Basic Maths 1

SECTION ONE - NUMBERS

TVDes of Number

1

UARE NUMBERS:

(1x1)(2x2) (3x3) (4x4) (5x5) (axa) (7x7) (axa) (9x9) (10x10) (11x11)(12x12) (13x13) (14x14) (15x15)

1 4 9 18 25 38 49 84 81 100 121 144 189 198 225...' ' ~ ' "" ' "" ' "" ' "" ' "" \ 4 \ 4\ 4 \ 4 " 4 " 4

3 5 7 9 11 13 15 17 19 21 23 25 27 29, .,.".".,...

Note that the DIFFERENCESbetween the sguare numbers are all the ODDnumbers.

2) CUBE NUMBERS: ,

They'reoalled CUBE NUMBER9 beoause __ --"yVthey're like the volumes of this pattern of oubes. 4x4x4=84

1 8 '27 84 1'25 '218 343 51'2 7'29 1000...Admit it. you never knew maths could be this exciting did you.

3) TRIANGLE NUMBERS: . ... ...... .........................To remember the triangle numbers you have to

picture in your mind this inoreasingpattern of triangles,where each new row has one more blob than the previous row.1 3 8 10 15 21 28 38 45 55... 0" 4 " 4 " 4" 4 " 4 " 4 ' 4 ' 4 ' 4 ' 4' 4

2 3 4 5 a 7 a 9 10 11 12It's definitely worth learning this simple pattern of differenoes, as well as the formulafor the nth term (see P.8) which is: nth term = Y2n (n + 1)

2 4 8 18 32...

5) PRIME NUMBERS:

2 3 5 7 11 13 17 19 23 29 31 37 41 43...

Prime numbers only divideby themselvesand 1 (Note that 1 is NOTa prime number).Apart from 2 and 5. ALLPRIMES ENDIN1. 3. 7. OR 9.90 POSSIBLE IJrimesare: 71. 73. 77. 79 101.103. 107. 109 241. 243. 247. 249... etc.

However. nof all of fhose are primes. and working out which are and which aren't is a little bit fricklJ- see next page for all the details on how to find primenumbers.

1) Write down the next five numbers in each of the sequences on this page.2) Write down an expression for the nth term for each of the first 3 sequences on this page.


2

Prime Numbers

1) BasicallAnd that's the best way to think of them. (Strictly.they divideby themselvesand 1)So Prime Numbers are all the numbers that DON'Tcome uR in Times Tables:

2 3 5 7 11 13 17 19 23 29 31 37

As you can see, they're an awkward-lookingbunch (that's because they don'tdivideby anythingl). For example:

TheonlVnumbers that multiplyto give1 areTheonlVnumbers that multiplyto give 31 are,

In fact the onl~ to get ANYPRIME NUMBERis

1 x 11 x 31

I

1 x ITSELF

2) Thev All End in 1, J, 7 or 91) 1is NOTa prime number

2) The first four prime numbers are 2. 3. 5 and 1

3) 2 and 5 are the EXCEPTIONSbecauseall the rest end in 1~ 1 or 9

4) But NOT ALLnumbers ending in ~ 1 or 9are primes, as shown here:

(Only the circled ones are primes)

@@@Q)@@@@)21 @ 21 @@) 33 @ 39

@)@@4951 851~

@83@89

J) How to Find Prime Numbers - a veO'simple methodFor a chosen number to be a prime:

il)mfiim1}. I

~m__ti1NIl:m !HJjjT'

Olm:\!ldhmmllllBmtnr'I " "1 I'x~. ',;_ ','"" '~,- '---~-".~--'-,---",+,,---,. -, '.'-'" ", -'-.- ., --' , . ~ "- \ \

'" If \ I I

I - SOrr,efhi"8 "k /, / I~ "Decide whether or not !l33 is a prime number." _ YO~~fhe"o"_c~,::~~~orr,es~p//have fo sfa/'f Paper,_

-:::- fhe squar by esfirr,afi _1) Does it end in either 1, 3. 7 or 91 Yes / I / ~ro ,of_seeP. 14

"8-I \ ",,-

2) Find its square root: -J'233 =15.'284 \ \ "-

3) List all Drimes which are less than this square root: 3, 1, 11and 134) Divide all of these primes into the number under test:

233 + 3 = 11.8881 233 + 1 = 33.2851233 + 11 = '21.181818 233 + 13 = 11.923011

5) Since none of these divide cleanlv into 233 then it 19 a Drime number. Easy Peasy

Nowcover the page and write down everything you've just learned.1)Write down the first 15 prime numbers (without lookingthem up).2) Findall the prime numbers between a) 100 and 110 b) 200 and 210 c) 500 and 510

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SECTIONONE - NUMBERS

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3

Factorsand Prime Factors

~ IThe MULTIPLESof a number are simply its liMES TABLE:

E.g. the multiplesof 13 are 13 28 39 52 85 78 91 104

Factors The FACTORSof a number are all the numbers that DIVIDE

INTO IT. There is a special way to find them:

Example 1: "FindALLthe factors of 24"

Start off with 1 x the number itself. then try 2 x . then 3 xand so on. listing the pairs in rows like this. Try each one inturn and put a dash if it doesn't divide exactly. Eventually.whenyou get a numberreDealed, you SIOD.

