Cambridge IGCSE Mathematics Extended Practice Book

7 Perimeter, area and volume

7.2 Three-dimensional objects

t "OZ TPMJE PCKFDU JT UISFF EJNFOTJPOBM ćF UISFF EJNFOTJPOT PG B TPMJE BSF MFOHUI
CSFBEUI BOE IFJHIU
t ćF OFU PG B TPMJE JT B UXP EJNFOTJPOBM EJBHSBN *U TIPXT UIF TIBQF PG BMM GBDFT PG UIF TPMJE BOE IPX UIFZ BSF BUUBDIFE

UP FBDI PUIFS *G ZPV GPME VQ B OFU
ZPV HFU B NPEFM PG UIF TPMJE

Exercise 7.2

1 8IJDI TPMJET XPVME CF NBEF GSPN UIF GPMMPXJOH GBDFT
(a) (b)

×2 ×2 ×2
(d)
×6
(c)

×4 ×1 ×8

2 %FTDSJCF UIF TPMJE ZPV DPVME QSPEVDF VTJOH FBDI PG UIF GPMMPXJOH OFUT
(a) (b) (c)

3 4LFUDI B QPTTJCMF OFU GPS FBDI PG UIF GPMMPXJOH TPMJET
(a) (b)

(c) (d)

Unit 2: Shape, space and measures 43

7 Perimeter, area and volume

7.3 Surface areas and volumes of solids

t ćF TVSGBDF BSFB PG B UISFF EJNFOTJPOBM PCKFDU JT UIF UPUBM BSFB PG BMM JUT GBDFT
t ćF WPMVNF PG B UISFF EJNFOTJPOBM PCKFDU JT UIF BNPVOU PG TQBDF JU PDDVQJFT
t :PV DBO ĕOE UIF WPMVNF PG B DVCF PS DVCPJE VTJOH UIF GPSNVMB
V = l × b × h
XIFSF l JT UIF MFOHUI
b JT UIF CSFBEUI BOE

h JT UIF IFJHIU PG UIF PCKFDU

t " QSJTN JT B UISFF EJNFOTJPOBM PCKFDU XJUI B VOJGPSN DSPTT TFDUJPO UIF FOE GBDFT PG UIF TPMJE BSF JEFOUJDBM BOE
QBSBMMFM
*G ZPV TMJDF UISPVHI UIF QSJTN BOZXIFSF BMPOH JUT MFOHUI BOE QBSBMMFM UP UIF FOE GBDFT

ZPV XJMM HFU B

TFDUJPO UIF TBNF TIBQF BOE TJ[F BT UIF FOE GBDFT $VCFT
DVCPJET BOE DZMJOEFST BSF FYBNQMFT PG QSJTNT

t :PV DBO ĕOE UIF WPMVNF PG BOZ QSJTN JODMVEJOH B DZMJOEFS
CZ NVMUJQMZJOH UIF BSFB PG JUT DSPTT TFDUJPO CZ UIF
EJTUBODF CFUXFFO UIF QBSBMMFM GBDFT ćJT JT FYQSFTTFE JO UIF GPSNVMB
V = al
XIFSF a JT UIF BSFB PG UIF CBTF BOE l JT UIF

MFOHUI PG UIF QSJTN :PV OFFE UP VTF UIF BQQSPQSJBUF BSFB GPSNVMB GPS UIF TIBQF PG UIF DSPTT TFDUJPO

t 'JOE UIF WPMVNF PG B DPOF VTJOH UIF GPSNVMB
V = 1 πr2h
XIFSF h JT UIF QFSQFOEJDVMBS IFJHIU 5P ĕOE UIF DVSWFE
3
TVSGBDF BSFB VTF UIF GPSNVMB
TVSGBDF BSFB πrl
XIFSF l JT UIF TMBOU IFJHIU PG UIF DPOF

t area of base × h
'JOE UIF WPMVNF PG B QZSBNJE VTJOH UIF GPSNVMB
V = 3
XIFSF h JT UIF QFSQFOEJDVMBS IFJHIU

t 'JOE UIF WPMVNF PG B TQIFSF VTJOH UIF GPSNVMB
V = 4 πr3 5P ĕOE UIF TVSGBDF BSFB VTF UIF GPSNVMB
TVSGBDF 3
BSFB πr2

Exercise 7.3 A

6TF π = GPS BOZ TIBQFT JOWPMWJOH DJSDMFT JO UIJT FYFSDJTF


Tip 1 $BMDVMBUF UIF TVSGBDF BSFB PG FBDI TIBQF (b)
(a)
%SBXJOH UIF OFUT PG UIF 0.5 mm
TPMJET NBZ IFMQ ZPV XPSL
PVU UIF TVSGBDF BSFB PG 18.4 m
FBDI TIBQF
0.4 mm 12 m
Remember,
1 m2 = 10 000 cm2 1.2 mm

8m

14 m

(c) (d) 4 m

12 mm

1.5 cm

2 " XPPEFO DVCF IBT TJY JEFOUJDBM TRVBSF GBDFT
FBDI PG BSFB DN2

(a) 8IBU JT UIF TVSGBDF BSFB PG UIF DVCF
(b) 8IBU JT UIF IFJHIU PG UIF DVCF

3 .ST /JOJ JT PSEFSJOH XPPEFO CMPDLT UP VTF JO IFS NBUIT DMBTTSPPN
ćF CMPDLT BSF DVCPJET XJUI EJNFOTJPOT DN × DN × DN

(a) $BMDVMBUF UIF TVSGBDF BSFB PG POF CMPDL
(b) .ST /JOJ OFFET CMPDLT 8IBU JT UIF UPUBM TVSGBDF BSFB PG BMM UIF CMPDLT
(c) 4 IF EFDJEFT UP WBSOJTI UIF CMPDLT " UJO PG WBSOJTI DPWFST BO BSFB PG N2

)PX NBOZ UJOT XJMM TIF OFFE UP WBSOJTI BMM UIF CMPDLT

44 Unit 2: Shape, space and measures

7 Perimeter, area and volume

Tip 4 $BMDVMBUF UIF WPMVNF PG FBDI QSJTN

ćF MFOHUI PG UIF QSJTN JT (a) (b) 4 cm 2 cm (c) (d) 40 cm
UIF EJTUBODF CFUXFFO UIF 80 mm
UXP QBSBMMFM GBDFT 8IFO 45 mm 3 cm 65 mm A = 28 cm2
B QSJTN JT UVSOFE POUP JUT 50 mm 8 cm
GBDF
UIF MFOHUI NBZ MPPL
MJLF B IFJHIU 8PSL PVU UIF (e) (f ) 20 mm (h)
BSFB PG UIF DSPTT TFDUJPO 10 cm 8 cm (g)
FOE GBDF
CFGPSF ZPV
BQQMZ UIF WPMVNF GPSNVMB

4m 1.25 m

12 cm 25 cm 1.25 m 1.25 m

1.2 m1.2 m 12 cm

5 " QPDLFU EJDUJPOBSZ JT DN MPOH
DN XJEF BOE DN UIJDL
$BMDVMBUF UIF WPMVNF PG TQBDF JU UBLFT VQ

6 (a) 'JOE UIF WPMVNF PG B MFDUVSF SPPN UIBU JT N MPOH
N XJEF BOE N IJHI

(b) 4BGFUZ SFHVMBUJPOT TUBUF UIBU EVSJOH BO IPVS MPOH MFDUVSF FBDI QFSTPO JO UIF SPPN NVTU
IBWF N3 PG BJS $BMDVMBUF UIF NBYJNVN OVNCFS PG QFPQMF XIP DBO BUUFOE BO IPVS
MPOH MFDUVSF

7 " DZMJOESJDBM UBOL JT N IJHI XJUI BO JOOFS SBEJVT PG DN $BMDVMBUF IPX NVDI XBUFS UIF
UBOL XJMM IPME XIFO GVMM

8 " NBDIJOF TIPQ IBT GPVS EJČFSFOU SFDUBOHVMBS QSJTNT PG WPMVNF NN3 $PQZ BOE ĕMM JO
UIF QPTTJCMF EJNFOTJPOT GPS FBDI QSJTN UP DPNQMFUF UIF UBCMF

Volume (mm3) 64 000 64 000 64 000 64 000
Length (mm)
Breadth (mm)
Height (mm)

Exercise 7.3 B

6TF ɀ GPS BOZ TIBQFT JOWPMWJOH DJSDMFT JO UIJT FYFSDJTF


1 'JOE UIF WPMVNF PG UIF GPMMPXJOH TPMJET (JWF ZPVS BOTXFST DPSSFDU UP UXP EFDJNBM QMBDFT

(a) 1.2 cm (b) (c) (d) (e)
20 m 2.7 cm 36 mm 40 mm
3.5 cm 3.5 cm 3 cm
8 cm 12 cm

Unit 2: Shape, space and measures 45

7 Perimeter, area and volume

2 (JWF ZPVS BOTXFST UP UIJT RVFTUJPO JO TUBOEBSE GPSN UP UISFF TJHOJĕDBOU ĕHVSFT
ćF &BSUI IBT BO BWFSBHF SBEJVT PG LN

(a) "TTVNJOH UIF &BSUI JT NPSF PS MFTT TQIFSJDBM
DBMDVMBUF
(i) JUT WPMVNF
(ii) JUT TVSGBDF BSFB

(b) * G PG UIF TVSGBDF BSFB PG UIF &BSUI JT DPWFSFE CZ UIF PDFBOT
DBMDVMBUF UIF BSFB PG MBOE
PO UIF TVSGBDF

Mixed exercise 6TF ɀ GPS BOZ TIBQFT JOWPMWJOH DJSDMFT JO UIJT FYFSDJTF


1 " DJSDVMBS QMBUF PO B TUPWF IBT B EJBNFUFS PG DN ćFSF JT B NFUBM TUSJQ BSPVOE UIF PVUTJEF
PG UIF QMBUF

(a) $BMDVMBUF UIF TVSGBDF BSFB PG UIF UPQ PG UIF QMBUF
(b) $BMDVMBUF UIF MFOHUI PG UIF NFUBM TUSJQ
2 8IBU JT UIF SBEJVT PG B DJSDMF XJUI BO BSFB PG DN2

3 $BMDVMBUF UIF TIBEFE BSFB JO FBDI ĕHVSF

(a) 50 mm (b) 120 mm (c)

170 mm 150 mm 192 mm 6 cm
2 cm
40 mm 5 cm

320 mm

(d) 5 cm (e) 5 cm (f) 1 cm 7 cm
8 cm
4 cm 5 cm 2 cm

6 cm 6 cm 3 cm 6 cm
12 cm 3 cm 5 cm

(g)
12 cm

30°
10°

46 Unit 2: Shape, space and measures

7 Perimeter, area and volume

4 MNOP JT B USBQF[JVN XJUI BO BSFB PG DN2 $BMDVMBUF UIF MFOHUI PG NO
M 12 m N

10 m
P

O
5 4UVEZ UIF UXP QSJTNT

20 mm
A 40 mm

120 mm
B

20 mm
15 mm

(a) 8IJDI PG UIF UXP QSJTNT IBT UIF TNBMMFS WPMVNF
(b) 8IBU JT UIF EJČFSFODF JO WPMVNF
(c) 4 LFUDI B OFU PG UIF DVCPJE :PVS OFU EPFT OPU OFFE UP CF UP TDBMF
CVU ZPV NVTU JOEJDBUF

UIF EJNFOTJPOT PG FBDI GBDF PO UIF OFU
(d) $BMDVMBUF UIF TVSGBDF BSFB PG FBDI QSJTN

6 )PX NBOZ DVCFT PG TJEF DN DBO CF QBDLFE JOUP B XPPEFO CPY NFBTVSJOH DN CZ
DN CZ DN

7 'JOE UIF EJČFSFODF CFUXFFO UIF WPMVNF PG B DN IJHI DPOF XIJDI IBT B DN XJEF CBTF
BOE B TRVBSF CBTFE QZSBNJE UIBU JT DN XJEF BU JUT CBTF BOE DN IJHI

8 5FOOJT CBMMT PG EJBNFUFS DN BSF QBDLFE JOUP DZMJOESJDBM NFUBM UVCFT UIBU BSF TFBMFE BU CPUI
FOET ćF JOTJEF PG UIF UVCF IBT BO JOUFSOBM EJBNFUFS PG DN BOE UIF UVCF JT DN MPOH
$BMDVMBUF UIF WPMVNF PG TQBDF MFę JO UIF UVCF JG UISFF UFOOJT CBMMT BSF QBDLFE JOUP JU

Unit 2: Shape, space and measures 47

8 Introduction to probability

8.1 Basic probability

t 1SPCBCJMJUZ JT B NFBTVSF PG UIF DIBODF UIBU TPNFUIJOH XJMM IBQQFO *U JT NFBTVSFE PO B TDBMF PG UP

o PVUDPNFT XJUI B QSPCBCJMJUZ PG BSF JNQPTTJCMF

o PVUDPNFT XJUI B QSPCBCJMJUZ PG BSF DFSUBJO
o BO PVUDPNF XJUI B QSPCBCJMJUZ PG PS 1 IBT BO FWFO DIBODF PG PDDVSSJOH

t 2
1SPCBCJMJUJFT DBO CF GPVOE UISPVHI EPJOH BO FYQFSJNFOU
TVDI BT UPTTJOH B DPJO &BDI UJNF ZPV QFSGPSN UIF
FYQFSJNFOU JT DBMMFE B USJBM *G ZPV XBOU UP HFU IFBET
UIFO IFBET JT ZPVS EFTJSFE PVUDPNF PS TVDDFTTGVM PVUDPNF

t 5P DBMDVMBUF QSPCBCJMJUZ GSPN UIF PVUDPNFT PG FYQFSJNFOUT
VTF UIF GPSNVMB

t &YQFSJNFOUBM QSPCBCJMJUZ JT BMTP DBMMFE UIF SFMBUJWF GSFRVFODZ

Exercise 8.1

1 4BMNB IBT B CBH DPOUBJOJOH POF SFE
POF XIJUF BOE POF HSFFO CBMM 4IF ESBXT B CBMM BU SBOEPN
BOE SFQMBDFT JU CFGPSF ESBXJOH BHBJO 4IF SFQFBUT UIJT UJNFT 4IF VTFT B UBMMZ UBCMF UP SFDPSE
UIF PVUDPNFT PG IFS FYQFSJNFOU

