rеfrisеratЪr. It

rэduсtion

Еf

2

5olo

increase

2001, the average price of a new house in a partiоular town in the Midlands was Е104 500.

In

over the next year, priсes of now houses in the town inоreased by about 5%. Еstimate the avorage priсe of a new house a year later.

Solutlon

й

we know that the prices rose ьy

*

Еstimate of the average price of a new house a Уear ]ater

=

Sinсe 5% is the same as

over this period.

3o, price rise = =t5225 Adding to the original priоe, We get:

Ж

Е1o4 500 + Е5225 = Е109 725.

So muсh for сalсulating simplе pеrсеntagе inсreasеs using pеnсil and papеr only. Unfortunatеl5 most perсеntagе сalсulations arе

morе сompliсated than this and rеquirе a сalсulator. The mеthod for сalсulating pеrсentagе priсе сhanges is еxplained in thе nеxt two ехamples.

E)иiJ|PLE Fo|lowing

3 a

6%o

inсrease

I

f

So, the new priсe = 1.06 x 82.1= 87.026p.

Тhis rounds to 87.0p, whiоh сonfirms the previous answer from

budget announсement on petro| tax, garages

inсreased a|l their pump prices bу

'i

It may not be obvious to you where the 1.06 comes from. lt is helpful here to think in terms of hundredths. Before the 6% price inсrease we have iffi of the given аmount. Adding 6Yo wi|| increase this to ]# + *.= l*ш, which equa|s .t.06.

the two-staged method.

60/o.

ЕXAмPLЕ сurrent petro| priсes Unleaded premiurn, 82.1 per litre.

4

15% rэduсtion

This time the soсks sale is rather |ess inviting. As you сan see, the

reduсtion now is on|у 15Yol

socкs

All priсes to go up by 6% at midnight tonight.

lmprove your soсks life!

у.s6 apair Now 15% off!!

What is the new priсe of un|eaded premium petrol at this garage?

Solution An inсrease of 69lo means an inсrease of six hr.lndredths, or, in other words an inсrease of 0.06 of the original priсe. A possib|e way of proсeeding here is to perform the calсu|ation in two stages. First' find the priсe inсrease and then add it on to the originaI priсe. As you wi|l see shortIy, there is a quiсker, one. staged method, br.rt you will find this method easier to follow if you fiБt work through the two stages exp|ained below.

Stage

1:

Finding the price inсrease Тhe priсe increase = 0.06 x 82.1p = 4.926p Petrol priсes are usually quoted to one deсimaI p|aсe,

so this priсe inсrease wouId be rounded to the

Stage

2:

nearest tenth of a penny, i.e. 4.9p.

Adding on the priсe inсrease Тhe new priсe = 82.1p + 4.9p = 87.0p (Notiсe that the priсe is written as 87'0p, rather than 87p in order to stress that the priсe has been stated aсоurate to one deсimal p|aсe.)

As was suggestсid above, if all you want to find is the new priсe, this two-staged method is unneоessari|y сomplex. Тhe whole p]ocess оan be reduсed to a sing|e stage, by multiplying the o|d priсe by .1.06.

What is the sa|e priсe of a pair of these soсks?

Solution

a сonvenient fraсtion' it makes sense to do this сalсulation on a сaIсulator. As fior Example 3, I will first do it the long-winded, two-staged way and then more direсtly using the one-staged method.

Stage1: Find the price deсrease The priсe deсrease = 0.15 x 2.50 = Е0.375 Тhis сaп be rounded up to the next penny, i.e. Е0.38.

Stage Subtraсting the price reduсtion Тhe new Price = 12.50 - Е0.38 = Е2.12 Again, sinоe 15% is not easily сonverted into

2:

As was the сase with Example 3, the who|e proсess сan be reduоed to a single stage, by mu|tiplying the old price by 0.85.

Again, it is helpful to think in terms of hundredths. Before the 15% priоe deсrease we have ].ff of the given amount. Subtraс.ting 15% will deсrease this to i# - i*' = loo* whiоh equats 0.85.

So the new priсe = 0.85 x 82.50 =P'125' After rounding, this confirms the previous answer from the twostaged method.

You will nеed somе prасtiсе at сalсulating perсеntagе inсrеasеs and dесrеasеs, so have a go at Еxеrсisе 6.3 now.

ЕxЕRсIsE

6.з

CalсuIаting perсentаge increases.аnd

decreаses (i) A taЬlе normally sells at d42. How muсh will it сost цrith a

(ii)

30% reduсtion?

Chесk my garagе bill.

Fullserviсe repairs аnd

17.5% Тоta| (inс. VAТ) VAт

@

parts

Е186.40

Гтг{j6,т|

Can you spot where this сhild has gone wrong?

t I

{

t

The problem is that she has stаrtеd from a truе faс that 70'Ь = and built up a rule тyhiсh doesnt work for any other fraсtion.

This is probably the most сommon misapprehension about

perсеntages. If you still havе proЬlеms with this, thе сhanсes arе thаt thеy сan be traсed baсk to a fuzzinеss about fгaсtions. You may know tllat 20Y" is morе than 5Y". Howеvеr, it is not the сase that *'is morе than l. If you think baсk to сhaPter 04 and the idea of a fraсtion being a sliсe of сakе, thеn imaginе a сakе сut into twenty еqual sliсes. Еaсh of thesе sliсеs is a twеntieth of the сakе and is thеrefore a very small sliсe indeеd. one fifth, on thе othеr hand is a large sliсe.

ГgжЕl

*

= 5o/o

(iii) If you earn t230 pеr wееk, rмhiсh would you prеfеr? A risе of.: a 6Y" or b {,L2 per цrееk? (iv) Your taxable еarnings are {884 this month. How muсh of this will you have left after paying 33Y" in stoppages?

Persistent problems with percentages will bе no surprisе to you to bе told that a lot of сhildrеn's timе in sсhool tаkеs plaсе with onе еye.,shut and thе other staring out of thе window. Мany сhildrеn сomе away from a It

lеsson in pеrсentagеs (or whatеvеr) with only a fеw piесеs of the

Shopping during thе salеs is an opportunity to сhесk out somе of thesе ideas. For example, l off is a bettеr disсount tЬan 10Yo off. Also rеmеmber that 40Y" off thе priсе of somеthing-fairly сheap like a paсket of envelopes rеPresеnts only a small saving in асtual monеy' цrhereas thе sаmе Perсеntage reduсtion from, say' the priсе of a housе reprеsrnts a huge saving.

jigsaw аnd have to somеhow fill in thе геst of thе piсture themsеlvеs. Unfоrtunately thеy don't always gеt it right ...

Fractions

Deсimals

Perсentages

fi

9

Perhaps thе most important thing you nеed to grasp is thаt fraсtions, deсimals and pеrсentagеs аrе геally thе sаmе thing. I

havе found that drawing thrеe numЬer linеs one aЬovе thе othеr is a helpful rмay of еmphasising thеsе сonneсtions, as shown on thе prеvious pаge.

Thе arrows shorм that 3, 0.6 and 60Y" have the samе valuе.

Percentаgе cheсklist Yоu should now be ablе tо: . rca|ize that сonvеrting to pеrсеntagеs makes it еasiеr to сomparе fraсtions о link pеrсеntages to сommon and dесimal fraсtions . .o,,uЪft frorn-perсеntages to (simple) fraсtions and dесimals and viсе vеrsa, e.g,750Ь =1= 0.75 . еxprеss somеthing as a pеrсеntage of somеthing еlsе, е.g. 6 is 25Y" of 24 . сalсulatе pеrсеntagе inсreases and deсreases.

Practiсe exerсise 1 Whiсh is biggеq 8"/" or

2 3

*

Study thе newspaper сцttings on Page 74 andthen аnswеr thе

following quеstions. a From thе fitst сutting' thе final sentenсе еnds by stating thаt .typiсal housing сosts ... would pass the d4000 mark in OсtoЬer'. Usе thе information from the rеst of the

Ь

6.| Frа.ction

t 3 4

-z_ 10

I J20

Ь?

a going up? b сoming down? с nеithеr?

Vhаt is 20Y" of {80?

4 5 Vhat is 1.0% of 20Y" of {80?

6 Hеrе afе somе еgg priсеs bсforс

1)

чgs

Largе.еggs

(per hаIf dozen)

Neul priсе (per hаlf dozеn)

7l

75

B4

instеаd

of. L7.5"/o?

Why do сhildrеn's sweеts tеnd to suffеr grеatеr inflation than mоst of thе othеr things wе buy?

is сorrесt.

Decimаl 0.5

0.7s 0.7 0.2 0.05 0.6

0.375

frасtion

(Ю

Perсеntаgе

s0% 75% 70"/o

20%

5%

60%

371%

6.2 Commеnts in thе text.

6.3

(i)

Thе reduсion in priсе is 30% o' t. тh'"" tеnths of {'42 сan bе found on your сalсulator by pressing еithеr 3 Гтl 10 rЯ 42=-l oг 0.3 t! 42Г1;1 both of whiсh givе the сorrесt аnswеr {12.60, So фе rеduсеd priсe is [42 _ f|2.60 or f,29.40.

(Nofa A quiсkеr way of doing thЬ is to say that a 30Y" reduсtion will bring thе priсe down to 70Yo ol the old priсе. on thе сalсulator you would press 0.7 E] 42ra, whiсh givеs thе answer {29.40.')

(ii)

Thе VAT is inсorreсt. Using a сalсulator:

88

Vhiсh has had thе grеatеr priсe inсrеasе: small or large еggs? Мy garаgе bill has сome to {,120.93 and inсludes VAТ at t7.5%. what rмould the bill bе:

a rмithout VAT? b if vAт werе ratеd at 25уo

E

and aftеr a priсе risе.

old priсе

fфre

Answers to exerсises for chapter

Whiсh is biggец 15"Ь ot *?

If thе rate оf inflation drops from 5Y" to 4Yo, arc priсеs:

Small

artiсlе to сheсk that this

Turning to thе sесond artiсlе on thе samс Pagе' еstimatе thе numbеr of rеsidеnts in Campodimеlе who arе aged betwееn 75 and 99 vears old.

Prеss 186.40 гг| 0.175 ГТI to givе the answer {'32.62.

So I'vе bееn overсharged by €10.

Notе: thе dirесt mеthod for сhесking the final Ьill is to press 186.40 |тI 1.175 t = t

(iii) 6% of {230 = {13.80, whiсh is a biggеr rise than {12. (iv) I will have 67Y" of d884 lеft, whiсh is {'592.28.

Answers to praсtice exercise

Гтl

8

ГП

199

г=l

2 fr is apprоximatеly еqual to 6.7уo, so |5"Ь is bigger than 3

15a.

The annual ratе of inflation mеasures how muсh average

priсеs have risеn ovеr a Уeat.If that ratе is a positivе numbЪr (suсh as 4oЬ or 5Y", fot ехample), this mеans that priсes havе risen. So, еvеn though thе ratе of inflation has fallеn, the сurrеnt tate of 47o shows that priсеs arе still rising, Ьut not quitе as quiсkly as thеy Werе ovеr thе previous yеar.

4 20Y" is l. One fifth of {80

) LIY" of 20"Ь

=

ry

=

{!6.

is one tеnth of 20"/" = 2Y"

2"/" of {80 = {80

xh

= d7.60

Thе solution is summarizеd in thе taЬlе bеlow.

(p)

(p)

Price iпсreаse (p)

Perсentаge inсreаse (rounded to 1 decitпаl pldcе)

71

75

4p

*x100=5.6Y"

84

88

4p

*x100=4.8Y"

Оld priсе Nеul priсe Small

еggs Largе

еggs

Ь

To сalсulatе thе bill inсlusivе of VAT at 25уo' wе multiply thе nеt bill Ьv 1.25.

So, although Ьoth еggs have sееn thе samе aсtual priсе risе (4p in еaсh сase), thе small еggs havе shown thе grеater

pеrсentаgе risе.

7 a This

сalсulation is slightly hardеr than thе othеrs, as it involvеs working baсkwards aftеr the pеrсеntage inсrеasе has beеn addеd. Thе story linе of thе solution is as follows: Lеt thе bill without VAT bе thought of as 100"/o аnd thе

bill with VAT (сosting t12О.93) as 11'7.5Y". So wе must dividе thе total bill by L|7.5 and thеn multiply Ьy 100.

x

1:25 = t 128.65.

Whеn perсеntage inсrеasеs arе аppliеd to thе priсс of goods, it is gеnеrally thе сasе that fraсtions of pеnсе get roundЪd up.

For examplе, if' a Ьat of сhoсolatе сosting 36p is subjeсtеd io a L0-"/" inсrеase, thе truе priсe should Ье 39,6p. However, as shopkееpеrs саnnot сhargе 0.6 of а pеnnь this priсе is likely to Ье roundеd up to 40p. This represеnts a loss of 0.4p to thЬ сustomеf and, a loss of 0.4p reprеsеnts a muсh grеatеr prоportion of somеthing сosting 40p than of an itеm сosting' t1,02.92

ь= 12ь% and so is biggеr than8Y".

No/e: You сan use your сalсulаtor to сonvеrt to a pеrсеntagе by prеssing 1

t120j3 Х #Ъ = 702.92 (rounded to thе nеarеst pеnny).

So the nеt bill (i.е. not inсluding the VAT) = {102.92. i.e.

saу, {'40. Sinсе thе things that сhildrеn buy (сomiсs, sweeй'

еtс.) tеnd to Ье сhеap, сhildrеn losе out from thеsе rounding lossеs morе than adults.

a Еarlier in

the artiсlе, thе сlаim was madе that housing сosts would soon Ье 25 per сent highеr than thеy wеre i Уear ago.-To сhесk an inсrеase of 25% from a starting

уa!u3,o|f'з|99'J'q'' |.2s Г->13200 Г=l

thе following On your сalсulator:

-

This сonfirms thе rеsult of t'4000 mentionеd in the last sеntеnсe.

Altеrnativеly, you might bе ablе to do thе сalсulation in yоur hеad, as follows.

25Y" is onе quartеr' and onе quarter of {3200 is {800.

Addingwе gеt d3200 + {800 = {4000.

Finally, howevеr, you also nееd to сhесk that the 25Y" inсreasе rеlated to thе full onе yеar pеriod for whiсh it applied. Sinсe thе сomments wеrе madе in Fеbruary 1995, the pеriod from (thе prеvious) Ьprt| 1994 to (ttrе nеxt) Oсtobеr 1995 сovеrs morе than onе full yеar (aсtually aЬout 18 months)' so thе сlaim doеs sееm justifiеd.

b This

artiсle talks about .morе thаn 10 pеr сеnt' of 900

pеople.

Sinсе 10% = 16a, onе tеnth of 900 = 90. So therе arr morе thаn 90 residеnts agеd Ьеtween 75 and 99 years old.

It is sаid of frogs that thеy sort all othеr animаls thеy mееt into just thrее сatgoriеs.

If it is small, thеy еat it.

If it is largе, they run arмay from it.

And if it is аЬout thеir own sizе, thеy matе with it.

I.фink it,s fair to say that, in gеneral, humans аrе slightly mоrе disсriminating! Any aсtivity rмhiсh involvеs making judgЬmепts aЬout thе sizе of things сan Ье сallеd measuring. Alйough frogs may not Ье еngagеd in higlrly sophistiсated mеisuring herе, thёy аre trУ.ing to undеrstand thе Ьignеss or smallnеsJ оf thingЬ around thеm.

What do we measure? Мost pеoplе tеnd to think of mеasuring as using weighing sсalеs or a tapе.!Иеighing mеasurе. But what sort of thing do thеsе dеviсеi tеll us aЬout? sсales tеll us about wефt and a tapе mеasurе

aЬout lеngth. Thеsе typеs

dimensians.

o !) a з

of

mеаsurеmеnt are сalled

Dimеnsions of mеasurе arе mеasurеd in сеrtain measuringunits.

For ехаmplе, -weight may bе mеasurеd in kilograms, grams, ounсеs' pounds, and so оn, whilе lеngth may Ье mеasurеd in сеntimеtrеs' mеtrеs' inсhеs, milеs' and sO on.

Th9r9 ar9, of соursе,-many dimеnsions othеr than length and weight whiсh wе nееd tO mеasurе' for examplе: tеmpеraturе, timе, area, capaсiry anglе, volumе, spееd ...

Е .т П

Еxеrсisе 7.1wi|| givе you a сhanсе to think aЬout thеsе dimеnsiоns and also аЬout the units in whiсh thеy arе usually mеаsurеd.

ExERсlsЕ 7.1 Dimensions and units Completе thе tаble (thе first two havе beеn donе for you).

э

GI

Question

ln this chapter you will loаrn: о about meаsuring dimensions

. .

and units how to round numbers how to сonveЁ un.ts of mgаsu]э.

How How How How How

How How How How

heavy is your laundry? |ong is the сurtain rail? hot is the oven? far is it to London? fast can you run? |ong does it take to сook? muоh does the jug hold? big is your kitсhen? big is the field?

Dimension

of measure

Likelу unrts of measure

weight length

сm or in

kg or lb

ExERоlsE 7.2 Being awaпe of the dimensions of meаsure Hеrе is a list of еight сommon dimеnsions of mеasure..

Length (L), Area (A), Volumе (V), !Иеight (!r), Timе (T), ъmperaturе (T.), Capaсф (C) and Speеd (S). Маkе a notе of whiсh of them arе likely to be important in thе following everyday aсtivitiеs. (Note: therе may Ье sevеral dimensions involvеd in еaсh aсtivify.)

As you seе, I havе inсluded trvo lots of units in thе examplеs abovе Ьeсаusе both arе in сommon usagе. They arе knolмn as thе mеtriс system of units and thе imperial systеm of units.

Bесausе many peoplе are сonfused by thеsе various units of mеasurе' they arе еxplained in somе detail later in thе сhaptеr.

Why do we measure?

Thе rеason Wе measurе is that, quitе simply, we live in a morе сomplex world than a frog. Although words likе .largе' and .small'are sometimes good enough for somе partiсular Purposеs (.Give me somе of thе lаrge app|es', .['d like a smаll hеlping', and so on), oftеn wе need to be more prесisе. Hеrе is an eхamplе whеre the word .large' provеd inadеquatе during a national rail strike.

Еuerydау аctiuities

o Baking о Buying

.

о

a сakе

and laying a carPet Chесking the сhildrеn's shoеs Setting out on a journey in good timе

A vегy largе proportion of thе staff didnt show up for work.

(Union spokesperson)

How do we measure?

A largе proportion of thе staff showеd up for work.

is basiсally a waу of desсriЬing things. Desсriptions 'Меasuring сan сomе in two basiс forms. First, thеrе.arе desсriptions of quаlitу and thesе tеnd to bе madе with words. For еxamplе, yоu may dеsсriЬе yorrr various friеnds as happ5 carefree, moody, thoughtful, sensitivе, and so on. Thеsе arе not thе sorts of dеsсriptions that еasily lend thеmselves to being reduсed to numЬеrs. Desсriptions of quапtitу, on thе other hand, do involve numbеrs. For ехamplе, Ann is 1.59 m tall, Donal is 73 .Whеn yеars old, Chris has 5 сhildren, and so on. pеoplе talk aЬout mеasurеment' thеy ate usually thinking about

(Managеmеnt spokеspеrson)

Thе use of thе word .largе, in thеsе quotations is highly dubious.

How large would thе proportion havе to bе for you to сonsider it .large' _ 10уo,25Y" or perhaps 70%? ... [t is intеresting that both sides in thе dispute have Ьeen dеliЬeratеly vaguе about thе exaсt figurеs and prefеr instеad to givе a gеnеral impression.

Somеtimеs, howеvеr, a gеnеral impression is simply not good еnough, and somеthing more prесisе is nееdеd. For еxamplеj you may have sеen signs on thе motorwаy advising drivers of .large' vеhiсles to stop at thе nеxt emеrgenсy phonе and сontaсt the poliсe. [f you are driving a |orry, how would you know whеthеr this rеfегrеd to you? Rest assurеd that thе small print Ьеlow thе sign goes on to eхplain that: .Largе means 11, 00" (3.3 m) wide or over' Thе rеason that wе tеnd to mеasurе with numbеrs is to help us makе dесisions and сomparisons fairly and aссuratеly. Carеful mеasuring helps us bakе .thе pеrfeсt сake'еvery time, lay well-

fitting сarpets with

a

minimum of waste, сhесk that the

сhildrеn's shoеs don't pinсh and so on. In Еxerсisе 7.2 уou arc

askеd to think about thе mеasuring dimеnsionЬ involvеd in

thеse sorts of еvеryday tasks.

I 1

t I

mеasuremеnt based on numbers, Ьut not alrмays.

Somе mеasuring sееms to fall bеtwееn quаliry and quantity. For еxamplе, you might dеsсribе somеthing as bеing largе or small,

fast or sloщ сhilly or warm. Thеsе arе ways of indiсating

whereаЬouts on somе sort of sсаlе (rеspесtivеly thеy rеfеr to

sizе, speed and tеmpеrаture). Yеt, although thеy makе no mention of numbers, thеsе dеsсriptions are a soгt of mеasurе.

Suсh words сan Ье rаnked into a mеaningful order and thеy thеn produсе what is сallеd an ordering sсalе. on thе othеr hand, words whiсh dеsсribе, saь an еmotion or a сolour, do not normally rеlatе to a useful sсalе. Thus, you сant say that

.сurious' is Ьiggеr than .еxсitеd' or that .rеd' is morе than .bluе'.

Suсh words arе simply dеsсriptions' Еxеrсisе 7.3 wi|| give you praсtiсе at using an ordеring sсalе.

ЕxЕRGlsE

7.3

How aссurately shou|d we measure?

Using and ordering scale

Hеrе arе fivе words usеd in desсribing how likelу sоmеthing is to happеn:

Thе aссuraсy with whiсh wе measurе dеpеnds еntirеly on what and why wе are measuring. A Ьrain srrrgеon and a tree srrrgеon

likel5 impossiblе, douЬtful, сertain, highly improbablе

have diffеrеnt nееds for aссuraсy when s;iwing up thеir

respeсtivе .patients'. A nursе wеighing out drugs will ехеrсise greater сarе and preсision than a grееngroсer wеighing out potatоеs. A саlсulator сan somеtimes givе a falsе sеnse of thе aссur1сy of an answеr. As the ехamplе below shows, it may givе

Thеse dеsсriptive words сan bе writtеn in ordеr of likеlihood, from lеast likely to most likеly, thus: impossible, highly improЬablе, doubtful, likеl5 сertain

a result showing eight-figurе aссuraсy but thе numbеrs on whiсh thе саlсulation was performеd may be only approximatе.

Supposе you wish to rеplaсе the fеnсе in your gardеn. Thе lеngth of fenсing nееdеd is, sа5 21 m and еaсh panеl of fеnсing is 1.8 m in length.

Prеssing 21 Гт1 1.8 Г;l on yorrr сalсulator will proЬably

Now, hеrе arе somе for you to do.

Rank the following sets of rмords into usеful ordering sсales.

1

Thеsе fivе words аrе usеd in desсribing ways of travelling on foot: jog, stop, sprint, rмalk, amblе

2

Thеsе arе thе dеvеlopmental stages that ЬaЬies usually go

produсе thе аnswеr L1.666666 (for rеasons that will bе explainеd shortly, on somе сalсulators thе answеr will bе shown as L1.666667). Fot this sort of сalсulation, it is plainly silly to givе an answer to eight figurеs. If the last siх digits of your answеf arе eithеr dubiоus or unnесеssary then disposе of thеm.

Howеvеr, you havе to Ье a bit саreful how you do this.

NumЬеrs сan bе shоrtenеd so that you finish with а suitablе number of digits (say thrее). This is сalled giving your answеr

throuф;

walk, lie, sit up, stand, roll ovеr

3

.сorreсt

Thesе words arе oftеn writtеn in sequenсe on an еlесtriс iron:

T7ith thе aЬovе еxamplе, сalсulating thе numЬеr of panеls of fеnсing rеquirеs that you buy a wholе numbеr of panеls, so the answеr will be given сorrесt to fwo signifiсant figurеs. tn this сase 1',!'.666666 would Ье roundеd up to 12 panеls. Nole: You would still nееd 12 panеls еvеn if thе answеr on thе сalсulator was 11.333333!

wool, linеn, silk, сotton, rayon

To summarize, thеn, measuring сan take thе following thrеe basiс forms: . цrords alonе . words whiсh сan Ьe rankеd in ordеr . numЬеrs.

This proсеss of simplifying

unnесеssarily aссuratе

mеasrrrеments to a nеar approхimation is сallеd rounding. Some еxamplе аre givеп in TaЬlе 7.1 below.

Thе typеs of mеasuring sсalе whiсh thеsе three appгoaсhеs usе atea

. . .

to thrее signifiсant figurеs' (or .to 3 sig. figs.,, for shоrt).

Measurement

words ordеring sсale numЬer sсalе.

Although all three typеs of sсale are helpful in providing an interesting variety of dеsсriptions and сomparisons, it is thе third of thesе, mеasuring with numbегs, whiсh is thе most important in mаthеmatiсs.

Rounded ta

3 sig. figs.

4.18345926 371.41429 o.0142419 74З12,692

4.18 З71

11.6666

11.7

0.o142 74300

tablв 7.1 rounding to 3 signifiсant figures

I!? ш

й

с f

Notiсe that thе third and fourth еxamplеs in Table 7.1 Ьave produсеd answers whiсh сontain not thrеe Ьut fivе figurеs.

Howevеr, thе two zеros at the Ьeginning of 0.0142 and the two zеros at thе еnd of 74300 arе not сonsiderеd to bе signifiсant figurеs. They are only thеrе to give thе overall magnitude of thе numЬer. It makеs sеnsе to do this as othеrwise thе number 74372.692 would Ьe roundеd to743,whiсh is сlеarly nonsеnsе!

The last example in the aЬove tab|e, \|.6666, is different fiom thе othеrs in thе following rеspeсt. As you сan sее' its third digit has bеen rounded up from a 6 to a 7. TЬe сluе tо why it has Ьееn rounded up сan be found Ьy looking at the fourth digit in thе original numbеr: thе 6. Sinсе it is Ьiggеr than 5, the 6 in the tеnths сolumn is rounded цp to a 7, !лd this is thе reason that some сalсulators produсе thе answеr 11.666667 for the fеnсе panel сalсulation. Suсh сalсulators have been dеsфеd so that thеy round up thе final digit displayеd whеn thе nеxt digit would have beеn a 5 or grеater. In order to be аble to do this, thеse сalсulators nееd to proсеss thеir сalсulations to greаter aссuraсy than the eight figures that they display, whiсh aссounts

for why thеy tеnd to be slightly more expеnsive than

thе

сalсulators whiсh don't rouпd. Ехеrсisе 7.4 gives you praсtiсе at rounding.

By the way, dont worry if this еxplanation of rounding sounds сonfusing - it is еasiеr to do than to rеаd about!

ЕxERсlsE 7.4 Rounding pract.ce

Round thе folloйng numЬеrs to four signifiсant figures.

Number Ansulеr to 4 sig. figs.

4124.7841 4125 38.4163

1 2 3 291.7412 4 39042.611 5 39048.619 6 з8.41з| 7 446.982 I 0.142937 9 1317.699

10

30s0.1491

Therе arе many praсtiсal situations whеrе сarеful measurеmеnt is еssential. for еxample, drеss-making' сarpеntry wефing out parсеls to сalсulate thе сost of postagе, and so on. Howеvеr, in othеr situations' an estitnаte based on еxрrienсe and сommon sensе is oftеn good enouф. For еxample, when returfing a lawn,

you may wish to mеasurе its arеa fairly aссurаtely using a.tape measurе' but if you dесidе to sееd it, simply paсing it out to еstirnate thе area may bе suffiсiеnt. Еstimatiоn is a skill whiсh gtеatltУ improves with prасtiсе. I somеtimеs find it hеlpful to

imaginе evеryday objeсts of a standard sizе tо hеlp me makе an еstimatе. Fоr еxample:

Estirпаtе о

estimating hеight or distanсе

Hеlpfшl imаgе

- a door is roughly 2 -

mеtres

high a running traсk is 400 mеtrеs around a standard milk bottle holds onе prnt a bаg of sugar wеighs 1 kg

о

еstimаting сapaсirylvоlumе

-

. .

estimating wеight estimatin8 air tеmpеraturе

- typiсаl Winter tempеraturеs 0.с - 10.с - summеr tempеraturеs 20.С - 30.с - spring/autumn tеmреratures ].0"с * 20"с

And now, as promisеd еarlier in thе сhaptец wе turn to thе units that arе used in measuring.

lmperial and metriс units Until about 1970, measurеmеnt in thе UK was largely donе with

impеrial units' Sinсe thеn, horмevеr, the British population has at lаst orмned up to thе faсt that they havе thе samе numЬеr of

fingеrs аnd thumbs as thе rеst of thе world, and have. .gone deсimаl'. The deсimаlization of money in t971was сarriеd out quiсkly and еffесivеly. As a rеsult, most pеoplе mastеred thе nеw

сoinagе within days. It was also intеnded that the familiar impеrial units of lеngth, wеight аnd сapaсity be фasеd out within a fеw yеars. This сhange, сallеd tnetriсаtioп,was to havе sм/ept awаy thе most familiar of the measuring units - fееt' inсhes, yаrds, pounds, stonеs' pints, gallons and so on _ in

favour оf mеtres, kilоgtams, litrеs and thе like. Indeеd, during thе |970s, many сhildrеn lеarnt only the mеtriс units in sсhool on the assumption that the old impеrial units would sоon bе siх fеet (sorry 1.83 mеtrеs) under. Unfоrtunatеly, howеvеr, the сhangе-over was so half-hеartеd that' at thе present timе of writing, wе are still regulаф using Ьoth systеms (and having fun trying to сonvеrt from one to the othеr!). Sinсe the 1980s, сhildrеn have bееn taught both systems in.sсhool.

