Cambridge IGCSE Mathematics Core and Extended 3ed

Stat istics

a) Construct a cumulative freq ue ncy table for each type of
o r a n ge.

b) Drawacumulativefreq uencydiagram for each type of
orange.

c) Calculate the median mass for each type of orange.
d) Using your graphs estimate:

i) the lower quartile,
ii) the upper quartile,
iii) the inte r-q uartile range
for each type of orange.
e) Write a brief report comparing the two types of orange.

5. Two compet ing brands of battery arc compared. A hundred
batteries of each brand are tested and the duration of
each is recorded. The results of the tests arc shown in the
cumulative freque ncy diagrams below.

1$' 100 1$' 100 '""lY .
~• ~ 00
10 20 30 40
g g Duration{ll)

;® ; 00

~s ® ~s w

um u~

0 10 20 30 40 •
Duretion(h)
0

a) The manufacturers of brand X claim that on average
their batteries will last at least 40% longer than those of
brand Y. Showing yo ur method clearly, decide whether
this claim is true.

b) The manufacturers of brand X also claim that their
batteries are more reliable than those of brand Y. ls this
claim true? Show your working clearly.

39 Cumulative frequency

Student assessm ent I

I. ll1irty students sit a Maths exam. The ir marks are given as
percentages and arc show n in the table below.

I-M ark 120 - 30 - I 40 - 1so - I 60 - I 70 - 1ao- I 90-100 I

I._Frequency l 2 3 I S l 7 l 6 l 4 1 2 1 I

a) Construct a cumulative frequency table of the above
res ults.

b) Draw a cumulative frequency diagram of the res ults .
c) Using the graph, estimate a value for:

i) the median,
ii) the upper and lower quartiles,
iii) the inter-q uartile range.

2. 400 students sit an !GCSE exam. Their marks (as
percentages) are shown in the table below.

Mark(%) Frequency Cumulative frequency
31 - -40 21
-41 - 50 55
51 - 60 125
61 - 70 74
71 - 80 52
81 - 90 45
91 - 100 28

a) Copy and comple te the above table by calculating the
cumulative frequency.

b) Draw a cumulative frequency diagram of the results.
c) Using the graph, estimate a value for:

i) the medianresult.
ii) the upper and lower quartiles,
iii) the inter-q uartile range .

Sta t istics

3. Eight hundred students sit an exam. ll1eir marks (as
percentages) are shown in the table below.

Mark(%) Frequ ency Cumulative frequency
1-1 0 IO
JO
11-20 40
2 1- 30 50
3 1--40 70
-4 1-50
5 1- 60 100
6 1- 70 240
7 1-80 160
8 1-90
9 1-1 00 70
JO

a) Copy and complete the above table by calculating the
cumulative freq ue ncy.

b) Draw a cumulative frequency diagram of the results.
c) A n 'A ' grade is awarded to a s tudent at or above the

80th percentile. What mark is the minimum req uireme nt
foran'A 'grade?

d) A 'C' grade is awarded to any stude nt between and
including the 55 th and the 70th percentile. What
marks form the lower and upper boundaries of a 'C'

grade?
e) Calculate the inter-q uartile range for this exam.

Mathematical investigations
and ICT

• Heights and percentiles
The graphs below show the height charts for males and females
fr om the age of2 to 20 years in the United States.

Stature-forage percentiles:

"lii:111111"Boys,2to20years

2 3 4 5 6 7 8 9 1011121314151617181920
Age(years)

Note: Heights have been given in both centimetres and inches.

Stat istics

" 68

2 3 4 5 6 7 8 9 1011121314151617181920
Age(years)

1. From the graph find the height corresponding to the
75thpe rce ntilefor 16year-oldgirls.

2. Find the height which 75% of 16 year-old boys arc
taller than.

3. What is the median height for 12 year-old girls?
4. Measure the heights of students in your class. By carrying

o ut appropriate statistical calculations, write a report
comparing your data to that shown in the graphs.
5. Would all cultures use the same height charts? Explain yo ur

Topic 9 Mathematica l investigations and ICT

• Reading ages
Depe nding on their target audience, newspapers, magazines
and books have different levels of readability. Some are easy to
read and others more difficult.