Red //// //// ////

White //// //// //// ///

Green //// //// //// //

(a) $BMDVMBUF UIF SFMBUJWF GSFRVFODZ PG ESBXJOH FBDI DPMPVS
(b) &YQSFTT IFS DIBODF PG ESBXJOH B SFE CBMM BT B QFSDFOUBHF
(c) 8IBU JT UIF TVN PG UIF UISFF SFMBUJWF GSFRVFODJFT
(d) 8IBU TIPVME ZPVS DIBODFT CF JO UIFPSZ PG ESBXJOH FBDI DPMPVS

2 *U JT +PTI T KPC UP DBMM DVTUPNFST XIP IBWF IBE UIFJS DBS TFSWJDFE BU UIF EFBMFS UP DIFDL
XIFUIFS UIFZ BSF IBQQZ XJUI UIF TFSWJDF UIFZ SFDFJWFE )F LFQU UIJT SFDPSE PG XIBU IBQQFOFE
GPS DBMMT NBEF POF NPOUI

Result Frequency
4QPLF UP DVTUPNFS
1IPOF OPU BOTXFSFE 44
-Fę NFTTBHF PO BOTXFSJOH NBDIJOF
1IPOF FOHBHFE PS PVU PG PSEFS
8SPOH OVNCFS

48 Unit 2: Data handling

8 Introduction to probability

(a) $BMDVMBUF UIF SFMBUJWF GSFRVFODZ PG FBDI FWFOU BT B EFDJNBM GSBDUJPO
(b) * T JU IJHIMZ MJLFMZ
MJLFMZ
VOMJLFMZ PS IJHIMZ VOMJLFMZ UIBU UIF GPMMPXJOH PVUDPNFT XJMM

PDDVS XIFO +PTI NBLFT B DBMM

(i) ćF DBMM XJMM CF BOTXFSFE CZ UIF DVTUPNFS
(ii) ćF DBMM XJMM CF BOTXFSFE CZ B NBDIJOF
(iii) )F XJMM EJBM UIF XSPOH OVNCFS

8.2 Theoretical probability

t :PV DBO DBMDVMBUF UIF UIFPSFUJDBM QSPCBCJMJUZ PG BO FWFOU XJUIPVU EPJOH FYQFSJNFOUT JG UIF PVUDPNFT BSF FRVBMMZ

MJLFMZ 6TF UIF GPSNVMB

P(outcome) = number of favourable outcomes
number of possible outcomes

'PS FYBNQMF
XIFO ZPV UPTT B DPJO ZPV DBO HFU IFBET PS UBJMT UXP QPTTJCMF PVUDPNFT
ćF QSPCBCJMJUZ PG
IFBET JT P(H) = 1

t 2
:PV OFFE UP XPSL PVU XIBU all UIF QPTTJCMF PVUDPNFT BSF CFGPSF ZPV DBO DBMDVMBUF UIFPSFUJDBM QSPCBCJMJUZ

Tip Exercise 8.2

*U JT IFMQGVM UP MJTU UIF 1 4BMMZ IBT UFO JEFOUJDBM DBSET OVNCFSFE POF UP UFO 4IF ESBXT B DBSE BU SBOEPN BOE SFDPSET
QPTTJCMF PVUDPNFT TP UIF OVNCFS PO JU
UIBU ZPV LOPX XIBU UP
TVCTUJUVUF JO UIF GPSNVMB (a) 8IBU BSF UIF QPTTJCMF PVUDPNFT GPS UIJT FWFOU

(b) $BMDVMBUF UIF QSPCBCJMJUZ UIBU 4BMMZ XJMM ESBX

(i) UIF OVNCFS ĕWF (ii) BOZ POF PG UIF UFO OVNCFST

(iii) B NVMUJQMF PG UISFF (iv) B OVNCFS <

(v) B OVNCFS < (vi) B OVNCFS <

(vii) B TRVBSF OVNCFS (viii) B OVNCFS <

(ix) B OVNCFS >

2 ćFSF BSF ĕWF DVQT PG DPČFF PO B USBZ 5XP PG UIFN DPOUBJO TVHBS

(a) 8IBU BSF ZPVS DIBODFT PG DIPPTJOH B DVQ XJUI TVHBS JO JU
(b) 8IJDI DIPJDF JT NPTU MJLFMZ 8IZ

3 .JLF IBT GPVS DBSET OVNCFSFE POF UP GPVS )F ESBXT POF DBSE BOE SFDPSET UIF OVNCFS
$BMDVMBUF UIF QSPCBCJMJUZ UIBU UIF SFTVMU XJMM CF

(a) B NVMUJQMF PG UISFF (b) B NVMUJQMF PG UXP (c) B GBDUPS PG UISFF

4 'PS B ĘZ ĕTIJOH DPNQFUJUJPO
UIF PSHBOJTFST QMBDF USPVU
TBMNPO BOE QJLF JO B
TNBMM EBN

(a) 8IBU JT BO BOHMFS T DIBODF PG DBUDIJOH B TBMNPO PO IFS ĕSTU BUUFNQU
(b) W IBU JT UIF QSPCBCJMJUZ UIF BOHMFS DBUDIFT B USPVU
(c) 8IBU JT UIF QSPCBCJMJUZ PG DBUDIJOH B QJLF

Unit 2: Data handling 49

8 Introduction to probability

5 " EBSUCPBSE JT EJWJEFE JOUP TFDUPST OVNCFSFE GSPN POF UP *G B EBSU JT FRVBMMZ MJLFMZ UP
MBOE JO BOZ PG UIFTF TFDUPST
DBMDVMBUF

(a) 1 <
(b) 1 PEE
(c) 1 QSJNF

(d) 1 NVMUJQMF PG
(e) 1 NVMUJQMF PG


6 " TDIPPM IBT GPSUZ DMBTTSPPNT OVNCFSFE GSPN POF UP 8PSL PVU UIF QSPCBCJMJUZ UIBU B
DMBTTSPPN OVNCFS IBT UIF OVNFSBM A JO JU

8.3 The probability that an event does not happen

t "O FWFOU NBZ IBQQFO PS JU NBZ OPU IBQQFO 'PS FYBNQMF
ZPV NBZ UISPX B TJY XIFO ZPV SPMM B EJF
CVU ZPV NBZ OPU
t ćF QSPCBCJMJUZ PG BO FWFOU IBQQFOJOH NBZ CF EJČFSFOU GSPN UIF QSPCBCJMJUZ PG UIF FWFOU OPU IBQQFOJOH
CVU UIF UXP

DPNCJOFE QSPCBCJMJUJFT XJMM BMXBZT BEE VQ UP POF

t *G " JT BO FWFOU IBQQFOJOH
UIFO "′ PS A
SFQSFTFOUT UIF FWFOU " OPU IBQQFOJOH BOE 1 "′
= o 1 "


Exercise 8.3

1 ćF QSPCBCJMJUZ UIBU B ESJWFS JT TQFFEJOH PO B TUSFUDI PG SPBE JT 8IBU JT UIF QSPCBCJMJUZ
UIBU B ESJWFS JT OPU TQFFEJOH

2 ćF QSPCBCJMJUZ PG ESBXJOH B HSFFO CBMM JO BO FYQFSJNFOU JT 3 8IBU JT UIF QSPCBCJMJUZ PG OPU
8
ESBXJOH B HSFFO CBMM

3 " DPOUBJOFS IPMET TXFFUT JO ĕWF EJČFSFOU ĘBWPVST ćF QSPCBCJMJUZ PG DIPPTJOH B
QBSUJDVMBS ĘBWPVS JT HJWFO JO UIF UBCMF

Flavour Strawberry Lime Lemon Blackberry Apple
P(flavour)

(a) $BMDVMBUF 1 BQQMF

(b) 8IBU JT 1 OPU BQQMF

(c) $BMDVMBUF UIF QSPCBCJMJUZ PG DIPPTJOH 1 OFJUIFS MFNPO OPS MJNF

(d) $BMDVMBUF UIF OVNCFS PG FBDI ĘBWPVSFE TXFFU JO UIF DPOUBJOFS

4 4UVEFOUT JO B TDIPPM IBWF ĕWF BęFS TDIPPM DMVCT UP DIPPTF GSPN ćF QSPCBCJMJUZ UIBU B
TUVEFOU XJMM DIPPTF FBDI DMVC JT HJWFO JO UIF UBCMF

Club Computers Sewing Woodwork Choir Chess
P(Club)

(a) $BMDVMBUF 1 OPU TFXJOH OPS XPPEXPSL

(b) $BMDVMBUF 1 OPU DIFTT OPS DIPJS

(c) * G TUVEFOUT IBWF UP DIPPTF B DMVC
IPX NBOZ XPVME ZPV FYQFDU UP DIPPTF TFXJOH
(d) * G GPVS TUVEFOUT DIPTF DIPJS
DBMDVMBUF IPX NBOZ TUVEFOUT DIPTF DPNQVUFST

50 Unit 2: Data handling

8 Introduction to probability

8.4 Possibility diagrams

t ćF TFU PG BMM QPTTJCMF PVUDPNFT JT DBMMFE UIF TBNQMF TQBDF PS QSPCBCJMJUZ TQBDF
PG BO FWFOU
t 1PTTJCJMJUZ EJBHSBNT DBO CF VTFE UP TIPX BMM PVUDPNFT DMFBSMZ
t 8IFO ZPV BSF EFBMJOH XJUI DPNCJOFE FWFOUT
JU JT NVDI FBTJFS UP ĕOE B QSPCBCJMJUZ JG ZPV SFQSFTFOU UIF TBNQMF TQBDF

JO B EJBHSBN QPTTJCJMJUZ EJBHSBNT BSF VTFGVM GPS EPJOH UIJT

Tip Exercise 8.4

ćJOL PG UIF QSPCBCJMJUZ 1 %SBX B QPTTJCJMJUZ EJBHSBN UP TIPX BMM QPTTJCMF PVUDPNFT XIFO ZPV UPTT UXP DPJOT BU UIF
TQBDF EJBHSBN BT B NBQ PG TBNF UJNF 6TF ZPVS EJBHSBN UP IFMQ ZPV BOTXFS UIF GPMMPXJOH
BMM UIF QPTTJCMF PVUDPNFT
JO BO FYQFSJNFOU (a) 8IBU JT 1 BU MFBTU POF UBJM

(b) 8IBU JT 1 OP UBJMT

FAST FORWARD
Tree diagrams are also probability 2 +FTT IBT UISFF HSFFO DBSET OVNCFSFE POF UP UISFF BOE UISFF ZFMMPX DBSET BMTP OVNCFSFE POF
space diagrams. These are dealt UP UISFF
with in detail in chapter 24.
(a) % SBX B QPTTJCJMJUZ EJBHSBN UP TIPX BMM QPTTJCMF PVUDPNFT XIFO POF HSFFO BOE POF
ZFMMPX DBSE JT DIPTFO BU SBOEPN

(b) )PX NBOZ QPTTJCMF PVUDPNFT BSF UIFSF
(c) 8IBU JT UIF QSPCBCJMJUZ UIBU UIF OVNCFS PO CPUI UIF DBSET XJMM CF UIF TBNF
(d) 8IBU JT UIF QSPCBCJMJUZ PG HFUUJOH B UPUBM JG UIF TDPSFT PO UIF DBSET BSF BEEFE

3 0O B TDIPPM PVUJOH
UIF TUVEFOUT BSF BMMPXFE UP DIPPTF POF ESJOL BOE POF TOBDL GSPN
UIJT NFOV

Drinks: DPMB
GSVJU KVJDF
XBUFS
Snacks: CJTDVJU
DBLF
NVďO

(a) %SBX B QPTTJCJMJUZ EJBHSBN UP TIPX UIF QPTTJCMF DIPJDFT UIBU B TUVEFOU DBO NBLF
(b) 8 IBU JT UIF QSPCBCJMJUZ B TUVEFOU XJMM DIPPTF DPMB BOE B CJTDVJU
(c) 8 IBU JT UIF QSPCBCJMJUZ UIBU UIF ESJOL DIPTFO JT OPU XBUFS

Unit 2: Data handling 51

8 Introduction to probability

8.5 Combining independent and mutually exclusive events

t 8IFO POF PVUDPNF JO B USJBM IBT OP FČFDU PO UIF OFYU PVUDPNF XF TBZ UIF FWFOUT BSF JOEFQFOEFOU

– %SBXJOH B DPVOUFS BU SBOEPN GSPN B CBH
SFQMBDJOH JU BOE UIFO ESBXJOH BOPUIFS DPVOUFS JT BO FYBNQMF PG
JOEFQFOEFOU FWFOUT #FDBVTF ZPV SFQMBDF UIF DPVOUFS
UIF ĕSTU ESBX EPFT OPU BČFDU UIF TFDPOE ESBX

t *G " BOE # are JOEFQFOEFOU FWFOUT UIFO 1 " IBQQFOT BOE UIFO # IBQQFOT
1 "
¨ 1 #
PS
1 " BOE #
1 "
¨ 1 #


t .VUVBMMZ FYDMVTJWF FWFOUT DBOOPU IBQQFO BU UIF TBNF UJNF

– 'PS FYBNQMF
ZPV DBOOPU UISPX BO PEE OVNCFS BOE BO FWFO OVNCFS BU UIF TBNF UJNF XIFO ZPV SPMM B EJF

t *G " BOE # are NVUVBMMZ FYDMVTJWF FWFOUT UIFO 1 " PS #
1 "
1 #

t 8IFO UIF PVUDPNF PG UIF ĕSTU FWFOU BČFDUT UIF PVUDPNF PG UIF OFYU FWFOU
UIF FWFOUT BSF TBJE UP CF EFQFOEFOU

– 'PS FYBNQMF
JG ZPV IBWF UXP SFE DPVOUFST BOE UISFF XIJUF DPVOUFST
ESBX B DPVOUFS XJUIPVU SFQMBDJOH JU BOE UIFO
ESBX B TFDPOE DPVOUFS
UIF QSPCBCJMJUZ PG ESBXJOH B SFE PS B XIJUF PO UIF TFDPOE ESBX EFQFOET PO XIBU ZPV ESFX

ĕSTU UJNF SPVOE :PV DBO ĕOE 1 " UIFO #
CZ DBMDVMBUJOH 1 "
¨ 1 # HJWFO UIBU " IBT BMSFBEZ IBQQFOFE


Exercise 8.5

1 /JDP JT PO B CVT BOE IF JT CPSFE
TP IF BNVTFT IJNTFMG CZ DIPPTJOH B DPOTPOBOU BOE B WPXFM
BU SBOEPN GSPN UIF OBNFT PG UPXOT PO SPBE TJHOT ćF OFYU SPBE TJHO JT $"-$655"