Convеrsions Ьеtween mеtriс and impеrial units tеnd to involve

rathеr awkward numbеrs. For еxample' therе arе .aЬout' 39.370078 inсhes in one mеtre! Not surprising|у, a number of

half-bakеd approximations have appeared like the metriс yard, the metriс foot and еven thе mеtriс briсk. Havе a look at the tablе below and you will sее just how .approximatе' somе of thе approxrmatlons arе.

Unit Metriс yaгd

,Truе'uаlue

= 39 in

39.370078 in 2.54 cm 30.48 сm 1609.344 m

Mеtriсinсh=2сm

Mеtriс foot = 30 сm Меtriс mile = 1500 m

mi|limetre

-

(mm)

10 -+ оentimetre _ 100 .-r metre

(cm)

lmpeialunits inсh

-

(in)

12

*

foot

-3

(ft)

--+

yard (yd)

-

(m)

-

1760

--+

1000 -+ ki|ometre (km)

mi|e

tablв 7.2 meаsuring |ength in imperia| аnd metriс uпits

Тhe numbers on the arrows tell you how to сonvert from onе unit to anothеr. Thus, there arе 10 mm in 1 сm, 100 сm in 1 m, and so on. [f you want to know how many mm arе in 1m, thеn muфlу thе two numbеrs 10 аnd 100 (i.e. therе are 1000 mm in 1 m). It will help you undеrstand and rеmеmber thе mеtriс units whеn you rеаlizе that: . for еaсh dimеnsion thеrе is a bаsi.с шnit - thе basiс unit for length is thе mеtrе . all the other units get thеiг name frоm thе basiс unit: е.g., Ьесause ceпti- means onе hundrеdth (й), thеn a сeпtitпеtre is one hundredth of a metrе.

Tab|e 7.3 will help you work out thе othеrs.

MILLI- one thousandth

сENтl- one hundredth DECI. one tenth

ф

(#J

ф

KILO- one thousand (1000) tab|в 7.8 metriс prвfixes

Area

a one-dirnensionаl (1-D) measurеmеnt Ьесausе it involves only onе direсtion. Problеms involving surfaсes (size of paper' сarpеts' сurtain matеrial' lаwns, ...) are И,uo-dimensionаl (2-D). Sometimеs rме dеsсribе arеа simply by stating thе lеngth and thе brеadth. For ехample, сurtain matеrial is bought by thе metrе (length) but vrе also nееd to know that thе roll is 1 m 20 Lеngth is

TaЬ|e 7.4, givеn nеar thе еnd of this сhapteц summarizеs most of the mеtriс and imperial units that you arе likely to nееd. I will explain how to usе it by foсusing on the most basiс mеasurе of all - lеngth.

Metric urtits

Сonvеrting betlueеn metriс and impеrial units is a littlе triсkier.

If you don't nееd to be too aссuratе' it is hеlpful to remember that a twеlvе-inсh rulеr is аlmost еxaсtly 30 сm long. Dividing 30 Ьу L2, it follows that onе inсh is roughly еqual to 2.5 сm.

When you nеed to bе mоrе aссurate' usе thе сonvеrsion 1 inсh = 2.54 сm, and also usе a сalсulator!

сm wide. If you arе buying paint, on thе othеr hand, thе instruсtions on thе tin may say somеthing likе: .... сontеnts suffiсiеnt to сovеr 35 m,'. Thе unit dеsсriЬed as a,m,,, or a .squarе mеtrе' is thе Ьasiс metriс unit of area. It means еxaсtly what it says. One m2 is thе arca of a square' 1 m Ьy 1 m, as illustrаtеd bеlow.

Thе arеa of this ]. m x 1 m square is equal to 1 nf.

Thеre should bе еnоugh paint in thе tin to сovеr 35 of thеse squarеs.

Similarly, a ft, (squarе foot) is thе area of a squarе 1' ft by 1 ft, and so on for thе various othеr units of area. Howеver, itЪ all not quitе as еasy as it sоunds. Havе a go at Еxеrсisе 7.5 now and sее if you саn avoid thе traps that people often fаll into.

ЕxЕRоIsE

7.5

Area trаps a 'flow many squarе fееt (ff) are thеrе in onе squarе yard (yd)? b How many сm2 arе thеre in onе m,?

с lVhat is thе arеa of this rесtangle?

d If you doublе the dimensions of this reсtangle (i.е. doublе thе length апd the Ьreadth), what do you do to thе arеa?

Volume

If you ask most pеople about thе word .volumе', they will tеll you that it is thе knoЬ on thе TV sеt whiсh makеs it go loud and quiеt. .Volume', as usеd in mathеmatiсs, is rather diffеrеnt. It dеsсribеs an amount of spaсe in tbree dimensioпs (3.D). If you think of a Ьox, its volumе will depend on the thrее dimеnsions lеngth, Ьrеadth and hеight.

Unlikе thе words .lеngth' and,area,,.volumе' is not a word in .W.е vеry сommon еveryday rrsagе. tend, instеad, to usе tеrms likе:

Holu big is thе briсk? or !Иhat is the size of thе Ьox?

Hоwevеr, thе troublе with words likе .big' and .sizе' is that they don,t nесеssarily refеr tо volumе. In faсt thеy сan Ьe сallеd upon to dеsсribе aлу of a number of dimеnsions. For еxamplе, thе sizе of a pеnсil might mеan its lеngth. Thе sizе of a pieсe of papеr might mean its area. The sizе of a bag of sugar might еvеn mеan its wеight, and so on.

Pеople who work

in the building tra8е

bесomе skillеd at

еstimating amounts of еarth and сonсrеtе. Thеy usually mеasure thеsе volumеs in so many .сuЬеs'. A .сuЬе' usually rеfеrs to сubiс metrе or a сuЬiс foot. A сuЬiс mеtrе is the amount of spaсе takеn up by a сuЬе measuring 1m by 1m Ьy 1m.

Anothеr measure whiсh dеals with thrее dimеnsions is саpaсity.

Thе diffеrеnсe bеtweеn capacitу аnd volume is that capacitу dеsсribеs a сontainеr and is a measrrre to shoцr how muсh thе vеssеl holds. For еxample, wе talk about thе сapaсity of a sauсepan' buсkеt, Ьottle, etс. The units of capacitу, whiсh arе inсludеd in TaЬlе 7.4, are normally only usеd with liquids. Нavе a look now at ThЬlе 7.4, whiсh shows somе of thе most сommon mеtriс and impеrial units for length, arеa, volumе, сapaсity and wеight.

Metic units

Length

mil|imetre (mm)

Area

centimetre, (сm,)

Volume

Capacity

Weight

-

10

--+

сentimetre (сm)

(m) -1000--+ kilometre (km)

heсtare

-

-

10 000 + metr# (m,)

100

-

--+

metre

10 000 ->

сentimetreз (сm.) _ 1 000 000 * metre. (m.) millilitre (ml) - 1000 * litre (l) gram (g) - 1000 kilogram (kg) -1000 + tonne

-

lmperial units

Length

Area

Volume

Capacity

Weight

inсh (in) - 12--+ foot (ft) - 3 --+ yard (yd) - 1760 --+ mi|e inсh,(in,) - 144 --+ foot, (ft) - 9 + yаrd, (yd,) - 4в40 * aсre - 640 _+ square mile in" - 1728 --+ ft. - 27 Уd" fluid ounсe -2o- pint-2 _+ quart-4 + gal|on ounce (oz) -16 + pound (b) - 14 --+ stone - 112 --> hundredweight (сvut) - 20 --+ ton

.

tаble 7.4 metriо аnd imperiaI units

You will proЬаbly nееd praсtiсe at using thеsе tables, so do Еxerсisе 7.6 now.

ЕxЕRсlsЕ 7.6 Getting fami|iar w.th the units of

measuпe

Similarly a сubiс сеntimеtrе (сс) is thе amount of spaсе taken up

bva1сmсubе.

a How b How

с

How

f

Horдr

d How е How

many many many many many many

millimеtrеs are thеrе in a mеtrе? сentimеtrеs are thеrе in a kilomеtrе?

grams are thеre in a tonnе? inсhеs аrе there in a milе? square inсhеs arе thеre in a square yard? ounсеs are thеrе in a ton?

Unfortunatеly it isn't enough to bе able to сonvert units within thе mеtriс оr thе impеrial systеm sеparatеly. Somеtimеs it is nесessary to сonvert betwеen thе two. Thblе 7.5 shows some of the most сommon сonvеrsions betwееn mеtriс and impeгial units.

Accurate convedon

use and undеrstand standard units of thеsе mеasures (сentimеtrе, kilogram ...)

Rough 'n, readу conversion

Lengith 1m=39.37in

1 metre is just over a yard very long stride)

(a

capaсity

Weight

Speed

1 in = 2.54 сm

.1

1 |itre = 1.76 pints

1 litre = 11 pints (a large bottle

1 gallon = 4.54 litres 1 kg = 2.2

pounds

1 pound = 0.454 kg = 45i4 g 100 kmph = 62.1 mph 100 mph ='t61 kmph

inсh = 2*.сm

of orange squash) 1 gallon = 4* litres

1 kg = just over 2 Ib

- a bag

of sugar 1 lb = just under I kibgram

Skmph=5mph

table 7.5 сonvеrting bеtween mеtriс аnd imperiа| units

Again, it would bе a good ideа to praсtisе some of thеsе сonvrrsions noщ so have a go at Еxеrсise7.7.

ExЕRGlsE

7.7

Praсtice exercise

Usе TaЬlеs 7.4 and 7.5 (and a сalсulator whеrе appropriate) to answеr thе following: a !Иhiсh of thesе would Ье a rеasonablе weight for an adult?

60 kg, 600 kg, 6 kg b Is it bеttеr valuе to Ьuy a 25 kg bag of potatoes of a 56 lb bag for thе samе money? с !Иhat is thе hеight of your kitсhеn сеiling from the floor, in

d e

f

mеtres?

Somе Frеnсh roads havе 90 kmph spееd limits. Vhat is this rouфly in mph?

Is * litrе of bееr morе or lеss than а pint?

If wе Ьought milk by the * litrе, how many bottlеs would you havе to Ьuy to havе roughly 7 pints?

о

estimatе lengths, wеights and so on, in terms of thеsе units, to know when a measrrrеmеnt is aЬout right and to bе awarе what sort of aссuraсy is appropriatе usе various mеasuгing instrumеnts (tapе mеasurе, rulеr, wеighing sсales, Ьalanсe, measuring jug, thеrmomеtеr, сloсk) Ьe awarе of сompositе units (milеs per hour, priсe per gram, and so on).

As a footnotе,

I feel that I should raisе thе issue of my usе of thе

word .weight' throughout this сhapter. Striсtly spеaking,

I

should talk about .mass, rather than weight. The wеight of an objeсt is a mеasurе of the forсe of gravity aсting on it and this will vary depending on whеre thе objeсt is in relation to thе еarth. Мass is the .amount of mattеr' whiсh thе oЬjeсt сontains and, whеrеver its positiоn, this will not vary. Most sсientists feel that this distinсtion is сritiсal but, provided your mathematiсs is сonduсted mostly on thе Еarth's surfaсe, I wouldnt let it worry vou too muсh!

Summary

This сhapter started by looking at thе .what', .Why' and .how' quеstions of mеasurеmеnt.

Vhаt do wе mеasurе? lеngth, area, volume, сapaсiц.wеight, timе, tеmpеraturе, anglе, speеd, ...

Houl do wе mеasurе? - using words alonе

-

using words rankеd in order (an ordеring sсalе) using numbers (a number sсalе)

Finallg hеrе is a сhесklist of thе Ьasiс skills of mеasuring whiсh you will neеd.

whу do wе mеasurе? To hеlp.makе deсisions and

Meаsuring сheсklist

Thе final part of the сhaptеr lookеd at thе сommon metriс and impеrial units of mеasurе and at how wе сan сonveft within and betwееn the two svstems.

о

bе familiar with thе сommon mеasurеs (lеngth, wеight, arеa, volume, сapaсit5 timе, spееd, tеmpегаturе)

сomParrsons.

Answers to exerсises for Ghapter o7 7.t

Quеstion

7.6

Dirnension Likelу uпits o| meаsure o| meаsure

How hеavy is your laundry? wеight kg or lЬ How long is thе сurtаin rail? lеngth сm or in Нow hot is thе ovеn? tеmpеrature dеgrееs ("C or

London? lеngtЫ distanсе How fast сan you run? spееd How long doеs it takе to сook? timе How muсh doеs thе jug hold? capacitу How Ьig is your kitсhеn? volume How big is the fiеld? arеa How far is it to

7.2 Еuerydау аctiuitiеs

. Baking

a сakе

oF)

km or milеs km hr or mph minutеs

сс, pint or m3 or ftз

floz

heсtarе or

acte

Мeаsuring

dimensions

сarpеt shoеs

о Buying and laying a о Chесking thе сhildrеnЪ о Sешing out on a journеy in good

timе

!И; T;

L;

A

L; C L; Ц

T.; С

S

7.3 stop, amblе, walk, jоg, sprint lie, roll ovеr, sit up, stаnd, walk rayon' silk, wоol, сotton, linеn

7.4

1 2 3 4 5 6 7 8 9 10

7.5

Number

4124.784t з8,41'63 291.7412 39042.611 39048.6L9 38.4137 446.982 0.'t42937 1'31'7.699 30s0.1497

Апsшer to 4 sig. figs 41.25

38.42 291..7

39040 390s0

38.41.

447.0

0.1429

1З1'8

3050

a 9 ft = 1Уd, b 10000сm2=tцz с Arеa=6mx4m=24rff d

doubling thе lеngth and the brеadth makеs thе arеa four timеs as big.

7.7

a Numbеr of millimetrеs in a mеtrе = 10 Х 100 = 1000 Ь Numbеr of сеntimеtrеs in a kilomеtre - 100 x 1000 = 100 000

с

d е

NumЬеr of grams in a tonnе = 1000 x 1000 = 1 000 000 Numbеr of inсhеs in a milе = 12 x 3 х 7760 = 63 360 Numbеr of squarе inсhes in a squarе yard = L44 x 9 =

f

Numbеr of ounсеs in a ton = |6

1,296

501.760.

х

74

x

712

x 20

a 60 kg would bе a rеasоnable weight for an adult. b A 25 kg Ьag wеighs 25 x 2.2 = 55 lb. So, buying a 56 Ьag for the samе monеy is a slightly bешer dеal.

сA d

е

f

=

|Ь

typiсаl height of a kitсhеn сеiling from the floor is

roughly 2Ь-З m.

90 x 5 + 8 = 56 mph, roughly.

1 litre = L.76 pints. So, sinсе } |itre = $ = 0.88 pints, this is lеss thаn onе pint.

7 pints = h = approхimatеly 4 litrеs, or 8 half litrе bottlеs.

opеn a newspapеr or watсh thе news on TV and you will be еxpесtеd to makе sеnsе of a tan3e of сharts and graphs and to

proсеss statistiсal faсts and figurеs. For еxample, hеrе statistiсal faсt.

Did you know that eight out of tеn

is

a

advеrtisеrs are

prepared to mislеad thе publiс a little in ordеr to sеll thеir

produсt? Furthеrmore, thе othеr two are preparеd to mislеad thе publiс a lot! onе of thе troubles with statistiсs is that there is suсh sсope for dесеption. For examрle, I just made up thе figures quotеd aЬovе out of my hеad. But writing thеm .in Ьlaсk and whitе' somеhoиr seems to lеnd сrеdibility to so-сallеd .faсts'.

Dесеption сan oссur not only through thе quoting of inсorrесt information. Еqually сommon is .dirty dеaling' by means of thе inсorrесt display of сorrесt informаtion. This сhapter dеals with thе сhaшs and displays that arе most сommonly usеd and misusеd in thе mеdia - barсharts, pieсharts, linе graphs and tablеs _ as wеll as sсaffеrgraphs. It providеs еxamplеs of whеre they arе usеd, what thеy mеan, how they arе intеrpretеd аnd how thеy arе somеtimеs misusеd to сrеatе a falsе imprеssion.

GI a r+ т g) q)

+ 5 a+ a

т'

Barсharts and pieсharts Barсharts and piесharts are usеful whеn rме Want to сomParе diffеrеnt сatеgoriеs. Barсharts (sometimеs сalled bloсk graphs) сonsist of a sеt of Ьars set еithеr vеrtiсally or horizontally. Thе

-l

hеight (or lеngth) of eaсh bar is an indiсatiЬn of its sizе. PiеЪharts also allow diffеrеnt сategories to bе сomparеd but hеrе thе sizе of eaсh item is rеprеsеntеd Ьy thе sizе of its sliсе on a piе. This сhaptеr exаminеs differеnt typеs of barсharts аnd piесharts.

o tl

35

ц) П

30 25 20 o/o

15 't0

ln this сhapter you will learn: о how to draw a variety of

statistical graphs and diagrams о how to spot misleading graphs.

c 0

Packod

|unch

Pаid sсhoo|

mea|

treв sоhool

meаl

Other

|lgure 8.1 verticа| bаrchаrt showing the Iunсhtime mеa|s of pupi|s source: soсia/ т,Элds 25' Figuв з..l3'

сso

Thе main strеngth of aЬarchart is that сolumns plaсеd sidе by

United кingdom

Figurе 8.1, for еxamplе, you сan see аt a glanсе that thе most

Germany

sidе or plaсеd onе aЬovе thе other are еasiеr to сomparе. So, in

pоpular form of lunсhtimе mеal of pupils for thе уear in quеstion was a paсkеd lunсh. You сan also sее thаt roughly

Belgium

Portugal Franоe

twiсе as many сhildrеn took a paсkеd lunсh as ate a free sсhool

Italy

mеal.

Sometimеs barсharts сan Ье usеd to show сatеgoriеs from morе

than onе sourсе on thе same graph. Thе most сoпlmon way of doing this is to usе a соmpound Ьarсhart, as shown in Figurе 8.2.

lreland Denmаrk Spain Greeсe Netherlands

50

Percentages

45 40

30

oЬ

8.3

horizonta| barоhаrt showing the seawater bathin0 аreas not сomp|ying With mandatory сoliform standards: Ec сomparison, 1993 flgшre

sourc6: soо,a/ тrёnds 25, Figura l

25

1.1

7'

сso

Males (%)

20

6Yo

15 10

0

Underweight Desirable

overweight

Е E Е Бl

obese

ligure 8.2 сompound barchаrt showing body mass by gender

This сompound barсhart shows two for the priсе of onе * both male and fеmalе data arc reprеsentеd on the samе graph.

Somеtimes you mаy rмish to draw a barсhart whеrе thе сatеgory namеs arе rather long. !Иith a vеrtiсal barсhart thеrе simply isnt еnough spaсe to writе thе names Ьеnеath еaсh bar and so a

tomales

Underweight Dвsirаьte Overweight

oьesв

(o/o)

horizontal barсhart may Ье prеfеrrеd. This is illustrated in Figurе 8.3.

Piесharts, as thе name suggеsts, shows thе information in thе form of a pie. The sizе of eaсh sliсе of the piе indiсates its valuе.

For examplе' thе two piесharts in Figure 8.4 dеpiсt the samе

data that wеrе usеd for thе сompound barсhаrt in Figurе 8.2. In

Figurе 8.4, thе data for malеs and fеmalеs have Ьеen kept sеparatе, with onе piесhart drawn for еaсh. The piесharts show thе numbеr of malеs and femalеs who fall into the four main сatеgoriеs of body mаss (undеrweight, dеsirable, ovеrwеight and obеsе).

tlgшlв

8.4

pieсharts showing the body mass indiсes of mа|rs and femа|es soutЕg: sociа/ rrerlds 25' Figure 7.6'

сso

An important fеaturе of a piесhart is that it only makes sеnse if thе various sliсеs whiсh make up thе сomplеtе pie, whеn takеn

togеthe(, aсtually rеprеsent something sеnsiblе. For ехample, thе piесhart in Figurе 8.5 shows, for diffеrеnt types of sсhool, thе various avetage pupiУtеасhеr fatios; in other urords the avеrage numЬеr of pupils Рer teaсher. As thе figure title 6uggеsts, this is

rathеr silly drawn as a piесhаrt аnd would havе Ьееn muсh bеffer drаwn as a barсhart.

Tуpe of school Nurserv Primary Sесondary Nоn-maintained Spесial tl g

ш

rв

8.

б

a

s

i

I

speсia| schoo|s

Auerаge 21.9

|0.4

a Ь

с

8.1

n

g

p u p

illtraфe

r.::н,jlJ#llll11:

J"l

Interpreting graphs

[n Figure 8.1., еstimate thе pеrсеntage of pupils falling into

the four сatеgoriеs of thе lunсhtimе mеals. Use your еstimates to сheсk that thеy сovеr all of thе pupils in thе survеy.

From Figurе 8.1, estimatе the perсentage of pupils who ate a sсhool meal, whеthеr paid or frее.

From Figurе 8.2, would you say that it was mеn or wоmеn whо tended to shoцr gteater еxtrеmes of wеight?

d From Figurе

8.3, whiсh сountry had the worst rесord in terms of сomtrlianсе with these sеawater Ьаthing rеgulations?

For whiсh сountriеs did roughly |0"Ь oI thеir

seawatеr

Ьathing areas not сomply тdth thеse rеgulations?

8.4, whiсh сatеgory of urеight (undеrwеight, dеsirablе, ovеrwеight or obеsе) сontains roughly a quartеt of the fеmalеs surveyed? For whiсh сategory of weiфt arе thёrе roughly half as mаny again fеmalеs as malеs? For whiсh сategory of wеight are thеrе roughly half аs many again rnalеs

е From Figurе

f

as fеmalеs?

to know how two

different

so on.

5.8

It is oftеn too еasy to сast your еye vaguely ovеr a graph and murmur' .oh, yеs, I sее.'Thе ne)с exеrсisе asks you to lingеr on thе various graphs that you have lookеd at so far and to makе surе that you rеally do undеrstend them.

Е)(ERсlsЕ

Somеtimes, howеvеr, wе wish

о Is a сhild,s health linked to the sizе of thе family's inсomе? о How has a partiсular plant 8rown over timе? о Are lung сanсer and hеart disеasе linkеd to smoking? and

Seсondаry

L5.4

sh .Wi

the sliсеs of the piе.

thеir diet?

21..5

ly piechа rt

Thе graphs you have lookеd at so far havе bееn hеlpful if you want to makе сomparisons - barсharts allow you to makе сomparisons based on thе hеights of the bars, while сomparisons within pieсharts arе basеd on the relativе sizes of

mеasurеs arе rеlated to еaсh other. For еxamplе: о Doеs a pеrson's Ьlood pressure rеlatе to thе fat intakе in

Non.mаintаined

rаtio

Scattergraphs and line grаphs

Еxplain Ьriefly in your orvn lvords why the pieсhart in Figurе 8.5 is sill5 and make a rough sketсh of what it would look likе rеdrawn as a barсhart.

To answeг thеsе sorts of questions, whiсh look at two differеnt measures togetheц we neеd a two.dimensional gгaph. This usually takes the form of eithеr a sсaftеrgraph or a linе graph.

The figurеs in Table 8.1 show the сontriЬutions to, and rесeipts from, thе ЕC budgсt in 1993. Thе samе information is also portrayed in thе sсattеrgraph shown in Figure 8.6. By analysing the sсattergraph, it is possiЬlе to eхplore the relationship betweеn сontributions and rесeiрs. For еxample, .Do сountriеs пrhiсh givе thе most tеnd to reсeivе thе most?', and so on.

Countrу

Germany Franоe Italy Unitэd Kiпgdom Spain Netherlands Belgium Dвnmark Greeсe Portugal lreland Luxembouщ

Contributions Е billion 14.9

9.0 8.0 5.9 4.O 3.1

1.9

0.9 0.8

Recepts 2 Ыllion 5.6 8.2

6.8 3.5 6.5 2.1

1.9 1.2

01

4.0 2.6

0.5

2.з

o.2

0.3

ta!|o 8.1 сontributioпs to, and reсeipts from, the Eс budget' 1993 sou!сo: soсiа, 7i6nds 25' Figu@ 6.21'

cso'

1995

In ordеr to draw a sсattеrgraph of this information' yorr must plaсе onе of thе mеasurеs on onе of thе axes, one on the othеr and mark on еасh axis a suitaЬlе sсalе. As you сan sее from Figurе 8.6, I havе сhosеn to plaсе .ContriЬutions, to thе ЕC on thе horizontal axis and .Reсеipts' on the vеrtiсal axis. Еaсh сountry is plottеd as a sеparatе point. For еxamplе, thе point

сorrеsponding to Gеrmany is shown on the еxtrеme right of the graph. Following thе down arrow to thе СontriЬutions aхis, you сan sее that this point lines up with thе va|ue {,1,4.9 Ьillion.

Rеading aсross to thе Rесеipts axis from thе samе point, thе сorrеsponding valuе is {5.6 Ьilliоn.

ЕкЕRоlsЕ

8.2

Prаctising sсаttergraphs

Do you think that сountries whiсh havе a high rаtе of marriage also tеnd to hаvе a high divorсе ratе? Havе a look first at Tablе 8.2 and thеn at thе сorrеspоnding sсattеrgraph in Figurе 8.7.

a b

с

Chесk that you undеrstand how thе pоints havе bееn plottеd and try to matсh еaсh point up to its сorrеsponding сountry. !Иhy do you think thеrе arе no data for Irеland in thе divorсе ratе сolumn? How has thе point сorrеsponding to Irеland bееn shown on thе graph?

Ovеrall, doеs thе sсattеrgraph show a сlеar rеlationship Ьеtwееn a сountryЪ marriagе ratе and divorсе ratе?

Maniage Country Germany 5.6 Franсe 4'7 lta|y 5.з United Kingdom 5.4 Spain 5.5 Netherlands 6.2 Belgium 5.8 Denmark 6.2 Greeсe 4.7 Portugal 7.1 lreland 4.5 Luxembouщ 6'4 table

0246810121416 оontributions (tb)

8.6 scattergraph showing the re|ationship between net сontributions аnd net reсeipts to the ЕU (bаsed on.n'o*ij:,:*T.}),o",u,Figure6.2'l,сso ligure

Thе pattеrn of pоints on a sсattеrgraph hеlps tо rеvеal thе sort of relationship Ьеrwееn thе two mеasures in quеstion. For еxamplе, in Figurе 8.6 you сan seе that thе points liе in a fairly сlear pattеrn running from Ьottom lеft to top фht оn thе graph.

This rеflесts thе not-too-surprising faс that сountries whiсh make small сontributions (likе Irеland and LuхеmЬourg, for examplе) also tеnd to gеt small rесeipts. Similarly, thе major сontriЬutors' likе Gеrmany and Franсе, arе also two of thе largеst rесеivеrs of monеy from thе fund. You will probably nеed to spеnd somе time сonsolidating your undеrstanding of a sсattеrgraph, so do Ехеrсise 8.2 пow.

8.2

rate

Divorce rate 1.7

1.9 0.5 3.0 o.7 2.O

2.2

2.5 0.6 1.3

1.8

marriage and divorce rаtes: Eс сomparisons,

popu|ation)

.|992 (rate per 1000

source: social тrends 25, Figure 2.:lз,

оso

Linе graphs, likе sсattеrgraphs arе two-dimensional, so again wе will Ье dеaling with twо mеasurеs at a timе and еxamining thе rеlationship Ьetweеn them. A line graph is one of thе most сommon typеs of graph and indееd is what most pеoplе think of whеn wе usе thе word .graph'.

Figurе 8.8 shows a partiсular typе of linе graph, known as a tiпe graph. It is so сallеd for thе oЬvious rеason that thе mеasurе on thе horizontal аxis is time.

ЕxЕRclsE 8.3 lnterpreting line graphs a How havе lеvеls of marriagе аnd divorсе

b

с

altеrеd in Grеat Britаin ovеr thе 22 уears bеtwееn 197О and 1,992? Еstimatе thе numbеr of marriagеs in L992 in Grеat Britain.

Notiсе that thе points markеd on thе two linе graphs havе bееn takеn ovеr fivе-yеar intеrvals. In thе original line graphs (printеd in thе puЬliс ation Soсiаl Ttends 2S|, the pоints wеrе basеd on data takеn in onе.yеar intеrvals. How do you think thе sizе of thе intеrval might affесt thе ovеrall shapеs of thе line graphs?

u.c

Misleading graphs

0

Marгiage rate

If you rеad through nеwspapеrs and magazinеs, it isnt diffiсult to spot graphs whiсh arе mislеading. Thе graph Ьеlow соntains a finе сollесtiоn of disastеrs! Sеe how mаny you сan spot.

ligurв 8.7 sсаttergraph based on the dаta in tаb|e 8.2

Unemployment graph for the North

-x_E

Mаrriаges

-

Oivorосs

t|gule 8.8 а timr graph showiлg marriages and divorсes in Greаt Britаin sourcс Adaрt€d fiom fuial тreds 25' Flgulв 2.14' сso

|igшre

8.9

spot the errors on this grарh!

In ordеr to makе sеnsе of this graph and to sеe somе of the distortions it сontains, you rеally nеed tо havе а look at thе dаtа from whiсh it was drawn. Thеsе are givеn in Tablе 8.3.

Rate (/o) 8.6 10.6 11.2 11.2 11.7

Year 1990 1991

1992 1993 1994 tabIe

8.2

As you сan seе' now thе inсrеasе is not nеarly so dramatiс and it is сlеar еxaсtly what thе figurеs rеfеr to.

Onе final point about the sсalе on the vеrtiсal axis not starting at zero. In faсt it is aссeptaЬlе to draw а vеrtiсal sсalе starting from a numbеr othеr than zero, providеd an indiсation is madе

unemployment rates (%) for the N0rth of Еng|and (1990 t0 1994)

Thе graph shown in Figurе 8.9 сertainly looks dramatiс.