So the FACTORSOF 24are 1.2.3.4.8.8.12.24

1 X 242 X 123x84x8

This method guarantees you find them ALL. And don't forget 1 and 24!

Examl!le 2: "Find the factors of 64"

1x 84Checkeach one in turn. to 2 x 32see if it dividesor not. Use

~s:::-- 3 x -

your calculator if you are ~ 4 x 18not totally confident. 5 x -

8x-7x-8 x 8 ___ The8 hasrepealedso slop here.

Findin

90 the FACTORSof 84are 1.2.4.8,18.32.84

Prime Factors- The Factor TreeAnv number can be brokendown into a string ofPRIME NUMBERS all multiplied toeether- this is called "Expressingit as a product of Drimefactors". and to be honest it's pretty tedious - butit's in the Exam.and it's not difficultso long as lJouknow what it is.

Themildlyentertaining"FactorTree"method isbest. where you start at the top and split your numberoff into factors as shown. Eachtime you get a primeyou ringit and you finallyend up with all the primefactors. whichyou can then arrange in order.

90. "As a product of primefactors". 420 = 2x2x3x5x7

Thentry these withoutthe notes:1) List the first 10 multiples of 7 and of 9. What is their Lowest Common Multiple (LCM)?2) List a/l the factors of 38 and 84. What is their Highest Common Factor (HCF)?3) Express as a product of prime factors: a) 990 b) 180.


4

Fractions

TheFractionButton: I ElUse this as much as possible in the calculator paper. It's very easy. so make sureyou know how to use it - you'll lose a lot of marks if you donlt:1) To enter Y4press 0 El 112) To enter 1%press D 6111 GIll3) To work out ~ X %press D mUD 11m 11 114) To reducea fraction to its lowest termsenter it and then press11

e.g. 71'2' 11El 16a c=IID = %5) To convert betweenmixedand top heavlJfractions press. GI

e.g. 2 % fJ m 11m 11 11m which gives 1%

Doina Fractions Bv HandYou're not allowed to use your calculator in the Non-Calculator Exam (unsurprisingly).Frighteningly. you'll have to do them "blJhand" instead. so learn these 5 basic rules:

3/ X 4/ - 3x4/ - 12//5 /7 - /5x7 - /35

3/ . v - 3/ X 3/ - 3x3/ - 9//4 -;-/3 - /4 11 - 14xl - /4

4) Addin subtractin - frau ht.i) First get the bottom lines the same (get a "common denominator")

2 I 2x5 Ix3 10 3 (multiply each fraction by the same number top andE.g. 3 + 5" = 3x5 + 5x3 = 15 + 15 bottom, but a different number for each fraction)

ii) Add or subtract TOP LINES ONLY but only if the boHom numbers are the same.e.g. 1~5+%5=1%6 or %+~ =%. ~-~ =~

5) Finding A FRACTIONOF sornethin2..=just rnultiRlY..Multiplll bll the to p . divide bll the bottom: Jlof£380=Jlx£380=£3240=£182

~ ~ ~ 20 20 20

I Finally- Checking. I

ALWAY9 check your answer.

1) With your calculator:

a) 1/2 x 3/4 b) 3/5 + 2/9 c) 1/3 + 2/5 d) Findx: 2 % = % e) Findy: 1%8= %

2) By hand: a) '2/3 x 4/5 b) 4/5 + 3/10 c) 5/8 - 2/8 d) Express38/84 in itssimplest form. e) 3 % - 2 3/4 f) 2~ x 3 ~ ID 2% + 1Xo.


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5

PercentaaesYou shouldn't have any trouble with most percentage questions, especially types 1 and 2. Howeverwatchout for type :3 questions and make sure you know the proper method for doing them. "Percentage change"can also catch you out if you don't watch all the details - using the ORIGINALvalue for example.

"Rnd x% of y" - e.g. Find15%of £48 =:}0.15 x48 =£8.90

"Express x as a percentage of y"e.g. Give40p as a percentage of £3.34 =:} (40 + 334) x 100 =12%

~ I - IDENTIFIEDBY NOT GIVING THE "ORIGINAL VALUE"Theseare th~ type most peopleget wrong- but only because they don't

recognise them as a type 3 and don't applythis simplemethod:

Example: I Ahouseincreasesinvalueby 20% to £72.000.Findwhat it was worth before the rise.