(a) % SBX VQ B TBNQMF TQBDF EJBHSBN UP TIPX BMM UIF PQUJPOT UIBU /JDP IBT
(b) $BMDVMBUF 1 5 BOE "

(c) $BMDVMBUF 1 $ PS - BOE 6

(d) $BMDVMBUF 1 OPU - BOE 6


2 " CBH DPOUBJOT UISFF SFE DPVOUFST
GPVS HSFFO DPVOUFST
UXP ZFMMPX DPVOUFST BOE POF XIJUF
DPVOUFS 5XP DPVOUFST BSF ESBXO GSPN UIF CBH POF BęFS UIF PUIFS
XJUIPVU CFJOH SFQMBDFE
$BMDVMBUF

(a) 1 UXP SFE DPVOUFST

(b) 1 UXP HSFFO DPVOUFST

(c) 1 UXP ZFMMPX DPVOUFST

(d) 1 XIJUF BOE UIFO SFE

(e) 1 XIJUF PS ZFMMPX
JO FJUIFS PSEFS CVU OPU CPUI

(f) 1 XIJUF PS SFE
JO FJUIFS PSEFS CVU OPU CPUI

(g) 8IBU JT UIF QSPCBCJMJUZ PG ESBXJOH B XIJUF PS ZFMMPX DPVOUFS ĕSTU BOE UIFO BOZ

DPMPVS TFDPOE

3 .BSJB IBT B CBH DPOUBJOJOH GSVJU ESPQ TXFFUT BSF BQQMF ĘBWPVSFE BOE BSF CMBDLCFSSZ
ĘBWPVSFE 4IF DIPPTFT B TXFFU BU SBOEPN BOE FBUT JU ćFO TIF DIPPTFT BOPUIFS TXFFU BU
SBOEPN $BMDVMBUF UIF QSPCBCJMJUZ UIBU

(a) CPUI TXFFUT XFSF BQQMF ĘBWPVSFE
(b) CPUI TXFFUT XFSF CMBDLCFSSZ ĘBWPVSFE
(c) UIF ĕSTU XBT BQQMF BOE UIF TFDPOE XBT CMBDLCFSSZ
(d) U IF ĕSTU XBT CMBDLCFSSZ BOE UIF TFDPOE XBT BQQMF
(e) : PVS BOTXFST UP B

C

D
BOE E
TIPVME BEE VQ UP POF &YQMBJO XIZ UIJT JT UIF DBTF

52 Unit 2: Data handling

8 Introduction to probability

Mixed exercise 1 " DPJO JT UPTTFE B OVNCFS PG UJNFT HJWJOH UIF GPMMPXJOH SFTVMUT

)FBET 5BJMT
(a) )PX NBOZ UJNFT XBT UIF DPJO UPTTFE
(b) $BMDVMBUF UIF SFMBUJWF GSFRVFODZ PG FBDI PVUDPNF
(c) 8IBU JT UIF QSPCBCJMJUZ UIBU UIF OFYU UPTT XJMM SFTVMU JO IFBET
(d) + FTT TBZT TIF UIJOLT UIF SFTVMUT TIPX UIBU UIF DPJO JT CJBTFE %P ZPV BHSFF (JWF B SFBTPO

GPS ZPVS BOTXFS
2 " CBH DPOUBJOT SFE
FJHIU HSFFO BOE UXP XIJUF CBMMT &BDI CBMM IBT BO FRVBM DIBODF PG

CFJOH DIPTFO $BMDVMBUF UIF QSPCBCJMJUZ PG
(a) DIPPTJOH B SFE CBMM
(b) DIPPTJOH B HSFFO CBMM
(c) DIPPTJOH B XIJUF CBMM
(d) DIPPTJOH B CMVF CBMM
(e) DIPPTJOH B SFE PS B HSFFO CBMM
(f) OPU DIPPTJOH B XIJUF CBMM
(g) DIPPTJOH B CBMM UIBU JT OPU SFE
3 5XP OPSNBM VOCJBTFE EJDF BSF SPMMFE BOE UIF TVN PG UIF OVNCFST PO UIFJS GBDFT JT SFDPSEFE
(a) $BMDVMBUF 1

(b) 8IJDI TVN IBT UIF HSFBUFTU QSPCBCJMJUZ 8IBU JT UIF QSPCBCJMJUZ PG SPMMJOH UIJT TVN
(c) 8IBU JT 1 OPU FWFO

(d) 8IBU JT 1 TVN

4 +PTI BOE $BSMPT FBDI UBLF B DPJO BU SBOEPN PVU PG UIFJS QPDLFUT BOE BEE UIF UPUBMT UPHFUIFS
UP HFU BO BNPVOU +PTI IBT UISFF DPJOT
UXP D DPJOT
B DPJO BOE UXP D DPJOT JO IJT
QPDLFU $BSMPT IBT GPVS DPJOT
B DPJO BOE UXP D QJFDFT
(a) %SBX VQ B QSPCBCJMJUZ TQBDF EJBHSBN UP TIPX BMM UIF QPTTJCMF PVUDPNFT GPS UIF TVN PG

UIF UXP DPJOT
(b) 8IBU JT UIF QSPCBCJMJUZ UIBU UIF DPJOT XJMM BEE VQ UP
(c) 8IBU JT UIF QSPCBCJMJUZ UIBU UIF DPJOT XJMM BEE VQ UP MFTT UIBO
(d) 8IBU JT UIF QSPCBCJMJUZ UIBU UIF DPJOT XJMM BEE VQ UP PS NPSF
5 " GBJS EJF JT SPMMFE UISFF UJNFT BOE UIF OVNCFS SFWFBMFE XSJUUFO EPXO 8IBU JT UIF
QSPCBCJMJUZ UIBU B QSJNF OVNCFS XJMM CF XSJUUFO EPXO UISFF UJNFT
6 1BUSJDL LFFQT IJT TPDLT MPPTF JO B ESBXFS )F IBT TJY EBSL POFT
GPVS XIJUF POFT BOE UXP
TUSJQFE POFT 8IBU JT UIF QSPCBCJMJUZ UIBU IF UBLFT PVU UXP TPDLT BOE UIFZ BSF B QBJS

Unit 2: Data handling 53

9 Sequences and sets

9.1 Sequences

t " OVNCFS TFRVFODF JT B MJTU PG OVNCFST UIBU GPMMPXT B TFU QBUUFSO &BDI OVNCFS JO UIF TFRVFODF JT DBMMFE B
UFSN 51 JT UIF ĕSTU UFSN
510 JT UIF UFOUI UFSN BOE 5n JT UIF nUI UFSN
PS HFOFSBM UFSN

t " MJOFBS TFRVFODF IBT B DPOTUBOU EJČFSFODF d
CFUXFFO UIF UFSNT ćF HFOFSBM SVMF GPS ĕOEJOH UIF nUI UFSN PG BOZ
MJOFBS TFRVFODF JT 5n = a + n o
d
XIFSF a JT UIF ĕSTU WBMVF JO UIF TFRVFODF

t 8IFO ZPV LOPX UIF SVMF GPS NBLJOH B TFRVFODF
ZPV DBO ĕOE UIF WBMVF PG BOZ UFSN 4VCTUJUVUF UIF UFSN OVNCFS JOUP
UIF SVMF BOE TPMWF JU

You should recognise these Exercise 9.1
sequences of numbers:
1 'JOE UIF OFYU UISFF UFSNT JO FBDI TFRVFODF BOE EFTDSJCF UIF SVMF ZPV VTFE UP ĕOE UIFN
square numbers: 1, 4, 9, 16 . . .
(a)

(b)

(c)

(d)



cube numbers: 1, 8, 27, 64 . . . (e) ¦
¦
¦
¦ (f) 1
1
(h)



(g)



triangular numbers: 1, 3, 6, 10 . . . 42

Fibonacci numbers: 1, 1, 2, 3, 2 -JTU UIF ĕSTU GPVS UFSNT PG UIF TFRVFODFT UIBU GPMMPX UIFTF SVMFT
5, 8 . . .

(a) 4UBSU XJUI TFWFO BOE BEE UXP FBDI UJNF

(b) 4UBSU XJUI BOE TVCUSBDU ĕWF FBDI UJNF

1

(c) 4UBSU XJUI POF BOE NVMUJQMZ CZ 2 FBDI UJNF
(d) 4UBSU XJUI ĕWF UIFO NVMUJQMZ CZ UXP BOE BEE POF FBDI UJNF

(e) 4UBSU XJUI
EJWJEF CZ UXP BOE TVCUSBDU UISFF FBDI UJNF

3 8SJUF EPXO UIF ĕSTU UISFF UFSNT PG FBDI PG UIFTF TFRVFODFT ćFO ĕOE UIF UI UFSN

(a) 5n = n +
(b) 5n = n
(c) 5n = n ¦
(d) 5n = n ¦
(e) 5n = n ¦ n
(f) 5n = ¦ n

4 $POTJEFS UIF TFRVFODF









(a) 'JOE UIF nUI UFSN PG UIF TFRVFODF
(b) 'JOE UIF UI UFSN
(c) 8IJDI UFSN PG UIJT TFRVFODF IBT UIF WBMVF 4IPX GVMM XPSLJOH
(d) 4IPX UIBU JT OPU B UFSN JO UIF TFRVFODF

54 Unit 3: Number

9 Sequences and sets

5 'PS FBDI TFRVFODF CFMPX ĕOE UIF HFOFSBM UFSN BOE UIF UI UFSN

(a)



(b) −
¦
¦
¦
(c)




(d)



(e)




6 ćF EJBHSBN TIPXT B QBUUFSO NBEF VTJOH NBUDITUJDLT

n=1 n=2 n=3

(a) %SBX B TFRVFODF UBCMF GPS UIF OVNCFS PG NBUDITUJDLT JO UIF ĕSTU TJY QBUUFSOT
(b) 'JOE B GPSNVMB GPS UIF nUI QBUUFSO
(c) )PX NBOZ NBUDIFT BSF OFFEFE GPS UIF UI QBUUFSO
(d) 8IJDI QBUUFSO XJMM OFFE NBUDIFT

9.2 Rational and irrational numbers

t :PV DBO FYQSFTT BOZ SBUJPOBM OVNCFS BT B GSBDUJPO JO UIF GPSN PG a XIFSF a BOE b BSF JOUFHFST BOE b ȶ
b

t 8IPMF OVNCFST
JOUFHFST
DPNNPO GSBDUJPOT
NJYFE OVNCFST
UFSNJOBUJOH EFDJNBM GSBDUJPOT BOE SFDVSSJOH EFDJNBMT
BSF BMM SBUJPOBM

tt :*SPSVBU DJPBOOB DM POOVWNFSCU FSSFTD DVBSOSJOOPHU ECFFD XJNSJBUMU FGOSB JDOUJ UPIOFT GJPOSUNP UbaI F * SGPSBSUNJP Oba BM OVNCFST BSF BMM OPO SFDVSSJOH
OPO UFSNJOBUJOH
EFDJNBMT

t ćF TFU PG SFBM OVNCFST JT NBEF VQ PG SBUJPOBM BOE JSSBUJPOBM OVNCFST

In 1.2, the dot above the two Exercise 9.2
in the decimal part means it is
recurring (the ‘2’ repeats forever). 1 8SJUF EPXO BMM UIF JSSBUJPOBM OVNCFST JO FBDI TFU PG SFBM OVNCFST
If a set of numbers recurs, e.g.
0.273273273..., there will be a dot (a) 3 , 16, 3 16, 22 , 12, 0.090090009. . ., 31 , 0.020202. . .,
at the start and end of the recurring 87 3
set: 0.273.
(b) 23, 45, 0.6, 3 , 3 90, π, 5 1 , 8, 0.834

42

2 $POWFSU FBDI PG UIF GPMMPXJOH SFDVSSJOH EFDJNBMT UP B GSBDUJPO JO JUT TJNQMFTU GPSN

(a) 0 4 (b) 0 74 (c) 0 87
(d) 0.114 (e) 0.943 (f) 0.1857

Unit 3: Number 55

9 Sequences and sets

9.3 Sets

t " TFU JT B MJTU PS DPMMFDUJPO PG PCKFDUT UIBU TIBSF B DIBSBDUFSJTUJD
t "O FMFNFOU ∈
JT B NFNCFS PG B TFU
t " TFU UIBU DPOUBJOT OP FMFNFOUT JT DBMMFE UIF FNQUZ TFU \^ PS ∅

t " VOJWFSTBM TFU ℰ
DPOUBJOT BMM UIF QPTTJCMF FMFNFOUT BQQSPQSJBUF UP B QBSUJDVMBS QSPCMFN
t ćF FMFNFOUT PG B TVCTFU ⊂
BSF BMM DPOUBJOFE JO B MBSHFS TFU
t ćF FMFNFOUT PG UXP TFUT DBO CF DPNCJOFE XJUIPVU SFQFBUT
UP GPSN UIF VOJPO ∪
PG UIF UXP TFUT
t ćF FMFNFOUT UIBU UXP TFUT IBWF JO DPNNPO JT DBMMFE UIF JOUFSTFDUJPO
PG UIF UXP TFUT
t ćF DPNQMFNFOU PG TFU A AҔ
JT UIF FMFNFOUT UIBU BSF JO UIF VOJWFSTBM TFU GPS UIBU QSPCMFN CVU OPU JO TFU A
t " 7FOO EJBHSBN JT B QJDUPSJBM NFUIPE PG TIPXJOH TFUT
t " TIPSUIBOE XBZ PG EFTDSJCJOH UIF FMFNFOUT PG B TFU JT DBMMFE TFU CVJMEFS OPUBUJPO 'PS FYBNQMF \x x JT BO JOUFHFS


x ^

Tip Exercise 9.3 A

.BLF TVSF ZPV LOPX UIF 1 4BZ XIFUIFS FBDI PG UIF GPMMPXJOH TUBUFNFOUT JT USVF PS GBMTF
NFBOJOH PG UIF TZNCPMT
VTFE UP EFTDSJCF TFUT BOE (a) ∈ \PEE OVNCFST^
QBSUT PG TFUT (b) ∈ \DVCFE OVNCFST^
(c) \