Howеvец although unеmplo1тnent ratеs in thе North of Еngland did risе ovеr this pеriod, thе risе was not as drаmatiс as suggеstеd by this rеpresеntation. There arе a numbеr of еrrors and mislеading featurеs of thе graph. Lеt's go through them in turn. о Thе title is not very hеlpful. Thеrе is no сlеar еxplanation of what region the graph rеfers to (.thе North' сould rеfer to anywhеre), or to what is bеing mеasurеd (the title should state thаt thеsе arе pеrсеntagеs).

The аxes are not laЬеllеd. Thе vertiсal axis should show сlearly that thе figurеs arе .Pеrсеntagеs' аnd thе horizontal axis should say .Yеar'.

Thеrе is no scale markеd on thе vеrtiсal axis, so yоu havе no idеa what thеsе figurеs arе.

Thе sсale on thе vеrtiсal axis has beеn сut in order to makе thе graph look stееpеr. on сlosеr inspeсtion of thе сorrеsponding data shown in Tаble 8.3, you сan sее that thе aхis runs from 8.6"Ь to 1I.7oЬ, but this has not bееn madе сlеar on the graph.

Finаlly, thе horizontal aхis has bееn dеliЬеratеly squashеd up to make the graph look stеeper.

A more сorrесt vеrsion of thе gгaph is shown in Figurе 8.10.

12

Sometimes the axis break is shown like

this...

... and

sometimes it is shown like this ...

Figure 8.11 shows thе samе graph drawn with thе vеrtiсal axis starting at 8уo but with thе Ьrеak in thе axis addеd to alert the rеadеr to this potеntial sourсе of сonfusion.

1992

Year

|igшrе

10

8'.t1 а grаph demonstrаting аn axis breаk

To end this sесtion оn mislеading graphs, hеrе is onе of the most сommon typеs of distortion. Havе a look at Figurе 8.12 and sее if you сan spot how it might givе a falsе imprеssion.

I

Е6

c

с

1992

Yoar

llgure 8.10 line graph showing unemployment rates (7o) for the North ol England (1990-1994)

on thе axis that this has bееn donе. Thе most сommon mеthod is to mark a Ьrеak on thе axis. as shown bеlow.

sort of vеstеd intеrеst. You will find it hеlpful to ask yoursеlf, ..What uЪ the vеstеd interеst, and thеrеforе what imprеssion is this graph dеsignеd to сonvеy?'

Answers to exerсises for Ghapter 08 8.1

Number of disсs so|d in 1987 = 18.6 mi||ion Number of disсs soId in 1993 = 93 mi||ion |igulв 8.12 sаles of c'mpaсt disоs in the UK'

.|987

Thе graph in Figurе 8.].2 shows up

x::.::::,,*,""

a

23and25

favouritе triсk of

advertisers, whiсh is to make differеnсеs look bфer than thеy aсtually arе. Cеrtainl5 CD salеs inсrеasеd grеatly in thе UK bеtwееn L987 and 1'993 _ by five timеs, in faсt (thе 1993 figarc of 93 million is fivе timеs as great as tЬe L987 figurе of 18.6 million). Howevеr, not only has thе 1993 disс Ьeеn drawn fivе times as tall as thе ].987 disс, but it is also fivе times as widе. Thе ovеrall imprеssion оf the largеr disс, thereforе, is that it has an аrea whiсЬ is fivе times fivе, i.е. twеnfy-fivе timеs as grеat as that of thе smallег disс. Advеrtisеrs arе ablе to еxploit thе faсt that most of us work on impressions, not faсts!

Summary This сhaptеr has сovеrеd four of the most сommon typеs of graphs: barсharts, piесharts, sсattergraphs and line grаphs. Thе final sесtion dеalt with mislеading graphs and а list was providеd of somе of thе соmmon ways in whiсh graphs сan be drawn in an unhelpful or dеlibеratеly distortеd way.

ovеr thе next fеw wеeks, why dont you look out for some more еxamplеs of mislеading graphs in nеwspapers and magazines. Мost pеoplе who prеsеnt informаtion to thе publiс havе somе

a

Еstimates from Figurе 8.1 arе аs follows: o/ /o Luпсhtime meаl з2 Paсkеd lunсh 29 Paid sсhool mеal 1'6 Frее sсhool mеal 23 othеr Thе pеrсentage of pupils who atе a sсhool mеal, whethеr paid or freе is 29"Ь + |6Y" = 45"Ь.

It sееms that womеn tеnd to show grеatеr еfirеmеs of wеight. This сonсlusion сan Ье drawn from looking at thе .Undеrwеight' and .oЬеsе' Ьars, whiсh arе Ьoth taller for fеmalеs than for malеs.

Thе Unitеd Kingdom sееmеd to havе thе worst rесord in this rеspесt.

Thе two сountriеs for whiсh roughly 10% of thеir sеawatеr Ьathing arеas failed to сomply wеrе Portugal and Franсе.

Rоughly a quartеr of thе {еmalеs (aсtually 26"/')

feI|

into

the .Ovеrwеight' саtеgоry.

Half аs many again fеmalеs as malеs (9Yo comparеd with 6%| fe|| into the .Underwеight' category.

Roughly half as many again mаlеs as fеmalсs (40%

.Overweight' сatеgory. сomparеd witЬ 26%'t fеll into thе Thе piесhart in Figurе 8.5 fails to mеet a Ьasiс сondition of a piесhart in that thе сomplеte piе doеsn't rеprеsent anything meaningful. In this сasе' thе сompletе piе сorrеsponds to thе sum of thе various avеragе ratios in the differеnt фpеs of sсhool, and who сares about that!

Thе data would bе morе hеlpfully drawn as a barсhart, as shown in Figurе 8.13.

Nursory Primary

Seсondаry Non.maiпtained

specia|

тур0 o' schooI

llgшrв

8.13

vertiоаI barсhart showing average pupi|/teaсher ratios by typ0

of sоhooI

8.2

a No сommеnts. Ь Divorсе was illеgal

in Irеland n 1992 so therе сan be no offiсial divorсе rate. Thе point сorresponding to lreland

с 8.3

a b

с

has bееn ignorеd on thе sсattеrgraph.

Thеre is little еvidenсе of any сlear Pattern in thеsе points linking divorсе and marriаgе ratеs. Маrriage lеvels have stеadily fallen while divorсe levеls have donе the revеrsе ovеr the period.

Thе numbеr of marriages in |992 in Grеat Britain is estimаted to bе rouф|у 360 thousand. \Vhеn graphs are plottеd Ьased on data takеn at one-year intеrvals, as opposеd to evеry five years, thеre arе likely to Ьe morе suЬtle сhanges in direсtion. Мy graphs given in Figure 8.8 are aсtually rathеr сrudе, еaсh being сonstruсtеd by joining up six points rлrith straiф lines. Нowеver, supposing that, say, in |987, therе was a sudden blip in thе graph, this п,ould simply not have shown up on Figuге 8.8.

o Пr Е э

GT ц)

{r o 't

3

C П

g)

In this chapter you will learn:

. .

why algebra might be useful some of the rules of algebra о how a|gebra сan be used to prcve things r about sprcadsheets with a сomputer.

ls algebra abstraсt and irrelevant?

For many studеnts, thе arrival of algеbra in thеir sсhool lеssons was the point at whiсh they felt they partеd сompany with mathеmatiсs. Algebra has а reputation of bеing hard, largely bесause many pеoplе sее it as abstrасt and irrеlеvant to thеir livеs.

Lеt us first сonsider whеther algеbra is abstraсt. Thе simplе

answеr to this сhargе is, .Yеs it is!,. Algebra is сertainly abstraсt, for that is thе point ot a|geЬta. Thе word .aЬstraсt' mеans .taken away from its familiar сontext'. Thе rеason that algеbra is suсh a powеrful tool for solving problems is that it enables сomplеx

ideas to Ьe rеduсеd to just a few symbols. Naturall5 if algebra

is to Ье useful to you' you nееd to undеrstand what thе symЬols mеan and how they arе relatеd. Assuming this is the сasе, еxpressing somеthing as a brief mathеmatiсal stаtement (whiсh might bе a formula or an еquation) allows you to strip away the dеtails, to forget aЬout thе сonte)ff from whiсh it was takеn and foсus on the essential undеrlying rеlationship. Of сoursе, there arе often situations rмhеre уou don't want to strip away thе сontехt (in questions of human rеlationships, for еxamplе).

Clеarly you wouldnt wish to usе algеbra for suсh problems.

Nеxt, lеt's еxaminе the сharge of algebra bеing irrelеvant. Мost people beliеve that thеy never use algebra. Yet in many jobs, partiсularl5 say, in mediсinе and enginееring, formulas arе сruсial for сonvеrting units, сalсulating drug dosagеs' setting maсhinеs сorreсtly for diffеrеnt tasks, and so on.

Inсrеasingly, large organizations and govеrnment institutions usе formulas for deсiding on and desсribing thеir funding arrangemеnts. To takе the ехamplе of еduсation, a formula is usеd to definе how muсh money is alloсated to sесondary and primary sсhool budgets on the Ьasis of thе numbеr and age of pupils. Therе are many questions that immеdiаtely arisе. For examplе: о [s it a tair waу of alloсating money? о Is it right that tЬe шеighting (i.e. thе rеlativе amount) for seсondaгy agе сhildren is muёh grеatеr than for primary age сhildгen or should all сhildrеn bе alloсatеd thе samе amount' rеgardless of age? о How сan you find out urhat thе rеlativе wеightings arе for сhildrеn of diffеrent agеs? о who should deсidе thеsе sorts of question and саn intеrested Paгеnts and tеaсhегs enter thе dеbatе?

Thе point I wish to make in introduсing this сhaptеr is that pеoplе саn dеЬatе this question oпlу if theу understаnd шhаt а formulа is sауing.If yоu dont understand basiс algеЬra, othеr pеople will bе making suсh dесisions for you and you will havе no idеa whethеr or not they arе aсting in your bеst interests.

Beforе starting to ехamine any formulas, wе Ьеgin Ьy looking at somе of thе Ьasiс featurеs of algеЬra - horлr it is usеd as a

shorthand way of ехprеssing somеthing' and somе of thе

сonventions rеgarding how it is writtеn.

Algebra as shorthand Thеrе arе many situations in еvеryday lifе whеrе it is сonvеniеnt to adopt a shorthand - usually in thе form of aЬЬrеviations or spесiаl symЬols - in ordеr to spееd things up. I am awarе that'

for many people, thе symЬols in algеbra seem to

сarrsе

сonfusion rаthеr than Ье an aid to еffiсiеnсy. Howеver, thе idеa of using a shorthand isn,t just сonfinеd to mathеmatiсs.

Somе еxamplеs arе givеn in Еxеrсisе 9.1 for yоu to intеrprеt. Thеn,

in Еxеrсisе 9.2 уou arе askеd to еxaminе somе shorthands.

mathеmatiсal

ExERсlsE 9.1 Shorthands in everyday life

EXAMPLЕ a b

с

d

det. hse, lge gdns, gd deсs,

Source

Meaning

FсH. . . .K1, P1 , M1'

c4Е l<2,. ' ' PAS, MoТ' fsh' good runner NYWJМ seeks same with view to B&D' S&M' etс.

Sее if you сan idеntify the sourсe of thеsе shorthands and what

thеy mеan?

ExЕRсlsE 9.2 Shorthаnd in mathemаtiсs Hеrе arе somе mathеmatiсal sentenсes. Rеwritе eaсh onе in mathematiсal shorthand. Thе first one has bеen donе for vou.

a b

с

d e

f

Loпghand Thrее multiplied by two and a half Thе sum of twеlvе, and four and thrее quarters Thе sum of the squarеs of threе and four Four timеs the diffеrеnсе of ninе and threе Five plus four, all dividеd by the produсt of fivе and four

Shorthаnd

3 x2Ь

сurtain material tends to сome in various .standard lengths' and in impеrial units, 48 inсhes happеns to Ье onе of thеsе standard

Thе numbеr of inсhеs, 1, is found by multi plying thе numbеr of mеtrеs, М,ЬУ 39.37.

lengths.)

As you сan sее from thesе еxamples, mathematiсs is full of shorthand notation. For еxamplе: о instеad of writing numbеrs out as words, .onе, two, thrее, and so on', we havе savеd timе by invеnting thе numerals, .1,

2,3, etс., о rathеr than say .multipliеd by' or .addеd to', we usе

Using a сalсulator, thе answer is 47.244 inсhеs, a rеsult rryhiсh you might round up to 48 inсhеs. (As аn asidе, roundingup may be appropriate here for two rеasons. Firstl5 if you arе buying somеthing likе сuшain matеrial or a сarPеt' it is always Ьeffer to have a littlе bit too muсh than to bе a littlе bit short. Sесondly,

Now givе some thought to bееn writtеn.

horлr thе formulа, I = 39.37

х М, has

First, notiсе that I have introduсеd thе abbrеviаtions = and x to savе the trouЬlе of writing out the words .еquals' and .timеs'.

You may feel that this shorthand hаs rеduсed the formula to its

thе

Ьarе еssеntials, but is aсtually possiЬlе to dispensе with the x altogеther and writе thе formula even morе Ьriеfly, as follows:

Мorе еxamplеs of notation will

bе еxplainеd in thе nеxt sесtion.

For now, lеt us foсus on Part f of Еxеrсise 9.2, as it dеmonstrates a kеy fеature of a|geЬra.

This dеmonstratеs an important сonvеntion in algebra, namеly that writing two lеttеrs togetheq or a numbеr and a lеttеr togеther, impliеs that.thеy arе multiplied. For еxamplе: ,аb, means а times b .4y'means 4 timеs y

Thе solution to this еxamplе, whiсh is given on Page 131, is rеpеatеd below for сonvеniеnсе.

and so on.

о

symbols x and + squaring is rеpresеnted by writing a small two abovе and to the right of the number or lеttеr that is being squared. Thus, fivе squarеd, or 5 timеs 5, сan Ье writtеn as 5 x 5 or as 52.

r:3937 х

*LtхчgJ

ж

Beforе rеading on, makе sure that you сan usе this formula. For еxamplе, supposе you havе just bought сurtain material with a drop (thе drop is typiсally thе distanсе from сurtain rail to thе window sill) of 1.20 mеtrеs and you want to know rмhat that is in inсhеs. Simply replaсе t|te М in thе formula by 1.20 and сalсulatе thе сorresponding value of /, as follows:

I = 39.37М

'L.76L'means 1.76 times

A seсond aspeсt of the formula worth noting is that I have usеd the lеttеrs I and М to rеpr€sеnt' respeсtively, thе number of

inсhеs and thе number of metrеs. In algеbra, the lеttеrs whiсh we happеn to сhoosе to rеpresеnt numЬеrs arе quitе arbitrary.

Thus, I сould havе rмrittеn thе formula as' say' Y = 39.37X,witll Y reprеsеnting the numЬеr of inсhеs and X thе numbеr of mеtrеs. Horмever, it is usually a good idеa to relatе еaсh lettеr to the quantity that it rеprеsеnts as this will hеlp you to remembеr what thе various lеttеrs stand for. For that rеason' I used the initial lеttеrs of Inсhеs and Мetrеs, i.е. / аnd М, in this formula.

Thе most сommon lеttеrs usеd in bаsiс sсhool algebra tеnd to be:

X, У, d, the number of iпсhes, /

L

b

and n.

Thеre is no obvious rеason for this сhoiсe, with thе possible еxсеption of the z, whiсh сan Ье thought of as rеprеsrnting somе unknown zumbеr.

The neхt sесtion dеals with formulas in praсtiсal сontеxts and how to use them to do сalсulations.

Galсulating with formuIae Drug dоsages neеd to bе сarеfully сalсulatеd and mеаsurеd out.

Giving too littlе оf thе drug mеans that thе patiеnt doеsnt gеt thе full Ьеnеfit, but giving too muсh соuld bе highly dаngеrous.

The proЬlеm is grеatly сompliсatеd whеn thе drug is to Ье administеred to a сhild, Ьесause сlearly a dose thаt woцld bе suitablе for аn adult wоuld bе too muсh for a young сhild.

Thеrе nееds to bе a way of adjusting thе dosаgеs depending, pеrhaps, on thе body wеight or thе agе of thе сhild сonсernеd.

Onе suсh formula, basеd on the сhild,s age, is as follows.

F=1.8С+32 i.e. the temperature in

degrees Fahrenheit is found by...

Hеrе is an еxample of thе formula in opеration.

C = D xLдttz reprеsеnts Child dosagе, D rеprеsеnts аdult Dosagе and' A is thе Аge of thе сhild. \Vrittеn out in longhand, this formula mеаns the following: Child dosagе = Adrrlt dosаgе x я#', whеrе

2 oven temperatures

A typiсa| сooking temperature for an oven is 180.C. What is this in "F?

Solution Applying the formula: The temperature in degrees Fahrenheit,

1

Let us take an example where the adult dose of сough mediсine is 10 mg of |inсtus. What would be the appropriate dosage for a

F

= 1.8 =

сhild of 6 years of age?

Now hеrе arе somе for you to try.

Solution First let us write down what we know.

Е)(ЕRclsЕ9.4 Temperatureсonversion a A warm summеr day,s tеmpегatrrrе would

With these numbers to hand, We are ready to сalсulate the childЪ dose, using the formula.

b

D=10,A=6

C=D x

*,

=

1oХ*,

= 10 Х

*

= 3* mg

Now hеrе arе somе to try for yoursеlf. ЕxЕRсlsE 9.3 Gаlсulating dosages Using thе formula С = D Х #гlъ сalсulate thе dosagеs for the following situations. a An adult prеsсription of a сertain drug is 24 micrograms (рg).

What would bе an appropriatе dosе for a сhild agеd 10

b

EXAIIлPLE

С

EXAIT'IPLE

... multiplying the temperature in degrees Celsius by 1.8 and ...

yеars?

An adult prеsсription of anothеr drug is 200 рg. lD(hat would bе an appropriate dose for a сhild aged 4 yеars?

.sИе

now movе on to anothеr formula, this timе for сonvеrting tempеratufеs. Tеmpеraturеs in dеgrееs Cеlsius сan be сonvеrtеd to dеgreеs Fahrеnhеit with thе following formula.

с

x

180 + 32

356"F

Ье something like 30"C. t0Иhat would this Ье in dеgrеes Fahrеnhеit?

Thе boiling point of watеr is 100"с. TИhat is this in .F? Thеrе is only onе tеmpеraturе whiсh is thе samе in dеgrеes F as in degrеes C. Try to find it.

(Hint:It is a tempеraturе wеll bеlow fueezingpoint.)

Тhе next example of a formula is сonсеrnеd with сalсulating а 'phonе Ьill. Ъlephonе bills are usuаlly сalсulatеd еaсh quarter on thе basis of a fixеd sum for thе rеntal of the line plus a variaЬle сost basеd on the сalls you makе. For еxamplе, my last bill of f,97.61was madе up of a rеntal of {20.|6 аnd a furthеr 4.20 pеnce pеr unit usеd. The formula for this сan bе written as follows.

С=20.16+0.042U whеre C is the Chargе in pounds and U is thе numЬеr of Units usеd.

Notiсе that thе сhargе rate of 4.20 pence pеr unit has bееn rеrмrittеn in the formula in pounds (i.e. as 0,042) in ordеr tо matсh with thе units of thе rеntal rмhiсh rлras also еxprеssеd in

what if I produсеd another twenty еxamplеs, or a hundred, or

Е)(AпiPLE 3 I used 1844 units |ast quarter. ls the quarter|y оhaщe for my

gеt round to сheсking to bе оnly take one of thе onеs you didnl.lrhilе arithmеtiс is usеful for wrong to blow your thеory aPart. doing сalсulations witЬ pаrtiсular numbеrs, algebra is nееdеd for making geпerаlizаtiozs. The mathеmatiсal pгoof of this gеnera|ization (that thе sum of two odd numbеrs is always еven) is outlinеd and еxplainеd Ьеlow. But bеforе launсhing in to it, you nееd to spеnd a fеw minutes thinking aЬout how wе might rеprеsеnt еvеn and odd numbеrs algeЬraiсally.

pounds.

teIephone biIl сorreсt?

Solution

Тhe сhaщe is сalсulated as foIlows: C = 2O.16 + (0.042 x 1844) = fl97.6'l (rounded to the nearest penny).

This confirms the bilI which

I

reсeived as being correсt.

еvеn a million? That would bе quite impressivе, but unfortrrnatеly

providing lots and lots of spесial сasеs would not сut muсh mustard with a mаthematiсian. !Иhy, then, is it so hard to prove a numеriсal rеsult to bе always true? The rеason is that you сan't try all thе infinitе numbеr of possible сases, and it would

An aside on even and odd numbers Еxеrсisе 9.5 givеs you the opportunity to try somе of thesе for yoursеlf.

ExЕRсlsЕ

9.5

More bills

Calсulatе the quartеrly сhargе for a household whiсh usеd:

a b

944 units 3122 units.

Proving with algebra This final seсtion looks at an aspgсt of mathеmatiсs сlosе to thе hearts of mathemаtiсians - the idеa of prоof. Vithout algеbra, proving that a mathеmatiсal rеsult is true is quitе diffiсult. It is oftеn easy еnough to show that thе rеsult is truе for sеvеral pаrtiсulаr numbеrs but it is quite a diffеrеnt maffеr to say that уou hnoш it is truе tor аIl possiblе numbеrs. For еxample, is it thе сasе that adding two odd numЬеrs always produсеs an еvеn ansrмеr? .!Ие

сould take sоmе еxаmples аnd seе if it works. Thus: З + 7 = 10, whiсh is еvеn 5 + 11 = 16, whiсh is evеn 23 + 15 = 38, whiсh is еvеn 1'L| + 333 = 444, whiсh is еvеn.

So, it doеs sееm tо be true, Ьut have wе provеd it? Сеrtainly nоt! Сhесking only four еxamplеs doеs not сonstitutе a pгoof. But

If wе think of whole numbеrs as represеnted by, say, thе letter К, then wе сan writе even numЬеrs as 2K. You сan сheсk this out by giving K anу valuе you wish to think of. For еxamplе: whеn whеn whеn whеn

K = 3,2K = 6, whiсh is еven. К = 8,2K = 16, whiсh is еvеn. К = 13,2K = 26, whiсh is even. К = 50,2K = 100, whiсh is even.

and so on.

The reason wе know tЬat2K is always еvеn is that it сontains a f'actot 2, whiсh is еssеntially whаt an еvеn number is.

Similarly, if wе represent the wholе numbеrs by, say, thе lеttег L, any odd numbеr сan Ье writtеn with thе formula 2L + 7. Again, let's takе a few examplеs to сheсk this out. whеn when when when

L L L L

2L + 1 = 7,vyhiсh is odd. = 8, 2L + 1 = |7,цrhiсh is odd. = |3' 2L + 1 = 27,чrhiсh is odd. = 50,2L+ 1 = 101, whiсh is odd. = 3,

And again, from logiсal rеasoning we сan shoul that thе numbеr 2L + \ must bе odd. Thе ехplanation liеs in thе faс that thе number 2L + t is 1 more than the numbеr 2L, whiсh itself must be еvеn bесausе it сontains the faсor 2. A numbеr one grеater

than an еvеn numЬer is nесеssarily odd.

TИith thеse ways of rеprеsеnting еvеn and odd numbеrs at our

disposal, wе arе now rеady to provе the earlier rеsult aфbraiсally. I have rеstatеd it below in Examplе 4.

Spreadsheets

ЕxAI\,tPLЕ 4

Prove algebraiоa|ly the result that the sum of two odd numbers a|ways gives аn even number. |et both the two odd numbеrs be represented bу 2K + 1. However, the prob|em with doing this is that, whatever va|ue for K is оhosen' we find ourselves With two odd numbers with the same va|ue. Тhis is subt|y different from the problern we set out to prove. We need to allow the two odd numbers to be different, so, using different |etters, we сan let them be 2K + 1 and 2L + 1, respeсtive|y.

Their sum = (2K + 1) + (2t + 1) Simplifying, we get 2K + 2L + 1 + 1 = 2K + 2L + 2 Now, notiсe that 2K + 2L + 2 оan be written as 2{K + L + 1). +

L + 1 isawho|e

We have now proved the resu|t in general terrns. No mаtter what whole number values you think up for K and L, the general argument demonstrates that the result 2(K + L + 'l) will always be an even number.

If you arе unfamiliar with algeЬraiс rеasoning' you may nеed to rеad this proof through mоrе than onсе. Thеn, whеn you are

morе сonfidеnt, havе a trу at writing your own proof in yоur answеr to Ехеrсisе 9.6.

By the wa5 don't worry if you find this diffiсult. Мost pеoplе find algеЬraiс proofs hard to fathom and you would nееd a lot morе praсtiсе at working with algеЬraiс symЬols than has Ьееn providеd in this сhaptеr if you arе to pеrform proofs with сonfidеnсе. I havе inсluded it mostly to indiсate thе sоrt of things that mathеmatiсiаns spend their time on, and to givе you an insiфt into how algеbraiс symЬols сan Ье an aid in solving aЬstraсt problеms.

ЕxЕRcIsE

B

с

D

E

1

Solution one possibility might be to

Sinсethis numberсontains afaсtorof 2,andK numbel then 2{K + L + 1) must be even.

A

9.6

Proving with аlgebrа produсt [s it truе that thе of two odd numЬers is always аn odd numbеr? (Remindеr: produсt mеans .thе rеsult of multiplying

2

CellBS

3

A sprеadshеet is a сomputer tool that is usеd tо sеt out data in rows and сolumns on a sсrееn. The rows are numbеrеd

|,2,3

... down thе lеft-hand sidе, whilе the сolumns are labеlled A, B, С ... aсross thе top. A typiсal spreadshееt might resеmЬlе the tablе shown above, еxсеpt thаt morе rorмs and сolumns are visiЬlе on the sсrееn at aгУ onе timе.

Еaсh .сеll' is a loсation whеrе information сan Ье storеd. Сеlls arе idеntifiеd by thеir сolumn and row position. For ехamplе, сеll B3 is indiсatеd in the taЬlе shown abovе; it is thе сеll in row 3 of сolumn B. Thе information you might want to put into еaсh сell will bе onе of threе basiс typеs: . Numbеrs. Thеse сan Ье either wholе numbеrs or dесimals, for example 7,120, 6.32. о words. Thesе сan еithеr be hеadings or explanatory teхt. o Formulaе. Thе rеal powеr of a spreadshееt is its aЬility to handle formulaе.

!Иhеn you еnter a suitablе formula into a partiсular сеll, this formula чrill сontаin onе or more rеfеrеnсes to othеr сells. Thе сеll will thеn display a value сalсulatеd from thе numbеrs сurrеntly storеd in thе сеlls rеfеrrеd to in this formula; this сould be an averagе or pеrhaps a row or сolumn total. If thеsе сell valuеs are altеrеd, the formula will instantly rесalсulatе on thе basis of thе updatеd valuеs and display thе new rеsult. To take

examplе, supposе I еntеrеd my height in mеtres into сеll A1. I сould thеn entеr a formula into сеll 81 for сonvеrting the height into сentimеtrеs (typе = A1"100).

a simplе (L.72

i|

.еntеr'thе formula and thе va|ae L72 is displayеd in B2. Now еnter any new height in metrеs into сеll Press the <ЕNTЕR> kеy to

A1 and thе value in B2

adjusts

to display thе height in

together'.)

сеntimеtrеs.

TЬst this out first with a fеw speсial сasеs and thеn try to provе it algеЬraiсally.

spгеadshеet is usеful for stоring and proсessing data whеn rеpeated сalсulations of a similar naturе arе requiгеd. Next to

A

word proсеssing' a sprеadshееt is thе most frеquently usеd tool in Ьusinеss. It is аlso ехtrеmеly usеful for housеholdеrs to help solvе proЬlеms that сrop up in thеir various lifе rolеs as соnsumеrs' tax payеrs' mеmbеrs of сommuniry organizations, еtс. For еxamplе, it саn Ье usеd tо invеstigatе quеstions suсh as: о How muсh rмill this journеy сost for diffеrent groups of

pеoplе?

Is my bank statеmеnt сorreсt? . Whiсh of thеsе Ьuys offеrs thе Ьеst valuе for monеy? o \Vhat is thе сaloriе сount of thеsе various mеals? о .!0Иhat would thеse valuеs look like sortеd in ordеr from smallest to biggеst? о How сan I quiсkly еxprеss аll thеsе figurеs as perсentages?

A sprеadshееt is a powеrful tool for саrrying out rеpеаtеd сalсulations. You simply perform thе first сalсulation and thеn a furthеr сommand will сomplеtе all thе othеr сalсulations automatiсally. Anоthеr advantage of a sprеadshееt ovеr penсil and papеr is its sizе. Thе grid that appеars on thе sсrееn is aсtually only a window on a muсh largеr grid. In faсt, most spreadshееts havе hundrеds of rows and соlumns, should you nееd to usе thеm. Мovеmеnt around thе sprеadshееt is also straightforward - you сan usе сеrtain kеys to movе to adjaсеnt сells оr to anothеr сеll of your сhoiсе.

о

onсе thе data has bееn entеrеd into thе sprеadsheet, thеre arе a varietу of options аvailablе for hеlping tо makе bеttеr sеnsе of thе figurеs. For ехamplе, соlumns or rows сan bе rе-ordеrеd or sortеd еithеr alphabеtiсally or aссording to sizе. Row and сolumn totals сan Ье quiсkly found and еntirе rorмs or сolumns сan Ье сonvеrtеd into pеrсеntagе form. A variety of summary valuеs сan bе саlсulatеd (mеan, modе, rangе and so on). Finally, sprеadshеets сontain powеrful graphing faсilitiеs that еnaЬlе you to display some or all of thе data as a piесhart, Ьargraph, sсattеrgraph and so on.