An INCREASEof 20% means

that £72.000 represents 120%of the originalvalue. If it was aDROP of 20%. then we wouldput "£72.000 = 80%" instead.and then divide by 80 on theLHS. instead of 120.

Alwavs set them out exact/v like this examDle. The trickiest bit is deciding the top %figureon the RHS- the 2nd and 3rd rows are always 1%and 100%

Percentaae ChanaeIt is common to give a change in value as a Dercentage.Thisis the formulafor doingso - LEARNIT. AND USE IT:

~. :;".' :~-~-:"'5-::'...~-t. ,;;<;~::;_,z-,~ ~;\j~' -. ...~ ~-..:-.~ -~.. _: .~_ ~-'-"~:

:U~DI~{.~~1:IIIIr.~",c ':'"~..'~m;;J~'" . . '. . " ~t~

~ ,.'_. ",.~,. ',P-

By "change". we could mean all sorts of things such as: "Profit". "loss". "appreciation".

"depreciation". "i~crease". "decreas~ I "error". "djscount". etc. For example."prnfit"

percentage 11profitll = .~ x 100.. _ Note the great importance, of us~ng the. ,/ orlglhal 4- ORIGINAL VALUEin this formula.

i The Acid Test' LEARNThe details for ~ and PERCENTAGE, . CHANGE,then turn over and write it all down.

1)A trader buyswatchesfor £5 and sellsthemfor £7. Findhisprofitas a percentage.2) A car depreciates by 30% to £14.350. Whatwas it worth before?3) Findthe percentage error in rounding3.452 to 3.5. Giveyour answer to 2 DP.


6

Rational and Irrational NumbersNormal human beings (like you and rne!) find the whole topic of "Rational and Irrational

Nurnbers" cornplete,lybizarre. Unforlunalelt;. the fanatics who write your Exarn papers are wildaboul Ihem and you're bound to get a question on it. The good news. as ever. is that if vou LEARNlhis Dage Ihoroughlv. you'll sail through it like a surnmer evening breeze...

RATIONALNUMBERS The vast majority of numbers are rational. They are always either:

1) A whole number (either positive (+ve). or negative (-ve» e.g 4. -5. -122) A fraction p/q. where p and q are whole numbers (+ve or -ve). e.g. Y4.-Y2.3/43) A finite or repeating decimal. e.g. 0.125 0.3333333333... 0.143143143143..

IRRATIONALNUMBERS are messy!

1) They are always NEVER-ENDINGNON-REPEATINGDECIMALS. 1t is irrational.2) Aeood source of IRRATIONALNUMBERS is RQUARE ROOTS AND CUBE ROOTS.

1) "Determine which of these numbers are RATIONAL and which are IRRATIONAL":-J2 J4 .J36 .J42

Youknow J4 is 2. and you know .J36 is 6. 'cos 6x6 = 36.

.J42 looks~ harder. Butget this:

The !ll1!JJnumbers with rational square roots are fHluare numbers.

So check... is 42 square'? Nope. So there you go. .J42 is irrational. Easy.

(If you're not sure if it's square or not. you could split it into primefactors.)Of course. if you've got a calculatorhandy. you could always stick the numbers in andsee if they are non-recurrinedecimals (irrational)or otherwise (rational).

2) ''Findan irrational number between 6 and 10"Since square roots are our main source of irrationalnumbers. you might well go for

..fi or -J8. Wellthey are both certainlyirrationalbut they are not between 6 and 10.

because..fi = 2.645 and -J8 = 2.828... SomethingIike.J4Qwillbe more likeit.

Since 62=36 and 102=100. possibleanswers are .J37..J38..J39 J97..J98..J99Allthese have irrationalvalues between 6 and 10. so they wouldall do as answers (exceptfor ..J49. ..J64 or.J81 - why'?)

j) Sometimes thel}'lIdo a nastl} question using leHers:E.g. if P is rationaland q is irrational.say whetherp + q and p q are rationalor irrational:

If you've got a calculator.you can picksome numbers and try it out. It.s not too hard to

figure out anyway. If p = 1 and q = -J2. then adding 1 to -J2 isn't gonna change the fact

that it's a non-reDeatin~decimal. And1 x -J2 =-J2 . so it's irrational. .

1) Answer the whole of Example 1,

SECTION ONE - Nvr.rBERS

2) Give three irrational numbers between 30 and 40.

7

Recurrin Decimals and SurdsThese two topics are quite closely related to rational and irrational numbers (how lovely).

Turnin Recurrin Decimals into FractionsI

As you willundoubtedly remember, RECURRINGDECIMALSare RATIONALNUMBERSso you shouldalso be able to turn them into FRACTIONS,i.e. alb where a and b are whole numbers.

This is actually very easy if you just learn the simple rules, and sinoe it's been given speoialmention in the syllabus you'd have to be pretty daft not to. There's two ways you oan do it:1) by UNDERSTANDING2) by just LEARNINGTHERESULT. Both ways are 0001.