^ ⊂ \QSJNF OVNCFST^
Sometimes listing the elements (d) \ ^ ⊂ \QSJNF OVNCFST^
of each set will make it easier to (e) \

^ ∩ \

^ \



^
answer the questions. (f) \

^ ∪ \

^ \



^
(g) A \

^
B \

^
TP A B
(h) *G ℰ \MFUUFST PG UIF BMQIBCFU^ BOE A \DPOTPOBOUT^
UIFO AҔ \B
F
J
P
V^

2 A JT UIF TFU \




^

(a) %FTDSJCF TFU A JO XPSET
(b) 8IBU JT O A

(c) -JTU TFU B XIJDI JT UIF QSJNF OVNCFST JO A
(d) -JTU TFU C XIJDI JT UIF TJOHMF EJHJU OVNCFST JO A
(e) -JTU B ∩ C
(f) -JTU CҔ

3 ℰ \XIPMF OVNCFST GSPN UP ^
A \FWFO OVNCFST GSPN UP ^
B \PEE OVNCFST
GSPN UP ^ BOE C \NVMUJQMFT PG GSPN UP ^
-JTU UIF FMFNFOUT PG UIF GPMMPXJOH TFUT

(a) A ∩ B
(b) B ∪ C
(c) AҔ ∩ B
(d) B ∩ C)΄
(e) A ∩ B΄
(f) A ∪ B ∪ C

56 Unit 3: Number

9 Sequences and sets

Tip 4 -JTU UIF FMFNFOUT PG UIF GPMMPXJOH TFUT

:PV DBO VTF BOZ TIBQFT UP (a) \x x ∈ JOUFHFST
o ȳ x ^
ESBX B 7FOO EJBHSBN CVU (b) \x x ∈ OBUVSBM OVNCFST
x ȳ ^
VTVBMMZ UIF VOJWFSTBM TFU JT
ESBXO BT B SFDUBOHMF BOE 5 8SJUF JO TFU CVJMEFS OPUBUJPO
DJSDMFT XJUIJO JU TIPX UIF
TFUT (a) \



^
(b) \



^
REWIND
Exam questions often combine Exercise 9.3 B
probability with Venn diagrams.
Revise chapter 8 if you’ve 1 %SBX B 7FOO EJBHSBN UP TIPX UIF GPMMPXJOH TFUT BOE XSJUF FBDI FMFNFOU JO JUT DPSSFDU TQBDF
forgotten how to work this out.
ℰ \MFUUFST JO UIF BMQIBCFU^
P \MFUUFST JO UIF XPSE QIZTJDT^
C \MFUUFST JO UIF XPSE DIFNJTUSZ^

2 6TF UIF 7FOO EJBHSBN ZPV ESFX JO RVFTUJPO UP ĕOE

(a) O C)
(b) O P΄)
(c) C ∩ P
(d) P ∪ C
(e) P ∪ C)΄
(f) P ⊂ C

3 *O B TVSWFZ PG TUVEFOUT
TFWFO EJE OPU MJLF NBUIT PS TDJFODF 0G UIF SFTU
TBJE UIFZ MJLFE
NBUIT BOE TBJE UIFZ MJLFE TDJFODF

(a) %SBX B 7FOO EJBHSBN UP TIPX UIJT JOGPSNBUJPO
(b) 'JOE UIF OVNCFS PG TUVEFOUT XIP MJLFE CPUI NBUIT BOE TDJFODF
(c) " TUVEFOU JT DIPTFO BU SBOEPN 'JOE UIF QSPCBCJMJUZ UIBU UIF TUVEFOU MJLFT NBUIT CVU OPU

TDJFODF

Mixed exercise

1 'PS FBDI PG UIF GPMMPXJOH TFRVFODFT
ĕOE UIF nUI UFSN BOE UIF UI UFSN

(a)



(b)



(c)




2 5n n o


(a) (JWF UIF ĕSTU TJY UFSNT PG UIF TFRVFODF
(b) 8IBU JT UIF UI UFSN
(c) 8IJDI UFSN JT FRVBM UP

3 8IJDI PG UIF GPMMPXJOH OVNCFST BSF JSSBUJPOBM

15 , 0.213231234. . ., 25, 7 , 0.1, − 0.654, 2, 22 , 4π

8 17 5

Unit 3: Number 57

9 Sequences and sets

4 8SJUF FBDI SFDVSSJOH EFDJNBM BT B GSBDUJPO JO TJNQMFTU GPSN

(a) 3 (b) 0.286

5 *O B HSPVQ PG TUVEFOUT
TUVEZ CJPMPHZ BOE TUVEZ DIFNJTUSZ TUVEFOUT TUVEZ CPUI
TVCKFDUT

B C B = biology
C = chemistry

(a) . BLF B DPQZ PG UIF 7FOO EJBHSBN BOE DPNQMFUF JU TIPX UIF JOGPSNBUJPO BCPVU UIF
TUVEFOUT

(b) )PX NBOZ TUVEFOUT JO UIF HSPVQ TUVEZ OFJUIFS CJPMPHZ OPS DIFNJTUSZ
(c) 8IBU JT n # ∩ $

(d) * G B TUVEFOU JT DIPTFO BU SBOEPN GSPN UIF HSPVQ
XIBU JT UIF QSPCBCJMJUZ UIBU IF PS TIF

TUVEJFT
(i) DIFNJTUSZ
(ii) CJPMPHZ
(iii) DIFNJTUSZ BOE CJPMPHZ
(iv) DIFNJTUSZ PS CJPMPHZ PS CPUI

(v) OFJUIFS DIFNJTUSZ OPS CJPMPHZ

58 Unit 3: Number

10 Straight lines and quadratic
equations

10.1 Straight lines

t ćF QPTJUJPO PG B QPJOU DBO CF VOJRVFMZ EFTDSJCFE PO UIF $BSUFTJBO QMBOF VTJOH PSEFSFE QBJST x
y
PG DPPSEJOBUFT
t :PV DBO VTF FRVBUJPOT JO UFSNT PG x BOE y UP HFOFSBUF B UBCMF PG QBJSFE WBMVFT GPS x BOE y :PV DBO QMPU UIFTF

PO UIF $BSUFTJBO QMBOF BOE KPJO UIFN UP ESBX B HSBQI 5P ĕOE y WBMVFT JO B UBCMF PG WBMVFT
TVCTUJUVUF UIF HJWFO

PS DIPTFO
x WBMVFT JOUP UIF FRVBUJPO BOE TPMWF GPS y

t ćF HSBEJFOU PG B MJOF EFTDSJCFT JUT TMPQF PS TUFFQOFTT (SBEJFOU DBO CF EFĕOFE BT

m = change in y
change in x

o MJOFT UIBU TMPQF VQ UP UIF SJHIU IBWF B QPTJUJWF HSBEJFOU
o MJOFT UIBU TMPQF EPXO UP UIF SJHIU IBWF B OFHBUJWF HSBEJFOU

o MJOFT QBSBMMFM UP UIF x BYJT IPSJ[POUBM MJOFT
IBWF B HSBEJFOU PG

o MJOFT QBSBMMFM UP UIF y BYJT WFSUJDBM MJOFT
IBWF BO VOEFĕOFE HSBEJFOU

o MJOFT QBSBMMFM UP FBDI PUIFS IBWF UIF TBNF HSBEJFOUT

t ćF FRVBUJPO PG B TUSBJHIU MJOF DBO CF XSJUUFO JO HFOFSBM UFSNT BT y x + c
XIFSF x BOE y BSF DPPSEJOBUFT PG QPJOUT
PO UIF MJOF
m JT UIF HSBEJFOU PG UIF MJOF BOE c JT UIF y JOUFSDFQU UIF QPJOU XIFSF UIF HSBQI DSPTTFT UIF y BYJT


t 5P ĕOE UIF FRVBUJPO PG B HJWFO MJOF ZPV OFFE UP ĕOE UIF y JOUFSDFQU BOE TVCTUJUVUF UIJT GPS c ćFO ZPV OFFE UP ĕOE
UIF HSBEJFOU PG UIF MJOF BOE TVCTUJUVUF UIJT GPS m

t :PV DBO ĕOE UIF DPPSEJOBUFT PG UIF NJEQPJOU PG B MJOF TFHNFOU CZ BEEJOH UIF x DPPSEJOBUFT PG JUT FOE QPJOUT BOE
EJWJEJOH CZ UP HFU UIF x WBMVF PG UIF NJEQPJOU BOE UIFO EPJOH UIF TBNF XJUI UIF y DPPSEJOBUFT UP HFU UIF y WBMVF PG
UIF NJEQPJOU

Tip Exercise 10.1

/PSNBMMZ UIF x WBMVFT XJMM 1 'PS x WBMVFT PG −


BOE
ESBX B UBCMF PG WBMVFT GPS FBDI PG UIF GPMMPXJOH FRVBUJPOT
CF HJWFO *G OPU
DIPPTF
UISFF TNBMM WBMVFT GPS (a) y = x + 5 (b) y 2x −1 (c) y 7 2x (d) y x − 2
FYBNQMF
−
BOE
:PV (h) 4 2x − 5y
OFFE B NJOJNVN PG UISFF (e) x = 4 (f) y = −2 1 (g) y 2x − 1
QPJOUT UP ESBX B HSBQI "MM (i) 0 x 2y −1 2 2
HSBQIT TIPVME CF DMFBSMZ (j) x + y = −
MBCFMMFE XJUI UIFJS FRVBUJPO
2 %SBX BOE MBCFM HSBQIT a
UP e
JO RVFTUJPO PO POF TFU PG BYFT BOE HSBQIT f
UP j
PO BOPUIFS
Remember, parallel lines have the
same gradient. 3 'JOE UIF FRVBUJPO PG B MJOF QBSBMMFM UP HSBQI a
JO RVFTUJPO BOE QBTTJOH UISPVHI QPJOU
−


4 "SF UIF GPMMPXJOH QBJST PG MJOFT QBSBMMFM

(a) y 3x + 3 BOE y = x + 3 (b) y 1 x − 4 BOE y 1 x − 8
2 2

(c) y 3x BOE y 3x + 7 (d) y 0 8x − 7 BOE y 8x + 2

(e) 2y 3x + 2 BOE y = 3 x + 2 (f) 2y 3x = 2 BOE y 1 5x + 2
2

(g) y = 8 BOE y = −9 (h) x = −3 BOE x = 1
2

Unit 3: Algebra 59

10 Straight lines and quadratic equations

5 'JOE UIF HSBEJFOU PG UIF GPMMPXJOH MJOFT

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

6 (4, 6) x (–2, 1) x 6
4 0 123 12 4
2 x –1 –2 –1 0 2
4 –2 –1 (1, –2)
–4 0 –3 (3, –3) –2 x
0 1234567

(e) y (f) y (g) y (h) y

4 7 6 4
2 6 3 3
5 2
4 –3 0 x 1
3 –3
–2 0 x 2 3 0
–2 1 –1
24 –8 –4 48
–4 x
0 –2

–3

Tip 6 %FUFSNJOF UIF HSBEJFOU m
BOE UIF y JOUFSDFQU c
PG FBDI PG UIF GPMMPXJOH HSBQIT

:PV NBZ OFFE UP SFXSJUF (a) 3x − 4 (b) y x −1 (c) y 1 x + 5 (d) y x
UIF FRVBUJPOT JO UIF GPSN 2
y = mx + c CFGPSF ZPV DBO
EP UIJT (e) y = x + 1 (f ) y = 4x − 2 (g) y = 7 (h) y 3x
2 4 5

(i) x + 3y = 14 (j) x + y + 4 = 0 (k) x 4 y (l) 2x 5 y

(m) x + y = −10
2

7 %FUFSNJOF UIF FRVBUJPO PG FBDI PG UIF GPMMPXJOH HSBQIT

(i) 9y
8

7

6

5

4

3

y y 2
(d) (h)
(a) 3 1x
3 –9–8–7–6–5–4–3–2–10 1 2 3 4 5 6

(c) 2 2 x (j) –2
1 x1 –3
–4

–3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 –5
–6

(b) –2 (e) –2 –7
–3 (f) –3
(g) (l) (k) –8
–9

8 'JOE UIF x BOE y JOUFSDFQUT PG UIF GPMMPXJOH MJOFT

(a) y x o (b) y o 1 x (c) y o x
2

(d) x + y (e) x y
2

60 Unit 3: Algebra

10 Straight lines and quadratic equations

9 8JUIPVU ESBXJOH UIF HSBQIT
DBMDVMBUF UIF HSBEJFOU PG UIF MJOF UIBU QBTTFT UISPVHI FBDI PG
UIFTF QBJST PG QPJOUT

(a) o
o
BOE


(b) o
o
BOE


(c) o

BOE


(d)

BOE


(e) o
o
BOE
o

(f) o

BOE



10 6TJOH UIF TBNF TFU PG QPJOUT BT JO RVFTUJPO
ĕOE UIF DPPSEJOBUFT PG UIF NJEQPJOU PG FBDI
MJOF TFHNFOU

10.2 Quadratic expressions
t " RVBESBUJD FYQSFTTJPO IBT UFSNT XIFSF UIF IJHIFTU QPXFS PG UIF WBSJBCMF JT UXP GPS FYBNQMF x

t :PV DBO FYQBOE NVMUJQMZ PVU
UIF QSPEVDU PG UXP CSBDLFUT CZ NVMUJQMZJOH FBDI UFSN PG UIF ĕSTU CSBDLFU CZ FBDI UFSN

PG UIF TFDPOE :PV NBZ UIFO OFFE UP BEE PS TVCUSBDU BOZ MJLF UFSNT

t *G UIF UXP CSBDLFUT BSF UIF TBNF UIBU JT
UIF FYQSFTTJPOT B TRVBSF

UIF GPMMPXJOH FYQBOTJPOT DBO CF BQQMJFE

o x ¦ y) x ¦ xy y
o x y) x xy y

t 'BDUPSJTJOH JT UIF PQQPTJUF PG FYQBOEJOH
J F QVUUJOH CBDL JOUP CSBDLFUT
t " USJOPNJBM JT B RVBESBUJD FYQSFTTJPO XJUI UISFF UFSNT 5P GBDUPSJTF USJOPNJBMT PG UIF GPSN x x
MPPL GPS UXP