Summаry A kеy point mаdе in thе introduсtion to this сhaptеr Was that it is silly to сritiсizе algеЬra bесausе it is aЬstraсt. Еssеntially, thе purposе of аlgеЬra

lЪ

in a

to bе aЬstraсt. Algеbra involves exprеssing

mathematiсal shorthand in thе form of symbols and lеttеrs. This has the еffесt of rеduсing thе proЬlеm to its barе еssеntials and allows you to sее and manipulatе its

rеlationships

main fеaturеs. Cеrtаin algеbraiс сonvеntions were еxplainеd (for еxample, that writing 4X actaa||y mеаns .4 times Х').

As thе title suggests, thе main aсtivities of thе сhaptеr involvеd using formulas and the neхt sесtion invited you to dip your toе into thе esoteriс world of mathеmatiсal proof. Тhе сhapter endеd with a look at a most usеful piесe of сomputer software

-

a sprеadshееt.

Answers to exerсises for Chapter 09 9.1

Ехаmple a dеt. hsе, lgе gdns, gd dесs, FсH ... b K1,

с

с4R K2'

P1, M1,

PAS,

MoT,

d NY!ИJМ

...

sееks same

housе

Knitting Nеwspapеr ad. for a сar

fsh

viеw to BEсD, SEсM,

Source Newspapеr ad. fot a

with еtс.

Pеrsonal ad. in a US nеwspaper

Мeапing

Dеtaсhеd housе, large gardеns,

good dесorations,

full сеntral heating ...

Knit 1, Purl 1, Мake 1, Cablе 4 forп,ard, Knit 2 Powеr assistеd

steеring, holds an

МOT сertifiсatе,

full sеrviсе history New York white

Jewishmalе...

Bondage

&

Disсiplinе, Sadism & Мasoсhism, еtс.

9.2

Longhаnd

Shorthаnd

с

12 + 41 3, + 42

a b

d е

f

Thrее multiplied by two and a half Thе sum ofЬеlvе, and four and thrее quarterc Thе sum of thе squares of thrеe and four Four times thе differenсе of nine and thrее Fivе plus fouц all dividеd by the produсt of fivе and four The numbеr of inсhеs, 1, is found by mulфlying thе numbеr of mеtrеs, М,ЬУ 39.37.

3

x2Ь

4(9

ii*

-

3|

I=39.37xМ

a А tеn-yеar-old,s dosage = 24 x#n = 24 x# = 1"О.9pg. Ь A fоur-year-old,s dosagе = 200 хudп = 200 Х = 5Qрg.

9.4 a The tеmpеrature in dеgrееs Fahrеnheit, F = 1.8 x 30 + 32 9.3

rсa

= 86"F. = 1.8 x 100 + 32

Ь

Thе tеmperature in degrееs FaЬrеnhеit,

с

Yоu might havе tried to find this tеmperaturе Ьy trial and

F

= 212"F.

еrror. Thе solution is _40. This сan bе сhесked Ьy puшing

thе value -40"с into the formula. and the rеsult -40"F сomes out.

Thus: тhe tеmperaturе in dеgrееs Fahrеnhеit, F = 1.B х_40 + 32

Thе answеr сan bе сalсulatеd dirесtly ", follows. Lеt thе unknown tеmpеraturе ="i#3,io.", T. If thе tеmpегaturе ToF = T"C, thеn they arе сonnесted by thе formula, as follows.

T=1.8T+32

Thе taЬlе bеlow summarizеs how йis еquation сan now bе solvеd.

AIgebrа

T

T-1.8T

-0.8т -o.8Т

r

-{.8

1

2

E,хplanаtion This is thе еquation to Ье solvеd =1.8T+32 1.87-1.87+ 32 Subtraсt 1.8T from both sidеs (1) = Simplify the tеrms in 7 Dividе Ьoth sidеs by -0.8 (2) Simplify Thе solution of thе еquation is

= 32 _ -

-O.8

=

-40

-* З2

-40

Thе intеntion hеrе is to сollесt thе Т tеrms on one side of the = and lеavе thе numЬеr on thе othеr sidе.

TЬe intеntion hеrе is to isolate thе T on its own.

Rеmеmbеr that thе oЬjесt of thе еxеrсisе is to find thе valuе of T.

9.5 The formula is: a С - 20.1'6 + (0.042 nearеst pеnny). = 20.t6 + (0.042 nеarеst pеnny).

ЬC

x

C = 20,,l'6 + 0.О42U 944) = d59.81 (rоundеd to thе

x 3L22)

=

{'15L.28 (roundеd tо thе

9.6 Proving with algеbra It is truе that the produсt of two odd numbers is аlways an

odd numbеr.

First herе arе somе speсial сasеs: 3 x 5 = 15, whiсh is odd 5 x 9 = 45, whiсh is odd 1'3 x 7 = 91, whiсh is odd I13 x 5613 = 634 269, whiсh is odd.

Now we move on to a general algеbraiс solution.

As Ьеforе, we сan lеt thеse two odd numbеrs Ье rеprеsentеd

Ьу2K+1and2L+7.

Thеir produg1= (2K + 1.) х (2L + 1) This саn Ье written as 2K(2L + 7) + L(2L + 1| implifying, wе gеt 4КL + 2K + 2L + t Ignoring the final tеrm, thе ,7' for the moment, notiсe that thе first three terms, 4KL + 2K + 2L сan Ьe wriffеn as

2(2KL+К+L).

Sinсе this numbеr сontains a factor of' 2, and 2KL + K + L is a whole numbеr, tЬen 4KL + 2K + 2L must Ьe еvеn.

Now add thе final .1' and it follows that 4KL + 2К + 2L + 1 is odd.

This сhapter providеs a fеw suggestions fоr numbеr puzzles аnd aсtivitiеs whiсh should hеlp to amusе and еntertain on a long journеy of a Wet wееkеnd.

1 Number plate games Gamеs with сar numbеr platеs сan bе рlayеd anywherе neat a road or сar park or on a long journеy. Sее if you сan spot numbеrs whеrе:

(i)

the digits add to 10 (е.g. T163 WV) all thе digits are even (е.g. s426 ЕJs) (iii) all the digits are odd (е.g. Х715 FLo) (iv) all thе digits arе prime numЬеrs (e.g.T725

(ii)

WxC)

!Иith praсtiсe' most pеoplе quiсkly gеt good at looking for and spotting paffеrns in numbers, in whiсh сasе thе game сan bе madе morе сhallenging. For ехample, try to spot numЬеrs

q) э

whiсh are thе produсt of two primes (е.g.Y247 produсt of 13 and 19, whiсh arе both primе).

g

т'

The renowned |ndian mathematiсian, Srinivasa Ramanujan was onсe visited by the British mathematiсian G. H. Hardy. Hardy remarked that he had just trave|led in а taxi bearing the rather dull

сL N N сL П ПI

number

сubed).

Not everyone has quite the sаme fаscination and ski|l with number properties as Ramanujan. But you should not under-

сл ч

estimate the degree of interest and social caсhet you are |ikely to attraсt at dinner parties by passing on gems about the properties of сertain numbers. For examp|e' a perfeсt opening |ine during that awkward 'first introduced' phase at a party might be, 'Did you know that our host's telephone number is the first six digits of the deоimаI expansion of pi, baсkwards?'

Пl

э сл

We|l, this sort of chat.up |ine сertain|y seems to work for me!

2 Pub сriсket

з

a

''1729.

'On the contrary', said his friend, 'it is a very interesting number.

It is the smallest number expressible as a sum of two cubes in two different ways' (10 сubed p|us 9 оubed or ,t2 сubed p|us 1

o

o .I a GI q) o o

TTT 247 is the

In this сhapter you will learn:

.

about the fun side of maths with a colleсtion of puzzles and number games.

This is plаyеd on a сar or сoaсh journеy whеrе thе routе is likеly to Pass a numbеr of pubs.

Playеrs take turns to .bаt'. You sсorе aссording to thе numbеr of lеgs (human or animal) whiсh сan bе sееn on еасh pub sign that you drivе past. For еxаmplе, thе .Bull and Butсhеr' sсorеs six (four for thе Ьull and two for the Ьutсhеr), whilе the .White Hart' sсorеs four. If you go past a pub sign vrhiсh has no lеgs, thеn you arе .out' and thе nеxt playеr takes a turn at batting.

Starting with the thumb, numbеr thе fingеrs as shown. Now to multipl5 saу,7 and 8, touсh the 7 finget of one hand цrith thе 8 fingеr of the othеr (it doеsn't mattеr whiсh way round).

depiсtеd!)

Now thе answer to 8 x 7 canЬe found as follows: a Count thе numbеr of fingеrs beloш аnd iпсludiпg the

Popular signs in this gamе, Ьy thе way' arе thе .Coaсh and Horsеs' (with 24 or morе lеgs, dеpending on the numbеr of horsеs) and thе .Criсketеrs'Arms' (with up to thirty, depеnding оn rмhеthеr or not both batsmen аnd both umpirеs arе

3 Guess

.

my number

onе playеr piсks a numЬеr betwеen 1 and 100 and thе othеr player must guess it with as fеw quеstions as possiblе. Note thаt the quеstions must bе suсh that thеy rеquirе a Yеs/I.{o answеr.

Usеful quеstions afе onеs suсh as: .Is it lеss than 50?' or .Is it еvеn?'

Lеss usеful quеstions arе onеs suсh as: .Is it 26?'sinсе this еliminatеs only onе number at a timе.

4 The story ot

12

How many diffеreпt stories сan you make of 12? .Wеll, it is 11 + 1, 10 + 2, etc.

Dont forget it is also 12 + 0. .!Иhat

3,3 x 4 and 6 х2?

And don't forget |2 x 1. Now we'rе gеtting stuсk. about 4 x

Ah! it's 24 lots of a half, so it's 24 x l.

Wеll, if you'rе allowing fraсtions itЪ 1i Hey, this сould go on all nфt!

Мost pеоplе know thеir .timеs taЬlе' up to about 5. Howеver, with 6 and aЬove they may havе problеms. Hеre is a method whiсh providеs thе answеr to all produсts bеtwееn 6 and 10 (remеmbеr that in mathеmatiсs .produсt' mеans what you gеt

whеn yоu multiply), using the сheapеst digital сalсulator around Plaсе your hands in front of you as in the diagram, thumЬs uppеrmost.

So thе answеr is fifty-six.

Try it for somе othеr numЬеrs. !Йhy doеs it always work?

You will nееd to do somе algеbra to provе it works еvеry timе!

6 Magiс squares This is a magiс squarе.

I

1

6

3

5

7

4

I

2

+ 10*.

5 Finger tables

- your fingеrs.

b

touсhing fingеrs (in this сasе 5). This givеs thе numЬеr of tеns in the answеr. Мultiply thе numЬеr of fingеrs on еaсh hand аboue tЬe touсhing fingеrs (hеrе it is 3 x 2 = 6). This givеs the number of units in thе answеr.

I 7

6

A magiс squarе is so namеd Ьесause аll thе rows, сolumns and diagonals .magiсally' add to thе same tоtal (in this сasе, 15).

This is known as а 3 by 3 mаgiс squarе bесаusе thеrе arе thrее

rows and thrее сolumns. о Can you makе a diffеrеnt 3 Ьy 3 magiс square so that all thе rows, сolumns and diagоnals add to 15? о How аbout a 4 bу 4 magiс squarе ... or a 5 Ьy 5?

ILintzYoll may have spottеd that thе total of 15 for the 3 by 3 magiс square is thrее timеs fivе (5 is both the numЬеr in thе сеntrе sqrrare аnd thе middlе valuе in thе range of 1 to 9). Can you first of all work out what thе rows, соlumns and diagonals of thе 4 by 4 square should add to?

7 Magic triangle

Tеn birds sit on а roof. You makе a noisе to sсare thеm off, and all Ьut four of thеm fly away. Hour many arе lеft? е How old is a соin еngravеd with thе datе 88вс? f You havе threе pairs of diffеrеnt-сolourеd soсks in a drawеr, еасh soсk sеparatеd from its partner. Нow many singlе soсks do you take out, without looking, to Ье surе that you havе: (i) a matсhing pair of any сolour? (ii) a matсhing pair of a partiсular сolour?

You nеed seven сandle stubs to makе a nеrм сandlе. How many сandlеs will you Ьe ablе to make if you start with 49 h

i

stubs? .!Иhat

is thе fеwеst number of сoins you nеed to pay for somеthing сosting 85p without rесеiving сhangе?

Fill in thе missing signs in thе sums Ьеlow.

2Г14Гl 3=5 2г12Г12=3 8

Thе numbеrs 1,2 and 3 have Ьееn plaсеd at the vertiсеs (i.е. thе сornеrs) of thе triangle.

Can you plaсe two of thе following numbers 4, 5, 6, 7, 8 and 9 along еaсh sidе of thе trianglе so that thе four numbеrs on еaсh sidе (i.е. inсluding thе numbеrs at thе vеrtiсes) add up to 17?

8

u,t^op

эplsdn

Thе yеar L961reads thе same whеn turnеd upsidе down.

Whеn was the most rесеnt yеаr prior to 196,l, that reads thе same upsidе dorмn? !Иhen цdll Ье thе nеxt Уear tьat this works?

Еxplore what happеns whеn lеttеrs and numЬеr are turnеd upsidе down. For еxamplе, what digits bесomе lеttеrs of thе alphaЬеt when turnеd upsidе dоwn?

9 Logiсally speaking a What word is from this sеntеnсе? is to 1861 as 8901 is to what? с Sevеn numbеrs сan Ье sееn in this sеntеnсе' Ьut you know thrеe of thеm arе writtеn out baсkrмаrds.

ь 1981

What аrе thеy?

10 CaIсulator inversions Еntеr thе numbеr 53045 on your сalсulatоr. Now turn it upsidе down. Can you sее somеthing that hеlps you keеp your fееt off thе grоund?

Next, try dесoding thе following .mеssagе':

3145x1.0+1.23 204+4

0.5x0.5x2 257x3

Finall5 havе a go at the following сгossword puzz|e. Usе thе rеsults of еaсh сalсulation, upsidе down, to fit thе pazz|e.

11 Four4s Thе numЬеr 7 сan Ьe еxprеssеd using four 4s as follows:

7=4+4-* Еxpress thе numЬеrs 0, 1.,2... 10 using еxaсtly four 4s and any other оperation (for еxamplе' #,_,X, +). You аre also allowеd square root, V Hintz 2 сan be written as

ff.

12 The bells, the bells! a If it takes 15 sесonds for a сhurсh

bеll to сhimе 6 o'сlосk, how long dоes it take tо сhimе midnight? (It's not as easy as it sounds!)

bA

fenсе panel is 2 m long. How many fеnсe posts arе nееded to panеl a 16 m gap?

13 Return iourney

I plan to сomplеtе a rеturn jоurnеy (therе and baсk) in an Aсross

1 6 7 8

3x].000+45mayЬea

good fit (4) 5 x 1'111 - 18 is no morе (41

-

19 is usеful for troublеd watеrs (3) 25 + 11 Ьut plеasе spеak 9З

up! (2)

10 1'2З х 25 tot a quiсk gin?

(4)

12 Two sсore, for a surprisе

Down

2

0.65 + О.1234 is onе third of a poliсе offiсеr's grееting (s)

3 17х10x7 x3+3is anothеr possiЬility (4) 4 lLz x 37 is еmеrald in Irеland (4) 5 1111 x5-48forano-win situаtion (4) 9 Тwo fifths expressеd as a

deсimal is onе third of a Christmas greеting (2) |4 13 х 244з9 + 5000000, so 11 1101 x 7 dеsсribеs life in stop and buy somе (7) thе 10 aсross lanе? (4) 13 "l'9 x 2 х 193 саn bе found 16 # + 1 for thе Spanish (2) 17 {&Yeri|у, it sounds likе a ona1aсross(4) сolлr hath spoken (2) 15 Half сould bе a nеedle pulling thrеad (2) (21

аveragе time of 40 mph. Howеvеr, my оutwаrd journеy is slow and I сomplеtе that part at 20 mph. How fast must I travеl on the return journеy to avеragе 40 mph ovеrall?

14 Find the numbers

a Two Ь

с d

е

сonsесutivе numЬеrs add

tо give 49.

are thе

numЬers? .!Иhat Thrее сonsесutivе numЬеrs havе a total of 60. arе thе numbеrs? .What Two сonsесutive numbеrs havе a produсt of 600. arе the numbers? (Nore: Thе produсt is what you get when you multiply.)

Thrее сonsесutivе numЬеrs have a produсt of 1716. What arе thе numbеrs?

Two numbеrs havе a diffеrenсe of 15 and a produсt ot 54. .!Иhat arе thе numЬеrs?

15 Explore and explain the pаttern Try thеsе out on your сalсulator a 37 х 3, З7 x 33' З7 x 333, etс.

ь

.!Иhat

1',, 1'L,, 1,l'1z, LL1"L,,

....

(sa5 8,9 and 10) Take thrеe сonsесutivе numbеrs (8x10=) Мultiply thе first and third numbеr (9x9=1 Squarе thе middlе numbеr SuЬtraсt smallеr answеr from biggеr answеr (81-80=1.)

Thе rеsult is 1 Doеs this always work for thrее сonsесutivе numbеrs?

,l',2, Try to affange the digits 3,4 and 5 (eaсh usеd onсе only) to form two numbеrs so that thе sum of thе numbеrs is as largе as possiblе.

Now try to arrangе thеm so thеir sum is as small as possiЬlе. Ехplore thе same sorts of quеstions for multipliсation' ...

Reveвe the digits and subtraсt whiсhever the smaller from the bigger.

сropping up?

is to

|saу' 724

|724

-

427 = 297

1792 + 297 = 1O89

.!Иhy

doеs 1089 kеep

If not, thеn whiсh numЬеrs does it not work fоr.

.Why?

By the way, this puzzle сan be set up as a triсk to impress your friends, as follows.

Write down the figure 1089 on a pieсe of paper and seaI it in an envelope beforehand.

Then ask the friend to сhoose any three-digit number and perform the сa|сu|ation desсribed above.

Note: lf a zero oссurs in any part of the са|оulation, this must be сounted as we|l.

534-435-099

You will Ье givеn:

еithеr

a

As many gold piесеs as thе numbеr of minutеs you havе bеen alivе;

or

many gоld piесеs as the largеst numbеr you сan gеt on yоur сalсulator by prеssing just five kеys;

с

onе gold pieсe on thе first day of this month' two on thе sесond, four on thе third, eight on the fourth, and so on, еnding on thе last day of thе month. l0(ith thе hеlp of your сalсulatоц dесidе what yоu shоuld do.

19 lnitially speaking

Do you always get 1089?

For example:

options.

b As

Reverse the digits of the answer and add it the answer

T.y it for othеr thrее-digit numЬеrs.

maths tеxtbook, thе еvil, сunning and еxtrеmеly wеalthy empress Calсula оffеrs you a сhoiсe of onе of the following

or

17 1089 and all that number

18 GoId pieсes In exсhangе for yоur luсky сalсulаtor and this oh-so-prесious

16 Large and small sums

Take a three-digit

.t089.

Reversed, 099 becomes 990. Тhis gives 099 + 990 = .viсtim' |n order to give the triсk a|i|t|e,pzazz', ask your for some additional but totally irrelevant information (for example, date of birth' telephone numbei favourite сolour, and so on).

This puzzlе is еasiеst to еxplain with thе following еxamplе.

Solution Сlцe 3BМ,sHTR 3 Blind Мiсе, Sее How Thеy Run Now сomplеtе thе solutions Ьеlow.

If you gеt them all right, it should spеll out thе name of a film as well аs a сatсh phrasе whеn you rеad down thе сеntral boхes.

3 BLIND 101

10

1s

МIсЕ

E

ЕЕ Ho\и TНЕY RIJN

_------ П--

ш---- ПN

П--K-----ш-П--_3

Е---- ПN ш--ПTT{Е10

76Е 64ш 11 E 100

Iш

Ы

------

П--- E---- шN E --_ v,Е 3 П---- ш--- @----- --Е-------П-__

24

36П

шNшE---

ПN П-- --@N ш-П ПN ш -_П----- ш---

П шш М----

20 Nim Nim is one of thе oldеst rесordеd gamеs, possibly Chinеsе in origin, and is usually played Ьy two pеoplе. Therе are many versions of Nim, oпе of rмhiсh is dеsсribed bеlow.

Start with a pilе of matсhstiсks. Еaсh player, in turn, rеmovеs at lеast onе but not morе than six matсhеs. The winnеr is thе player who piсks up thе last matсh.

Hеrе is a typiсal gamе.

Start чrith 28 matсhеs in the pile.

A piсks up 4 matсhеs,|еaving24 B piсks up 5 matсhеs, lеaving 19 A piсks up 2 matсhеs, leaving 17 B piсks up 6 matсhеs, lеaving 11 A piсks up 3 matсhes, lеaving 8 B piсks up 1 matсh, |eaving7 A (rеalizing that defеat is just a matсh away) piсks up 1 matсh, lеaving 6 B piсks up all remaining 6 matсhes, thеreЬy winning gamе, sеt ... and matсh.

21 Guess the number Thе rulеs of this game аrе givеn at thе еnd of Chaptеr 05.

22 Calculator snooker

Рlауer А еnters any two-digit number. B takеs a .shot' Ьy pеrforming a multipliсation sum. To .pot' a ball, thе first digit of the answеr must bе сorrесt aссording to thе tablе shown.

(Thе degreе of aссuraсy сan Ье variеd aссоrding to expеriеnсе.)

Ball

Red

Yellow Green

Brown

Blue

Pink

BIaоk

Result needed

1

2...

3...

4...

5..

6...

7..

Sоore

't

2

3

4

5

6

7

othеrп,isе, thе rules arе similar to .rеal' snooker. Thеrе arе 10 (or 15) rеds and onе of еaсh of thе six .сolours'. A playеr must

sсorе in the order гed, сolour, rеd, сolour, and so оn, until all the gonе. (Nofe: Thе сolours are replaсed but thе reds arе rеds havе .sИhen

not.) thе last rеd has gonе, thе сolours are pottеd .in ordеr' and arе not rеplaсеd.

For ехamplе' one sequenсе of plays was: Player

Еnters

Jimmy Peta

69

Karen

Disp|аy

69

х2=

138

x5= x2=

690 1380

x5.5= x 1.6

Comments

7590

12144

=

x7=

85008

23 Place invaders plaуers)

Peta pots the first red.

She eleсts to go for blue ... ... and misses.

Karen pots the seсond red.

She eleсts to go for black ... ... and pots it. Тhe third red.

She eleсts to go for blaсk agаin ... ... and misses.

(a game for one or two

This gamе сan be playеd at differеnt lеvеls (1,2, 3, еtс.). Мovе on to a nеw lеvel whеn you find the game too easy.

P|aсe invaders

1

Еntеr a 3.digit numbеr into thе сalсulator (say 352). Thеse thrее digits arе rеmovеd onе at a timе by subtraсting to zero. Еxamplе: Stаrting numbеr 352

Kеy prеssеs

Eztr

Е s0Е !з00

!

Note: If you start with a S-digit numbеr, the game еnds with a display of 100000.

Display You сan make up your own numbers and let your pаrtnrr

3s0 300 0

Try thе following: 416, L4з, 741.

38

5, 5L2,

8 5

з, 264,,l"79,

9 54, 5

Plaсe invaders 5 The samе as Plaсe invaders 3, еxсеpt that you сan usе deсimals, e.g. 451.326 to be removеd Ьy subtraсtion in thе ordеr of 1',2, 3,4, 5, 6.

89,

PIaсe invaders 2 Thе same as Plaсе invadеrs 1 еxсеpt thаt thе digits must bе rеmovеd in asсеnding order. Еxamplе: Starting numbеr 352

Kеy prеssеs

EzЕ

fl зо0p E sоЕ

Notez If yоu make a mistakе, it is еasy to undo by adding baсk thе numЬer that you havе just subtraсted.

Answers for Ghapter 10 1-5 No сommеnts.

6 Hеrе is anothеr 3 by 3 magiс squarе

Display i'e. you remove thв

350

2, then the 3, and thеn the 5

50 0

Plaсe invaders 3 Thе samе as Plaсе invadеrs 2, еxсеpt that you сan usе numbеrs тyith more digits. Try 4-digit numЬеrs, then 5, 6' 7 and' 8.

Plaсe invаdeпs 4 Thе samе аs Plaсе invаdеrs 3, еxсеpt that you rеmovе the digits by addition, not subtraсtion. This timе thе gаmе will еnd with a 1 followеd by a string of zеros. Use as fеw goеs as possiblе. Еxamplе: Starting number 1736 Key prеssеs

E 4Е

Е 60Е

E 2ooE E s000Е

Display L740 1800

2000 10000

I

1

6

3

5

7

4

I

2

Now herе is a 4 by 4 magiс square.

16

3

2

13

5

10

11

I

I

6

7

12

4

15

14

1

\Pith this 4 bу 4 magiс squarе eaсh rоw, сolumn and diagоnal adds to 34. !Иhat makеs this one еvеn morе magiс is that еасh bloсk of four сornеr squarеs also adds to 34.

Hang on - what about thе four сепtral numbеrs ...?

lI 0=4+4-4-4 a 4+4 44 4 L=

2= а

4+4

Of

1+*

44

, Of 4 + + - +

4+4+4

4=4+#

5={4 +"'!4+1

6=4+# 7=#

-4or4+4-X

8=4+4+4-4

9=4+4+X

8a 9a

1881 b

.What

10=4+4+fr 12 a Answеr 33 seсonds.

It takеs 15 sесonds for 6 сhimes. Thеrе arе 5 intеrvals bеtwсеn thе first and thе siхth сhimе. Thеrеforе it must takе 3 sесonds per intеrval. Тwеlvе сhimes has 11

8008

word is missing from this sentеnсе? b 1981 is to 1861 as 8901 isto 8601. с Sеven numЬеrs сan bе sееn in this sеntеnсе' but you knoп, thrее of thеm arе writtеn out baсkwаrds. d Four Ьirds arе lеft. е No сoin сould havе beеn еngravеd with this datе.

f (i)

4

soсks

(ii) 6 soсks

8 Sеvеn сandlеs initially. But thеsе Ьutts' so the answеr is еight.

h Four сoins: 50p

10

+

20p

i 2Г:л4Г:l 3=5 2Г}-12Гт12 = З 8гт1 2Г12 =2 ЕLsiЕ is so iLL

b

will produсе sеvеn more

sесondly, rеmеmbеr that thеrе is аlways onе fеrмеr .spaсе' thaп therе arе .posts'.

+ 10p + 5p

or

8|т1 2fT1 2 =2

intеrvals, hеnсе 33 sесonds.

9 Posts arе nееdеd for 8 spaсes.

Both thе aЬovе questions refеr to a сommon .typе, of maths quеstion known as thе old .posts and spaсеs'triсk.

Thеrе are two things to rеmеmber in any .posts and spaсеs' sоrt of situation. Firstly, bе сlеar about whiсh you arе trying to сount' thе .posts' or the .spaсеs'. And

13 It сan't bе done! Supposе thе total distanсе (thеrе and baсk) is 40 milеs, thеn thе total journеy (thеrе and Ьaсk) must takе

ехaсtly onе hour. If thе outward journеy of 20 milеs is сomplеtеd at a speеd of 20 mph, thе onе hour is сomplеtеly usеd up!

14a 24 and25.

b L9,20 and21. с 24 and25.

d 1L,L2 and 13. е 3 and 18.

a

111., 1221,12321. Thеsе numbеrs arе palindromеs (i.e. thеy rеad thе samе bасkwards as forwards). ь 1, tz,l', L232,l', 1'234321. Again a palindromiс sеquеnсе similar, but not idеntiсal to Part a. с This rеsult works for all sеts of thrее сonsесutivе numbеrs.

16 Thе largеst sum is 5Ъ $a\ + 32| ,l.59 (125 + 34| Thе smallеst sum is Thе largеst produсt is 22 403 (521 x 43| Thе smаllеst produсt is 3185 (245 x 13).

15

17 This rеsult works for most Ьut not all numЬеrs. Try starting rмith somе рalindromiс numbеrs and seе what happens.

18 Let us сalсulate eaсh option in turn. a As many gold piесеs as thе numЬer of minutеs you havе Ьееn alivе.

b

Assuming that you arе' say' forty-fivе yеars old, thе requirеd сalсulation is: 45 х 365 x 24 х 60 = 23 652 000.

In othеr words, bеfwеen 23 million and 24 million.

As mаny gоld piесеs as thе largеst numbеr you сan gеt on your сalсulator by prеssing just fivе kеys. Мy bеst effort

3 BLIND MIсЕ ш ЕЕ 101 DАII\4AII D] шs

l

Hov

THЕY RI',N

3 МUsffiшввкs

pEсдпв EЕARs шN ls шol,Еs ПN сoEг сoI'JRsЕ pш 24 ЫrAсKB коs @ акво ПN v/Е 3 ffi шсs fl кoм @nшшг акв 10

l

l

99 Гт1g г=l This сomеs up with a Puny 891 gold piесеs. lf your сalсulator has a .square' funсtion, markеd k]l you will do muсh bешеr than this. So muсh so that evеn fivе key Prеssеs will produсе an answеr too large for most сalсulators to display and whiсh will therеfore rеsult in аn

TI]Е 10 @ oммашом E t.шs 36 ш INсFrЕs шN ш 76 BRoМBoNЕs ш [IЕ BIG PARАDЕ 64 ш QUARЕS с E Еss BoARD l1 Е LAYЕRs гo твдr fl вe'м 100 EЕ TI{Е BOILING POIN ш @г M дтпд

99 ЕI Е Е produсes an еrror

20_2з No сommеnts.

hеre was to prеss:

error mеssagе. For ехample:

с

19 Thе solution is .Somе like it hot'.

mеssagе

on

my

сalсulator.

Onе gold piесе on thе first day of this month, two on thе sесond, four on thе third, eight on thе fourth, аnd so оn, еnding оn thе last day of thе month.

This arrangеment may sound very lоw key, Ьut in faсt it will produсе an astronomiсally largе rеsult quitе quiсkly.