I. Decide on some factors that you think would affect the
readabilit y of a tex t.

2. Write down the names of two newspapers which you think
would have different reading ages. Give reasons for yo ur

There are established formulae for calculating the reading age

of different texts.

One of these is the Gunning Fog Index. It calculates the

reading age as follows:

i (1Reading age= + l~L)where

A = number o f words
n = number of sentences
L = number of words with 3 or more syllables

3. Choose one article from each of the two newspapers you
chose in question 2. Use the Gunning Fog Index to calculat e
the reading ages for the articles. Do the results support your
predictions?

4. Write down some factors which yo u think may affect the
reliability of your results.

• ICT activity
In this activity you will be collecting the height data of all the
stude nts in yo ur class and plotting a cumulative frequency
diagram of the results.

I. Meru; ure the heights of all the students in yo ur class.
2. Group yo ur data appropriately.
3. Enter your data int o graphing software s uch as Excel or

Autograph.
4. Produce a cumulative frequency diagram of the results.
5. From your graph find:

a) the median height of the students in yo ur class.
b) the inte r-q uartile range of the heights .
6. Compare the cumulative frequency diagram from yo ur class
with one produced from data collected from another class
in a different year gro up. Comment on any differences/
similarities between the two.

Index

A supplementary 258 calculations
acceleration 178 types 21-4 estimatinganswers 14-15
accuracy,appropriate 14 verticallyopposite 246 orderofoperations 26
acuteangles 214 withinparallellines 246 withfractions 34-7
addition answers,estimating 14-15 withupperandlowerbounds
apex,ofapyramid 298 17-2 1
ofalgebraicfractions 115 ApolloniusofPerga 213
offractions 35-6 approximation 12 calculator calculations
ofmatrices 397-9 Arab mathematicians 103 appropriateaccuracy 14
ofvectors 384 arc 216,292 orderofoperations 26-8
adjacentside,ofaright-angled Archimedes 213
capacity,metricunits 276,278
triangle 349 convertingfromoneunitto Cardano.Girolamo 381
al-Karaji 103 another 278-80 Cartesiancoordinates 315
al-Khwarizmi 103
al-Kindi -465 ofacircle 286-7 ofacircle 216
algebraic fractions ofaparallelogram 283-5 ofenlargement 415-17
ofarectangle 281 ofrotation 410,412
..•.,additionandsubtraction 115 ofa sector 294-6 chaostheory 441
simplifying 11-4. 115-18 ofatrapezium 284-5 chord.ofacircle 216,242
ofa triangle 281-3,370-1 cirdeproperties 242-3
alternate 246 ofsimilarshapes 235-7 cirdes
atthecentreofacircle 257 under a speed-time graph angleproperties 253-5,257-9,
atapointandonaline 245
betweenalineandaplane 181 -3 273
areafactor 231.235 area 286-7
37-4-6 arithmeticsequences 155-7 circumference 286-7
betweenatangentandaradiusof Aryabhata 3 tangentsfromanexternalpoint
averages 466--8
a circle 254-5 243
bisecting 223-5 backbearings 346-7 terminology 216
calculating size by trigonometry barcharts 4n circular prism see cylinder
base,ofatriangle 282 circumcircle,ofatriangle 226
359 bearings 346 circumference,ofacircle 286-7
corresponding 246 Bernoulli family of mathematicians classintervals 485
inaquadrilateral 128,249-5 1 columnmatrix 394
inasemi-circle 253-4 345 columnvectors,translation 386
inatriangle 128,2-48-9 Bhascara 3 commondenominators 116,117
inoppositesegmentsofacircle bonus 76 common difference.in an arithmetic
bounds,upperand lower 16-21
258 brackets sequence 155
inthesamesegmentofacircle commonratio,inageometric
andorderofoperations 26,
257-" JJ--4 sequence 162
obtuse 214 compasspoints 346
of elevation and depression expanding 104-5, 108 complementofaset 90
Brahmagupta 3 completingthesquare 142
357-9 compositefunctions 207--8
ofirregularpolygons 256
ofrotation 412
reflex 214
right 21-4