OVNCFST XIPTF TVN JT UIF DPFďDJFOU PG UIF x UFSN JO UIJT DBTF
BOE XIPTF QSPEVDU JT UIF DPOTUBOU UFSN JO UIJT
DBTF
'PS FYBNQMF
BOE ¨
TP UIF UXP OVNCFST BSF BOE BOE UIF GBDUPST BSF x
x


t :PV DBO GBDUPSJTF UIF EJČFSFODF PG UXP TRVBSFT ćF ĕSTU BOE MBTU UFSNT JO UIF CSBDLFUT BSF UIF TRVBSF SPPUT PG UIF
UFSNT JO UIF EJČFSFODF PG TRVBSFT ćF TJHOT CFUXFFO UIF UFSNT BSF EJČFSFOU JO FBDI CSBDLFU 'PS FYBNQMF
x ¦
x
x ¦


t :PV DBO VTF GBDUPSJTBUJPO UP TPMWF TPNF RVBESBUJD FRVBUJPOT 'PS FYBNQMF
x ¦ x o

o ĕSTU SFPSHBOJTF UIF FRVBUJPO TP UIBU UIF SJHIU IBOE TJEF FRVBMT
x ¦ x
o OFYU GBDUPSJTF UIF USJOPNJBM
x ¦
x ¦

o ĕOBMMZ
VTF UIF GBDU UIBU JG a ¨ b UIFO a PS b UP ĕOE UIF SPPUT ćFSFGPSF x ¦
TP x
PS x ¦


TP x ćFTF BSF CPUI TPMVUJPOT UP UIF FRVBUJPO

Exercise 10.2 A

Tip 1 &YQBOE BOE TJNQMJGZ

ćF BDSPOZN
'0*-
NBZ (a) (x + 2)(x + 3) (b) (x + 2)(x − 3)
IFMQ ZPV UP TZTUFNBUJDBMMZ
FYQBOE QBJST PG CSBDLFUT (c) (x + 5)(x + 7) (d) (x 5)(x + 7)

' o ĕSTU ¨ ĕSTU (e) (x 1)(x − 3) (f) (2x 1)(x + 1)

0 o PVUFS ¨ PVUFS (g) ( y 7)( y − 2) (h) (2x y)(3x 2y)

* o JOOFS ¨ JOOFS (i) (x2 + 1)(2x2 − 3) (j) (x 11)(x + 12)

- o MBTU ¨ MBTU (k) ⎛ 1 x + 1⎠⎞ ⎝⎛1 1 x⎞⎠ (l) (x 3)(2 3x)
⎝ 2 2

(m) (3x 2)(2 4x)

Unit 3: Algebra 61

10 Straight lines and quadratic equations

Remember the expansions for the 2 &YQBOE BOE TJNQMJGZ
square of a sum or a difference.
(a) (x + 4)2 (b) (x − 3)2 (c) (x + 5)2
The difference of two squares (d) ( y − 2)2 (e) (x + y)2 (f) (2x y)2
always gives you two brackets (g) (3 2)2 (h) (2x 3y)2 (i) (2 5)2
that are identical except for the (j) (4 6)2 (k) (3 − x)2 (l) (4 2x)2
signs, so you may be able to write (m) (6 3y)2
down the answer to an expansion
just by inspection. 3 8SJUF JO FYQBOEFE GPSN 5SZ UP EP UIJT CZ JOTQFDUJPO

Tip (a) (x 5)(x + 5) (b) (2x 5)(2x 5)

"MXBZT MPPL GPS BOE SFNPWF ( )( )(c) (3 7 y)(7 y 3) (d) x2 y2 x2 + y2
BOZ DPNNPO GBDUPST ( )( )(e) (4 3x)(3x 4) (f) x3 y2 x3 y2
CFGPSF ZPV USZ UP GBDUPSJTF ( )( )(g) 4x2 y2 2z2 4x2 y2 2z2
RVBESBUJD FYQSFTTJPOT ((h) 2x4 2y)(2x4 2y)
ćFO
SFNFNCFS UP JODMVEF (i) (5y )(4xy2 4xy2 5y)
BOZ DPNNPO GBDUPST ZPV ( )( )(j) 8x3 y2 7z2 8x3 y2 7z2
IBWF SFNPWFE JO UIF ĕOBM
BOTXFS Exercise 10.2 B

Tip 1 'BDUPSJTF GVMMZ (b) x2 + 7x 12
(d) 4 5x + x2
:PV DBO DIFDL ZPVS (a) x2 + 4x 4 (f) x2 9x 8
BOTXFST BSF DPSSFDU CZ (c) x2 + 6x 9 (h) 3 4x + x2
FYQBOEJOH UIF GBDUPST *U JT (e) 15 + 8x + x2 (j) x2 7x 8
VTVBMMZ FBTJFS UP GBDUPSJTF (g) x2 8x 15 (l) x2 4x 32
JG ZPV XSJUF UIF UFSNT (i) 26 − 27x + x2 (n) −12 + x + x2
JO EFTDFOEJOH PSEFS CZ (k) x2 + 3x 10
QPXFS
ĕSTU (m) 12 − 7x + x2
(o) −54 + 3x + x2
Tip
2 'BDUPSJTF GVMMZ
3FNFNCFS UIF QBUUFSO

a o b a b
a o b
(a) 5x2 15x 10 (b) 3x2 18x 24
ćF PSEFS JO XIJDI ZPV (c) 3x3 12 2 9x (d) 5x2 15x 10
XSJUF UIF CSBDLFUT EPFTO U (e) x3 x2 + 20x (f) x4 y + x3 y − 2x2 y
SFBMMZ NBUUFS a b
a o b
(g) x3 5x2 14x (h) 3x2 15x 18
a o b
a b
(i) −2x2 + 4x + 48 (j) 2x2 2x − 112

3 'BDUPSJTF GVMMZ (b) 16 − x2 (c) x2 − 25
(e) 9x2 4 y2
(a) x2 − 9 (h) 121y2 144x2 (f) 81 − 4x2
(d) 49 − x2 (k) 200 − 2x2
(g) x2 9 y2 (n) x2 y2 − 100 (i) 16x2 49 y2
(j) 2x2 18 (q) 1 81x4 y6
(m) 25 − x16 (l) x4 y2
(p) 25x10 − 1 25x2 64w2
(o) y4 − z2

62 Unit 3: Algebra

10 Straight lines and quadratic equations

Exercise 10.2 C

Remember to make the right- 1 4PMWF UIF GPMMPXJOH RVBESBUJD FRVBUJPOT
hand
side equal to zero and to factorise (a) x2 3x 0 (b) 8 2 32 0
before you try to solve the
equation. (c) 6x2 12x (d) −16x − 24x2 = 0

Quadratic equations always have (e) x2 − 1 0 (f) 49 − 4 2 0
two roots (although sometimes
the two roots are the same). (g) 8 2 2 (h) x2 + 6x 8 = 0

(i) x2 + 5x 4 = 0 (j) x2 4x 5 = 0

(k) x2 x − 20 = 0 (l) x2 + 8x 20

(m) x2 + 15 8x (n) −60 − 17x = −x2

(o) x2 + 56 15x (p) x2 − 20x + 100 = 0

(q) 5x2 20x 20 0

Mixed exercise 1 'PS FBDI FRVBUJPO
DPQZ BOE DPNQMFUF UIF UBCMF PG WBMVFT DSBX UIF HSBQIT GPS BMM GPVS

FRVBUJPOT PO UIF TBNF TFU PG BYFT

(a) y 1 x
2

x −1
y

(b) y 1 x + 3
2

x −1
y

(c) y = 2

x −1
y

(d) y 2x − 4 = 0

x −1
y

2 %FUFSNJOF UIF HSBEJFOU BOE UIF y JOUFSDFQU PG FBDI HSBQI

(a) y 2x −1 (b) y + 6 x (c) x y = −8
(e) 2x 3y = 6 (f) y x
(d) y = − 1
2

3 8IBU FRVBUJPO EFĕOFT FBDI PG UIFTF MJOFT

(a) B MJOF XJUI B HSBEJFOU PG BOE B y JOUFSDFQU PG −
1 2
(b) B MJOF XJUI B y JOUFSDFQU PG 2 BOE B HSBEJFOU PG − 3

(c) B MJOF QBSBMMFM UP y x + 8 XJUI B y-JOUFSDFQU PG −
4
(d) B MJOF QBSBMMFM UP y 5 x XIJDI QBTTFT UISPVHI UIF QPJOU
−


Unit 3: Algebra 63

10 Straight lines and quadratic equations

(e) B MJOF QBSBMMFM UP 2y 4x + 2 0 XJUI B y JOUFSDFQU PG −
(f) B MJOF QBSBMMFM UP x + y = 5 XIJDI QBTTFT UISPVHI


(g) B MJOF QBSBMMFM UP UIF x BYJT XIJDI QBTTFT UISPVHI


(h) B MJOF QBSBMMFM UP UIF y BYJT XIJDI QBTTFT UISPVHI −
−


4 'JOE UIF HSBEJFOU PG UIF GPMMPXJOH MJOFT

y
9
8
7
6
5
4
3
2
1x

–9–8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8 9

–2
–3
–4
–5
–6
–7
–8
–9

5 8IBU JT UIF FRVBUJPO PG FBDI PG UIFTF MJOFT

a
b
c
d)
y y y
y 6

4 4
2
2

–6 –3–3 3 6 x 7 –6 –3 –2 3 6x –10 –2 10 20 x
–6 –4 –2 –4
24 x –4 –6

e
f)
y y

53

–5 –3 35 x –4 –2 24 x
–5 (5,–5) –3

Tip 6 $BSPMJOF MJLFT SVOOJOH 4IF BWFSBHFT B TQFFE PG LN I XIFO TIF SVOT ćJT SFMBUJPOTIJQ DBO
CF FYQSFTTFE BT D = t
XIFSF D JT UIF EJTUBODF DPWFSFE BOE t JT UIF UJNF JO IPVST
UIBU TIF
5JNF JT VTVBMMZ QMPUUFE SVOT GPS
PO UIF IPSJ[POUBM PS
x BYJT CFDBVTF JU JT UIF (a) 6TF UIF GPSNVMB D = t UP ESBX VQ B UBCMF PG WBMVFT GPS

BOE IPVST PG SVOOJOH
JOEFQFOEFOU WBSJBCMF JO
NPTU SFMBUJPOTIJQT *O (b) 0 O B TFU PG BYFT
ESBX B HSBQI UP TIPX UIF SFMBUJPOTIJQ CFUXFFO D BOE t ćJOL DBSFGVMMZ
UIJT HSBQI ZPV XJMM POMZ
OFFE UP XPSL JO UIF ĕSTU BCPVU IPX ZPV XJMM OVNCFS UIF BYFT CFGPSF ZPV TUBSU
RVBESBOU :PV XPO U
IBWF BOZ OFHBUJWF WBMVFT (c) 8SJUF BO FRVBUJPO JO UIF GPSN PG y = mx + c UP EFTDSJCF UIJT HSBQI
CFDBVTF $BSPMJOF DBOOPU
SVO GPS MFTT UIBO IPVST (d) 8IBU JT UIF HSBEJFOU PG UIF MJOF
BOE IFS TQFFE DBOOPU CF
MFTT UIBO LN QFS IPVS (e) 6TF ZPVS HSBQI UP ĕOE UIF UJNF JU UBLFT $BSPMJOF UP SVO

(i) LN (ii) LN (iii) LN

64 Unit 3: Algebra

10 Straight lines and quadratic equations

(f) 6TF ZPVS HSBQI UP ĕOE PVU IPX GBS TIF SVOT JO

(i) IPVST (ii) 2 1 IPVST (iii) 3 PG BO IPVS
2
4

(g) $ BSPMJOF FOUFST UIF 5XP 0DFBOT .BSBUIPO ćF SPVUF JT LN MPOH
CVU JU JT WFSZ IJMMZ

4IF FTUJNBUFT IFS BWFSBHF TQFFE XJMM ESPQ UP BSPVOE LN I )PX MPOH XJMM JU UBLF IFS

UP DPNQMFUF UIF SBDF JG TIF SVOT BU LN I

7 'PS FBDI QBJS PG QPJOUT
ĕOE

(a) UIF HSBEJFOU PG UIF MJOF UIBU QBTTFT UISPVHI CPUI QPJOUT
(b) UIF DPPSEJOBUFT PG UIF NJEQPJOU PG B MJOF TFHNFOU KPJOJOH FBDI QBJS PG QPJOUT

(i) o

BOE


(ii) o

BOE


(iii)

BOE


(iv) o

BOE


(v) o
o
BOE o



8 3FNPWF UIF CSBDLFUT BOE TJNQMJGZ JG QPTTJCMF

(a) (x − 8)2 (b) (2x 2)(x −1)

(c) (3x 2y)2 (d) (1 6y)2

(e) (2 3x)(3x 2) (f) (2 5)2

( )(g) 3x2 y 1 2 ⎛ 1 y⎞⎠ 2
⎝ 2
(h) x +

(i) ⎛ x + 1⎞ ⎛ x − 1 ⎞ (j) ⎛ 1 − 2⎞⎠ ⎛ 1 + 2⎞⎠
⎝ 2⎠ ⎝ 2 ⎠ ⎝ x ⎝ x

(k) (x 2)2 (x − 7)2 (l) −2x (x − 2)2 + 8x2

(m) 2x (x 1)2 − (7 2x)( 2x)

9 'BDUPSJTF GVMMZ

(a) a3 4a (b) x4 − 1
(c) x2 x − 2 (d) x2 2x 1
(f) x2 + 16x + 48
(e) (2x 3y)2 − 4z2
x2 (h) x2 5x 6
(g) x4 − 4
(j) 2x2 14x 24
(i) 4x2 4x − 48 (l) 3x2 15x 18

(k) 5 20x16

10 4PMWF GPS x

(a) 7x2 42x 35 0
(b) 2 2 8=0
(c) 6x2 18x 12
(d) 3x2 6x = −3
(e) 4x2 16x 20 0
(f) 5x2 20x 20 0

Unit 3: Algebra 65

11 Pythagoras’ theorem and similar
shapes

11.1 Pythagoras’ theorem
t *O B SJHIU BOHMFE USJBOHMF
UIF TRVBSF PG UIF MFOHUI PG UIF IZQPUFOVTF UIF MPOHFTU TJEF
JT FRVBM UP UIF TVN PG UIF