Thе Ьеst rмay to gеt an imprеssion of thе еffесts of douЬling is to sеt your сalсulatorЪ сonstant to multiply Ьy 2 and thеn kееp prеssing Г;l. What you will sее should bе somеthing likе thе following:

Day

Amount

1 2 3 4 5 6 1 2 4 8 16 З2

12 ... 16 ...2048 ... 32768 ...

Bеforе you gеt to thе еnd of thе month you will proЬaЬly find that thе сalсulator has ovеr.strеtсhеd itsеlf and produсеd an еrrof mеssagе!

The answer, thеrеfore, is that the .bеst' option to сhoose

depends on what sort of features your сalсulator has * for ехample, how many figurеs it displays, whiсh kеys

it providеs and so on. But whiсhеvеr сalсulator yоu usе, thе third option is сеrtainly a good onе to go for!

шN

oN l

ПN l

l

l

Едко

Nolм that you havе сarefully rеad through evеry pagе of thе prесеding tеn сhaptеrs (wеll, maybe you skippеd a fеw pagеs!),

yоu might likе to tаke stoсk of what you havе learnt Ьy trying to answer thе quеstions in this diagnostiс quiz.

A quiz, or tеst' сan

bе taсklеd in many diffеrent ways. If you gеt еvеry quеstion right, you may fееl good аbout yoursеlf, but you probably havеnt lеarnt аnything from it. If you gеt all thе quеstions wrong' you will proЬaЬly fееl prеtty deprеssеd and unaЬlе to еxploit thе lеarning opportunities offеrеd by thе ехperienсе. I hopе that you will bе somewhеrе in bеtwееn. This quiz is not dеsignеd to triсk you or to makе you fееl dеprеssеd.

Having said that, you arе vеry unlikеly to find all thе quеstions еasy or to gеt rvеry quеstion right.

Hеrе are somе guidelinеs for taсkling the quiz.

You should Ье prеparеd to usе your сalсulator for еvеry

quеstion, ехсеPt for Quеstion 1 wherе you are askеd not to.

Rеad еaсh quеstion сarеfully bеforе you do it, so that you

a.

аns|Цеr eхасtlу the question thаt bаs been аskеd. For еxamplе, if it asks you to writе numbеrs in ordеr from smallсst to largеst, dont givе your answеrs from largеst to

fl

g) GT

smallеst. Ье afraid to look things up in еаrliеr сhaptеrs of thе book if you havе forgoffеn, say, how to сonvrrt milеs into kilomеtrеs. This isnt a tеst to Ье takеn undеr еxamination

о Don't

сonditions and you arеnt eхpеФеd to rеmemЬr аll thе formulas and сonvеrsions in your hеad.

o a + o э

So, pleаse givе the quiz your vеry Ьеst shot. It is dеsignеd to tаkе about onе hour, Ьut Ье prеparеd tо takе longеr thаn that if you

nееd to. !7hеn you havе donе all that you сan do, thеn work thrоugh my solutions at the еnd of thе сhaptеr. As you will seе, I havе inсludеd dеtailеd сommеnts aftеr thе solutions in ordеr thаt you сan .turn your еrrors into lеarning oppoтtunitiеs'. For

Пl

еaсh оnе thаt you answеrеd inсorrесtly, аsk yoursеlf thе following questions:

. .

.сI

C fl

N

.wHЕRЕ AND wнY HAvЕ,I GoNЕ !trRoNG?' ..W}IAт сAN I LЕARN FRoМ TнIs?'

Good luсk!

ln this chapter you will leаrn:

.

how muсh maths you have already learnt by reading this book!

Quiz 1

a the

Try thеsе сalсulations without using your сalсulator. a (i) 3 x 14; (ii) -5 - 17; (фi) -20 + 4; (iv) 15

Ь

с

FI 2L +

(i)

8,;

1*;

2x; 0i) (ii) {81; 3Ь

_

x (iii){(3, (iii)

5*

3;

(iv) 33

+ 75.

x

5

+ 4,)-

8

Еxprеss thе following as dесimal numbеrs.

a20+7+Й+т€г+*о Ь 60-3+*+й- й L

1000

spеed of a car trаvelling on thе outsidе |ane of, a motorway in thе UK b thе speеd of somеone having a Ьrisk walk с thе spееd of a top 100 m runnef d thе spееd оf а supеrsoniс jеt airсraft. .It is possiblе to fit the world's populatiоn on thе Islе of TИight.'

Usе thе follorмing faсts tо сheсk this сlaim. о Thе Islе of !Иight hаs an area of 381 krn,. о It is possiЬlе to squееzе about 10 аvеragе-sized

pеoplе into one

In thе numbеr 13.873, tЬe ,7' reprеsеnts thе number of hundrеdths.

.What

a

ь c

does the,7, rePrеsеnt in thе following numbеrs?

271.93

1'1.724 0.001.1.7

9

roughly 6 billion.

Nаme

с two fifths or 0.5? d 10 pеr сеnt or an еighth? е 8 pеr сеnt or a tenth?

RNLI

thrее quartеrs or 70 pеr сent?

0.06 or onе twеntiеth?

Using suitab|e metriс units, estimate thе following: a the hеight of a сhair sеat from thе floor Ь thе width of a сookеr с thе wеight of a nеw-born baЬy d thе distanсе from London to Birmingham e the сapaсity of a doorstеp milk bottlе f thе wеight of a lеtter

g h

i i

thе tеmperature insidе a domеstiс rеfrigеrator

two teaspoonfuls of liquid thе thiсknеss of а {1 сoin the tеmpеrаturе on a hot summеr's day in London.

Рlaсе thе following in oгdеr of sizе from srnallеst to lаrgеst a 450m! half a litrе; one pint; 75 сentilitrеs Ь l.i metrеs; 18 сm; 1.2 km; 300 mm с half a wееk; four days; 95 hours; 0.01 of a уear.

Using both itnpеriаl аnd metriс units, estimаte the

following:

thе timе of writing, is

Thе tablе bеlow lists, in thousands of pounds ({000)' thе voluntary сash donations to the top 10 IJK сharitiеs in a partiсular yеar.

!Иhiсh is bфer:

a Ь

m2.

о The worldЪ population, at

National Trust Oxfam Savе the Childrеn Fund Impеrial Canсer Rеsеarсh Fund Canсеr Researсh Campaign Barnardos Hеlp thе Agеd Salvation Army

NsPсс

a b

с

Inсome ({'000) 78 74s 58 972 56 229

53 48 45 36

866 395

352 452

331,41

32 30З 30 818

Rеrлrritе thе tеn inсomes, roundеd to thе nеarеst {million.

Using the roundеd figurеs, sketсh a horizontal barсhart to reprеsеnt thе earnings of thе top five сhаritiеs in a partiсular yеar. Еxplain why а pieсhart would not Ье аpprоpriatе for

dеpiсting thе еarnings of thе top fivе сharities.

10 In a сеrtain уear' an еstimatеd 19.2 million international visitors саmе to Britain and spent {9.2 billion. Thе piесhart bеlow shows thе еstimаted annual tourist spending, brokеn down by whiсh part of thе world thе tourists сamе from.

Australia/NZ

(whеrе r reprеsеnts her annual inсomе and

0ther

tax аllolмаnсеs)

а

Africа

b

Мidd|e East

с

Far Еast

A rеprеsеnts hеr

Ехplain in your own rлrords how, aссording to the tax formulа, annual tax on еarnings is сalсulatеd.

Calсulatе how muсh Pеtra чdll hаvе to pay in tax:

(i)

ovеr thе yеar

(ii) еaсh wееk. (Givе your answеr to thе nеarеst penny.)

Assuming thеrе arе no othеr stoppagеs from hеr wages' сalсulаtе horм muсh Pеtra will reсеivе еaсh wееk after thе weеkly tax bill has bееn paid. (Givе your answеr to thе nеarеst pеnny.)

Solutions to the quiz

North Ameriсa

a b

с

From thе piесhart, еstimate thе annual spending by tourists from Еuropе.

From whiсh onе of thе regions listеd hеre did roughly {2 billion of the tourist rеYеnuе соmе?

If Britain wеre aЬlе to attract an extra 10 million visitors, how muсh morе rеvеnuе might it bе ablе to generatе? Мakе a note of any assumptions that you have madе in doing this сalсulation.

t1 A survеy of toy priсes was tаkеn in thrее largе storеs. Thе pгiсes (in {) of thrее toys arе summarizеd Ьеlow.

Toy

A

Toy B Toy C

Hаmleуs lohn Leulis

1,2.99 9.25 29.00 22.7s 7.99 5.95

Toуs

Я Us

12 Petra

earns {,6200 pet Уeat doing pаrt-time woгk. Shе pays tax at thе Ьаsiс ratе of 20Y" and hеr tах allowanсes arе {37s0.

Thе annual amount that shе has to pаy in Taх, T, сan bе сalсulatеd from thе following formula.

T=0.2(t-A\

с 62.341,

3a

4

tens

b tеnths

с hundrеd thousandths

'a

b

с

d е

4.97

Ьuying thеm at the most еxpеnsivе priсe?

From your аnswеr to part с, сalсulate yоur total savings as a perсеntage of thе total сhеapеst priсе.

(i) 42; (фl -22i {фi) -5; (iv) * or 0.2 (i) 3l; (ii) i; (iii) 161; (iv) 181 (i) 64; (ii) 9 (or -9); (iii) 5 (or -5)

27.364 b 57.8|З

9.29 22.99

On the Ьasis of thеsе priсes: a whiсh of thе thrеe storеs sеems to be thе most eхpеnsivе? b whiсh storе is the сheapest? с If you bought all threе itеms at thе сhеapеst priсе on offеr, how muсh would you havе savеd сomparеd with

d

1 a b c 2a

5

thrее quartеrs

0.06 0.5

an еighth a tеnth

For this quеstion' you must

get both thе

сorrесt answet аnd

thе сorrесt units.

а

b

aЬout 45 сm (I will aссept answеfs bеtween 40 and 50

сm)

(I will aссept answers bеtwееn сm) 70 50 and с about 4 kg, or 4000 g (I will aссеpt answеrs bеtween 3 and 5 kg) d aЬout 175 km (I will aссеpt answеrs bеtweеn 150 and 200 km) е aЬout 570 ml (I will eссept answеrs Ьеtwееn 500 and 600 ml) f about 40 g (I will aссеPt answеrs bеtwеen 10 and 60 g) g bеtwееn OoC and 5oC

about 60 сm, or 600 mm

h aЬotltl2 ml

i

6a 7

i

(I will aссеpt answеrs Ьеtwееn 10 and 15 ml) about 3 mm (I will aссеpt answеrs bеtwееn 2 and 4 mm) about 30.C (I will aссеpt answеrs bеtwееn 25 and 35.C)

450 ml, half a litrе, onе pint, 75 сеntilitrеs Ь 18 сm, 300 mm, 1i m' 1.2 km с half a wеek, 0.0]. of aУear' 95 hours, four days.

a Ь

с d

thе speеd of a сar on thе outsidе lane of a motorway in thе

Imperiаl

Меtriс

110-145 kmph

thе spееd of a top

mph 20_25 mpЬ

5-8 kmph

airхaft

aЬove 760

thе spееd of somеonе having a brisk wаlk 3-5

on thе

Fund

mph

piесhart would not bе appropriatе bесausе thе сomЬinеd inсomе from thеsе fivе сharitiеs togеthеr, whiсh wоuld соrrеspond to the сomplеtе piе, doеs not

30-40 kmph

rеPrеsеnt аnything mеaningful.

aЬovе 1200 kmph

10a

NSPсC

ЬA

thе neаrest f'tnillion)

79 59 56 54 48 45 36

3з З2 31

horizontal Ьarсhart showing thе еarnings of thе top fivе сharitiеs in thе yеaг in quеstion.

Roughly d4.5 billion and {5b.)

(I

will aссept anything Ьеtweеn {4b

b North Amеriса

{5 Ьillion. This сalсulation assumеs that thе еxtra 10 million visitors spеnd at thе samе rate as do

с Roughly

Thе roundеd inсomеs arе as follorмs.

Nаme Inсome (rounded to

Savе the Children Fund Impеrial Canсеr Rеsеarсh Fund Canсеr Rеsearсh Campaign

80

A

basis of tеn people pеr m,, the Islе

National Trust oхfam

0 10 20 304050 60 70 Inс0me {Еm)

a

Barnardos Hеlp thе Agеd Salvation Army

RNLI

Save the Children

of !Иight will hold roughly 3.8 billion pеoplе.

RNLI

Oxfam

Reseаrсh Fund

70-90 mph

It is not possiЬlе.

.Е б ()

lmperia| саnсer

UK

100 m runnеr thе spееd of a supеrsoniс jet

Nati0nаI тrust

lla

сurrent visitors.

Hamlеys is thе most еxpеnsive. b John Lеwis and Toys ЯUs are vеry similar in priсе, with Toys Я Us having the slight edgе (a total priсе of {37 .25, сomparеd with John Lеwis' {37.95). Мost еxpеnsivе priсes = {12.99 + d29.00 + {'7.99 = {,49.98

Lеast eхpеnsivе priсеs = {9.25 + {'22.75 + {'4.97 {,36.97

=

{'49.98 _ {"36.97 = f'1З.01. d Pеrсеntagе saving = tj*s Х 100 = З5.2oЬ

Saving

12a b

=

Thе annual tax bill, T, сan be found as follows.

SuЬtraсt thе tax allowanсеs, А, from annual inсomе, and multiply thе rеsult Ьу 0.2.

(i)

d4e0

(i\\ f,9.42

с {,109.81..

1,

Detailed сomments on the solutions

Nехt, add the fraсtion pafts: * + i. RеmеmЬеr that in ordеr to add fraсtions with diffеrецt dеnominators, you must rеwritе thеm as equivalеnt fraсions vrhiсh have thе samе dеnominatoц whiсh in this сasе is еasiеst

3xl4i

la

This сan bе wriшеn out and сalсulatеd as follows:

?+*=

tr

1,4

Finall5 add thе fraсtion total to thе whole numbеr

42 Solution 42

Solution

x3

(ii) Adding and subtraсting with nеgativе numbеrs сan bе сonfusing and it is somеtimеs a good idеa to writе thе саlсulation out on a number linе. as follows:

-25

-15

nеgativе sign, _.

Nеxt vou do the сalсulation with thе numЬеrs.

20 + 4 mеans T, giving thе rеsult 5.

So thе solution is -5. (iv) 15 + 75

This сan Ье writtеn as *.

Thе fraсtion сan now bе simplifеd to the simplеst еquivalеnt fraсtion by dividing thе numеrator and thе dёnominator Ьy 15, giving thЪ solution * or 0.2.

A сommon mistakе hеrе is to do the division the wrong way round, giving ff = 5.

1,tr

First add thе rлrholе numbеrs: 2 + 1

ztr

=3

+1=

3tr.

First subtraсt thе wholе numЬеrs: 3 _ 2 = 7. down what still has to bе саlсulatеdz 1'Ь -1 Notiсе that you сan't just subtraсt thе fraсtion parts dirесtly bесausе thе fraсtion bеing suЬtraсtеd (tfuее quarters) is biggеr than thе fraсtion it is Ьеing subtraсtеd from (a half). The way round this is tо Ьorrow thе wholе numЬer part' thе 1, and turn it into quartеrs аlong with the fraсtion parts' as follows: Solution = 1+-1=Я-1=1

.sИritе

Multiply еaсh part separatеly by 3 and thеn add thе

-22. (фi) -20 + 4 Herе you arе dividing a negativе numbеr (-20) Ьy a positivе one (4). First you must dесidе on thе sign of thе answеr (i.е. whеthеr the answеr is positivе or nеgativе). Beсausе thе two numЬеrs arе of diffеrent sф, the result must bе nеgative, so rлrritе down thе

2Ь +

3t-

(iii) s* x 3

-10

You start at -5 and thеn suЬtraсt. 1'7.In other wоrds, you movе 17 stеps to thе lеft. This gives thе rеsult of

Ь (i)

donе using quartеrs.

So, thе fraсtions bесomе

= 3.

rеsults together.

5x3=15 \х3=1,L

15 + 1,Ь=

(iv) 33 x 5

1,6Ь

Мultiply еaсh part sepаratеly Ьy 5 and then add the rеsults togеthеr.

3x5=15 Zх5=ч =3*

15+3*=18]

(i) 8.,or

8 squarеdmеans 8 x 8 = 64 (ii) {s1, or the squarе root of 81, mеans finding thе numЬer whiсh, whеn squarеd, gives 81. Thе most

oЬvious аnswef is 9' Ьeсause it satisfiеs this сondition = 81. Howеvец if you givе thе quеstion a little furthеr thоught you may notiсe that thеrе is anothеr possiblе answеl, namel5 -9. You сan сhесk this by

- i.e.9,

squ4ring *9, thus:

(-9\'=-9x-9=81

(iii)r(T ++)

First. work out what is insidе thе Ьraсkets:

(з,iц,)=9+16=25.

Next, find thе squarе root of 25. Using thе samе reasoning as in part (ii), this givеs thе two possiblе solutions, 5 or -5.

2 a20+7+fr

+,'!+тfiт

Thеsе numbеrs havе bееn arrаngеd in a fаmiliar pattеrn - tеns' units, tenths, hundrеdthso and so on. Thus the number сan Ье wriffеn down dirесt|у as 27.364.

b 60-3+*+L*-#

Тhis quеstion is similar to part a but sliфtly сompliсatеd Ьy thе two valuеs whiсh are suЬtraсtеd. Thеrе is no singlе сorrесt way of doing this, Ьut my approaсh was to Ьrеak

it down as follorмs:

6О-З=57 -

с

-т*,г

',Solution #o1

-

4o = # = # * .iйт й + -т#г = 57.81,3

= lоooД = 57 +* +

Division by 1000 has thе еffeсt of moving thе dесimal plaсе three plaсеs to thе lеft. Тhе numbеr 62341, has an invisiblе dесimal point after thе 1 (i.е. ,6234L.,, Thus, Yt6r = 62341

Thе only сommеnt hеrе is that you nееd to keеp in mind thе sеquenсе of dесimal plaсes, whiсh arе as follows: ... thousands hundreds tеns units о tenths hundredths thousandths ...

It is diffiсult to сomparе numbеrs rмritten as fraсtions and thе bеst strategy is to сonvert thе fraсtions to еithеr dесimals

or pеrсеntagеs.

a

Сonvеrting tfuее quartеrs to a pеrсеntage: Thrее quartеrs as a perсеntage = i x 100 = 75Y", whiсh is largеr than 70oЬ.Incidentall5 if you wеrе unablе to find thrее quartеrs of 100 in your hеad, usе your сalсulator' as

Alternatively, press 1 tt s E 169 Г=l Converting a tenth to a pеrсеntagе: A tenth as a pеrсеnta$€ = 19f x 100 = 10%, whiсh is largеr than 8"Ь.

Alternatively, prеss 1 E 10 Гп 100 Гn Therе arе no сomments on this quеstion exсеpt to suggеst

е

that you сould develop your estimation skills Ьy guеssing some mеasurеs around thе housе, thеn gешing out a tapе mеasurе' weighing sсаles, thermomеter, and so on and сheсking how good yorrr guеssеs werе. It is surprising hour quiсkly thеsе skills do improvе with praсtiсе.

As is thе сasе for all quеstions about сomparison of mеasurеs' the kеy thing is to сonvеrt all thе mеasurеmеnts to thе samr units. Onсе this has Ьееn donе, plaсing thеm in ordеr of sizе Ьесomеs a trivial task. Suitable сonvеrsions arе sеt out bеlow for thе measurеs' written here in thе ordеr in whiсh they were originally givеn in thе quеstion. a 450 ml;500 ml; 568 ml; 750 ml ь 175 сm; 18 сm; 1 200 000 сm; 30 сm с 84 hours;96 hours;95 hours; 87.6 hоurs.

As with most еstimatiоn quеstions' thrrе is no single сorrесt

mеthod, as eaсh person draws on their own past knowlеdgе and еxpеriеnсe. a Мy Ъxpеriеnсе of motorway driving is that trаffiс on thе outside lanе seems to travеl at around 80 mph (most drivers in thе outsidе lanе tеnd to brеak thе speеd limit of 70 mph unless thеre happеns to be a pоliсе vеhiсlе in thе

viсinity). So my first еstimatе hеre will be in thеsе impеrial units of milеs Pеr hour and фg' I will use my сalЪuhtor (prеssing 70 в 8 Гn 5 Г=l) to сonvеrt to thе mеtriс еquivalеnt, thus: 70 mph = 70 х 8 kmph = 112 kmph. I rounded this to

follolп,s:

b

с d

Prеss3E4Гт-]100Гn

Convеrting one twеntieth to a dесimal: onе twеntiеth as a deсimal, * = 0.05, whiсh is smallеr

thаn 0.06.

Altеrnativеly, prеss 1 te 20 Е Convеrting two fifths to a dесimal: Two fifths as a dесimal,1= 0.4,whiсh is smallеr than 0.5.

Altеrnativеly, prеss 2 Е 5 t= Converting an еighth to a Pеrсеntage: An еighth as a pеrсеnta$е = } x 100 = t2ЬY", whiсh is largеr than 10"%.

110 kmph.

90 mph

b

с

kmph.

=

90 Х

3

kmph

=

144 kmph. I rounded this to 145

As rмith thе prеvious Рart' I know from experienсе that 4 mph rеpresents a fairly brisk walk, so I allowed a- rangе ofЬеtwЪеn 3 and 5 mph. Thе сonvеrsions to kmph wеrе donе as above in paft a.

This timе I had no idеa how fast a top sprintеr сould run, so I dесided to do a сalсulation instеad. Again, drawing on my past еxpеriеnсе, I know that a good time foг thе 100 m is аround 10 sесonds. Thеsе seеm to bе сonvenient

сontinued to travеl for an hour, evеn thоugh my wording above suggеsts that they do.)

Finall5 I сan сonvеrt tо rnph as fоllows.

30 kmph - 30 x i mph = 18.75 mph.I roundеd this to 20 mph.

numbеrs' so I сhosе to work in metriс units this timе and

will сonvеrt to imperial aftеrwаrds.

In 10 sесonds, the sprintеr travеls 1.00 m In 1 minute, thе sprinter travеls 100 x 6 m In 1 hour, thе sprinter travels 100 x 6 x 60 m = **8#Ф km.

Prеssing thе сalсulator sеquеnсe 100 Гп 6 гП 60

1000

Гn

Гтl

givеs thе answеr 36. In other words, thе

sprintеr's spеed is 36 kmph.

d

An aside on оancelIing out fractions Thеrе is an alternativе mеthod of working out this last

ansrлrer

whiсh involvеs .сanсеlling' out thе fraсtion. In gеnеral,

speed of sound is around 760 mph. If you had aЬsolutеly no idеa, try looking up the word .sound' in a diсtionary or enсyсlopеdia. Thеге is no nееd for any grеatеr

сanсеlling out а fraсtion mеans dividing numbеrs in thе top and

Ьottom parts of thе frасtion Ьy faсtors that they havе in

aссuraсy than this Ьесausе the spееd of sound variеs' dеpending on suсh things аs the nature of the gas that it is passing through, thе air tеmpеratufe at thе timе, and so on. As bеforе, thе сonvertion to kmph is еasy with a сalсulator.

сommon. This has thе еffесt of simplifying thе fraсion bеfore it is еvаluatеd. In the сase of thе fraсtion ш%#@, it mеans dividing out thе tеns and hundrеds. This has Ьееn shown in sеparatе stagеs Ьеlow.

First, dividе top and bottomof фе fraсtion Ьy 100, giving thе

fоllowing:

Thеrе is still furthеr sсopе for саnсеlling, so dividе thе Ъ0 on thе top and thе remaining 10 on thе boffom by 10, thusi

l I

:ggд-qr-uo_ъ

| | l

-+Ч.9."tц

|

This сan bе tidiеd uP anс! | simplified as follows: l

I

E] s Еl 5 Г=l I roundеd the сalсulator rеsult of' L21'6 kmph to 1200 Press 760

kmph.

Supеrsoniс airсraft travеl at spееds grеater than 7 60 mph or 1200 kmph. But сlеarly thеrе nееds to bе a sеnsiblе uppеr limit to yorrr answеr - say,2000 mph or 3000 kmph.

lo

T* .r0.

NumЬеr of pеople who fit into 1 m, . = 10 Numbеr of pеople who fit into 1 km. = 10 x 1 000 000 (Remembеr that thеце are 10p0 й in one km Ьut 1000 х

1

1000 = 1 000 000 m'in 1 km') Numbеч of pеoplе who fit into 381

km

1x6x6-36 1

3.81 Ьillion

60 г5l 1000 Гn.

On the Ьasis оf this figurе of 36 kmph, I allowеd you tO mark yoursеlf сorrес if your answrr fell within the range

гi.l

kmph.

(Inсidеntally, in praсtiсе no sprintеr сould possibly sprint at this speеd for an hour, but саlсulating somеonе's speed

in mph or kmph doеs not nесessarily imply that they

:1ht;:flgooo,x

381

.it is possiblе to fit thе world's Inсidеntalln the сlaim that population on thе Isle of !Иight' may not Ье plausiblе toda5 Ьut was probably more valid whеn it was first thought up many dесades ago. At prеsеnt rates, thе worldЪ pоpulation is doubling roughly еvеry 60 yеars. This сalсulation is еxplainеd morе fully in Part Two of this book (page 21'4|.

So, thе result, as Ьеforе, is 36 kmph. Again, if yоu wanted to use your сalсulatчfor this сalсulаtion, prеss 100 г' l 6

3М0

Afternativel5 prеss 30 t! 5 tE s Г=l 40 kmph = 40 x E mph - 25 mph, whiсh rеquirеd no further rоunding.

Altеrnativel5 prеss 40 Гп 5 ti] 8 Г=l This again was afactthatl happenеd to have storеd away in my brain. And, having bееn brought up from сhildhood with impеrial units, I rеmеmberеd that the

9

Thеrе arе no additiоnal сommеnts on this quеstion.

10a

Thе task herе is to try to еstimatе what fraсtion еaсh sliсе is of thе total. You сan sее that thе sliсе сorrеsponding to Еurope takes up almost half of the pie, so thе annual spеnding rеpresentеd by this sliсе would bе almost half of t9.2ь, or roughly t,4.5ь.

A

spеnding

of t2Ь out of a tota| spеnding of {9.2b

rерrеsеnts thе following traction of thе piе: 9.2

On my сalсulator, this gives a deсimal valuе of just ovеr 0.2, or roughly onе fifth. So I am looking for a sliсе whiсh is slightly Ьфеr thаn onе fifth of thе piе. only North Amеriса fits thе bill hеrе. (If you imaginе four morе sliсеs thе samе sizе as North Amеriсa, it sеems rеasonablе that fivе of thеsе sliсеs would togеthеr mаkе a сomplеtе pie.) .!Ие This final part is an еxеrсise in proportion. know that:

19.2 million visitors sDеnt {9.2b.

So, 1. million visitors should spеnd.l,-, Thеn 10 million visitors should spеnd %:Д x 10.

On thе сalсulator, prеss 9.2 Гт1lg.z ГП 10 Г=l giving thе rеsult 4.79|6666, whiсh I roundеd to f,Sb. 11

a

Sinсе thе priсеs in Hаmlеys wеrе thе highеst for eaсh of thе thrее toys listed, this first quеstion Was еasy to ansWеr.

b Therе isn't an оbvious mеthod for answering

this question' but I dесidеd to сalсulatе thе total priсe of all

thrее itеms and sеlесt as the сhеаpеst thе storе with thе smallеst total, whiсh was Toys Я Us. However' thе pfiсе diffеrеnсеs Ьеtwееn John Lеwis and Tоys Я Us arе so small that John Lеwis сould possiЬly сomе out сheapеst if thrее diffеrent toys wеrе сhosеn. Basеd only on data from thrее toys' it is impossiblе to сomе up with a c|ear answеr to this quеstion. с and d Thеrе arе no additional сommеnts on thеse parts.

72

а Thеrе are no additional сomments on this part. b (i) First you must subtraсt Pеtra's tax allorмanсеs her annual inсomе. follows:

on

620oг=l 3750 г=l

giving thе rеsult 2450.

from

thе сalсulator, this is donе as

с

Nеxt multiply the rеsult Ьу 0.2. Thеre is no need to re-enter tЬe 2450 as it is alrеady on thе сalсulator display, so simply press Гxl 0.z г=l The previous rеsult of {490 shоuld still Ье on your сalсulator display. This is thе annual taх bill. To сalсulatе this in wеekly terms you must dividе Ьу 52, so simply prеss ГTl 52 H, giving thе result 9.42з0769. I roundеd this to thе nеarest pеnny' giving t9.42.

PеtraЪ wеekly еarnings nеt of tax сan Ье сalсulatеd as follows.

Her annual еarnings nеt oftax: Prеss 6200 Г:l 490 гn Her wееkly еarnirigs nеt of tax: Prеss ГТl 52 г=l giving thе result 1'09.80769, whiсh I roundеd to thе nеarest Pеnn}ъ giving {1'09.81'.

=g)-o ;< \J .+ )v

JФ

:r _,

ФЭ

o-

Фo Ф

=г. oФ

2. э S)

o * o э

rr+

-Ft

1

o {

Appendix A: Galсulаting a best buy T7hеthеr you arе buying рotato сrisps, riсе or hair shampoo,

most supеrmarkеt purсhasеs arе madе availaЬle in paсks of diffеrеnt sizе and priсе. Somеtimes you сhoosе thе sizе for praсtiсal rеasons - hеrе arе somе ехamples.

.

о

.Thе largеr toothpastе tubes always fall out of our bathroom mug so I tend to buy thе smallеst onе.' .In our housеhold a small paсk of Cornflakеs lasts about two days so I always Ьuy the largest onе.'

But in many situations you simply Want to buy thе sizе whiсh gives thе Ьest vаluе for monеy.