Index

compoundinterest 81-3.163--4 differenceoftwosquares 109 expressions 110
concyclicpoints 258 directproportion 52--4, 167 bydifferenceoftwosquares
cone directvariation 167-9
directednumbers 10 109- 10
rotationalsymmetry 241 directionofrotation 412 bygrouping 108--9
surfacearea 305--6 discount 77 in solving quadratic equations
volume 301-5 distanceseespeed,distanceand
constantofproportionality 167 138--9
constantofvariation 167 division of quadratic expressions 110-11
conversiongraphs 174-5 ofaquantityinagivenratio simple 105--6
correlation 479-84 54-6 factors 5--6
cosine 352-3 offractions 36--7 Fermat.Pierrede 315,441
cosinecurve 360--1 rules 33 Fermat's L.astTheorem 315
cosinerule 367-70 Rbonacci (Leonardo Pisano) 385
costprice 77 dodecagon 218 formulae
cube,rotationalsymmetry 241 doubletime 76 forthetermsofanarithmetic
cubenumbers 5
cube roots 9-10 earnings 76--7 sequence 155--6
cubicfunction 193 Egyptianmathematics 275 transformation(rearrangement)
cubicsequences 157--61 elements
cuboid 107, 112- 14
planesofsymmetry 240 ofamatriK 394 fraction.proper 30
rotationalsymmetry 241 ofaset 87--8 fractionalindices 69-71
surfacearea 288-9 elevation.angle 357-9
volume 290 elimination,insolvingsimultaneous algebraic 124- 5
cumulativefrequency 491 -3 fractions 30-1
currencyconversions 75,174 equations 131 -2
curves.gradients 195--7 emptyset 88 additionandsubtraction 35--6
cyclicquadrilateral 258 enlargement 415-2 1 algebraic 114- 18
cylinder changingtodecimals 37-8
rotationalsymmetry 241 negative 419 equivalent 34
surfacearea 288-9 equations improper 30-1
volume 290 multiplicationanddivision 36--7
constructing 128-31, 136--8 simplest form (lowest terms)
D ofa line through two points
dao 34-5
332-3 Frenchmathematicians 315
continuousordiscrete 473, 485 ofa straight line 320-5 frequency
correlation 479-84 simplelinear 127
decagon 218 solving by graphical methods cumulative 491-3
deceleration 178 relative 447- 9
decimalplaces(d.p.) 12- 13 197-200 frequencydensity 486
decimals 31-2 EuclidofAlexandria 213 frequencytables 4n
changingtofractions 38,39--41 EudoxusofAsiaMinor 213 functions
terminatingandrecurring 4. Euler.Leonhard 345 definition 203
evaluation evaluating 203-5
30--41 inverse 205--6
deductionsfromgrossearnings 76 offunctions 203-5
denominatorofafraction 30 ofnumericalexpressions 110 G
depreciation 78 events,combined 452 Gauss,CarlFriedrich 441
depression.angle 357-9 exchangerates 75 geometricfigures,constructing
Descartes.Rene 315 exponential equations 65
determinant,ofamatriK 403--4 graphicalsolution 211 221-3
diameter,ofacircle 216 exponentialfunctions,graphs 195 geometricsequences 162-5
gradient
factorisation
and evaluation of numerical ofadistance--timegraph 176
ofastraightline 195,3 16--19
ofcurves 195--6
ofparallellines 326--7
ofperpendicularlines 333--6