TRVBSFT PG UIF MFOHUIT PG UIF PUIFS UXP TJEFT ćJT DBO CF FYQSFTTFE BT c2 = a2 + b2
XIFSF c JT UIF IZQPUFOVTF BOE

a BOE b BSF UIF UXP TIPSUFS TJEFT PG UIF USJBOHMF

t $POWFSTFMZ
*G c2 = a2 + b2 UIFO UIF USJBOHMF XJMM CF SJHIU BOHMFE
t 5P ĕOE UIF MFOHUI PG BO VOLOPXO TJEF JO B SJHIU BOHMFE USJBOHMF ZPV OFFE UP LOPX UXP PG UIF TJEFT ćFO ZPV DBO

TVCTUJUVUF UIF UXP LOPXO MFOHUIT JOUP UIF GPSNVMB BOE TPMWF GPS UIF VOLOPXO MFOHUI

t :PV DBO ĕOE UIF MFOHUI PG B MJOF KPJOJOH UXP QPJOUT VTJOH 1ZUIBHPSBT UIFPSFN

Tip Exercise 11.1 A

ćF hypotenuse JT UIF 1 $BMDVMBUF UIF MFOHUI PG UIF VOLOPXO TJEF JO FBDI PG UIFTF USJBOHMFT
MPOHFTU TJEF *U JT BMXBZT
PQQPTJUF UIF SJHIU BOHMF (a) x (b) 15 cm (c) 13 mm
3 cm 8 cm x
y
5 mm

4 cm

(d) (e) y (f ) x
x 24 cm 1.2 cm 0.4 cm 0.6 cm

0.5 cm

(g) 26 cm (h)
x 11 cm
y 4 cm
7.3 cm

7 cm

66 Unit 3: Shape, space and measures

11 Pythagoras’ theorem and similar shapes

Tip 2 'JOE UIF MFOHUI PG UIF TJEF NBSLFE XJUI B MFUUFS JO FBDI ĕHVSF

*G ZPV HFU BO BOTXFS UIBU (a) 70 mm (b) (c) z
JT BO JSSBUJPOBM OVNCFS
80 mm x 6 cm
SPVOE ZPVS BOTXFS UP
UISFF TJHOJĕDBOU ĕHVSFT 20.2 cm
VOMFTT UIF JOTUSVDUJPO UFMMT y
ZPV PUIFSXJTF
120 mm 4 cm 8 cm

(d) 5 mm 8.2 cm y (f )
5 mm x (e) 9 cm
15 cm
9 cm 16 cm 5 cm z

13 mm 12 cm 16 cm

3 %FUFSNJOF XIFUIFS FBDI PG UIF GPMMPXJOH USJBOHMFT JT SJHIU BOHMFE 4JEF MFOHUIT BSF BMM JO
DFOUJNFUSFT

(a) A (b) D (c) H (d) J
7
8 15 9 8 13
G I 12
B C F 9 L
10 12 E 11 5
K

4 $BMDVMBUF UIF MFOHUI PG UIF MJOF TFHNFOU KPJOJOH FBDI PG UIF GPMMPXJOH QBJST PG QPJOUT

(a) (−2, −2) BOE (2, 2) (b) (−3, −1) BOE (0, 2)
(c) (−4, 5) BOE (0, 1) (d) (2, 4) BOE (8, 16)
(e) (−1, −3) BOE (2, −3) (f) (−2, 0) BOE (4, 3)

Tip Exercise 11.1 B

8JUI XPSEFE QSPCMFNT
1 " SFDUBOHMF IBT TJEFT PG NN BOE NN $BMDVMBUF UIF MFOHUI PG POF PG UIF EJBHPOBMT
EP B SPVHI TLFUDI PG UIF
TJUVBUJPO
ĕMM JO UIF LOPXO 2 ćF TJ[F PG B SFDUBOHVMBS DPNQVUFS TDSFFO JT EFUFSNJOFE CZ UIF MFOHUI PG UIF EJBHPOBM /JDL
MFOHUIT BOE NBSL UIF CVZT B DN TDSFFO UIBU JT DN IJHI )PX MPOH JT UIF CBTF PG UIF TDSFFO
VOLOPXO MFOHUIT XJUI
MFUUFST 3 ćF TJEFT PG BO FRVJMBUFSBM USJBOHMF BSF NN MPOH $BMDVMBUF UIF QFSQFOEJDVMBS IFJHIU PG
UIF USJBOHMF BOE IFODF ĕOE JUT BSFB

4 " WFSUJDBM QPMF JT N MPOH *U JT TVQQPSUFE CZ UXP XJSF TUBZT ćF TUBZT BSF BUUBDIFE UP UIF
UPQ PG UIF QPMF BOE ĕYFE UP UIF HSPVOE 0OF TUBZ JT ĕYFE UP UIF HSPVOE N GSPN UIF CBTF
PG UIF QPMF BOE UIF PUIFS JT ĕYFE UP UIF HSPVOE N GSPN UIF CBTF PG UIF QPMF $BMDVMBUF UIF
MFOHUI PG FBDI XJSF TUBZ

Unit 3: Shape, space and measures 67

11 Pythagoras’ theorem and similar shapes

5 /JDL IBT B N MBEEFS UIBU IF VTFT UP SFBDI TIFMWFT ĕYFE UP UIF XBMM PG IJT HBSBHF )F XBOUT
UP SFBDI B TIFMG UIBU JT NFUSFT BCPWF UIF HSPVOE 8IBU JT UIF GVSUIFTU EJTUBODF IF DBO
QMBDF UIF GPPU PG IJT MBEEFS GSPN UIF XBMM

11.2 Understanding similar triangles
t 5SJBOHMFT BSF TJNJMBS XIFO UIF DPSSFTQPOEJOH TJEFT BSF QSPQPSUJPOBM BOE UIF DPSSFTQPOEJOH BOHMFT BSF FRVBM JO TJ[F
t *O TJNJMBS ĕHVSFT
JG UIF MFOHUI PG FBDI TJEF JT EJWJEFE CZ UIF MFOHUI PG JUT DPSSFTQPOEJOH TJEF
BMM UIF BOTXFST XJMM CF

TBNF :PV DBO VTF UIJT QSPQFSUZ UP ĕOE UIF MFOHUIT PG VOLOPXO TJEFT JO TJNJMBS ĕHVSFT

Tip Exercise 11.2

8PSL PVU XIJDI TJEFT BSF 1 ćF QBJST PG USJBOHMFT JO UIJT RVFTUJPO BSF TJNJMBS $BMDVMBUF UIF VOLOPXO MFUUFSFE
MFOHUI JO
DPSSFTQPOEJOH CFGPSF ZPV FBDI DBTF
TUBSU *U JT IFMQGVM UP NBSL
DPSSFTQPOEJOH TJEFT JO (a) (b) 12 mm
UIF TBNF DPMPVS PS XJUI B 4.47 cm β
TZNCPM α
4 cm 9 mm
2 cm x 2 cm 6 mm
α
1 cm 4 mm 8 mm

β
y

(c) β 5 mm (d) 8 cm y y
6 mm y 2 mm x z

α 4 mm 10 cm x
β 8 cm

α 3 mm

(e) (f )

8 cm y z 4 cm 8.5 cm 43°
x
αβ αβ 8 cm y
10 cm 6 cm
43° α
5 cm α
9.5 cm

(g) β (h) y
8 cm x βα xα

27 cm y 40 cm β 25 cm 15 cm
β
α
α 21 cm 30 cm
7 cm

68 Unit 3: Shape, space and measures

11 Pythagoras’ theorem and similar shapes

2 &YQMBJO GVMMZ XIZ Δ ABC JT TJNJMBS UP Δ ADE
A

BC

DE
3 /BODZ JT MZJOH PO B CMBOLFU PO UIF HSPVOE
N BXBZ GSPN B N UBMM USFF 8IFO TIF MPPLT VQ

QBTU UIF USFF TIF DBO TFF UIF SPPG PG B CVJMEJOH XIJDI JT N CFZPOE UIF USFF 8PSL PVU UIF
IFJHIU PG UIF CVJMEJOH

building

30 m 3 m Nancy
4m

11.3 Understanding similar shapes

t ćF SBUJP PG DPSSFTQPOEJOH TJEFT JO TJNJMBS TIBQFT JT FRVBM ćF MFOHUIT PG VOLOPXO TJEFT DBO CF GPVOE CZ UIF TBNF
NFUIPE VTFE GPS TJNJMBS USJBOHMFT

t ćFSF JT B SFMBUJPOTIJQ CFUXFFO UIF TJEFT PG TJNJMBS ĕHVSFT BOE UIF BSFBT PG UIF ĕHVSFT *O TJNJMBS ĕHVSFT
XIFSF UIF
SBUJP PG TJEFT JT a b
UIF SBUJP PG BSFBT JT a2 b2 *O PUIFS XPSET
UIF TDBMF GBDUPS
2 BSFB GBDUPS

t 4JNJMBS TPMJET IBWF UIF TBNF TIBQF
UIFJS DPSSFTQPOEJOH BOHMFT BSF FRVBM BOE BMM DPSSFTQPOEJOH MJOFBS NFBTVSFT
FEHFT
EJBNFUFST
SBEJJ
IFJHIUT BOE TMBOU IFJHIUT
BSF JO UIF TBNF SBUJP

t *G UXP TPMJET A BOE B
BSF TJNJMBS UIFO UIF SBUJP PG UIFJS WPMVNFT JT FRVBM UP UIF DVCF PG UIF SBUJP PG DPSSFTQPOEJOH
volume A 3
volume B ⎛ a⎞
MJOFBS NFBTVSFT *O PUIFS XPSET = ⎝ b⎠ *O BEEJUJPO
UIF SBUJP PG UIFJS TVSGBDF BSFBT JT FRVBM UP UIF TRVBSF

surface area A ⎛ a⎞ 2
surface area B ⎝ b⎠
PG UIF SBUJP PG DPSSFTQPOEJOH MJOFBS NFBTVSFT *O PUIFS XPSET =

Exercise 11.3

1 'JOE UIF MFOHUI PG FBDI TJEF NBSLFE XJUI B MFUUFS JO UIFTF QBJST PG TJNJMBS TIBQFT "MM
EJNFOTJPOT BSF JO DFOUJNFUSFT

(a) (b)

x 24 45 x
15 25 y 30

30 18

Unit 3: Shape, space and measures 69

11 Pythagoras’ theorem and similar shapes

2 *G UIF BSFBT PG UXP TJNJMBS RVBESJMBUFSBMT BSF JO UIF SBUJP
XIBU JT UIF SBUJP PG NBUDIJOH
TJEFT

3 ćF UXP TIBQFT JO FBDI QBJS CFMPX BSF TJNJMBS ćF BSFB PG UIF TIBQF PO UIF MFę PG FBDI QBJS JT
HJWFO 'JOE UIF BSFB PG UIF PUIFS TIBQF

(a) (b)

12 cm 18 cm

area = 113.10 cm2 69 mm 23 mm

area = 4761 mm2

4 ćF UXP TIBQFT JO FBDI QBJS CFMPX BSF TJNJMBS ćF BSFB PG CPUI TIBQFT JT HJWFO 6TF UIJT UP
DBMDVMBUF UIF MFOHUI PG UIF NJTTJOH TJEF JO FBDI ĕHVSF

(a) (b)

1 cm x 3m x

area = 10 cm3 area = 40 cm3 area = 16.2 m3 area = 405 m3

5 " TIJQQJOH DSBUF IBT B WPMVNF PG DN *G UIF EJNFOTJPOT PG UIF DSBUF BSF EPVCMFE
XIBU
XJMM JUT OFX WPMVNF CF

6 5XP TJNJMBS DVCFT
A BOE B
IBWF TJEFT PG DN BOE DN SFTQFDUJWFMZ

(a) 8IBU JT UIF MJOFBS TDBMF GBDUPS PG A UP B
(b) 8IBU JT UIF SBUJP PG UIFJS TVSGBDF BSFBT
(c) 8IBU JT UIF SBUJP PG UIFJS WPMVNFT

11.4 Understanding congruence
t $POHSVFOU TIBQFT BSF JEFOUJDBM JO TIBQF BOE TJ[F 5XP TIBQFT BSF DPOHSVFOU JG UIF DPSSFTQPOEJOH TJEFT BSF FRVBM JO

MFOHUI BOE UIF DPSSFTQPOEJOH BOHMFT BSF FRVBM JO TJ[F

t 5SJBOHMFT BSF DPOHSVFOU JG BOZ PG UIF GPMMPXJOH GPVS DPOEJUJPOT BSF NFU

o UISFF TJEFT PG POF USJBOHMF BSF FRVBM UP UISFF DPSSFTQPOEJOH TJEFT PG UIF PUIFS 444


o UXP TJEFT BOE UIF JODMVEFE BOHMF PG POF USJBOHMF BSF FRVBM UP UIF TBNF UXP TJEFT BOE JODMVEFE BOHMF PG UIF PUIFS 4"4


o UXP BOHMFT BOE UIF JODMVEFE TJEF TJEF CFUXFFO UIF BOHMFT
PG POF USJBOHMF BSF FRVBM UP UIF DPSSFTQPOEJOH BOHMFT
BOE JODMVEFE TJEF PG UIF PUIFS "4"


o UIF IZQPUFOVTF BOE POF PUIFS TJEF PG POF SJHIU BOHMFE USJBOHMF BSF FRVBM UP UIF IZQPUFOVTF BOE DPSSFTQPOEJOH
TJEF PG UIF PUIFS SJHIU BOHMFE USJBOHMF 3)4


70 Unit 3: Shape, space and measures

11 Pythagoras’ theorem and similar shapes

Remember, the order in which you Exercise 11.4
name a triangle is important if you
are stating congruency. If Δ ABC is 1 State whether each pair of triangles is congruent or not. If they are congruent, state the
congruent to Δ XYZ then:
condition of congruency.
AB = XY
(a) B X (b)
BC = YZ
x° x° 35° 35°
AC = XZ

A CY Z

78° 67° 78° 67°

(c) X M N (d) D P Q
y x

Y Z O y
(e) A x FR

E N
(f) M

B C D P O
(g) D
87° QY (h)
P RX 87° A C
E
Z

B

Tip 2 MNOP is a trapezium. MP = NO and PN = MO. Show that Δ MPO is congruent to Δ NOP
giving reasons.
When you are asked
to show triangles are MN
congruent you must
always state the condition Q
that you are using.