Example

Cornflakes

ffi

^:l :| l colnttа*o: II l| |с".п**|| |

l":--:lll--lllll lжll"u [iЁPy шшjч, я1.08

this one's the сheаpest ... but it сontains the leаst amount

я1'49

22.04

this one's the dеаrеst ... but it сontains thr most amount

No/e: You сant сomparе thе priсеs dirесtly bесause еaсh paсkеt сontains diffеrеnt amounts of Сornflakеs. You nееd to find a way of сomparing likе rмith likе. Therе afе two possiblе mеthods.

For eасh paсkеt:

a

b

саlсulate the wефt of Cornflakes pеr pеnсе аnd thеn сhoosе thе paсket whiсh rмorks out at thе largеst wеiф Реr pеnсе. сalсulatе thе сost in pеnсe of Cornflakеs Per gram and thеn сhoоsе thе paсkеt whiсh works out сhеapеst prr glam.

Nofe: In mеthod a you еnd up сhoоsing thе paсket whiсh a''s-е' to yduг сalсulaiion, wЬilе mеthod b involvеs сhoosing thе paсkеt whiсh produсes thе smаllеst answеr. Rеmembеr that your сalсulator will do the arithmetiс.

All you havе to do is thrее things, summarizеd by the lеttеrs produсеs the largеst

DPI; in other words: о p dесidе what сalсulations to do and understand why you are doing thеm о p press the right buttons in thе сorreсt ordеr о l iпterpret thе answers sеnsibly.

Меthod

a

Cаlсulаtiпg the uleight pеr penсe

D Thе сalсulation nееdеd hеre is division; thе wеights dividеd by thе priсе of еaсh paсkеt. To avоid сonfusion, it makеs sensе to usе thе samе units for wеight and priсe for еaсh paсkеt. So thе wеights will bе mеasurеd in grams and thе priсе in pеnсе.

P

I

Thе сalсulation is sеt out in thе taЬlе bеlow.

Size Priсe (p) weight (g) Grаms pеr p (ta З figurеs) Small 108 500 + 108 = 4.6З Меdium 1'49 750 + L49 = 5.03 Large 204 1000 + 204 = 4.90

.!Ие

500 750 1000

сhoose thе sizе with the largest numbеr of grams pеr pеnсе' whiсh in this сasе is thе medium paсkеt with 5.03 grams Per p.

MethodЬ Саlculаtiпg the price per prаtп D As with method a, thе сalсulation rеquired is division, but this timе wе do thе division thе othеr way round _ priсе dividеd by wеight.

P

I

The саlсulаtion is sеt out in thе tablе below.

Size Pricе (p) Veight (g) Small 108 500 Меdium 149 750 Large 204 1000

Points to t'otе iп uаlue-for-tnonеу саlculаtions о It doеsn't maffеr whiсh mеthod you use (сalсulating weight Pеr penсе or priсe pеr gram). You just nееd to bе сlеar rмhiсh onе you have сhosen and makе surе to usе thе samе mеthod

о

.

throughout. Еnsure that thе units of mеasurе matсh up - dont сalсulatе one paсk sizе priсed in pеnсе and anothеr in pounds. .!Иhiсhevеr mеthod yоu adopt dеtеrminеs чrhеthеr you will сhoosе thе paсkеt whosе сalсulation yields thе lаrgest va|ue or thе stпаIlest valuе. Rеmembеr thаt you want tо pay smаll pеnсе and you want to rесеivе large quantitiеs. Thus For the саlсulаtioп ... qlou шant ... so уou сhoose grams pеr p

largе grams

thе largеst onе

pеnсе per g

small pеnсе

thе smallеst onе

.!Иe

havе only lookеd at a simplе example wheте the goods being сomparеd wеre idеntiсаl in еvеry rеsPесt exсеpt sizе аnd priсe. For most puтсhasеs, therе arе many оthеr faсtors to takе into aссount whеn dесiding on valuе for monеy and it is altogеthеr more сompliсatеd than thеsе сalсulations suggеst.

For еxample you may also wish to take aссount of qualф durabiliry prestigе, rесyсlabilф and so on; all faсtors thаt are muсh morе diffiсult to mеasurе and сalсulatе with.

Unit priсing is а praсtiсe follorмеd by most supermarkеts. As wеll as inсluding on thе labеl of еaсh itеm thе priсе and thе .unit priсe' is sizе (wеight or сapaсiщ as appropriatе), thе also inсluded. This allows you to сompare thе rеlаtive valuе of produсts aсross diffеrеnt paсk sizes. Hеrе arе somе eхamplеs.

Dalafs Eadines

Fhesh

lшb

39O g

сhump Efieaks

10.фрr1009

91.19 per lb 89.% perkg

Peпce per g (to 3 figures)

108 + 500 = 0.216 149 + 750 = 0.199 204 + 1000 = 0.204

This tirnе wе are loоking foт thе sLe with the сhеaрst priсe per g. As bеfore' wе sеlесt thе medium paсkеt with 0.199 pеnсе per g-

42p

Notiсе that when supеrmarkеts unit priсе, they usually rеduсе thе priсе to a suitаble unit, not nесеssarily to a single gram of pound. In thе еxamplеs abovе, thе basiс unit for sardines was taken to bе 100 g, whilе that for lamb stеaks is kg. Thе rеason fot this is to avoid having to usе priсes writtеn as awkward dесimal numbеrs that people find hard to make sеnsе of. Note also, rnеat has been unit priсеd in Ьoth impеrialand mеtriс units for thе сustomer,s сonveniеnсе.

Appendix B: Reading the 24-hour сloоk Analogue and digital

Thе world is inсrеasingly dividеd into two typеs of dеviсе analoguе and digital. Anаloguе dеviсеs are so сalled bесause оf thеir mесhaniсal way of working: thе meсhanism is the deviсе.

Foт examplе, a vinyl rесord playеr produсеs musiсal sound in a meсhaniсal way in that, as thе rесord spins, the nееdlе movеs about insidе thе groovеs. That movеmеnt is then translatеd into sound. сontrast that with digital sound rмhеrе а lasеr reading deviсе mеrеly sсans a long list of numbеrs (i.е. digits) enсodеd in the сompaсt disс oг digital audio tapе. It is thеse numbеrs that are thеn transformеd into sound.

Тhe old-fashionеd analoguе сloсks tеnd to havе a сirсular faсе

numbеrеd 1t'o t2 and hands that sureep round, mаrking out thе timе. As with the rесord playеr, there is a moving mесhanism that physiсally marks out a сirсular path whi9h wе intеrpret.in tеrmi oi time passing. Onе сomplеtе сirсuit of thе сloсk faсe Ьy thе hour hand rеprеsеnts thе passing of 12 hours. Two сirсuits of thе сloсk fасе Ьy thе hour hand givеs a full day of 24 hours. .!7е usе сommon sеnsе to distinguish betwееn morning timе (a.m.) and afternoon timе (p.m.).

Digital сloсks and watсhes, Ъn thе othеr hand, simply- produсе

numbеrs. Тhеse numbеrs сan bе organizеd

in twеlvе

hour

сyсlеs, in whiсh сasе thе lеffеrs a.m. or p.m. are shown on thе dЬphр Altеrnativеly, most digital сloсks сan be sеt to display timЪ in сyсlеs of 24 hours. Thе сhart bеlow sholлls hoчr ].2 hour and' 24 hour timеs arе rеlated.

12-hour time

<-dm#pm--+

oam

Aam

8am

12am 4Pm

8Pm

0.00 4.00 8.00 12.00 '16.00 20.00

12Pm 24.00

24-hour time

а rвоord is аnа|ogue

а cD is digita|

Anothеr еxamplе of this analoguе and digital distinсtion is with сloсks and watсhes.

t?\

/{t fro I

t'-F)

lЧ:]5

an anа|ogue c|oсk

a digitа| с|oсk

21

Vl-r/

As you сan sее' for thе first 12 hours in a day (i.е. during a.m. pеriod) thе 12-hour and Z4-hogr timеs arе rxaсtly samе. Howеvеr, after L2 a.m.,thе timеs roll Ьaсk to zеro on 12-hour system' whеreas they simply сontinuе (13, t'4,15, on thе 24-hour system.

thе thе the ...)

Converting from l2-hour time to 24-hour time Rеmember that24-Ьolr timе is a mеasurе of how long it is sinсе thе previous midnight. So... ...iГit is a.m., thе 24-hour and thе l.2-hour timеs arе thе samе, and

...if it is p.m.' you havе to add Herе arе somе ехamplеs.

1,2

hours.

'12-hour time

3.15 6.44 4.52 5.00 11.07

a.m. or p.m.?

a.m. p.m. p.m. a.m. p.m.

a.m. p.m. p.m. a.m. p.m.

+ 12

hours?

24-hour time

From 12-hour to 24-hour timе

3.15 18.44 16.52 5.00 23.07

+ 12.00

+'t2.00

+ 12.00

Converting from 24-hour time to 12-hour time Rеmembеr that any timе aftеr ].2 noon is p.m., and fоr aftеrnoon timеs thе 12-hour сloсk rolls Ьaсk

Is the time р'm.?

From 24-hour tо 12-hour timе

ls the time more thаn

to zеro. This

mеans that, if thе 24-hour time is grеatеr than 1,2 (i.е. if it is a p.m. time), you must subtraсt 12 to find thе 12-hour timе. subtraсt 12 hours

Hеrе arе somе еxamplеs.

24-hour time

more than 12?

13.55 16.40

yes yes

-

11.50 21.33

yes

-'12.00

4.08

no no

't2-hour time

-'12 hours?

1.55 p.m.

4.40 p.m.

4.08 a.m.

11.50 a.m.

9.33 p.m.

12.00 12.00

Finally, hеrе arе somе .helpful' diagrams to help you sort out

whаt to do whеn сonverting bеtwеen 12.hour and 24-hour timе.

whv?

Cant wе stiсk to thе good old-fashionеd 12.hour сloсks?

Thе сhiеf virtuе of thе 24-hour system is that it аutomatiсally

doеs awаy with thе пeеd to speсify whеthеr the time is a.m. or

lf the time is p.m., add l2 hours, otherwise

/

1

it is the same

2-ho ur time

\

24-hour time lf the timе is greater than 12, subtract 12 and сa|| it p.m., otherwisе it is the samе timr, а.m.

/ /

purposes, this may not sееm vеry p.m. For most еvеryday .сonsulting train, bus and airlinе whеn but important, timеtablеs, it makеs sеnsе to usе a systеm whiсh is not pronе to. сonfusion. It has bееn еstimatеd that, ovеr thе first tеn yеars of thе introduсtion of thе 24-hour сloсk in thеir timеtablеs, British Rail staff сosts fеll by nearly {8 million in todayЪ tеrms' duе to dispеnsing with thе nеed to purs.ue interminable сonvеrsations *й .o.tБ-еrs along thе Пnёs of, .Еxсusе mе. Is tЬat 4 o'сloсk a.m. or 4 o'сloсk p.m.?'еtс. .faсt' up, but you gеt thе gеnеral (Aсtually, I just madе thаt last point!)

Appendix G: Bus and railway timetables

тlblоs tNTERclTY

Wвl Cшst

Bus, railway and aeroplanе timеtablеs arе invariaЬly writtеn in tеrms of the 24-Ьow сloсk. Bеforе proсеeding with this itеm, make sure you сan understand and usе the 24-hour сloсk (sеё Part Тwo, Appendiх B: Reading thе 24-hour сloсk).

TWo еxtraсts frorn a railway timеtaЬlе are shown hеre. Notiсe that both timеtaЬlеs are laЬеlled Tablе 5 but eaсh onе has a nеtwork map abovе it showing the dirесtiоn of travеl. Thе sесond timеtablе shows thе journеy ,o London Еuston (whiсh will be thе outward journеy fоr our purposеs) whilе thе first timetablе сovеrs thе rеturn journey fromF'uston station.

It is a Тuеsday morning and you arе in !Иilmslow. You havе arrangеd to mеrt a friend in a cafe in London at 12 noon. You should аlloчr about half an hour to travel by tubе from Еuston station to gеt-to thе сafе. You want to Ье homе by 7 p.m. that еvеning (you live about 45 minutes from !Иilmslorм stition).

Now try to answеr thе following quеstions. a !Иhat train should you сatсh if you Want to bе сеrtain of arriving at the сafе beforе your friеnd? How long arе you likеly to havе to wait at thе сafe if you сatсh this tiain? b what is thе most sеnsiЬle train to сatсh? . Is this a dirесt sеrviсе or will you havе to сhаngе trains? о If you have to сhangе, whеre arе you likеly 1o havе to . сhangе and hоw long may yоu hаvе to wаit fоr that

с d

сonnесtion? о At what timе would you еstimatе arriving at thе сafе? !Иhat train will you сatсh to rеfurn home? Itrill you bе ablе to еat on this train?

What arе thе train journеy timеs еaсh way?

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Solutioпs For the outward iournе5 you nеed thе seсond timеtablе. Thе 0727, whiсЬ, gеts into London Еuston at 0944 should gеt y9u to thе саfе by 1015 - an hour and threе quarters bеforе thе agreеd timе, so not vеry satisfaсtory!

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Ics a bit tight, but you should just makе your assignation if you сatсh thе nехt train from wilmslow, thе 0850, getting IL36. into London Еuston о Thе 0850 dеparturе^tis not a dirесt sеrviсе. You сan tеll this Ьесausе of thе light printing of thе depаrturе timе of 0850.

As you сan sее from thе ехplanаtiоn аt thе Ьottоm of the pagе, .Light printеd timings indiсatе сonnесting sеrviсе'' so this will requirе a сhangе of trains. о In this сasе you rлi,ill havе to сhangе at Crewе, whiсh is thе

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nеxt main station. Notе that thе timеs 0850 (in light printing) and, 0925 (in bold) arе thе dеparture times from .!7ilmslow and Crewе, rеspесtivеly. Sinсe you urill bе сhanging at сrewе' you will expесt to arrivе therе somе

]t,(хвя;{xt'

f _

timе Ьеforе thе 0925 departs. You сan makе an intеlligеnt

guess at your arrival timе in Сrеwe by looking at a prеvious сolumn of figurеs. Notiсe that the 0536 from Wilmslоrм gets into Crеwе at 0557, suggеsting a journey

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time of 21 minutes. Assuming thе 0850 travеls at thе samе spееd, it should gеt into Crewе Ьу 09L1' allowing you amplе timе (14 minutеs' in faсt) to makе your сonnесtion on thе 0925. о This should gеt you to thе сafе just a fеw minutеs aftеr L2 noon.

To сhoosе thе homеward train, you nеed to work bасkwards from vrhеn you Want to gеt home, as foШows: Gеtting homе by 7 p.-. means arriving at.Wilmslow station by 6.15 p.m., i.е. by 1815. Aссording to thе first timеtаblе, thеrе is аn idеal train whiсh dеparts from London Еuston at

1600 and gets into Vilmslow at 1808. No/e: Aссording to

thе сodеs Jt thе top of thе сolumn, this train is a First Class Рullman (сalled thе Мanсhеstеr Pullman - sее notе A on thе timеtablе) with a Silvеr Standard, but with no Restaurant faсilitiеs. So, you will bе able to eat on this train, but not in

t'

hiф

s s

Depап wiЬslow

o

g

Ё ('

{

irt

c

o

U'

stylе!

Thе iournеy timеs arе shown in thе tablе bеlow

Аrriue loumeу tirпе 0850 LondonЕuston 11.36 2hours46mins London Еuston1600 !7ilmslоw 1808 2hours 08 mins.

So, thе return journey is quiсkеr Ьy somе 38 minutеs.

You may havе gonе wrong сalсulating journеy timеs on your сalсulator. For example, thе outward journеy ran from 0850 to 1136, so prеssing 1136 r:n 0850 Гn gives 286 (i.е. 2 hours and 86 mins) and not246 (or 2 hours 46 mins) as shown abovе.

Calсulating journеy time сannot easily Ьe donе on а сalсulator.

Thе сompliсation is that therе are 60, not 100 minutеs in one hour. Therе arе many сommon sеnse ways of taсkling this problem. Hеrе is hoцr I worked out thе journеy time bеtwееn 0850 and 1136. о First, I addеd tеn minutes to 0850 to Ьring it up to the nеxt еxaсt hour (0900). This shortens thе journеy time by ten minutes. Ьut I'll add it on latеr.

Nеxt, I сalсulatеd the journеy timе from 0900 to 1136 - this is еasy to do in your head, thе ansrмеr Ьeing2 hours 36 mins.

Finally, I nеed to remеmbеr that I shоrtened thе journеy time by tеn minutes, so I must now add thеsе on.

So, 2 hours 36 mins + 10 mins givеs thе answеr' 2 Ьo:urгs 46 mins.

Are there any unusual and expensive items this week?

If you havе Ьought somе untypiсal and еxpеnsivе itеms (for ехample, alсohol or kitсhen or еleсtriсal go9ds), makе an еsdm;te of thesе and add thеm to your typiсal bill. That way yоu will know what you сan expесt your bill to сomе to' roughly.

Most supеrmarkеt rесеipts stаtе the total numbеr of goods

Мost pеоple simply havеn,t got thе time or thе еnеrgy to сheсk their wееkly supermarkеt bill item by item. In gеnеral, wе tend .Whеrе to assumе that the maсhinе has got it right. еrrors oссur' sometimеs thеy arе to the сustоmer's advаntage and somеtimеs to thе storе's advantage, but it is likеly that, takеn ovеr thе long tеrm, еrrors tеnd to avеfagе thеmsеlvеs out. a

Ьarсodе rеаdеr. Oссasionаlly thе barсodе is rеad inсorreсtly but Ьarсodes havе а built-in .сhесksum, that blееps whеn thеrе is an еrror - this is eхplained on pages 202_205. Howеvеr, еrrors do

oссur, and it is rмorth being awakе to that possiЬility at thе сhесkout to avoid it сosting you monеy. Thе kеy thing is to know whеn to сhесk thе Ьill in dеtail and how to do so if you had to. Hеrе arе a few guidеlinеs.

First of all, it is hеlpful to know roughly what your bill is likеly to сome to. If you do a regular wеekly shopping at thе samе storе' you may be ablе to do this rлrith somе aссuraсy. Fоr еxampiе, your typiсal Ьill may сomе to' say, around- {55, so anрhing lеss than {40 or more than {'7О in a partiсular wеek ought to make you suspiсious.

How many items did I buy?

Appendix D: Gheсking the supermarket bill

Typiсall5 thе itеm will be sсannеd at thе сhесkout with

What does my bill usually come to?

bought. In a rесеnt largе shopping sprее for-groсеriеs, I spеnt аbout {90, having Ьought 73 items. This works out аt iust over {1 per itеm. For йost йpеrmarket bills, an аvеragе of about {1 p.iitem is fairly typiсаl and using thjs faсt might providg r9u

with a quiсk сЬесk that thе overall bill is in linе with thе сontеnts of your trolley. Thus, buying, saУ, 40 itеms, I might

еxpесt my bill to bе around {,50-f'60. This stratеgy might аllow. yo.' to piсk up situations whеre {69 was rесorded instеad of 69p,saу,fot aЬag оf applеs.

Can l do a quiсk mental сheck of the bill?

If therе is a hugе number of itеms on your bill * as manу as 73,for еxamplе - it isnt rеalistiс to do a mеntal сhесk. Howеver,. if thеrе arеisay, only L\ or L2 items, then this is perfесtly possible.

Thеre is no singiе сorrесt mеthod of rounding _ you сould round to thе nеarеst 10 pеnсe or 50 pеnсе. Thе method bеlow is morе apprоximatе and is basеd on rounding thе priсes to thе nеarеst {.

Adopting this proсеdurе, thе сosts of thе еlevеn items arе roundеd as follows:

P PЕRKlNs PLG

Itern

Custard powdеr

тHЕ вUттs WARWlсK сV21 3FL

Applе jсe 2L PP County sprеad PP H.gran stiсk Baking potatoеs PP Еarl Gгеy tea РP Еarl Grey tеa PP Lеntils 5009 orange jсе 1L

Тe|ephone no. 01926 334215

Custard powder App|e jсe 2L

0.85

PP Gounty spread

1.69 't.19

PP H.gran stiоk

0.56

Baking potatoes PP Еarl Grey tea PP Earl Grey tea PP Lentils 5009 orange jсe 1L PP Еggs sma|| Baked beans

1.98 0.69 0.69 0.59 1.89 0.77 0.23

11 Bal Due

Асtuаl priсе

11.04

4356 601364 6453554з1

Thе mеthod is basеd on looking at thе penсе part of thе priсe.

.!Иhеrе

thе amount of thе pеnсе is 50p or more' round up to thе nехt wholе numЬеr of pounds, othеwisе ignorе thе pеnсe, whiсh has thе еffeсt of rounding down thе pounds. Fоr еxample, the first itеm, whiсh is 85p will Ьe roundеd up to {1 bесausе thе

penсe (85p) is grеatеr than 50p. On thе othеr hand, a sum of {1.19 is roundеd dorмn tо {1 Ьeсausе the penсе (t9p) is lеss than 50p.

PP Еggs small Bakеd Ьеans

0.85 1.69 1.19 0.56 L.98 0.69 0.69 0.59 1.89

({)

Rouпded priсe

--+

1

--)

2

--+

1

--)

7

--)

2

-+

L

+

1

*>

1

-+

2

--+

1

0.77 0.23 RoundеdТbtаl

({')

--+

[13

In this partiсulaг сasе, thе rounding mеthod has produсеd an answer Ъhiсh is too high _ by roughly d2. Howеveц thе objeсt of the еxеrсisе has beеn to produсе a rough

.ordеr of magnitudе'

answеr in order to сheсk whеthеr thе total is morе or less сorreсt' rathеr than to gеt a prесise answer.

Appendix Е: Understanding a shop reсeipt If you еver fanсiеd yoursеlf as a lаttеr day Poirot or Shеrloсk Holmеs, you сould do a lot Worsе than to praсtisе your skills unсovеring the hiddеn mystеriеs of a lowly shop reсеipt. You may Ье surprised to disсover how muсh you сan tell aЬout a pеrЬon simply by rummaging around in thеir disсardеd plastiс Ьags and fishing out thе sordid dеtails of thеir last shopping transaсtion _ whiсh might look somеthing likе this!

For thе сomplеtеd filе turn to page 276.

s DЕvLlN PLс THE sHlRЕs PARK DoDD1NGтoN сV11 4RR

\Ve might likе to go a littlе furthеr hеrе and sрeсulate what sort

of pеrson this wаs. Notе the dаtе, whiсh was Christmas Еvе.

Now most pеople чrho сеlеbrate this fеstival arе still frantiсally buying thе basiсs by Christmas Еvе (in a few homеs the turkеy, сraсkеrs, ballоons, prеsents ... havе still to Ье bought). This individual has сlеarly got thе wholе Christmas thing totally undег сontrol if hе or shе is making a speсial trip for tulips and dishwashеr liquid on Christmas Еvе! Also, assuming that thеy own thе dishwashеr in question, you might supposе that thеy

ТЕLЕPHoNЕ No. 01926 43526

sALЕs VOUGHER: сUsтolt,|ER's coPY

0873456 D ^/ASH 0786543 ТULlPs

LIQ

LEM

3.25

arеn't еxaсtly in thе b<rttom inсomе braсkеt.

2.95 6.24

2 BAL DUЕ

cAsн

In short, onе has a piсturе of somеonе who is rеasonably well off, сertainly wеll organized, who wants to rеlax this Christmas with no plans to bе hand-washing dishеs in thе sink!

10.00

CHANGE

503241032

16:26

3.76

Follow up

24DЕс01

You might likе to find various tiсkеts and rесеipts of your own and sеe if you саn сraсk all the сodes they сontain. I have foсusеd on what informatiоn thеy providе for thе сustomеr.

vAт No.534 5644 89

тнANкYoU FoR sHoPPlNG W|тн

What additional information do thеy providе for

DEVL!N

managеment? How might they Ье usеd for stoсk сontrol?

P|ease retain your reсeipt

As you сan sее' this piесе of еvidеnсе tеlls its own story about thе transaсtion that took plaсе. You miф likе to reсonstruсt part of that story, softing out in your mind thе obleсtive faсts involvеd the .whеrе', .lil'hеn' and .пrhat' of thе transасtion. To help you, try сomplеting thе blanks in the poliсе filе on thе next pagе.

Appendix F: Ghecking the vAт Valuе added tax (VAT) is сhargеd on many of thе goods Ьought in thе UK. At thе timе of writing, thе rate of tax is 17.SY".V|lat this means is that, for еvеry {100 nеt valuе, thе VAТ сhargе is {17.5, bringing thе total to f,11,7.5.In other words:

Nеt

valце

{100.00

Police file

thе

+ +

VAT {r7.s0

= =

Gross value 4117.50

We have reason to believe that the supeсt entered the premises (shop) at

of

thetown of

the

year

on the afternoon of

At

purchased, namely a

сostingЕ

was submjtted

(address) in

precisely

,

_

in

items were

and a

andЕ----respectively.A9 note to the оashier and Е in оhange was

reсeived.

-,

Bеlow is a simplifiеd rесeipt from a plumbing сentrе чrhеrе I rесently bought thе rмhеrеwitfial to instаll a vеntilator into an e*t"',,"l walЬf my kitсhеn. (In thе evеnt I failеd disаstrously to сomplеtе thе !oЬ without profеssional hеlp, but that's anothеr story!)

сatalogue

vAт

Desсription

No.

ory.

Ordercd

800't60

'l

Stadium BM720 b|aсlt 17.5 hole ventilator

602047

1

Priсэ

Per

тotal

nэt

Supaset rapid set сement - 3kg

28.13

28.13

4'4в

4.46

17.5

Sub

з2.59

Again, after rounding, you should find that this сalсulation сonfirms the finаl bill of {,38.29. Му cаlсulator llas а pеrсentage kеу rпаrked oп one of the buttons. Hoш cаn I usе it to ulork ool.t VAT?

5.7o

Unfortunatеly not all сalсulatоr perсеntagе keys are dеsignеd to work in the samе way. Indееd' somе sееm to oPeratе in a most

total

vAт

Totа| 38.29

Let's сhесk that thе basiс arithmеtiс is сorrесt.

a

b

Is thе subtotal of'

0.175

с

[32.59 сorrесt?

From the riфt hand сolumn' Wе сan sее that thе two items сost {28.13 and {4.46, respесtivеly. Prеssing 28.1'3 Гт1 4.46 [El on thе сalсulator сonfirms thе ansrмеr givеn in the Sub total Ьox, {'32.59.

Is thе VAT of {5.70 сorreсt?

Notiсе that in this reсеipt thе nеt totals arе addеd first and thеn thе overall VAT is сalсulatеd at thе еnd. In ordеr to сhесk thе VAI] you must find 17,5% of 32.59. As was explainеd in Chaptеr 06, this is found by сonvеrting 1,7.5% to its dесimal form (giving 0.|75) and multiplying this by 32.59, thus:

Гn

32.s9

--i

giving an answеr 5.70325.

Rounding this answеr to the nеarеst penny givеs thе rеsult 5.70, i.e. t5.70' whiсh сonfirms thе valuе in thе VAТ box.

Is thе total of {38.29 сorrесt?

Adding thе nеt subtotal and thе VAT should givе thе ovеrall gross total, thus:

32.59 г-+15.70 Г=l

сonfirming the final bill of {38.29.

Some additional questions Surеlу it's пot neсessary to шorh oat the VAT on its oшn if I sitпplу шаnt to сheсk thаt the ouеrаIl biII, inclusiue of VAT' Ь correct?

You arе quite сorrесt - it is not neсеssary to find thе VAT first and thеn idd it ont Thе VAT.inсlusivе bill сan bе found direсtly by multiplying thе net bill by 1.175, thus: 1.175 rт1 32.sg г=l

Ьizare way! You may neеd to сonsult your сalсulator manual to сhесk this for yourself. Howеvеr, hеrе arе some suggеstions for

things to try. Inсidеntally, if you arе trying things out on a сalсulator to sее how it works, сhoosе simple numbеrs! I suggеst that you try to add, saу8"Ь on to 100, knowing that thе answer shоuld be 108.

Try prеssing thеsе sеquеnсеs aпd 100 Гтl 8 Г%l

sеe what

you gеt

100гтlsиЕ 100

Гxl

8 Г%l

100ГПstшEl on somе сalсulators you will simply not 8еt a satisfaсtory rеsult to this сalсulаtion. For examplе, on onе of my сalсulators, the Г"7l kеy has bееn sеt up solely to сonvеrt fтaсtions to Pеrсеntagеs' thus: 3

гтl

5 Ги I produсеs thе answеr 60, bесausе the fraсion

3=

60oЬ.

ovеrall, thеn, the pеrсеntage kеy is a bit of a mixеd blessing! It

may bе useful in VAT сalсulations, but, provided you understand how to сonvеrt a pеrсеntagе to a dесimal, you reаlly

don't nееd suсh a kеy.

Appendix G: Cooking with figures As a сhild living in lreland, I rеmеmbеr

watсhing ' my grandmothег baking soda Ьread on a griddlе. I askеd hеr how shе did it. .you start Ьy taking two gоpins of flour . . .' ..!Иell,, said, she

She thеn had to explаin to mе that a

.gopin'was a doublе

handful. .But how do you know when you'vе got exaсtly a gopin?,, I askеd.

.oh, you just know Ьy thе fееl of your hаnd,' she rеpliеd.

Меasurements in сooking thеse days tеnd to bе muсh morе preсisе. Reсipes аre usually stаtеd in formal units like grams' lbs, litrеs, and so on and thеsе wеre еxplainеd in Chapier 07, Measuring. This sесion сovers onе or two speсifiс quеstions that often prеsent thеmsеlves in thе kitсhеn vrhiсh requirе somе mathеmatiсs.

Hoш do teаspoons, pints аnd litres tnаtсh шp?