Index

positive/negative 318 inter-quartilerange 466.49-4--6 seeolsocolumnmatrix;identity
gradient-interceptform 324,326 interest 79--83 matrix; row matrix; square
graphical solution intersectionofsets 90 matrix; transformation ma-
inverse,ofamatrix 404--6 trix;zeromatrix
ofequations 197-200 inversefunctions 205--6
ofexponentialequations 211 inverse proportion 56--7.169--70 mma
ofinequalities 149-52 inversevariation 169-70 definition 466
ofquadraticequations 191-2 irrationalnumbers -4--5,7~ forgroupeddata 469
of simultaneous equations ltalianmathematicians 381
median.definition 466
329-30 kite.properties 217 metricunits 276
graphs
Lagha.da 3 convertingfromoneunitto
exponentialfunctions 195 Laplace,PierreSimonde +II another 277~
forfindingsquareroots 9 laws of indices 62,123
inpracticalsiOJations 174 length,metricunits 276 midpoint,ofalinesegment 331-2
ofquadraticfunctions 189-9 1 line mirrorline 408
ofreciprocalfunction 192-3 mixednumbers 30-1,35
types 193--4 bisecting 223-5 modalclass 469
Greekmathematicians 213 ofbestfit 480 mode.definition 466
groupeddata,mean 469 ofsymmetry 240 multiples 7
groupedfrequencytables 473 seeolsostraightline multiplication
line segment
height(altitude),ofatriangle 214, calculatinglength 330-1 in solving simultaneous equations
282 midpoint 331-2 134
linearequations.solving 127~
heptagon 218 linearfunction,straight-linegraph ofamatrixbyascalar 400
hexagon 218,251 ofamatrixbyanothermatrix
hexagonal-basedpyramid 298 193
highest common factor (HCF) 7, linearinequalities 143 400-2
linearprogramming 152-3 ofavectorbyascalar 384
JS locus(loci) 265~ offractions 36--7
Hindumathematicians 3 lowest common multiple (LCM) 7. rules 33
histograms 484--8
Huygens,Christiaan +II 116 N
hyperbola 192, 193 naturalnumbers 4
hypotenuse 349 M negativeindices 64,68--9.124
maps see scale drawings netpay 76
identitymatrix 4-03 mass.metricunits 276,278 nets 219
image 408 matrix (matrices) Newtonian universe +I I
increase and decrease numbers,types 4
additionandsubtraction 397-9 numeratorofafraction 30
byagivenratio 57-9 determinant 403--4
percentage 46--7 elements 394 0
index(indices) 62--4 inverse 404--6 objectandimage 408
algebraic 123 octagon 218
inequalities 16 andtransformations 426--7 Omar Khayyam I03
graphing 149-52 multiplicationbyascalarquantity opposite side.of aright-angled
inlinearprogramming 152-3
manipulation 143 400 triangle 349
representingonanumberline multiplication by another matrix order

2-4--5,143--4 400-2 ofamatrix 39-4--6
solving 143, 149 order 394--6 ofoperations 26.33
symbols 24.143.149 representing a transformation ofrotationalsymmetry 240
integers 4 ordering 24
423-5 outcomes +12
• overtime 76

Index

p practicalandtheoretical 4-42-6 scaledrawings 227-9
profit and loss 77-8 scalefactor 231,235
1t(pi)
approximationsfor 3 percentage 78--9 ofenlargement -415-17
irrationalnumber 5 pyramid 298--301 ofnegativeenlargement -419
PythagorasofSamos 213 scatter diagrams -479-84
parabola 189,193 Pythagoras'theorem 353--6 scientificnotationseestandardform
parallellines 21-4
Q area 294--6
angleproperties 2-46 quadratic equations definition 293
equations 326-7 ofacirde 216
parallelogram graphicalsolution 191-2 segment,ofacircle 216
area 283-5 solvingbycompletingthesquare sellingprice 77
constructing 222-3 semi-circle.anglein 253--4
properties 217 142 sequences
Pascal.Blaise 315.+II solvingbyfactorising 138--9 arithmetic 155--7
Pascal'sTriangle 315 quadraticexpressions,factorisation definition 155
pentagon 218,251 geometric 162-5
percentage increases and decreases 110-11 quadraticandcubic 157--61
quadraticformula 1-41 terms 155
46-7 quadratic functions
percentageinterest 79 complement 90
percentageofaquantity,calculating graphs 189-91 elements 87--8
parabola 193 empty 88
44-5 quadraticsequences 157--61 intersection 90
percentageprofitandloss 78--9 quadrilaterals 218,251 notation 87,88,90
percentages 32 angleproperties 2-49-51 problemsinvolving 93--4
types 217 union 91
fraction and decimal equivalents quantity universal 90
43-4 dividinginagivenratio 54--6 shapes.similar 231--4
expressing as a percentage of significantfigures(s.f.) 13
reverse -48 similarshapes,areaandvolume
percentiles -49-4 anotherquantity -45--6
perimeter,ofarectangle 281 quanwmmechanics +II 235-7
perpendicularbisector 223.2-42. quartiles -49-4--6 simple interest 79-80
simplification,usingindices 62-3,
265 radius,ofacircle 216
perpendicularlines 21-4,333-6 range -466,-49-4 123,125--6
pictograms -472 ratio,enlargement/reduction 57-9 simultaneousequations 131--6
piecharts -473-7 ratio method
piecework 76 graphicalsolution 329-30
placevalue 33 fordirectproportion 52,53 sine 351-2
planeofsymmetry 2-40 fordividingaquantityinagiven sinecurve 359--60
plans see scale drawings sinerule 366-7
polygons ratio 5-4 solids see three-dimensional shapes
rationalnumbers -4,7--8 speed,distanceandtime 175--6
angleproperties 251-3.256 realnumbers -4--5 speed--timegraphs 178--83
regular 218,251 reciprocalfunction.graph 192-3 ,phoco
similar 219,231--4 reciprocalofanumber 36-7
rectangle.properties 217,281 rotationalsymmetry 2-41
types 218 rectangular prism see cuboid surfacearea 297-8
positionvectors 387 reflection 408--10 volume 296-7
positiveindices 62,66.123 reflectivesymmetry 2-40 spread.measures 466
positiveornegativenumbersee rhombus.properties 217 square.properties 217
rotation -410-13 squarematrix 39-4
directed numbers rotationalsymmetry 240 squarenumbers 5.100
primefactors 6-7 rounding 12
primenumbers 5,100 rowmatriK 39-4
principal 79
prism 290-2
probability