P O
PN = MO

Unit 3: Shape, space and measures 71

11 Pythagoras’ theorem and similar shapes

3 *O UIF GPMMPXJOH ĕHVSF
TIPX UIBU AB JT IBMG UIF MFOHUI PG AC HJWJOH SFBTPOT GPS FBDI
TUBUFNFOU ZPV NBLF

C

B

A
D

E

Mixed exercise 1 " TDIPPM DBSFUBLFS XBOUT UP NBSL PVU B TQPSUT ĕFME N XJEF BOE N MPOH 5P NBLF TVSF

UIBU UIF ĕFME JT SFDUBOHVMBS
IF OFFET UP LOPX IPX MPOH FBDI EJBHPOBM TIPVME CF

(a) %SBX B SPVHI TLFUDI PG UIF ĕFME
(b) $BMDVMBUF UIF SFRVJSFE MFOHUIT PG UIF EJBHPOBMT

2 *O Δ ABC
AB = DN
BC = DN BOE AC = DN %FUFSNJOF XIFUIFS UIF USJBOHMF JT
SJHIU BOHMFE PS OPU BOE HJWF SFBTPOT GPS ZPVS BOTXFS

3 'JOE UIF MFOHUI PG UIF MJOF TFHNFOU KPJOJOH FBDI PG UIF GPMMPXJOH QBJST PG QPJOUT

(a) o

BOE

(b) o

BOE



(c)

BOE

(d) o

BOE



(e) o
o
BOE o



4 " USJBOHMF XJUI TJEFT PG NN
NN BOE NN JT TJNJMBS UP BOPUIFS USJBOHMF XJUI JUT
MPOHFTU TJEF NN $BMDVMBUF UIF QFSJNFUFS PG UIF MBSHFS USJBOHMF

5 $BMDVMBUF UIF NJTTJOH EJNFOTJPOT PS BOHMFT JO FBDI PG UIFTF QBJST PG TJNJMBS USJBOHMFT

(a) 4.5 cm (b) 87° 30°
α β 63°

1.5 cm x x
y
13.5 cm β 30°
α

4.5 cm 10.5 cm

(c) 36 cm
α α

x 15 cm

5 cm

72 Unit 3: Shape, space and measures

11 Pythagoras’ theorem and similar shapes

6 $BMDVMBUF UIF SBUJP PG UIF BSFBT PG FBDI QBJS PG TJNJMBS TIBQFT
(a) (b)

1.5 m 0.75 m

150 m

3.0 m 1.5 m 50 m

7 ćF UXP QBSBMMFMPHSBNT CFMPX BSF TJNJMBS ćF BSFB PG UIF MBSHFS JT DN2 'JOE UIF BSFB PG
UIF TNBMMFS QBSBMMFMPHSBN

18 cm

4.5 cm

area = 288 cm2

8 ćF UXP DVCPJET
A BOE B
BSF TJNJMBS ćF MBSHFS IBT B TVSGBDF BSFB PG NN2 8IBU JT
UIF TVSGBDF BSFB PG UIF TNBMMFS

AB

50 mm 80 mm
area = 60 800 mm2

9 " TRVBSF CBTFE QZSBNJE IBT B CBTF PG BSFB DN2 BOE B WPMVNF PG DN $BMDVMBUF

(a) ćF QFSQFOEJDVMBS IFJHIU PG UIF QZSBNJE
(b) ćF IFJHIU BOE BSFB PG UIF CBTF PG B TJNJMBS QZSBNJE XJUI B WPMVNF PG DN

10 5XP PG UIF USJBOHMFT JO FBDI TFU PG UISFF BSF DPOHSVFOU 4UBUF XIJDI UXP BSF DPOHSVFOU BOE
HJWF UIF DPOEJUJPOT UIBU ZPV VTFE UP QSPWF UIFN DPOHSVFOU

(a) D H (b) A DG

C A G x yC Fy xE x yH
(c) BE F B I

I

(d) A D G
6x

C 7 6 5
5
y Fy xE H
B 7
I x

Unit 3: Shape, space and measures 73

11 Pythagoras’ theorem and similar shapes

11 "O N MPOH XJSF DBCMF JT VTFE UP TFDVSF B NBTU PG IFJHIU x N ćF DBCMF JT BUUBDIFE UP UIF
UPQ PG UIF NBTU BOE TFDVSFE PO UIF HSPVOE N BXBZ GSPN UIF CBTF PG UIF NBTU )PX UBMM
JT UIF NBTU (JWF ZPVS BOTXFS DPSSFDU UP UXP EFDJNBM QMBDFT

8.6 m xm

6.5 m

12 /BEJB XBOUT UP IBWF B NFUBM OVNCFS NBEF GPS IFS HBUF 4IF IBT GPVOE B TBNQMF CSBTT
OVNFSBM BOE OPUFE JUT EJNFOTJPOT 4IF EFDJEFT UIBU IFS OVNFSBM TIPVME CF TJNJMBS UP UIJT
POF
CVU UIBU JU TIPVME CF GPVS UJNFT MBSHFS
35 mm

17 mm 140 mm
105 mm

35 mm

(a) % SBX B SPVHI TLFUDI PG UIF OVNFSBM UIBU /BEJB XBOUT UP NBLF XJUI UIF DPSSFDU
EJNFOTJPOT XSJUUFO PO JU JO NJMMJNFUSFT

(b) $ BMDVMBUF UIF MFOHUI PG UIF TMPQJOH FEHF BU UIF UPQ PG UIF GVMM TJ[F OVNFSBM UP UIF OFBSFTU
XIPMF NJMMJNFUSF

74 Unit 3: Shape, space and measures

12 Averages and measures
of spread

12.1 Different types of average
t 4UBUJTUJDBM EBUB DBO CF TVNNBSJTFE VTJOH BO BWFSBHF NFBTVSF PG DFOUSBM UFOEFODZ
BOE B NFBTVSF PG TQSFBE

EJTQFSTJPO


t ćFSF BSF UISFF UZQFT PG BWFSBHF NFBO
NFEJBO BOE NPEF
t " NFBTVSF PG TQSFBE JT UIF SBOHF MBSHFTU WBMVF NJOVT TNBMMFTU WBMVF

t ćF NFBO JT UIF TVN PG UIF EBUB JUFNT EJWJEFE CZ UIF OVNCFS PG JUFNT JO UIF EBUB TFU ćF NFBO EPFT OPU IBWF

UP CF POF PG UIF OVNCFST JO UIF EBUB TFU

o ćF NFBO DBO CF BČFDUFE CZ FYUSFNF WBMVFT JO UIF EBUB TFU 8IFO POF WBMVF JT NVDI MPXFS PS IJHIFS UIBO UIF

SFTU PG UIF EBUB JU JT DBMMFE BO PVUMJFS 0VUMJFST TLFX UIF NFBO BOE NBLF JU MFTT SFQSFTFOUBUJWF PG UIF EBUB TFU

t ćF NFEJBO JT UIF NJEEMF WBMVF JO B TFU PG EBUB XIFO UIF EBUB JT BSSBOHFE JO JODSFBTJOH PSEFS

o 8IFO UIFSF JT BO FWFO OVNCFS PG EBUB JUFNT
UIF NFEJBO JT UIF NFBO PG UIF UXP NJEEMF WBMVFT

t ćF NPEF JT UIF OVNCFS PS JUFN
UIBU BQQFBST NPTU PęFO JO B EBUB TFU

o 8IFO UXP OVNCFST BQQFBS NPTU PęFO UIF EBUB IBT UXP NPEFT BOE JT TBJE UP CF CJNPEBM 8IFO NPSF UIBO

UXP OVNCFST BQQFBS FRVBMMZ PęFO UIF NPEF IBT OP SFBM WBMVF BT B TUBUJTUJD

Exercise 12.1

1 %FUFSNJOF UIF NFBO
NFEJBO BOE NPEF PG UIF GPMMPXJOH TFUT PG EBUB

(a)






(b)








(c)











(d)








(e)








(f)









Tip 2 'JWF TUVEFOUT TDPSFE B NFBO NBSL PG PVU PG GPS B NBUIT UFTU

*G ZPV NVMUJQMZ UIF NFBO (a) 8IJDI PG UIFTF TFUT PG NBSLT ĕU UIJT BWFSBHF
CZ UIF OVNCFS PG JUFNT
JO UIF EBUB TFU
ZPV HFU (i)



(ii)



(iii)




UIF UPUBM PG UIF TDPSFT
ćJT XJMM IFMQ ZPV TPMWF (iv)



(v)



(vi)




QSPCMFNT MJLF RVFTUJPO
(b) $PNQBSF UIF TFUT PG OVNCFST JO ZPVS BOTXFS BCPWF &YQMBJO XIZ ZPV DBO HFU UIF TBNF

NFBO GSPN EJČFSFOU TFUT PG OVNCFST

3 ćF NFBO PG OVNCFST JT 8IBU JT UIF TVN PG UIF OVNCFST

4 ćF TVN PG OVNCFST JT 8IJDI PG UIF GPMMPXJOH OVNCFST JT DMPTFTU UP UIF NFBO PG
UIF OVNCFST

PS

Unit 3: Data handling 75

12 Averages and measures of spread

5 "O BHSJDVMUVSBM XPSLFS XBOUT UP LOPX XIJDI PG UXP EBJSZ GBSNFST IBWF UIF CFTU NJML
QSPEVDJOH DPXT 'BSNFS 4JOHI TBZT IJT DPXT QSPEVDF MJUSFT PG NJML QFS EBZ 'BSNFS
/BJEPP TBZT IFS DPXT QSPEVDF MJUSFT PG NJML QFS EBZ
ćFSF JT OPU FOPVHI JOGPSNBUJPO UP EFDJEF XIJDI DPXT BSF UIF CFUUFS QSPEVDFST PG NJML
8IBU PUIFS JOGPSNBUJPO XPVME ZPV OFFE UP BOTXFS UIF RVFTUJPO

Tip 6 *O B HSPVQ PG TUVEFOUT
TJY IBE GPVS TJCMJOHT
TFWFO IBE ĕWF TJCMJOHT
FJHIU IBE UISFF TJCMJOHT

OJOF IBE UXP TJCMJOHT BOE UFO IBE POF TJCMJOH 4JCMJOHT BSF CSPUIFST BOE TJTUFST

*U NBZ IFMQ UP ESBX B
SPVHI GSFRVFODZ UBCMF UP (a) 8IBU JT UIF NFBO OVNCFS PG TJCMJOHT
TPMWF QSPCMFNT MJLF UIJT (b) 8IBU JT UIF NPEBM OVNCFS PG TJCMJOHT
POF
7 ćF NBOBHFNFOU PG B GBDUPSZ BOOPVODFE TBMBSZ JODSFBTFT BOE TBJE UIBU XPSLFST XPVME
SFDFJWF BO BWFSBHF JODSFBTF PG UP
ćF UBCMF TIPXT UIF PME BOE OFX TBMBSJFT PG UIF XPSLFST JO UIF GBDUPSZ

'PVS XPSLFST JO $BUFHPSZ " Previous salary Salary with increase
5XP XPSLFST JO $BUFHPSZ #
4JY XPSLFST JO $BUFHPSZ $
&JHIU XPSLFST JO $BUFHPSZ %


(a) $BMDVMBUF UIF NFBO JODSFBTF GPS BMM XPSLFST
(b) $BMDVMBUF UIF NPEBM JODSFBTF
(c) 8IBU JT UIF NFEJBO JODSFBTF
(d) )PX NBOZ XPSLFST SFDFJWFE BO JODSFBTF PG CFUXFFO BOE
(e) 8BT UIF NBOBHFNFOU BOOPVODFNFOU USVF 4BZ XIZ PS XIZ OPU

12.2 Making comparisons using averages and ranges
t :PV DBO VTF BWFSBHFT UP DPNQBSF UXP PS NPSF TFUT PG EBUB )PXFWFS
BWFSBHFT PO UIFJS PXO NBZ CF NJTMFBEJOH
TP JU JT

VTFGVM UP XPSL XJUI PUIFS TVNNBSZ TUBUJTUJDT BT XFMM

t ćF SBOHF JT B NFBTVSF PG IPX TQSFBE PVU EJTQFSTFE
UIF EBUB JT 3BOHF MBSHFTU WBMVF o TNBMMFTU WBMVF
t " MBSHF SBOHF NFBOT UIBU UIF EBUB JT TQSFBE PVU
TP UIF NFBTVSFT PG DFOUSBM UFOEFODZ BWFSBHFT
NBZ OPU CF

SFQSFTFOUBUJWF PG UIF XIPMF EBUB TFU

Tip Exercise 12.2

8IFO UIF NFBO JT 1 'PS UIF GPMMPXJOH TFUT PG EBUB
POF PG UIF UISFF BWFSBHFT JT OPU SFQSFTFOUBUJWF
BČFDUFE CZ FYUSFNF 4UBUF XIJDI POF JT OPU SFQSFTFOUBUJWF JO FBDI DBTF
WBMVFT UIF NFEJBO JT NPSF
SFQSFTFOUBUJWF PG UIF EBUB (a)








(b)












(c)









76 Unit 3: Data handling

12 Averages and measures of spread

Tip 2 5XFOUZ TUVEFOUT TDPSFE UIF GPMMPXJOH SFTVMUT JO B UFTU PVU PG


ćF NPEF POMZ UFMMT ZPV
UIF NPTU QPQVMBS WBMVF
BOE UIJT JT OPU OFDFTTBSJMZ (a) $BMDVMBUF UIF NFBO
NFEJBO
NPEF BOE SBOHF PG UIF NBSLT
SFQSFTFOUBUJWF PG UIF XIPMF (b) 8IZ JT UIF NFEJBO UIF CFTU TVNNBSZ TUBUJTUJD GPS UIJT QBSUJDVMBS TFU PG EBUB
EBUB TFU
3 ćF UBCMF TIPXT UIF UJNFT JO NJOVUFT BOE TFDPOET
UIBU UXP SVOOFST BDIJFWFE PWFS N
EVSJOH POF TFBTPO