The Understanding Method:1) Find the length of the repeating sequenoe and mulfiplt;by 10, 100, 1000, 10 000 or

whatever to move it all up past the decimal point by one full reoeated lumo:E.g. 0.234234234... x 1000 = 234.234234..

2) Rubtraot the orillinalnumber, r, from the new one (which in this case is 1000r )i.e. 1000r - r = 234.234234... - 0.234234...giving: 999r = 234

3) Thenjust DIVIDEto leave r: r = 23%99' and canoelif possible: r =2~11

The "Just learning The Result" Method:The fraction always has the repeating unit on the top and the same number of nines on

the bottom - easy as that. Lookat these and marvel at the elegant simplicityof it0.4444444 = 4/9 0.34343434 = 34/99

0.124124124 = 124/999 0.14561456 = 1456/9999Always oheok if it will CANCel DOWN of oourse, e.g. 0.363636 = 36/99 = 12/33

Manioulatina Surds11sounds like something to do with oontrolling diffioult ohildren, but it isn't. Surds are expressionswith irrationalsquare roots in them. You MUSTUSE THEMif they ask you for an EXACTanswer.There are a few simple rules to learn:

1) J8 x.Jb = Jab e.g../2 x.J3 = .J2x3 =-J6 - alsoJb2 = b. fairly obviously

2) % =.m e.g.0/./2 =J% =J.4=2

3) J8 +.Jb - NOTHINGDOING... (in other words it is definitely NOT-Ja+ b)

4) (a + .Jb)2 = (a + .Jb)(a + .Jb) = a2 + 2a.Jb + b l NOTjust a2 + .Jb2)5) (a + .Jb)(a - .Jb) = a2 + a.Jb - a.Jb _.Jb2 = a2 - b

6) Express Y.J5 in the form a~7C where a and b are whole numbers.To do this you must "RATIONAL/REthe denominator", which just means multiplying

top and bottom by -J5: 3~Y.J5.J5= 3~% so a = 3 and b = 57) If you want an exaot answer, LEAVE THERURDR IN. As soon as you go using that

calculator, you'll get a big fat rounding error - and you'll get the answer WRONG.Don't say I didn't warn you...

1) Express 0.142857142857 as a fraction.


8

Findin the nth Term"The nth term" is a formula with "n" in it which gives you every term in a sequencewhen you put different values for n in. There are two different types of sequence(for "nth term" questions) which have to be done in different ways:

Common Difference Ji e: "dn +For any sequence such as 3, 7, 11, 15, where there is a COMMON DIFFERENCE:

~~~4 4 4 _.m

you can always find "the nth term" using the FORMULA:Don't forget:

I} "a" is simply the value of THEFIRSTTERMin the sequence.2} "d" is simply the value of THECOMMONDIFFERENCEbetween the terms.3} Toget the nth term, you just find the valuesof "a"and "d" from the sequenceand stick them in the formula.

You don't replace n thou~h - that wants to stay as n4} - of course YOU HAVE TO LEARNTHE FORMULA. but life is like that.

I ~: I "Find the nth term of this sequence: 5, 8, 11, 14 "ANSWER: I} The formula is dn + (a-d)

2} The first term is 5, so a = 5 The common difference is 3 so ~ = 33} Putting these in the formula gives: 3n + (5-3)

so the nth term = 3n + 2

ChanlIa +

If the number sequenoe is one where the differenoe between the terms is increasing or decreasingthen it gets a whole lot more oomplioated (as you'll have spotted from the above formula -whioh you'll haveto learn/). This time there are THREEletters you have to fill in:

"a" is the FIRST TERM,

"d" is the FIRST DIFFERENCE(between the first two numbers),"C" is the CHANGEBETWEENONE DIFFERENCEAND THE NEXT.

I ~: I "Find the nth term of this sequence: Q, 5, 9, 14 "~~~

ANSWER: I} The formula is "a + {n-I)d + Mz{n-I){n-Q)C" 3 5

2} The first term is 2, so a = 2 The first difference is 3 so d = 33} The differences increase by 1 each time so C = +1

Putting these in the formula gives: "Q + (n-I)3 + Mz{n-I){n-fl) x I"Which becomes: 2 + 3n - 3 + Y!2n!2 - 1Y!2n + 1

Which simplifies to: Y!2n2+ l%n = %n(n+3} so the nth term = Mzn{n+3J.

I} Find the nth ferm of the following sequences:a} 4.7. 10. 13 b) 3, 8, 13. 18 c) I. 3. 8. 10, 15, d) 3, 4, 7. 12....

SECTIONONE - N~ \'SEl?S

9


Calculator ButtonsThenext few pagesare full of lovely calculator tricks to save you a lot of button-bashing.There'sbasically two types of oaloulator- the old-style and the more fanoy two-line displayers.

2-line DisDlavCalculators: Calculators:

Theseones only display numbers. They do the Thesefancy ones are dead common now.calculation each time you press an operation key. They're really easy to use because you just type

B C_H_'" ....... 3Jmost calculations exactly as they're written.

BEtIl D 11px:y. .

mEtll t m. 1iri

D [-. ..-:",360J El ( 3: lO+SY . Y1Ylit == '51:/]

I_ TheDeletebutton IEl L.m_...)]Pressing the IB button deletes what you've typed,

11 SEMI-CANCELone key at a time Uust like on a computer), so it's

much quicker than pressing 11and re-typing theand E ALL CANCEL wholelot. UseIB or you'll be in BIGTROUBLEI

The. button only cancelsI Cursor I11l1llHY a a Ithe NUMBERYOUARE ENTERING.. clears the whole calculation. These cursor buttons a and a are pretty useful

If 'd0u use . instead of . for whenfor editin(!what you've typed in. (You'llprobablyfind

you hit the wrong key, you'll HALVE you overwritewhat was there before. but you can change

the time you spend correcting mistakeslthis with the !t!S key to insert.ratherthanoverwrite.)

1) Enterina Neaative Numbers90rne oaloulatorshavea. button whiohyou pressafter you'veenteredthe nurnber.Othersjust havea rninusbutton. whiohyou pressbeforeenteringthe nurnber.

90 to work out - 5 x - 8 you'd either press... 8. £1811Elor... ..£111..

Why oan't they alljust be the same... (Theexamplesin this book will usethe. button.)

2) TheMEMORYBUTTONS.L-B (StoreandRecall)I

(On some calculators the memory buttons are called III (memory in) and lID (memory recall)).

Contrary to popular belief. the memory function isn't intended for storing your favouritephonenumber.but in fact is a mightyusefulfeature for keepinga numberyou'vejustcalculated, so you can use it again shortly afterwards.

For something like 15+1IN40' you could just work out the bottom line first and stick it

in the memory. Press IDII...II and then liE(Or.. or.. or11)to keepthe result of the bottomlinein the memory.Thenyou simplypressm.EI.. and the answeris 0.7044.(Instead of Ill, you might need to type..or..or lID on yours.) Once you've practised with thememory buttons a bit. you'll soon find them very useful. They speed things up no end.

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10

Calculator ButtonsBODMAS and the BRACKETS BUTTONS3)

One of the biggest problems many people have with their calculator is not realisingthat italways works things out in a certain order, which is summarised by the word BODMAS,which stands for:

Brackets, Other, Division, Multiplication, Addition, Subtraction.This becomes of very pressing importance when you want to work out a simple thing like23+45S4x3 - it's no good just pressing m_1I1I1I1311 El - it willbe

completely wrong. The calculator will think you mean 23 + 40/84 x 3 because thecalculator will do the division and multiplication BEFOREit does the addition.