definition -4-42
ofcombinedevents 452

Index

square roots 7,8-9 three-dimensional shapes trigonometricratios 349
square-basedpyramid 298 nets 219 trigonometry,inthreedimensions
symmetry 2-40-1
rotationalsymmetry 241 371-3
standardform 65-9 three-figurebearingsystem 346 two-waytable 452
statistics,historicaldevelopment time 85--6
u
465 seeolsospeed,distanceandtime
straight line timeandahalf 76 unionofsets 91
timetables 85 unitary method
equation 320-5 transformation
shortest distance between two directproportion 52,53
andinversematrices 426--7 fordividingaquantityinagiven
points 214 combinations 422.427-30
straight-linegraphs 316 definition 408 ratio 55
of formulae 107, 112- 14 universalset 90
drawing 327--8 transformationmatrix(matrices)
subsets 88-9 vectorgeometry 387-9
substitution 106--7 423-5 vectors
combination 427-30
in solving simultaneous equations translation 414--15 additionandsubtraction 384
131,132 ofacolumnvector 386 magnitude 386
trapezium 217,284--5 multiplyingbyascalar 384--5
subtraction travelgraphs 176--8 seeolsocolumnvectors;position
ofalgebraicfractions 115 treediagrams 453-7
offractions 35--6 triangles vectors
ofmatrices 397-9 acute-angled 215 Venndiagrams 90-3
ofvectors 384 angleproperties 128,248-9 vertex.ofatriangle 282
area 281-3,370-1 Villani.Giovanni 465
surface area base 282 volume
ofacone 305--6 circumcircle 226
ofacuboid 288--9 congruent 215 convertingfromoneunitto
ofacylinder 288--9 constructing 221-2,272-3 another 278--80
ofapyramid 300-1 equilateral 215.251
ofasphere 297--8 height(altitude) 214,282 ofacone 301-5
isosceles 215 ofaprism 290-2
surveys 4n-9 obtuse-angled 215 ofapyramid 298-300
Swissmathematicians 345 properties 218 ofasphere 296--7
symmetry 240-1 right-angled 215,349 ofsimilarshapes 235-7
scalene 215
tallycharts 4n similar 216,231--4 w
tangent types 215--16
vertex 282 Wiles.Andrew 315
toacircle 243 triangular prism
toacurve,gradient 196 rotationalsymmetry 241 zeroindex 64,123--4
trigonometricratio 349-51 volume 290 zeromatrix 403
temperaturescale 10
termsofasequence 155
ThalesofAlexandria 213
Theano (Greek mathematician)

213