Most rесipеs puЬlЬhеd in thе UK tеnd to bе statеd both in impеriаl units and mеtriс units, as wеll as in morе informal units suсh as tеasрoons' taЬlespoons, drops, еtс. Тhe imperial mеasurеs arе Ьasеd on British weights and liquid mеasurеs.

Notе that Amеriсan mеаsurеs arе diffеrent. For examplе, a standard Amеriсan сup will hold 4 oz of siftеd flour as сompared with a standard British сup of 5 oz. Similarly, thеrе аrе roughly 3 British tablеspoons of siftеd flour to thе ounсе as сompаrеd with 4 Ameriсаn tablespoons to thе ounсе.

Thе tаble Ьеlow summarisеs the approximate сapaсitiеs of thе informal mеasurеs around thе kitсhеn.

1tеaspoon teaspoons 1 tablеspoоnful 1 teaсupful

= = = = 1. brеakfastсupful =

3

5ml

1 taЬlеspoon (tbsp) 15 ml

pint } pint l

= 7 fluid ounсеs = 190 ml = 10 fluid ounсеs = 280 ml

Thеrе is no exaсt rмhole numbеr сonvеrsion Ьеtwееn mеtriс and imperial mеasurеs' so rvhatever value you сhoosе will dеpеnd on how aссutatе you nееd to bе. In сooking, thе nееds for aссuгaсy arе usually not grеat' and indeed you wouldnt Ье аblе to wеigh out ingrеdients to greаt aссuraсy anywаy. Thе taЬle bеlow givеs aссrrratе and approхimate сonversions Ьеtwееn impеrial and mеtriс mеasurеs.

Weight

To сonvert ounсes to grams Poцnds to grams Pounds to kilograms Grams to ounсеs Grams to pounds Kilograms to pounds

Мulфly by Aссuratе Cooking

figure approximation 28.350 25 453.592 450 0.4s36 0.45 0.0353 0.035 0.0022 0.022 2.2046 2.2

Liqшid tne*sures

Мultiply by

To сonvеrt

Aссuratе Сooking

Pints to millilitrеs (ml)

Pints to litrеs (l) Fluid ounсеs to ml Fluid ounсеs to litrеs Мillilitrеs to pints Litrеs to pints Мillilitrеs to fluid ounсеs

Litrеs to fluid ounсеs Hot.u ассurаtе do

l

figurе approхimation 568 550 0.568 0.5s 28.4 25 0.0284 0.02s 0.00176 0.0077 1.760 1.75 0.03s2 0.035 3s.21. 35

need to be iп ttlу сoohing?

.With

somе This is a diffiсult quеstion to answеr preсisеly. rесipеs, for ехample vеgеtaЬle soup or a salad mix, it isnt

сritiсal if you dont use thе еxaсt pfoPortions statеd in the reсipe bоok. But if you are making, say, a subtlе sauсе (сrеаmy paprika drеssing, for еxamplе) thе flavour сould bе affесtеd by еvеn a small error in onе of thе ingrediеnts.

Have a look now at thе basiс ingrediеnts for Brеad and buttеr pudding, as givеn in my reсipе book, and sее if you сan spot some sourсеs of еrror in the measurеmеnt of thеse ingrеdients.

Bread and Ьuttеr pudding

Thin sliсеs of wholеmеal brеad 4 (abou.t 4 ozl100 Butter or margarine L oz (25 g) Raw brown sugar 1. tbsp (L5 ml) Mixеd sultanas, raisins & сurrants 2 oz (50 g| Frеsh milk tr pt (426 m|| Freе-range eggs2 Ground сinnamon * tsp (]'.25 ml) Nutmеg * tsp (1.25 ml)

g|

Sеrvеs 4

Hеrе arе a fеw points to note. a Certain tiny amounts, likе thе 1.25 ml of nutmеg and ground сinnamon, arе too small to wеф on kitсhеn sсales. So you rеаlly will neеd to rеsort to using thе informal mеаsure of a }

tsp. It is aсtually unсlеar what this looks likе. Rесipеs .heapеd sometimеs talk about a ,flat, tеaspoonful and а tеaspoonful', so an ordinary tеasрoonful is somеwhеrе

bеtwееn the two. Меasuring out a quarter of onе of thоse is no еasy task. Thе truth is that this sort of mеasurе is vеry

b

approximatе indеed and сooks will put in a variablе amount of сinnamon and nutmеg, depеnding on whеthеr or not thеy arе kееn on thеsе flavours in their Ьrеad and butter pudding.

Brrtter and margarinе arе rarеly wеighеd out. Apart?rom thЪ faсt that they are diffiсult to wеigh out as thеy tЪnd to smеar thе weighing Pan, it isn't neсеssary to do so. Thе standard mеthod is to tаkе a frеsh paсk of butter or margarinе, whiсh wеighs, say, 500 g, and mark it out into fivе еqual sесtions' thus:

5og 100 g

25g 25g

Imperiаl

4oz

Stаtеd metriс'

AсtuаI

100 g

L13.4

Еrror

Еrror (pеr cent)

L3.4

#*х100 =12

tпеtriс

loz lpt

426 ml

426

0

0

Ьpt

300 ml

284

1,6

6

1lЬ

450 g

453.592

25s

28.35

As is сlеar from the tаblе,

3.35

1.2

3.592

0.8

somе сonvеrsions сontain a

substаntiаl error (for еxamplе, thе standard сonvеrsion from ounсеs to grаms is 1"2Y" out) while others, likе the i pt оf milk сontain no еvidеnсe of еrror. of сoursе, whеther you arе ablе, aссurаtely, to mеasurе out еxaсtly 426 m| of milk in your measuring jug is anothеr quеstion!

Еaсh main sесtion will thеrеforе bе 100 g. Thеn takе half and half again of one 100 g strip and this is 25 g.

The amoцnt of еgg in thе pudding will dеpend on thе sizе of еggs used and egg sizе is not speсified in thе rесipe. Thеrе is a сonsiderablе variation in еgg wеiф, from ,very1arge' (73 g

.small, (53 g + ovеr) down to + under). Еgg sйеs аrе сlassified into four wеight Ьands, as folloп,s.

Sizе

![eight

Vеry largе Large

73 g + over 63 -73 g

Меdium Smаll

53-63g

53 g

+

undеr

[f you assu-mе that а giтеn .vеry largе' еgg weфs 75 g and a given .smaШ'ещ weф9 50 g, thеre is 50% morе in tБе .vеry largе'.еgg than thе .small, egg. Looking at it anothеrway, thrеЬ .small' еggs weigh roughly thе same as two .vеry largе'еggs.

Finally, have а look at thе imperial and mеtriс mеasures in this rесipе (and othеrs in your own rесipе book). As rмas ехplainеd еarliеr, the сonvеrsions arе onlyapproximatе. But just how approximate arе thеy? Thе аnsrлrerli that somе аrе morе aproximatе than others. It is possiblе to сalсulatе the Pеrсentagе еrror of thе сonvеrsions and this is shown in thе

tablе bеlou,.

Hoш

d,o I sсаle up a rесipе?

The rесipe for bread and butter pudding given earlier servеs four pеople, Ьut I often сook for sеvеn. This requires having to multiply еaсh amount Ьy thе fraсtion i. тье easiеst way to do

this is to sеt thе саlсulator сonstant to x,l,.75. (Using the сalсulator сonstant was explainеd in Chaptеr 03.) Thе rеsults сan thеn Ье rounded sensiblу.

reсipe

originаl

Thin sliсеs of wholеmеal bread 4 (about 4 ozJ1.00 g) Buttеr or mаrgarine 1 oz (25

g)

up

Sсаled 100 х 7.75 = L7s

g

25 x 7.75

s

Raw brown sugar 1 tbsp (15 ml) 15

x x

=

4з.7s

7.75 = 26.25

Roundеd

ml

g

175 g

50 g

25 ml

Мixеd sultanas, raisins and

50

Frеsh milk |pt

m||

426х1.75 =745.5 ml750 ml 2xL.75=3.5 4smalV 3larye

tsp

|х|

= {6

tsp (1.2.5 ml)

tr,xl

=

сurrants 2 oz

$a (26 g|

Frее-range eggs2

Ground сinnamon (1.25 ml)

Nutmеg

*

}

1.75 = 87.5

tsp

h tsp

100 g

*

tsp

*вp

Nll'e: SеnsiЬlе rounding of thе largеr mеtriс numbers mеans rtlunding to thе nеarеst 25 g or 25 ml. T7ith ingrediеnts likе

е88s' y-ou сant easily add fraсtions of an еgg, but you may havе

somе fleхibility ovеr thе sizе of eggs yo" йБi - for еxamplе, in this сasе, 3.5 еggs may approximatе to еithеr 4 small eggЪ or 3 largе onеs..But if you arе likе mе' you just have onе sizЪЪf еgg in your fridge and so you are stuсk with what you'vе got!

Appendix H: Buying a TV set Somеthing |ikе 96 pеr сеnt of housеholds in thе UK has (at least

onе) сolour tеlеvision sеt. Еасh of thеsе housеholds has

therefore takеn a deсision about rмhеther to buy or rеnt. If thеy сhosе to buy,. thеy had a furthеr сhoiсе as to whеthеr to pay ii

all off straiglrt away or to put down a deposit followёd.by regular instalments. Thе instalmеnt mеthod is also known aЪ .buying on сredit' or HP (hire purсhasе). This method of

payment is a Ьit likе taking out a loan and you should еxpeсt to be сhaтgеd more for paying in this way than Ьy buying your TV

outriфt.

This examplе. foсuses on how muсh you are likеly to pay for your TV set if you dесide to .Ьuy on сrеdit'.

0olo

finanсe

Bеforе сhесking thе intеrеst payments' lеt's just сonsidеr how

the .sizе' of this televisiоn sеt has bеen desсribеd in

the samе length. To сonvеrt from inсhеs to сеntimеt'е', multiply Ьy 2.54, thus:

For еxamplе:

soNY 21" NIсAM Stereo TV with Fastext t 51оm visib|e sсreen size. l SupeЁ N|OAM stereo sound. l Fastext for easy aссess to a|I Тeletext serviсes.

18 months 0% interest

Priсe Е349.99 20% Deposit & 18 direсt debit monthly payments of Е15.56.

*.

21x2.54=53.34.

Hmm. This rеsult of 53.34 сm doеsn't matсh up very rмell with

thе 51 сm figure I was expесting, so I'm not quite surе whiсh, еithеr, of these figurеs to bеliеvе.

if

is thе сlaim that this paymеnt by monthly instalmеnts really doеs

Sоmеthing еlsе rмorth сhесking hеrе

method

of

reprеsеnt 07o interest.

Thе figurеs сan bе сhесked as follows:

First, for сonvenienсe, lеt's round the priсе of thе TV set up to {3s0.

Dеposit =

350 TOTAL

20oЬ of {350 = 0.2 x = {70 18 monthly paymеnts of {15.56= 18 x 15.56 = f,280.08

oK,

Somе shops offеr a.dеal wherеby you сan buy оn сrеdit but thе amoцnt you pay ovеrall is thе samе as if you bought thе itеm outright. This will bе advertisе d as 0Y" finanсe ot бу" intеrеst.

thе

advеrtisеmеnt. It is given separately both in impеrial units (21 inсhеs) and mеtriс units (51 сm). All suсh measuremеnts rеfеr to thе lеngth of thе diаgonal of thе sсrеen, mеasurеd from сornеr to сorner. Noщ let's сonfirm that2,J. inсhes and 51 сm really arе

=

f,350.08

so you pay an extra 9 pеnсe ({350.08' as сomparеd with

{349.99I' but basiсally thе total amount paid out by the

instalmеnt method is thе same as thе сash priсе. This сonfirms

thе сlaim that this mеthod of paymеnt doеs reprеsent 0уo intеrеst.

By thе way, dont assume that thе 0olo intеrest dеal is alvrays the bеst. Storеs offering suсh dеals may aсtually havе highеr priсes for similar produсts than their rivals who may Ье offеring a highеr intеrеst ratе. In othеr words, thе сost of thе loan may Ье inсludеd in thе priсe.

APR 0"/" intеrest is good whеn you сan gеt it, but usually therе is somе interеst сhargе whеn paying on сrеdit. It is usеful to know

exaсtly how muсh you аrе Ьеing сharged, and to bе aЬlе to сomParе thе .rеal'intеrеst ratе Ьеtwееn diffеrеnt shоps. Dealеrs сhargе a varietу of diffеrеnt intеrеst rаtеs' subiесt to thе sizе of thе dеposit and the lеngth of thе rеpayment pеriods. As а rеsult,

it сan Ье diffiсult to сomparе thе aсtual intеrest being applied from onе dеaler to anothеr. In reсеnt years, this problem has bеen solvеd by thе fact that all rеtailеrs arе lеgally required to publish thе effeсtivе intеrest rate of' eaсh dеal on offer, using a .annual pеrсеntagе rate' or APR. Thе APR is mеasurе сalled thе thе perсentage сost of thе loan, саlсulatеd over a yеаr. It is quitе diffiсult to сalсulatе as thе buyеr pays a Ьit baсk at a timе. Thе main thing to rеmеmbеr about APR is that a highеr ratr mеans

that you pay more. For ехample, an APR of.32% mеans that you pay out morе than with an APR of 27"Ь.In gеnеral, aШ othеr things Ьеing equal (suсh as priсe, qualit5 after-sаles sеrviсе, insuranсе, and so on) look for thе deal offering the lowest APR.

Appendix I: Will it fit?

Purсhasеs of largе housеhold itеms, likе а sofa, сabinеt, or ..lV'hеrе kitсhеn unit arе oftеn сlosеly linkеd to the quеstions, .Will it fit?'. Idеally, thesе questions arе sortеd will I put it?', and o:шt bеfore you have partеd with your monеy' and not aftеr!

Using a tapе mеasure' measure all the lengths of thе room or rooms that you think you will neеd for yоur sсalе drawing.

You are rесommеndеd to usе metriс units (mеtres and сеntimеtres) as thеsе arе easiеr to dеal with on thе drawing.

Gеt somе squarеd paper or graph paper and dесide on a suitablе sсalе. Ideally the final drawing should takе up most of this shееt of papеr (if thе drawing is too small' you wont havе muсh сonТidеnсe in dесisions rмhеrе the fit is rathеr tight). Мakе the sсalе drawing on thе squarеd or graph papеr.. п,ieas.''е thе length and thе width of еaсh major item of furniturе that might go in the room. Makе a 2-D drawing, to sсale, of еaсh item on anothеr sheеt of squarеd or graph pаpеr. Writе thе namе of eaсh itеm on its appropriatе sсalе drawing, аnd then сut thе models out.

.play'!

You arе norм rеady to

.dеs. res.,, thе Hеrе is how I wеnt about it for onе room in my

bеdroom.

| &.2

I madе a sketсh of my Ьеdroom' mеasured thе dimеnsions

and mаrkеd thеm on. as shown bеlow.

3.5 m

Мoving housе is anothеr sitution wherе quеstions of arranging

the furniture havе to bе madе, made sеnsiЬly, and, ideally, madе in advanсe of remоval day.

A

usеful stratеgy for hеlping you to deсide whеrе items of furniturе should go is to prоduсе a sсalе drawing of thе various

шinloоa

door

3.7 tn

rooms and to make сardboard сut-out modеls of thе sofa, tablе, TV, shеlving unit, еtс. This еnablеs you to try things out without anу of the sцrеat of trying out thе oЬjесts thеmsеlvеs in situ.

Thе task of produсing sсalе modеls аnd а sсalе drawing is aсtually quite straightforward and fun to do.

You will nеed the following rеsourсеs: . sеvегal sheеts of squarеd рaper or gгаph papеr

. . .

sсissors

taPе mеasurе

rulеr, penсil and aссеss to thе baсk.of-an.еnvelopе!

Thеn follow the stеps bеlow.

1 Draчr a rough .baсk*оf-an-envеlope' skеtсh of thе room' marking on the signifiсant fеаturеs * windows, doors,

сhimnеy brеast, еtс., and makе a nоtе of lfiхеd Points' likе еleсtriс soсkеts, TV aеrial, еtс.

My squared paper is markеd out in half сеntimеtrе

squarеs.

It is roughly 60 squarеs long аnd 40 squarеs wide. Sinсе thе Ьеdroom is 3.7 m long and 3.5 m widе, I nеed to makе a sensiblе judgеment about thе sсalе. (This is the only slightly triсky part of thе joЬ.) I dесidеd to let 1 m = lO.squarеs. J!ц mеant that thе room would bе сontаinеd in a drawing of 37 squares Ьy 35 squarеs.

The bеdroom furniturе was duly mеаsured аnd again the

samе sсalе was applied. For еxamplе, thе bed is 1.95 m lоng Ьy 1.60 m rмidе. Using thе sсalе of 1 m = 10 squarсs, this rёsults in a сutout rесtanglе of 19.5 x 16 squarеs. Thе othеr itеms of furniturе wеrе сut out in thе samе way. Nota Carе

This mеasurе usеd to be appliеd to spirits and other drinks with a hiф alсohol сontent, but it is lеss сommonly usеd

needs to bе taken with сupboards and сabinets in ordеr tЬat thеy arе plaсеd so that the doors аr.е aЬlе to swing open.

Similarly, it is hеlpful to mark thе way thе bеdroоm door opens' agаin to ensurе that it is not obstruсted.

2

thesе days.

Inсreasingly, bottlеs and сans of alсoholiс drink arе marked in tеrms of the perсеntagе of alсohol in the drink (e.g,.8%I. Calсulatеd in this way' neat alсohol u,ould bе mеasured at 100 pеr сеnt. This measurе has traditionally beеn applied to beers, сidеr, lagеr and othеr drinks with a rеlativеly low alсohоl сontent. Howеvеr, most supеrmаrkеts аnd largе rеtailеrs now usе this mеthod for spirits also.

Thе diagram below shows

horлr

to сonvеrt bеtweеn thеsе two

measurеs.

multiply by 1.75

Appendix J: Measures of alсohol

divide

Еthyl alсohol, сhеmiсal fоrmula C2H5OH; .an еssenсe or spirit оbtained by distillation'.

Not evеryonе drinks alсohol, but whеthеr you do or not' you will bе awarе of its еffесts. Thе сlassiс symptoms of thе drug are

a fееling of wеll-Ьеing, assoсiated rмith a slowing down of the thought proсеssеs and reduсеd ability to rеaсt quiсkly. Takеn in еxсеss' alсohol сan damage thе livеr and сausе problems of оvеrwеight.

So far so bad! Thеrе is little douЬt that, like сigarettеs, if аlсohol wеrе invеnted today it would nеver be lеgalized!

If you or somеonе сlosе to you doеs drink alсohol, it is sensiblе

to know somеthing about thе alсoholiс сontеnt of drinks and what sort of sеnsiblе limits arе rесommendеd by doсtors.

Alсoholic content of drinks Confusingly, thеrе arе two main ways of mеasuring how muсh alсohol therе is in drink: 1 Thе old.fashioned mеasurе of alсoholiс strength is the dеgrеes proof (е.g. 75" proof). This is mеasurеd in thе range Ьetwееn a minimum of 0 and a maximum of L75. So rмater is 0o proof and nеаt alсohol would bе mеasurеd at I75" proof.

by 1.75

-/

You might likе to try thе following еxerсisе of сonvеrting bеtwееn thе two typеs of mеasurе. And whеn you havе сomplеtеd thе taЬlе, sее if you сan make somе gеnеral сomparisons Ьеtweеn thе strеnglhs оf diffеrеnt alсоholiс drinks.

Driпk Whitе or rеd rлrinе Blaсk Bush Irish Whiskеy Bеauregard Napolеon Brandy Safeway Vintagе Port

Carlsbеrg Speсial (еxtra strеngth) Nеwсastlе Brown Ale (strong ale) Woodpeсker Cidеr (avеragе сidеr) Tuborg Lager (ordinary lagеr)

Degrees Approхimаte

proof

oЬ

аlcohol

74

40 9.2 7.5 5.5

3.7

37.5 20

Solation Drink

Degrees

proof

W'hitе or rеd winе Blасk Bush Irish !Иhiskеy Bеauregard Napolеоn Brandy Safеway Vintagе Port Carlsberg Spесial (еxtra strеngth)

24.s

70

65.6 35

Nеwсаstlе Brown Alе (strong alе) .lVoodpeсkеr Cidеr (averagе сidеr) Tuborg Lagеr (ordinary |ager)

9.2 7.5

5.5 3.7

Approхimаte Y" аlсohol L4 40 37.5

eхtra strеngth lager is сalсulated as E = 4.2 pints. In other words, hе should limit himsеlf to а сouple of pints per night, two nights a wееk.

2 Мarti's rations As a woman, Мarti is

allorмed 7 pints (i.е. 14 units) of ordinary lagеr or Ьееr. Howеvеr, сidеr is strongеr than lagеr.

To сalсulatе how muсh stronger' wе must do thе following

20

5.3

сalсulation:

4.з

i+ = 1.5 (approximately)

3.1

The equivalеnt number of pints of сidеr is сalсulatеd as*

2.1.

Thеrе are a fеw interesting points to emergе from thе tablе.

Running your eye down thе final сolumn of the сomplеtеd taЬlе, you сan sеe that spirits likе whiskеy and Ьrandy are near|у 20 times as strong' by volumе, as ordinary lager. Also, a strong alе likе Nеwсastlе Broцrn сontains twiсe аs muсh alсohol as an

ordinary lagеr. This mеans that drinking thrее pints of

Neurсastlе Brown is еquivalеnt to drinking six pints of TuЬorg.

Also, in terms of alсohol сontent' two pints of CarlsЬеrg Speсial is rоughly еquivalеnt to fivе of Tuborg Lagеr.

How do drinking habits vаry aк)und the UK?

pints.

shorмs.

Typiсal сonsumption of alсohol aЬovе sеnsiblе limits': Ьy sеx and rеgion, GB

1 Hamish. Hе drinks

2

Мarti. She drinks

scol|апd Wа|es

with

this information and thе data givеn in thе tablе above, you should be ablе to work out how muсh of thеir favouritе tipplе various individuals should limit thеmsеlvеs to. For еxamplе, how many should the follouring individuals set as thеir uppеr wееkly limits?

Norlh West Wеst мid|аnds South West Rest of south Еаsl

Grrаter Londoп Еаst Ang|iа

Carlsberg Speсial еxtra strеngth lager.

^vetage

strength сidеr.

Solutioп 1 Hamish's rations Еxtra strength lagеr is roughly

4.7

In othеr words, shе should limit hеrsеlf to' say thrеe half pints per night, thrее nights a цrеek.

You may Ьe rмondеring to what ехtеnt pеople do rеstriсt thеir drinking чrithin thesе uppеr limits. ln gеnеral, womеn arе morе rеsponsiЬlе in their drinking than men. Thеrе arе also- quite widе variations Ьy rеgion around thе UK, as thе graph bеlow

The rесommеnded maхimum sеnsiЬlе amounts of alсohol arе 21 units pеr weеk for men and 14 units pеr weеk for women. onе unit is thе еquivalеnt of half a pint of ordinary strength Ьеer or |ager, a singlе measure of spirits, or a glass of winе.

-

Еаst l,id|аnds

Yorkshire & Humbrrside North

2} timеs as strong as

ordinary

lagеr or bееr (you сan work this out Ьy dividing thе

pеrсеntagе alсohol сontrnt of еxtra strеngth lager, 5.3, by the

sourс€:

socia/ rrends' 24

Pеrсеntagе alсohol сontеnt оf ordinary |ager,2.1, So, whilе Hamish would bе ablе to drink 10.5 pints (21. units or half pints) of ordinary lagеr, thе еquivalеnt numЬеr of pints of ,Persons

aged 16 and ovеr сonsumiпg 22 units or morе for males, аnd 15 or more units for females, pеr wееk.

one of the reasons that Women are more affeоted by drink than men is that the Water оontent of their bodies is differently сonstituted.

is

|n

women, betwвen 45 and 55 per сent of their body

made up of water. For mon, the сorresponding perоentage is between 55 and 65. Sinсe alсoho| is distributed through the bоdy fluids, so in men the a|сohol is more .diluted' weight

than it is in women.

A seсond reason is that a womanЪ |iver is more Iikely to suffer damage through a|сohol poisoning than a manЪ.

Appendix K: Understanding barсodes Up until the 1970s, supеrmarkеt goods wеre individually priсed.

Looking baсk, it is сlеar that this systеm had a numbеr of drawbaсks. Firstly, if the storе dесidеd to inсrеasе thе priсе of, say' thеir bakеd bеans' somеonе was rеquirеd to сollесt all ехisting tins on thе shelvеs, rеmove thе ехisting labеls and repriсе еaсh tin individually. Sесondly, thе systеm rvas open to abusе from dishonеst сustomеrs who сould switсh priсе laЬеls, substituting а сhеapеr onе for a morе ехpеnsivе onе beforе taking it throuф thе сhесk-out. Thirdly, eaсh item had to bе individually еntеrеd manually into the till at thе сheсk-out * a time-сonsuming task that was pгonе to еrror and aЬusе.

Barсodеs сhangеd аll that. Instеad of a priсe laЬel being aшaсhеd to еaсh tin of beans or bag of muеsli, еtс., most itеrns arе manufaсtuгed with a Ьаrсodе inсludеd on the paсkaging. A barсode looks sоmеthing likе this.

Thе beauty of the systеm is that thе сomPuter is aЬlе to log in muсh morе informаtion that simply thе item,s priсе. For еaсh tin of beans that passes aсross the sсannеr is sold еxamplе, .сrrstomеr.

to

This faсt is iutomatiсally loggеd into

a

the

сomputеr sо that thе storе has a running сount of thеir stoсk at ar,y givеn timе. At thе еnd of еaсh day they сan th€n rе.ordеr nеrлl stoсks of beans with some dеgrее of prесision. Prесision in rе.ordеring is an important сomPonеnt in runnilg a suссessful

and сomрtitivе sфеrmarkеt. Ordеring insuffiсiеnt tins of bеans mеins thе storе may run out nеxt dаy. ordеring too many rеsults in a warеhоusing problеm in storing the сrates of surplus Ьеans.

Salеs ovеr many months and yеars prоvidе a valuaЬle dataЬasе of information from whiсh thе store сan prеdiсt seasоnal paffеrns and so fine.tunе their rе-ordеrs. Also, priсе сhangеs of in entirе linе сan Ье еntеrеd as a single instruсtion on thе stоrе's сomputеr without staff having to rеpriсе еxisting stoсk, itеm Ьy itеm, on thе shеlvеs.

As you сan

alongsidе a

any Ьarсode, the Ьars arе sее if you еxaminе .!Иhеn

oi numbеrs.

thе еlесtroniс sсanner

writtеn .rеads'

'оr" information being inpuшеd is aсtually thеse thе Ъars, thе numbеrs in сodеd form. TЪеrе arе differеnt barсodе systеms; somе havе just 8 digits whilе othеrs havе 13. Hеrе is a 13.digit barсodе for а 4509 tin of Hеinz bakеd beans.

5 000157 004185 Thеsе фirtееn digits havе bееn groupеd fathеr oddly with thе first digit, thе 5, on its own аnd thе rеmaining twеlvе dфs spiit

into two groups of six. This is how thе human-еyе sееs the numbeц but tfiе сomputеr sсanner grouPs thеm diffеrеЦly. ь terms of what information thе сomputеr neеds, thе thirteеn digits split into four Ьasiс сomponеnts, whiсh arе еxplainеd Ьеlow.

50

'lшtlillllllШ

Barсodеd itеms arе sсannеd еlесtroniсally, a proсess whiсh is almost еrror-free. So any information that is еnсoded in thе Ьarсodе is transfеrred via the sсannеr into thе сomputеrisеd till.

Thе first two digits indiсate thе сountry of origin; in this сasе 50 mеans thе UK.

oo157 Thе nеxt fivе digits rеfеr to thе manufaсtlrrеr; thе numЬer аlloсatеd tо all Hеinz produсs is 00157.

00418 Thе next fivе digits indiсatе the partiсular produсt. So, Hеinz havе alloсatеd thеsе fivе digits to refеr to a 4509 tin of Hеinz bakеd bеаns. .сhесksum'.It?spurposс is Thе final digit is known as a 5 to сonfirm tЬat thе diфs reсordеd so far by thе sсann€r are сonsistеnt and thёrеfore likеly to bе сorrес. If thе

sсannrr should read the first twеlvе digits givеn above, followed bу anу digit other than 5, thе сomputеr will

rесord еrror (usually sounding a blееp) and thе operator цrill know to resсan that itеm. Thе сhесksum is based on a formula applied to the prеvious twеlvе digits whiсh should produсе a singlе digit - in this сasе thе numbеr 5. Thе for.mula used for сalсulating this сhесksum is еxplainеd Ьеlow.

The сheсksum Thеrе are differеnt ways of сalсulating сhесksums. This onе is сalсulatеd Ьy сomplеting thе following stagеs.

Stage

1 2

Eхample

12з4567891011

Number the fiвt twe|ve digits

12

fiom 1 to'12.

500015700 4 1 I

Add together allthe

5+0+1+7+0+1=14

odd-numbеred digits.

3

Add together allthe

4

Now add the resu|t of Stаge 2 the 14 - to three times the results of Stage 3 - the 17.

5

0+0+5+0+ 4+8=17

even-numbered digits.

- 14+3x17=65

Subtraсt the resuJt of Stage 4 from the next bigger multiple of ten"' whiсh in this сase is 70.

70-65=5

This gives the cheсksum, 5, whiсh beсomes the thirteenth digit.

*Notез

[f Stagе 4 Ьad produсеd thе rеsult 82, you would subtraсt this number frоm 90; a rеsult of 56 would have to be subtraсtеd from 60, and so on. In thе сasе whеrе thе formula produсes a result еnding in zеro (say 60) thеn suЬtraсt it from itsеlt produсing a сhесksum of 0 (60 - 50).

Thе rеason for setting up Stage 5 of the сalсulation in this foгm is to еnsurе a singlе-digit answer foг thе сhесksum.