The secret is to OVER-RIDE the automatic BODMAS order of operations usin& theBRACKETS BUTTONS. Brackets are the ultimate priority in BODMAS, which means anythin&in brackets is worked out before anythin& else happens to it. So all you have to do is:

1) Write a couple of pairs of brackets into the expression: (23+45)(S4x3)

..IIIIB EI2) Thenjust type it as it's written:

It's not too difficultto decide where to put the brackets in - just put thern in pairs around eachgroup of numbers. It's OKto have brackets withinother brackets too, e.e. ( 4 + (5+2)). As a ruleyou can't cause trouble by putting too many brackets in, so long as they always go in pairs.

4) The POWERSBUTTON .Thepowers button can be pretty handy for workingout if a number is rationalor irrational- likethis:

1) 144Y' ANS: press ... or to get 12, which is certainly rational.

2) 80-3/4ANS:press mllla.8mBIIEI whichgives 0.037383719... whichshows no signs of repeating so it willbe irrational.

3) V6~ ANS:press whichgives 1.449559327... which again is clearly irrational.

(If youdidn'tuse bracketshere.yourcalculatorwouldprobablyhavegiventhe wronganswer. That'sbecauseitdoesn'tknowhowmuchof the expressionto applythe )(Yto unlessyoumakeit clearwithbrackets. Tryit.Getinthe habitof usingbracketsand you'llsaveyourselfa lotof headachesl)

4) Express 49-Y' as a fraction in the forrn a/b.

ANS: press which gives 0.142857142.. and you might think it'sirrational but notice the repeat of 142 which suggests

it mat; be a repeating decimal and therefore rational. The. button reveals all.

5) The., ButtonIt basicallyturns numbersupside-downand this providestwo very usefulfunctions:1)MAKINGDIVISIONSABITSLICKER E.g. if you alreadyhave 2.3458328 in the displayandyouwantto do12+ 2.3458328, thenyoucanjust press .' whichdoesthedivision the \,\;ro~i .:.:>"_p and then nips it the ril1htway up!

2) ANALYSING~ECIMALS +0see ifthey rnightbe rational (i.e. sornethingsirnple)e.g. ifthe displayis O.142857W2 anC ~Q~ ~ . Elyou'llget 7. rneaningit was In before.

Il

Calculator Buttons

6) The STANDARDFORM BUTTON .~All you ever use this for is entering numbers written in STANDARDFORM into the

calculator. It would be a lot more helpful if the calculator manufacturers labelled it as 11because that's what you should call it as you press it: "Times ten to the power.." For

example to enter 8 x 103 you must only press BIIB, and NOT, as a lot of people do:

IIEl1I811. Pressing x 10 as well as EXP is horribly wrone, because the EXPalready contains the x 10 in it. This is why you must always say to yourself "Times ten tothe power.." each time you press the EXP button, to prevent this very common mistake.

TOREADA STANDARDFORMNUMBERFROMTHEDISPLAY: E.g. t 1.986 OS]

This must be written as 7.988 x 105 {NOT 7.9885} - YOU have to put the x IOnin yourself.

7) Convertin Time to Hrs Mins and Secswith.Here1sa tricky detail that comes up when you're doing speed distance and time: convertine

an answer like 2.35 hours into hours and minutes. What it definitely ISN'T is 2 hours and 35

mins - remember your calculator does not work in hours and minutes unless you tell it to,as shown below. You'll need to practise with this button, but you'll be glad you did.

1) To ENTERa time in hours. mins and secs

E.g. 5hrs 34mins and 23 secs, press 5 . 34 . 23 . Elto get[~=-- SO 3 ~~.t3 ).

2) Convertine hours. min and secs to a decimal time:Enter the number in hours, mins and secs as above.

Thenjust press. and it should convert it to a decimal like this ~S.Sl{(rSSSS-6).(Though some older calculators willautomaticall\j convert it to decimal when \jou enter a time in hours, minutes and secs.)

3) To convert a decimaltime as ou alwa s et from a formula into hrs mins and secs:E.g. To convert 2.35 hours into hrs, minsandsecs.

Simply press 2.35 El to enterthedecimal,thenpress...

The display should become r _.~_. "2°21° 0), which means 2 hours. 21 mins (and 0 secs).

The Ac,'d test ' LEARN our oaloulafor buffons. Practise untH'd0u can

. , answer all ofthese without having to refer back:

1) Explainwhat. and III do and give an example of using them.

2) How do you enter a) 88 b) 8 X 108 c) 50-4/5 d) ~4%23.3 + 35.8

3) Write down what buttons you would press to work this out in one go: 38 x 28.5

4) a) What would 3.4 x 108look like on the display? b) Givetwo uses for 8.5) a) Convert 4.57 hrs into hrs and mins.

b) Convert 5hrs 32mins and 23secs into decimal hrs.


12

Conversion FactorsConversion Factors are a m~hty powerful tool for dealinBwith a wide variety ofquestions. And what's more the method is real easy. Learn it now. /f's ace.

2) "If £1 = 7.75 FrenchFrancs. how much is 47.38 Francs in £ and p'!"

1) Obviously.Conversion Factor = 7.75 (The"exchange rate")2) 47.38 x 7.75 = £387.04

47.38 + 7.75 = £8.113) Not quite so obvious this time. but if roughly 8 Francs = £1. then 47

Francs can't be much - certainly not £387. so the answer must be£8.11Q

'~ map has a scale of 1:'QO.OOO.How big inreal life is a distance of 3cm on the maR'!"

To Convert 80&000cm to m:

1) Conversion Factor = 20 000

2) 3cm x 20000 = 80000cm (looks OK)3cm + 20 000 = 0.00015cm (not good)

3) 90 80.000cm is the answer.Howdo we convert to metres'!

1)C.F. = 100 (cm ~ m)2) 80.000 x 100 = 8.000,OOOm

(hmm)80.000 + 100 = 800m

(more likeit)3) 90 answer = 800m

I) Convert 2.3 Ic-r1 ft}..,. 2) Which is more. £34 or 280 frenoh franos? (Exchange rate = 7.75)3) A map is ~ +0a scale of 20m = 5km. A road is 8 km long. Howmany omwillthis be on the

rT1ap?\H + C.F.= 5+2. i.e. lom= 2.5 km)1I

SECT/O.;::\'£ - ..B£P.S

13

Metric and Imoerial UnitsMake sure you learn all these easy facts:

Metric Units1) Leneth2) Area3) Volume

4) Weight5) Bneed

mm, cm, rn, kmmm2, cm2, m2, km2,mm3, cm3, m3,

litres, mlg, kg, tonneskm/h, m/s

Imoerial Units1)Leneth2) Area

3) Volume

4) Weight5) Bneed

Inches, feet, yards, milesSquare inches, square feet,square yards, square milesCubic inches, cubic feet,

gallons, pintsOunces, pounds, stones, tonsmph

LEARN THESE TOO!

1 Foot = 12 Inches1Yard = 3 Feet1 Gallon = 8 Pints

1Stone = 14 pounds (Ibs)1Pound = 16 Oun"ces (Oz)

Metri~lmoerialConversionsYOU NEEDTO LEARNTHESE - they DON'Tpromise to give you these in theExam and if they're feeling mean (as they often are), they won't.

APPROXIMATE CONVERSIONS

1 kg = 2Y4Ibs 1gallon = 4.5 litreslm = 1yard (+ 10ro) 1foot = 30cm1litre = 13/4pints 1metric tonne = 1imperial ton1inch = 2.5 cm 1mile = 1.8km

or 5 miles = 8 km

UsinaMetric-lffloerial ConversionFactors1)Convert45mm into crn.2) Convert37 inches into cm.3) Convert5.45 litres into pints

CF =10, so x and + by 10, to get 450crn or 4.5crn. (Sensible)CF = 2.5, so x and + by 2.5, to get 14.8cm or 92.5crn.

CF = 1%, so x and + by 1.75, to get 3.11or 9.54 pints.

Tb"

A'd ~ t LEARNthe 21 Conversion factors in the shadede Cl ,es: boxes above.-Then turn over and write them down.

1) How many litres is 3Y2gallons? 2) Roughly how many yards is 200m?3) A rod is 48 inches long. What is this in cm?4) Petrol costs £2.83 per gallon. What should it cost per litre?5) A car travels at 85 mph. What is its speed in km/h?


14

Accura and Estimatinriate Accura

To decide what is appropriate accuracy, you need only rernernber these three rules:

1) For fairly casual rneasurerneht~FICA~_JFIGURE9 is rnost approp!!ifill,

EXAMPLES: Cooking - 250g (2 sig fig) of sugar, not 253g (3 9 F), or 300g (1 9 F)Distanoe of a journey - 450 rniles or 25 miles or 3500 rniles (All2 9 F)Area of a garden or floor - 330m2 or 15m2

EXAMPLES:Ateohnioalfigurelike34.2 milesper gallon, rather than 34 mpg.A length that willbe out to fit. e.g. Measure a shelf 25.80m long not just 280m.Any aoourate measurernent with a ruler: 87.50m not 700m or 87.540m

This is VEH'LEA9Y.so long as you don't over-oomplioate it.

1) ROUND EVERYTHINGOFF to nioe easy CONVENIENT NUMBERS2) Then WORK OUT THE AN9WER using these nioe easy numbers - that's it!

In the Exarnyou'll need to show all the steps. to prove you didn't just use a oaloulator.

EXAMPLE: Estirnatethe valueof 127.8+ 41.9 showingallyourworking.58.5 x 3.2

127.8+41.9 130+40 170Ans: ~ - - ~ 1 (" ~ " means "roughlv equal to" )58.5x3.2 80x3 180

Areas and Volumes

EXAMPLES: "Estimate the area of this shaDe and the volume of the bottle:"

Area ~ reotangle28m x 13m = 338rn2(or without a oaloulator:

30 x 10 = 300m2)

12'7bm

~_.

(~~==?: Volume~ ouboid. . I I, ': : :lOcm=4 x 4 x 10

'I I I5.2cm: :: = 1800m3, 'J

!..._~~ J-~~

Looks horrible - but it's OK if youknow your square numbers (P.1).

11,1"~r.m':':t!or:f.'i

EXAMPLE: "Estimate .J85 without using a calculator."

CD The square numbers @ The square roots are 9 and 10. so -J85 rnust be between 9 and 10. Buteither side of 85 85 is rnuoh nearer 81 than 100. so-J85 rnust be much nearer 9 than 10.are 81 and 100. 90 pick 9.1. 9.2 or 9.3. (Theanswer'sactually9.2195... if you'reinterested.)

Tb A.d ' ~ t LEARNthe 3 Rules for A ro riate Aocurao and 8 Rules

e Cl .es: ~.Thenturn overandwritethemalldown.1)Decidewhiohcategoryof accuracythese shouldbelongin and roundthem off aooordingly:a) Ajar of jarn weighs 34.58g b) A car's rnax speed is 134.25rnph 0) A cake needs 852.3g of flour2) Estimate the area of Great Britain in 9quare miles. and the volume of a milk bottle in crn3.3) Without your catcu!ator, estimate: a).J12. b).JI04. c).J52 d) vI30 .

SECTION OVE - .\,..,,,'8£/?S

Rounded Off ValuesYoushould be confident about rounding numbers off to a certain number of decimal places orsignificantfigures. If /Jets tricky when they start askin/Jabout the maximum and minimumvaluespossible for a given levelof accuracy in the rounding. There are three main aspectsspecificallymentioned in the syllabus, and this topic is very popular with the Examiners.

1) Findin er and Lower bounds of a Sinale Measurement

The simple rule is this: The real value can be as much as HALFTHE ROUNDED.UNIT above and below the rounded-off value