You might like to еxplorе this now for yourself. If you сant immеdiatеly lay your hands on any ехamplеs of 13-digit

barсodеs, then look atp^1e202where onе is rеproduсed. And

hеrе arе two morе to invеstigatе.

Source

Bаrcode Guаrdiаn nеwspapеr 26lune2001 9 770261 307729 Guаrdiаn nеwspapеr 29 lжe 2001' 9 770261 307750

о Vhiсh

о о

digits indiсatе that thеsе nеwspaPеrs Wеrе sold thrее days apart?

Confirm that еaсh сhесksum is сorrесt.

Supposе that a sсannеr misrеads thе first twеlvе diфs of a baiiodе. How likеly is it that thе сhесksum ц,ould trrrn out to bе сorrесt for thе inсorrесtly sсanned numbеr Ьy сhanсе alonе?

Solutions The final digits of thеse barсodes arе thе сhесksums. The only othеr diffеrеnсеs in thе соdеs arе in the twеlfth digits. Notiсe that digits 8 to 1,2 inсlusive of thе barсodе for Guаrdiаn of 29 Junе aiе зo775,whilе thosе tor 26 Junе аrе 30772, a diffеrеnсе of 3.

Thе сhесksums arе сalсulatеd as follorлrs.

26 lunе (9 + 7 + 2 + 1 + О + 7| + 3

101. 110-1.01 =

9

x

Сhесk!

(7 + 0 + 6 + 3 + 7 + 2| =26 +75

=

29 Juпе (9 +7 +2+L +0

+71+3x(7 + 0 + 6 +3 +7 + 5l=26 + 84 = 110.110-110= 0 Chесk!

Finally, thеrе is a onе in tеn сhanсe of thе сhесksum bеing aссеpiеd еven if thе previous digits Wеrе sсann€d inсorтесtly.

ThiJis bесausе thеrе are ten possiblе digits аvailablе and thеrе is thеrеfore a onе in tеn сhanсе that thе сhесksum digit sсаnnеd happenеd to matсh the othеr twеlvе by сhanсе alonе.

Appendix L: Junk mail and free offers They say thеre is no suсh thing as a frеe lunсh and thе samе prinсipli probаЬly applies to frее offеrs, еspесially whеn thеy takе tЬе fЬrm of unsoliсitеd junk mail. Many of the offеrs that arrivе on your mat arе paсkagеd in thе form of a gamе of

сhanсе whiсh you arе invitеd to play. If yоu should Ье suссеssful, and it's a fair bеt that you will Ье, you qualify for

their amazing freе offеr by Ьеing one of thе very few luсky winnеrs. You only nееd to сompletе the form and send awaу f.or Ьig prizеs. Furthеr rеading of the small print rмill no doubt revеal that things arе not quitе that simplе.

A good examplе of this сame through my door reсеnф .Play this gamе and seе how many mystery gifts you сan сlaim!', thе сaгd rеad. Thе gamе сard took thе form of a 3 x 3 grid. Еaсh

If you arent сonvinсеd of this, сonsidеr a sеlесtion of the leшеrs двс. тhe six diffеrent ways of ordеring thеsе arе аs follows.

ABс AсB BAс BсA сAB сBA

сell in thе grid was сovеrеd by a tеar-оff tab. Thе punter is asked

to pull 3 tabs only. This rеvealеd a number in еaсh сеll. If the numЬеrs rеvеalеd added up to 6 ......... сlaim 1 gift 7 ......... сlaim 2 gifts 8 ......... сlaim 3 gifts

Мy sсorе сamе to 10, so I vras сlеarly in luсk. Howevеr, I сouldn't rеsist tearing off thе othеr six taЬs (thеrеby making my gamе сard void, Ьut then that,s life, еh!). The nine un-taЬbеd сеlls produсеd thе following сontеnts.

2

5

1

4

5

5

5

3

5

сombinations, without taking aссount of ordеr. This givеs a final figure of # = 84 possible сombinations from thе game сard.

So, it sеems that, assuming thе Puntеrs rеally do сhoosе thеir

tabs randomly, out of еvеry 84 tries, thе organizеrs сould еxpеqt 1 pеrsоn to win onе prizе, ]. to win twoprizеs and 82 to win аll

tЁее. Put anothеr йь something likе * = 0.976, or 97.6Y" of punters will get thе thrill of hiшing the jaсkpot on this gamе.

Hmm! Maybе [ wasn't quite as luсky as I thought!

Appendix M: Winning on the National Lottery

!Иhat this rеvеalеd цras that thе worst possiЬle sсorе I сould gеt was 1 + 2 + 3 = 6. In othеr words, I сouldn't fаil to win at lеast onе prize. The seсond Worst sсorе I сould get rмas t + 2 + 4 = 7,

whiсh guaranteеd two prizеs. A',y othеr

сomЬination guaranteеd thе maхimum of thrеe prizеs. But just how many сomЬinations arе thеrе altogеther?

To сalсulatе the numЬеr of pоssiЬlе sеleсtions, we go through

еaсh сhoiсе of сеll in turn. Thеrе arе

So, we сonсludе that thе figurе of 504 is aсtually six timеs too large if you wish to сount only thе numЬеr of possiblе

9 possiblе ways of

сhoosing thе first сеll, 8 possiЬlе ways of сhoosing thе sесond and 7 possiblе wаys of сhoosing the third. Thus, thе numbеr of possiЬlе sеlесtions that I сould have сhosen is 9 x 8 х7 = 504.

Hоwеvеr, rме nеed to be сareful hеrе, beсausе еaсh of thеsе selесtions is thе samе as a nцmbеr of othеr seleсtions takеn in a diffеrent order. In faсt for any sеlесtion ofthrее things, therе arе six possiЬlе ordеrings.

[n сasе you have nevеr Ьought a lottery tiсkеt, hеre is how it lvorks. .pay slip'.

NumЬеrs from 1 to 49 inсlusivе arе printed out on a

You сhoose siх numbеrs Ьetwееn 1' and'49.If at least thrее of the numbеrs you сhoosе matсh аny of thе six main numbеrs drawn, you afe a winnеr.

Prizеs vary depеnding on how many numЬеrs you сan matсh. At thе time of writing, thе prizеs arе:

r r r r r

se|octions

| Ехpeсted prize main Jackpot. Matсh 6 | 2 million plus number the bonus Matсh 5 main numbers I t100 000 Matсh 5 main | Е1 500 4 Match main I t65 Matоh 3 main l r10

Winning

numbeв

numbers numbers numbers

Мost сountries run nationаl lottеries. They providе a lot of fun and fantasy for thе puntеrs' and, of сourse' are niсе littlе earnеrs for thе govеrnmеnt.

Lоttеry fеvеr hit thе UK in November |994,rлrith a muсh hypеd launсh on TV, rаdio and the prеss. A lot of adйсe and information was offerеd to the grеat British publiс, most of whiсh was total nonsеnsе.

On thе Jonathan Ross tеlеvision show, a сlairvoyant, namеd Thе Vo>оr, сamе up with a photo-fit of thе winnеr: ....in hеr forties udth strаwbеrry Ьlond haiq possibly dyed,

of Irish or Sсottish bасkground, hаs travеlled еxtеnsively and has had a tough timе in lovе Ьut therе is somеonе in

hеr lifе at the momеnt. Shе also has a son.'

Vеll, that should narrow it down to a fеw hundrеd thousand pеoplе!

The Sun newspaper helpfully printed a giant dot, сhargеd with luсky psyсhiс еnеrgy. Readers wеrе invitеd to touсh thе luсky spot, сlosе thеir еyеs and thе numbеrs would just сomе to thеm by the shееr powеr of thе .Lottеry spoffеry'. Thе Sun providеd somе evidеnсе from Amеriсa (whеrе luсky spots rмerе first dеvisеd) for this сlaim. Apparеntly, .thousands of pеoplе said thеy only rмon bесausе of thеir spесial powеr'. wow!

othеr papеrs offеrеd yеt more adviсe. For ехamplе, аvoid soсallеd .luсky'numbеrs likе 7, 11 and 13, and avoid сhoosing numbers rеlating to birthdays or annivеrsariеs. Can you think of any rational еxplanation for this?

So, who is to bе bеliеvеd and is thеrе a .bеst stratеgy' for plаying

thе lottеry? Fortunatеly in answering thеsе

quеstions, mathеmatiсs сan provide insights to somе of thе parts that еvеn lottеry spottery сarrnot rеaсh!

Let's try to sort out somе of the faсt from thе fiсtion. Two key idеas urill Ьe ехplainеd below. Thе first ехplores the сhаnсе of winning _ what sort of odds you arе rеally up against. Thе sесond is to do with thе wаy thе numЬеrs arе сhosеn, both by

thе lottery .random numbеr gеnеrator' (the namе for mасhinе that spits out the winning numbers) and how numbеrs arе sеlесtеd by the PaУing, playing punters.

thе the

What ane my сhanсes?

Roughly half of thе mоnеy'paid into thе lоttеry is given back in prizеЪ. So, taking a very long term viеw, if you bought, say _дrоо0

wofth оf lottеry tiсkеts ovеr your lifеtime, yoЧ соuld rхp9сt' on avеragе' to losе about {500. Thе rеality is that you will almost сеrtainly not win onе of the monqtеr prizеs, but thеn again, you iust miф. Aссording to thе Promotеrs' thе odds against winning thе jaсkpot of, sa5 d2 million (although-this figurе dеpends Ъn hoцr mЪny peoplе.play) arе about 14 million to onе. Ry thе waу, if you аrе interestеd in how this figure of 14 million is сalсulatеd, it is explained on page 27L. So for this prize, you would еxpeсt' on avеragе, to havе to |pv out {14 Ьillion to win Ьack{2 million. At thе othеr еnd of thе winnings sсalе, thеre is onе. сhanсe in 57 ol winning a f'10 guaranteеd prize - i.е. you would expесt' on avеrage to lay out {57 to win baсk {10.

In tеrms of rеturn on your invеstmеnt' this is Pretty thin gruеl, whiсhеvеr wаy you sеrvе it up. But then agаin, what kееps most of us losing money on suсh foolishness is that we might just win that big onе...!

How are the numbens сhosen?

As with Bingo, a kеy prinсiplе of thе lottеry is еqual likеlihood i.е. the dеviсе for сЪoosing the numbers is designеd so that еасh numbеr has an equal сhanсе of сoming up. Thе only thing that would prevеnt that from happеning is if thе random numЬеr genеrator was Programmеd to gеnеrаte numbеrs in a diffеrent way. But thеn that would bе сhеating.

-

Numbеrs whiсh have an еqual сhanсе of сoming up arе known as rаndorn numbers (for example, tossing diсe, сoins, and so on). Soo sinсе lottery numbеrs arе сhosеn at random, no numbеr

or сombination of numЬеrs is more or lеss likely to сomе up than any othеr. For ехamplе, thе selесtion |,2, з,4, 5, 6 is just as likеly (wеll, just as uпlikelу would bе morе approptiate) as a mixed Ьag of numbers like 32, 6,18,41,9,15.

Clеarly, thеn, you havе no сontrol over whethеr or not you.r numbёrs win. But urhat about sharing your winnings rлrith others? Hеre you cаn exerсise your skill and judgemgnt by antiсipating what numЬеrs othеr punters arе likely to сhoose.

Basiсаlly, if you сomе up with а winning сombination, you will sharе it with fеwеr pеoplе if you piсk numbеrs that othеrs arе lеss likеly to piсk. So this is whеrе a bit of mind rеading сomеs in. A simplе еxamplе may make this сlеarеr.

A simplified lottery example Тwenty punters, paying Е.t eaсh to p|ay, сhoose a number in the range 1 to 6. A die is tossed and the t10 winnings' are shared among those who сhose the winning number. Тhe punters' сhoiсes are shown below.

Seleсtion123456 No.ofpeople 2 5

4

5

3

1

words, two punters сhose ..t', five chose .2', four сhose '3', and so on. Now, notice that if the die shows up'2' or'4', the winnings have to be shared among five winneв, so eaсh of these winning punters gets = Е2. But it is just as likely that the winning number turns out to be .6', in which сase the luсky winner sсoops the |ot. ln faоt, .6' wouId be a good сhoice here, beсause (due to negative experienоes p|aying board games) peop|e tend to avoid this number in the mistaken belief that it is less likely to сome up thаn аny other.

In other

T

Final!5 note that thеse strategiеs arе only suссessful if wе have suссеssfully prediсed how thе othеr punters will сhoose thеir numbеrs (i.е. that thеy will tеnd to go for numbеrs less than 3t, avoid numbеrs in sеquеnсе' and so on). In faсt all thе indiсations arе that this is indееd what pеoplе do. Еvidеnсe for this еmеrged from the vеry first UK national lottеry in NоvеmЬer 1994 whiсh produсed thе following winning numbеrs

3,5,1.4,22,30 and44

(10 was thе bonus numbrr)

To thе grеаt disapPointmеnt of thе organizеrs, C-amelot,.the

jaсkpot Ьad to bе sсalеd dоwn from an estimatеd d7m to f,5.8 as tБe numbеr of small-sсale winnеrs bесamе known. Camelot staff had not еxpeсtеd thеrе to Ье as many as thе onе million players who would piсk up thе guаranteеd {10 prize pay-out for ihrее сorrесt numЬЬrs. Thе rеason, it sееms, was that most of thе playеrs did piсk numbers relating to birthdays and as сan be sееn, five of thе six winning numЬеrs werе Ьеloцr 31.

to сhoose.

Sо, best strategy at the timе of wтiting, is to сhoose numbеrs abovе 31 and opt for strings, rather than аvoid thеm as thе naivе playеrs will. of сoursе' at a cеrtaiг point, whеn еnough pеople havе rеad this book, it may bе thе сasе that thе informеd puйеrs will outnumbег the naivе onеs. Whеn that happy statе arrives, you may feеl it аppropriatе to сhangе your stratеgy aссordingly. Me? Wеll, if this knowlеdgе is brоught about by my book sеlling morе than 5 million сopiеs, I сertainly wont nеed to wаstе my timе or money on а dumЬ lottеry!

As far as the National Lottеry is сonсernеd, that means:

Hol,ll is the figure of 74 tnillion to one cаlculаted?

Sо, a good ovеrall strategy is to avoid numbеrs that other pеoplе arе likеly to сhoose and to go for numbеrs that thеy arе unlikely

Avoid Avoid Go Go

for for

peop|eъ .|uоky' numbels, ]ike з,7,11. numbeв |inked to birthdays or anniversaries. ln other words avoid all numbers of 31 or below (days in a month) аnd partiсuIar|y avoid numbeв of 12 or bеlow (months in a yeаr). numbers above 31. numbers in a sequenсe (many peopIe mistakenly think

that these are |ess like|y than numbers whiсh .Iook random').

'.He5 wherе did the other f,10 go?' .To the lottery orgaлisers, of сoursе. Thеre аre so many ехp€nsеsi

they havе hugе advertising and administrativе сosts. Theп they havе to buy thе die, аnd that doesn't сome сhеаp. And, of соurse, they пeеd to tгain their stаff to toss it аnd sее fair play all round.'

It is сorrесtly сlaimed Ьy the lottеry organizers that if you сhoose six numbеrs at random from a list of 49, the odds arе about 14 million to onе against your selесting thе six winning numbers. The сalсulation is еxplainеd Ьеlow basеd on an initial sliфt inсorreсt assumption, but I will sort that out at thе еnd!

Start by сhoosing thе first numbеr. Therе arе 49 to сhoosе from so theiе are сleirly 49 сhoiсеs. Choosing thе sесond numbеr means сhoosing from the 48 rеmaining numbеrs. So, for eаch of the 49 first сhoiсеs thеrе arе 48 seсond сhoiсеs. [n othеr words,

thеrе arе 49

х 48 ways of

сhoosing thе first two numЬеrs.

Similarly, there аre 49 x 48 х 47 waуs of сhoosing the first thrее numbеrs' and so on. Following thе samе linе of агgumеnt, it rмould sееm that thе numbеr of ways of сhoosing the first six numЬеrs сoтreсtly arе:

49x48x47x46x45x44

Unfortunatеl5 most Ьasiс сalсulators are unaЬlе to pеrform this сalсulаtion bесausе it produсеs a rеsult too big for thе сalсulator to displаy..!7ell, So will you takе my word for it that this is thе сorrесt you shouldn't! As' I suggеsted earlier, thеrе is an аnsrмеr? еrror in thе rеasoning hеrе, whiсh I rлrill now сorrесt.

Thе еrror is that I havе inсludеd еaсh сombinаtion of numbеrs many timеs, as a rеsult of whiсh this аnswеr is too large. To сonvinсe you of this, imagine that thе first sеt of six numbеrs you сhose wаs:

1,2,3,4,5 and6 This samе сomЬination of numbеrs сould аlso сrop up as 1, 3, 2,4, 5, 6 ot 6, 5,4,3,2,1' ot any ordеring you сan think of.

But just how many ordеrings are thеre? This quеstion rмas еxplored in Appendix L: Junk mail and frее offеrs. Thеrе we ordеred thrее things and found that thеrе rлrеrе six possiblе ordеrings. Thе number оf ways of ordering siх things is morе сompliсatеd and сan Ье сalсulatеd as follows.

Thеre arе six rмays of ordеring the first numbеr, fivе ways of ordering the sесond, four ways of ordеring thе third, and so on.

So, thе пumbеr of rмаys of ordеring six things = 6

2х|-720.

x5x4x3x

And now, bасk to thе plot. This last disсussion suggеsts that eаch сorпbiпation of' numbеrs сontainеd in thе сalсulation

49x48x47х46x45x44

=*-#*

Nour wе сan punсh this out on thе сalсulator and hopе wе gеt thе answer 14 million. Onе snag is that, if you сalсulatr thе top linе first bеforе dividing bу 720, you will almost сertainly сausе thе сalсulator to overflow. A snеaky way out of this is to dividе Ьy thе 720 sooner rathеr than latеr in thе сalсulation. For example, I prеssеd thе following:

49 rт1 720 г-;,148

гx--.l

47 Гт1 46

гт145 г x l 44

Is it safеr to travеl by road or by rail?

Мost people say that 'rail travеl is safеr, but is this rеally true?

A

useful start in answering this question is to look аt thе number оf deaths in a уeat due to rаilwаy and road aссidеnts.

Typiсal annual figures for Grеat Britain, supplied by фе offiсе of Population Censusеs and Survеys (oPсs) are as follows: Dеaths duе to railway aссidеnts Dеaths duе to road aссidents

70

4628

So, сlearl5 many more pеoplе die on thе roads than Ьy trаvelling оn a train * in faсt about sеvеnty timеs as man5 in thе year in quеstion. But does this mеan that rail ffavеl is sеvеnty

iimеs safеr than road travеl? Thе answеr is, not neсеssarily, Ьeсausе wе may not Ьe сomparing likе with likе. An important сompliсating faсtor is that many more peoplе travеl many morе kilometrеs by road than by rail. For еxamplе, privatе motor vеhiсlеs and taxis are usеd for around 90 pеr сеnt of distanсеs travеllеd in the UK. To аnswеr thе quеstion fairl5 wе nееd to take aссouпt of thе avеrage distanсеs travellеd by еaсh modе of transport and usе these to сalсulatе thе aссidеnt rаtes.Theseсan thеn Ьe с<rmpared dirесtly. So hеrе goes.

Thе fairеst figurе to usе here is thе numbеr of passеnger kilоmеtrеs for both road and rail. As thе name impliеs, onе

passеngеr is reсordеd whеn onе passrngеr travels onе. ^kilometrе. kilomеtrе If five passеngers еaсh travеl 10 kilometres, a tota| of 50 passеnger kilometres will be rесordеd.

is aсtually inсludеd 720 times.

Sо, the number of sеparate сombinations

Appendix N: Safe travel

r=1

This produсеs thе ansurer 13 983 816, whiсh isnt all that fat awаy from thе rеsult wе wеre hoping fot oI |4 million.

Typiсal annual rail passengеr .transport

usе

:

Typiсal annual road passengеr transport use

= 100

Ьillion passengеr km

= 590

billion passеngеr km

To сalсulatе thе aссidеnt ratе pеr Ьillion Passеngеr kilomеtrеs,. wе dividе the numbеr of dеaths in a yеar by thе numЬеr of

billion passengеr kilomеtrеs, thus: Rail passengеr dеath rаte Road passеnger dеath ratе

= =

ft

4!,2&

=

0.7

= 7.8

Using this сomparison, it sееms that road travеl is roughly ten times as dangerous as rail travel.

It is interеsting to look аt othеr forms of transport' based on this сompаrison of the number of dеaths pет billion passengеr kilomеtrеs. Тypiсal annual figurеs for Great Britain arе as follows.

Мodе

Rate per billion pаssеnger kilometres

Air .lVatеr

0.1

Rail

Bus or сoaсh

Car

Van Мotorсyсlеs

Pedal сyсlists

Foot

0.5 0.7 0.4

з.6

2.2 97.0 43.4 s3.4

.

)urcэ: D€partmвnt

of

тransрort

So, aссording to this mеasurе, air travеl reаlly is the safest form of transport and motorсyсling is vеry muсh thе most dangerous.

it should Ье

strеssеd that аll mеasurеs havе thеir drаrмbaсks аnd this is just onе possiblе mеasurе. A weakness in thе measure used hеrе is that it favours modеs of travеl whiсh arе fast over the slowеr mеthods. Thus, air travеl сan allow you to сovеr' say' a hundrеd kilometres in just a few minutes' цrherеаs you lмould takе days on foot to сover this sort of distanсе. As a rеsult, fоr a givеn numbеr of passеngеr kilomеtrеs, the exposurе to risk on foot is muсh grеatеr merely due to thе faсt that thеrе is a longеr pеriod of timе during whiсh an aссident саn happеn. Howevеr, providеd this aspесt is bornе

Howevеr,

in mind,

using thе numbеr

of dеaths per billion passenger

kilometrеs is proЬably thе fairеst сomparison availаblе.

Appendix O: World population It used to bе said that evеryone in thе world сould just fit onto thе Islе of Wiф if thеy all squeеzed up a Ьit. How сould you сhесk a сleim like this?

Clеarly, it is impossible to provе or disprovе suсh a сlaim with аbsolute сertainty - for one thing, the faсts and figurеs nееdеd arе simply not avаilablе with pеrfесt aссuraсy аnd thеre arе too many praсtiсal diffiсultiеs (are wе allowed to knoсk down all thе buildings and trеes, for еxample?). But thеre is somе fun to bе had in making sеnsiblе guessеs and doing аn .ordеr of magnitudе' сalсulation.

First, lеt us establish what informаtion is needеd tо makе the' сalсulаtion.

What is the сurrent population оf the world?

Clearly it is сhanging аll the timе, Ьut wе сan look it up in book. Aссording to thе UK govеrnmеnt a refеrеnсе ,Sociаl publiсation Treпds', the estimatе of thе world's population in 200]. was aЬout 6.2 billion.

Next, rлrhat is the area of thе Islе of Vight?

Aссording to tlle Маctпillаn Еnсусlopediа, this is 380 sq km (1,47 sq milеs).

Finallg wе nееd to make a sеnsiЬle guess as to how many pеoplе сould bе fittеd into onе square mеtrе of spaсе.

Assuming thеy arе all standing up (and Ьreаthing in) lеt's guеss that tсn pеoplе соuld Ье squееzеd into suсh a spaсе.

So, now for thе сalсulation.

Total arеa, in square metrеs

NumЬет of pеoplе who would fit into this

arеa

=

380

x 1000 000

x 1000 000 x 10 3.8 Ьillion

= 380 =

Sinсе this figurе is less than tЬe 6.2 billion whо arе еstimatеd to populatе thе globе, thе answer чrould appeat to bе thаt, evеn if аll thе treеs, housеs' сows and lamp-posts wеrе to bе removеd,

it simply сouldnt bе donе.

As a footnotе to this invеstigation, it is worth pointing out that this сlaim has Ьееn around for many dесadеs, over whiсh period thе worldЪ population has grown сonsidеraЬly. It is estimated that thе population is inсrеasing at a tate of roughly 1 8 per сent eaсh deсade. This mеаns that еstimates of future world populations, dесade by dесade, сan bе madе by multiplying thе. рrеsеnt estimatе Ьy 1.18. (If you arеnt surе whеrе this figurе of 1.18 сamе from, it is еxplainеd in Сhaptеr 06.)

For еxamplе, to make an еstimatе of thе population in 201'1'' multiply Ьy 1.18, thus: .Wе

6.2 billion Х 1.18 = 7.3 billion

сan makе baсkward projесtions dividing by 1.18 instеad of multiplying.

in a similar way,

but

An еstimate of the population in 7991', 1'981and so on сan bе found as follows: 1991 estimate = 6.2 billion + 1.18 = 5.25 billion 1.981 еstimatе = 5.25 billion + 1.18 = 4.45 Ьillion 1971 еstimate = 4.45 billion + 1.18 = 3.77 billion

So it rмould seеm that, in 1971', thе сlaim wasnt еntirеly prеpostеrous. And therе might еvеn havе bееn еnough rоom for thе сows!

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We have reason to believe that the suspeсt entered the premises o1 Devlin's shop at fte Shires Park in the town o1 Doddinфon on the afternoon ot 24th Deс in the year 2001 . At preсise|y 4 .26 p.m., 2 items wеre purсhased, name|y abottle of dishwasher liquid and a bunch of tulips, costing Е3.29 and €2'95 respeсtiveIy. A f 70 note was submitted to the сashier and €3.76

received.

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сhange was

Websites and organizations

fl

Name/resource A+B Books, who rмrite and publish mаthеmatiсs books for usе with a graphiсs сalсulator.

www.AplusB.сo.uk

1+

{r

fJ

с

Assoсiation of Ъaсhеrs of Мathеmаtiсs (UK) www.atm.org.uk/

Bournmouth University applеts (сomputеr animations)

Цr

Covеntry University Мathеmatiсs Support Cеntrс for

o

wrмrм.mis.сov.aс.uk/mаths-сеntrе/

-

dеmonstrating mathematiсal prinсiples http ://mathinsitе. bmth. aс. uk/htmVapplеts.html

Мathematiсs Еduсatioп

Mathеmatiсаl Assoсiation (UK) ww.w.m-a.org.ulс/

Mathpuzzlе сontains

resourсes wrмw.Мathpazz|e.coml

a

tan9e

of puzzlеs and othеr

maths

MathsNet сontains a wide range of puzzlеs, Ьooks and soffwarе

http ://wrмw. mathsnеt. net/

National Counсil of Tсaсhеrs of Мathеmаtiсs (USA) www.nсtm.org

opеn University Mathеmatiсs сoursеs

Www.opеn.aс.uk/maths

oundlе Sсhool sitе, with many useful rеsourсеs and links to othеr sites worldwidе http ://rмww.argonеt.сo.uk/oundlesсl/mlink.html UK National Statistiсs on line hшp ://www. statistiсs. gov.

uk/

- Mаthernаtiсаl Illiterаcу Consequenсes, Pеnguin, London. Rеal-rмorld examples of innumеraсg inсluding stoсk sсams, risk

Pаulos, John Allеn (1990) lnnumerасу

аnd its

pеrсеption and еlесtion statistiсs.

A сlаssiс text on mathеmatiсal proЬlеm solving that is wеll-known

P6|уa, G. (1990) Hot,l,l to Solue It, Pеnguin, London. around thе world.

Univеrsity of Plymouth Mathеmatiсs suppоft Мatеrials www.tесh. plym. aс. uk/mathУresourсеs/PDFlaЪ)Vmаthaid.html

Singh, Simon (2000) Thе Сode Booh, Fourth Еstatе, London. A history of сodеs and сiphеrs and thеir modеrn appliсations in еlесtroniс sесurity.

Reading list

Singh, Simon (1998) Fermаt's Lаst Theorеm, Folrth Еstatе' London. An aссount of Andrеw.!Иilеs' proof of Fermat's Last

Barrow, John D. (1993) Pi in the Skу: Counting, Thinhiпg апd Beiпg, Pеnguin, London. An еxploration of whеrе maths сomеs from and how it is pеrformеd. .!7yndham,

Еastawa5 Rob & Jerеmy (1998) Vhу Do Buses Cotпe in Thrеes?, RoЬson Books, London. Praсtiсal uses for various mathеmatiсal topiсs, inсluding proЬaЬilit5 Vеnn Diagrams and primе numbеrs.

Еastaway, Rob & Vyndhаm, Jеremy (2002| Holu long is а piеce of string?, Robson Books, London. Еxamplеs of mathеmatiсs in everyday lifе.

Flannеry Sаrah (2000| In Code: А Маthemаti.саI Journеу, Profilе Books, London. A сollесtion of problеms with solutions and explanations, Ьasеd on thе author's еxperienсеs of grоwing up in a mathеmatiсal homе.

Graham, Alan (2003) Teаch Yoursеlf Statistiсs, Hodder & Stoughton, London. A straightforward and aссessiblе aссount of thе big ideas of statistiсs with a minimum of hard mathеmatiсs.

Huntley, H.Е. (1970| The Diuine Proportion, Studу iп

МаthemаtiсаI Beаutу, Dovеr, Nеw York. Appliсations in art and naturе of thе .Goldеn Ratio'.

Ifrah, Georges (1998) The Uпiuеrsаl History of Numbers,TЬe Harvill Prеss' London. A dеtailеd book (translated from Frеnсh) аbout thе history of numbеrs and сounting from prеhistory to thе agе of the сomputеr.

Thеorеm, but also outlining somе problеms that havе intеrеstеd mаthematiсians ovеr many сеnturiеs.

Stеwart, Ian (L996| Frorn Hеre to Infinitу, oxford Univеrsity Press, oxford. An introduсtion to how mathеmatiсal idеas are dеvеloping today.

Stеwart, Ian (1'997) Does God Plау Dice?, Pеnguin, London. An intrоduсtion to thе theоry and praсtiсе of сhaos and fraсtals.

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deсimal fraction 48' 5o{1 deсimа|

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