E..g. If a len.gth is .given as 2.4 m to the nearest 0.1 m, the rounded unit is 0.1 m so the real value oould beanythin.g up to 2.4m z 0.05m .givin.g answers of 2.45m and 2.35m for the upper and lower bounds.

2) TheMaximum and Minimum Possible Valuesof a Calculation

When a calculation is done using rounded-off values there will be a DISCREPANCYbetween the CALCULATEDVALUEand the ACTUALVALUE:

EXAMPLE:A floor is measuredas being 5.3m x 4.2m to the nearest 10 cm.Thisgives an areaof 22.28 m2,but this is not the actual floor area because

the real values could be anything from 5.25 m to 5.35 m and 4.15 m to 4.25 m,

:. Maximum possible floor area = 5.35 x 4.25 = 22.7g]5 m2,

:. Minimum possible floor area = 5.25 x 4.15 = 21.7875 m2.

3) Maximum Percentage Error ,

Having found the two possible extreme values. the one which is FARTHESTfrom therounded value will give the maximum percentage error using this familiar formula:

E..g. for the above rectan.gle the max error is 22.7375 - 22.26 = 0.4775 so the maxpercenta.geerroris0.4775

22.28x 100 = 2.15%

4) Alas it is not alwavs so simple...In many formulas (especially in Exam Questions) it ISN'Tthe biggest input values that

give the maximum result. Consider z = x + X The maximum value for z will resultfrom the maximum value for x coupled with the minimum value for y.

So when the question looks more complicated, the safest method is to work out theanswer usinJJall four combinations and see which combinations give the maximum andminimum results.

1)x and y are rneasured as 2.32rn and O.45rn to the nearest O.01rn. T is given by T = (x - y)/y .Rnd the rnaxirnurnpossiblepercentageerror in T if the roundedvaluesof x andy areusedto calculateit.


=

16

Revision Summary for Section One

I know these questions seem difficult, but theVare the verv best revisionvou can do. The whole i

point of revision, remember, is to find out what vou don't know and then learn it until vou do.IThese searching questions test how much you know beHer than anvthine else ever can. They

follow the sequence of pages in Section One, so you can easily look up anything you don't know. I

Kee learnin these basic facts until DUknow them

1)What are square numbers, cube numbers, triangle numbers and prime numbers?2) List the first ten of each from memory. Then write down the first 5 powers of 2 and the

first 5 powers of 10.3) What are the three steps of the method for determining prime numbers?4) What are a) multiples, b) factors, c) prime factors?5) List the first five multiples of 13, and all the factors of 80.6) Give details of five different things you can do with the Fraction Button.7) Describe in words the method for each of the 4 rules for doing fractions by hand.8) Do your own example to illustrate each of the three types of percentage question.9) What is the formula for percentage change? Give two examples of its use.10) Name three different forms that a rational number can take, and give examples.11)Describe two forms that an irrational number can take, with examples.12) What is generally the best way of identifying a number as rational or irrational?13) Explain the process for finding an irrational number between say 14 and 19.14) Demonstrate the 2 methods for "doing" recurring decimals.15) Write down all you know about manipulating surds.16) What are the 2 formulas for finding the nth term of a sequence?17) What type of sequence does each formula apply to?18) Illustrate four extreme uses of the powers button.19) Explain exactly what BODMASis. Does your calculator know about it?20) Give a good example of where the brackets buttons should be used.21) Give a good example of where the memory buttons should be used.22) Should you use all these useful features when you're working things out?23) Which button is useful for analysing awkward decimals? Give an example.24) Which is the standard form button? What would you press to enter 6x108?25) Which button can be used to enter hours, minutes and seconds?26) Explain how to do so and also what to press to convert to a decimal time.27) What is the difference between decimal time and ordinary time?28) What are the three steps for using conversion factors? Give 3 examples.29) Give 8 metric conversions, 5 imperial ones, and 8 metric-to-imperial.30) Give three rules for deciding on appropriate accuracy.31) Give 2 rules for working out approximate answers to formulas.32) Give two rules for working out approximate areas and volumes.33) Estimate the following square roots: .Ji4,.J70, .J32, .J3534) How do you determine the upper and lower bounds of a rounded measurement?35) Explain how a calculated answer can have a range of possible values.36) How do you find the maximum possible error?37) How do find the maximum possible percentaBe error?38) What is the worst situation for this and how do you deal with it?


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Basic Maths 1