Runner A N T N T N T N T N T N T N T
Runner B N T N T N T N T N T N T N T

(a) 8IJDI SVOOFS JT UIF CFUUFS PG UIF UXP 8IZ
(b) 8IJDI SVOOFS JT NPTU DPOTJTUFOU 8IZ

12.3 Calculating averages and ranges for frequency data
t ćF NFBO DBO CF DBMDVMBUFE GSPN B GSFRVFODZ UBCMF 5P DBMDVMBUF UIF NFBO ZPV BEE B DPMVNO UP UIF UBCMF BOE

total of (score frequency) column
DBMDVMBUF UIF TDPSF ¨ GSFRVFODZ fx
Mean = total of frequency column

t 'JOE UIF NPEF JO B UBCMF CZ MPPLJOH BU UIF GSFRVFODZ DPMVNO ćF EBUB JUFN XJUI UIF IJHIFTU GSFRVFODZ JT UIF NPEF
t *O B GSFRVFODZ UBCMF
UIF EBUB JT BMSFBEZ PSEFSFE CZ TJ[F 5P ĕOE UIF NFEJBO
XPSL PVU JUT QPTJUJPO JO UIF EBUB BOE

UIFO BEE UIF GSFRVFODJFT UJMM ZPV FRVBM PS FYDFFE UIJT WBMVF ćF TDPSF JO UIJT DBUFHPSZ XJMM CF UIF NFEJBO

Exercise 12.3

1 $POTUSVDU B GSFRVFODZ UBCMF GPS UIF EBUB CFMPX BOE UIFO DBMDVMBUF

(a) UIF NFBO (b) UIF NPEF 3 (c) UIF NFEJBO (d) UIF SBOHF

3 3 22 22
33 3 22 2
5 32 33 32
3 2

2 'PS FBDI PG UIF GPMMPXJOH GSFRVFODZ EJTUSJCVUJPOT DBMDVMBUF

(a) UIF NFBO TDPSF (b) UIF NFEJBO TDPSF (c) UIF NPEBM TDPSF
%BUB TFU "

Score 23 5

Frequency

%BUB TFU #

Score

Frequency 25 22 23

%BUB TFU $

Score

Frequency

Unit 3: Data handling 77

12 Averages and measures of spread

12.4 Calculating averages and ranges for grouped continuous data

t $POUJOVPVT EBUB DBO UBLF BOZ WBMVF CFUXFFO UXP HJWFO WBMVFT
t 8IFO HJWFO B GSFRVFODZ UBCMF DPOUBJOJOH HSPVQFE EBUB JO DMBTT JOUFSWBMT
ZPV EPO U LOPX UIF FYBDU WBMVFT PG UIF EBUB

JUFNT ZPV POMZ LOPX JOUP XIJDI DMBTT UIFZ GBMM ćJT NFBOT ZPV DBOOPU XPSL PVU UIF FYBDU NFBO
NFEJBO PS NPEF


CVU ZPV DBO FTUJNBUF UIFN

t 5P FTUJNBUF UIF NFBO ZPV OFFE UP VTF UIF NJEQPJOUT PG UIF DMBTT JOUFSWBMT JO UIF UBCMF

– ćF NJEQPJOU JT UIF NFBO PG UIF TNBMMFTU BOE MBSHFTU TDPSFT JO UIF JOUFSWBM

– ćF MPXFTU BOE IJHIFTU TDPSFT JO B DMBTT JOUFSWBM BSF DBMMFE UIF MPXFS BOE VQQFS DMBTT MJNJUT

t – 0ODF ZPV IBWF GPVOE UIF NJEQPJOUT ZPV DBO FTUJNBUF UIF NFBO Estimated mean = sum of (midpoint × frequency)
ćF NPEBM DMBTT JT UIF DMBTT JOUFSWBM XJUI UIF IJHIFTU GSFRVFODZ sum off requencies

t ćF NFEJBO DMBTT JT UIF DMBTT JOUFSWBM JOUP XIJDI UIF NJEEMF WBMVF JO UIF EBUB TFU GBMMT 8PSL PVU UIF QPTJUJPO PG UIF
NFEJBO UPUBM GSFRVFODZ Ÿ UXP
BOE UIFO BEE UIF UPUBMT JO UIF GSFRVFODZ DPMVNO VOUJM ZPV SFBDI UIBU QPTJUJPO UP ĕOE

UIF NFEJBO DMBTT

Exercise 12.4

Tip 1 ćF UBCMF TIPXT UIF NBSLT m
PCUBJOFE CZ B HSPVQ PG TUVEFOUT GPS BO BTTJHONFOU

$MBTT JOUFSWBMT BSF PęFO Marks (m) Midpoint Frequency (f) Frequency × midpoint
HJWFO VTJOH JOFRVBMJUZ ≤ m 2
TZNCPMT ȳ m
≤ m 5
NFBOT WBMVFT HSFBUFS PS ≤ m
FRVBM UP VQ UP WBMVFT UIBU ≤ m
BSF MFTT UIBO ćF OFYU ≤ m
DMBTT JOUFSWBM IBT WBMVFT ≤ m
FRVBM UP PS HSFBUFS UIBO

TP B WBMVF PG XPVME 5PUBM
HP JOUP UIF TFDPOE DMBTT
JOUFSWBM (a) $PQZ BOE DPNQMFUF UIF UBCMF
(b) $BMDVMBUF BO FTUJNBUF GPS UIF NFBO NBSL
Tip (c) *OUP XIJDI DMBTT EPFT UIF NPEBM NBSL GBMM
(d) 8IBU JT UIF NFEJBO DMBTT
*O TPNF UFYUT
UIF
NJEQPJOU JT DBMMFE UIF DMBTT 2 ćF UBCMF TIPXT UIF OVNCFS PG XPSET QFS NJOVUFT UZQFE CZ B HSPVQ PG DPNQVUFS QSPHSBNNFST
DFOUSF
Words per minute (w) Frequency
78 Unit 3: Data handling ≤ w
≤ w
≤ w
≤ w
≤ w
≤ w
5PUBM

(a) %FUFSNJOF BO FTUJNBUF GPS UIF NFBO OVNCFS PG XPSET UZQFE QFS NJOVUF
(b) )PX NBOZ XPSET EP NPTU PG UIF QSPHSBNNFST NBOBHF UP UZQF QFS NJOVUF
(c) 8IBU JT UIF NFEJBO DMBTT
(d) 8IBU JT UIF SBOHF PG XPSET UZQFE QFS NJOVUF

12 Averages and measures of spread

12.5 Percentiles and quartiles
t 1FSDFOUJMFT BSF VTFE UP EJWJEF B EBUB TFU JOUP FRVBM HSPVQT *G ZPV TDPSF JO UIF UI QFSDFOUJMF JO B UFTU
JU NFBOT

UIBU PG UIF PUIFS NBSLT BSF MPXFS UIBO ZPVST

t 2VBSUJMFT BSF VTFE UP EJWJEF B TFU PG EBUB JOUP GPVS FRVBM HSPVQT RVBSUFST


– ćF MPXFS RVBSUJMF 2
JT UIF WBMVF CFMPX XIJDI POF RVBSUFS PG UIF EBUB MJF

– ćF TFDPOE RVBSUJMF JT BMTP UIF NFEJBO PG UIF EBUB TFU 22


– ćF VQQFS RVBSUJMF 23
JT UIF WBMVF CFMPX XIJDI UISFF RVBSUFST PG UIF EBUB MJF PCWJPVTMZ UIF PUIFS RVBSUFS MJF
BCPWF UIJT


t ćF JOUFSRVBSUJMF SBOHF
*23 23 o 2

Tip Exercise 12.5

:PV DBO ĕOE UIF QPTJUJPO 1 'PS FBDI PG UIF GPMMPXJOH TFUT PG EBUB DBMDVMBUF UIF NFEJBO
VQQFS BOE MPXFS RVBSUJMFT *O FBDI
DBTF DBMDVMBUF UIF JOUFSRVBSUJMF SBOHF
PG UIF RVBSUJMFT VTJOH UIFTF

SVMFT

Q 1 ( 1) (a) 55
4 (b) 3
(c)
Q2 1 ( 1) (d) 2 2 3 3
2
3 5 55 2
23 3 ( 1)
4

8IFO UIF RVBSUJMF GBMMT

CFUXFFO UXP WBMVFT
ĕOE

UIF NFBO PG UIF UXP TDPSFT

Mixed exercise 1 'JOE UIF NFBO
NFEJBO
NPEF BOE SBOHF PG UIF GPMMPXJOH TFUT PG EBUB

(a)
(b)
(c)

2 ćF NFBO PG UXP DPOTFDVUJWF OVNCFST JT ćF NFBO PG FJHIU EJČFSFOU OVNCFST JT

(a) $BMDVMBUF UIF UPUBM PG UIF ĕSTU UXP OVNCFST
(b) 8IBU BSF UIFTF UXP OVNCFST
(c) $BMDVMBUF UIF NFBO PG UIF UFO OVNCFST UPHFUIFS

3 ćSFF TVQQMJFST TFMM TQFDJBMJTFE SFNPUF DPOUSPMMFST GPS BDDFTT TZTUFNT " TBNQMF PG SFNPUF
DPOUSPMMFST JT UBLFO GSPN FBDI TVQQMJFS BOE UIF XPSLJOH MJGF PG FBDI DPOUSPMMFS JT NFBTVSFE JO
XFFLT ćF GPMMPXJOH UBCMF TIPXT UIF NFBO UJNF BOE SBOHF GPS FBDI TVQQMJFS

Supplier Mean (weeks) Range (weeks)
A
B
C

8IJDI TVQQMJFS XPVME ZPV SFDPNNFOE UP TPNFPOF XIP JT MPPLJOH UP CVZ B SFNPUF
DPOUSPMMFS 8IZ

Unit 3: Data handling 79

12 Averages and measures of spread

4 " CPY DPOUBJOT QMBTUJD CMPDLT PG EJČFSFOU WPMVNF BT TIPXO JO UIF GSFRVFODZ UBCMF

Volume (cm3) 23 5
Frequency

(a) 'JOE UIF NFBO WPMVNF PG UIF CMPDLT
(b) 8IBU WPMVNF JT NPTU DPNNPO
(c) 8IBU JT UIF NFEJBO WPMVNF

5 ćF BHFT PG QFPQMF XIP WJTJUFE BO BSU FYIJCJUJPO BSF SFDPSEFE BOE PSHBOJTFE JO UIF HSPVQFE
GSFRVFODZ UBCMF CFMPX

Age in years (a) Frequency
≤ a
≤ a
≤ a
≤ a
≤ a
≤ a
≤ a
5PUBM

(a) &TUJNBUF UIF NFBO BHF PG QFPQMF BUUFOEJOH UIF FYIJCJUJPO
(b) *OUP XIBU BHF HSPVQ EJE NPTU WJTJUPST GBMM
(c) 8IBU JT UIF NFEJBO BHF PG WJTJUPST UP UIF FYIJCJUJPO
(d) 8IZ DBO ZPV OPU DBMDVMBUF BO FYBDU NFBO GPS UIJT EBUB TFU

6 ćF OVNCFS PG TUVEFOUT BUUFOEJOH B DIFTT DMVC PO WBSJPVT EBZT XBT SFDPSEFE












(a) 'JOE UIF NFEJBO PG UIF EBUB
(b) 8IBU JT UIF SBOHF PG UIF EBUB
(c) 'JOE 2 BOE 23 BOE IFODF DBMDVMBUF UIF *23
(d) $PNQBSF UIF SBOHF BOE UIF *23 8IBU EPFT UIJT UFMM ZPV BCPVU UIF EBUB

7 " UFBDIFS BOOPVODFT UIBU BMM TUVEFOUT XIP TDPSF CFMPX UIF UI QFSDFOUJMF JO B UFTU XJMM IBWF
UP SFXSJUF JU 8IFO UIF SFTVMUT BSF PVU
PVU PG UIF TUVEFOUT IBWF UP SFXSJUF UIF UFTU

(a) 8IBU EPFT UIJT UFMM ZPV BCPVU UIF TDPSFT
(b) 8IBU EPFT UIJT UFMM ZPV BCPVU UIF QFSGPSNBODF PG UIF DMBTT PWFSBMM

80 Unit 3: Data handling

13 Understanding measurement

13.1 Understanding units
t 6OJUT PG NFBTVSF JO UIF NFUSJD TZTUFN BSF NFUSFT N

HSBNT H
BOE MJUSFT M
4VC EJWJTJPOT IBWF QSFĕYFT TVDI BT

NJMMJ BOE DFOUJ UIF QSFĕY LJMP JT B NVMUJQMF

t 5P DPOWFSU GSPN B MBSHFS VOJU UP B TNBMMFS VOJU ZPV NVMUJQMZ UIF NFBTVSFNFOU CZ UIF DPSSFDU NVMUJQMF PG UFO
t 5P DPOWFSU GSPN B TNBMMFS VOJU UP B MBSHFS VOJU ZPV EJWJEF UIF NFBTVSFNFOU CZ UIF DPSSFDU NVMUJQMF PG UFO

To change to a smaller unit, multiply by conversion factor

× 1000 × 100 × 10

km m cm mm
kg g cg mg

kl l cl ml

.. 1000 .. 100 .. 10

To change to a larger unit, divide by conversion factor.

t "SFB JT BMXBZT NFBTVSFE JO TRVBSF VOJUT 5P DPOWFSU BSFBT GSPN POF VOJU UP BOPUIFS ZPV OFFE UP TRVBSF UIF
BQQSPQSJBUF MFOHUI DPOWFSTJPO GBDUPS

t 7PMVNF JT NFBTVSFE JO DVCJD VOJUT 5P DPOWFSU WPMVNFT GSPN POF VOJU UP BOPUIFS ZPV OFFE UP DVCF UIF
BQQSPQSJBUF MFOHUI DPOWFSTJPO GBDUPS

Exercise 13.1

Tip 1 6TF UIF DPOWFSTJPO EJBHSBN JO UIF CPY BCPWF BT B CBTJT UP ESBX ZPVS PXO EJBHSBNT UP TIPX
IPX UP DPOWFSU
.FNPSJTF UIFTF
DPOWFSTJPOT (a) VOJUT PG BSFB (b) VOJUT PG WPMVNF
NN = DN
DN = N 2 $POWFSU UIF GPMMPXJOH MFOHUI NFBTVSFNFOUT UP UIF VOJUT HJWFO
N = LN
NH = H (a) LN = N (b) DN = NN
H = LH (c) N = DN (d) N = LN
LH = U (e) N = NN (f) DN =
NM = MJUSF N
DN3 = NM
3 $POWFSU UIF GPMMPXJOH NFBTVSFNFOUT PG NBTT UP UIF VOJUT HJWFO

(a) LH = H (b) LH = H
(c) LH = H (d) H = LH
(e) H = LH (f) H =
UPOOF

Unit 4: Number 81

13 Understanding measurement

4 *EFOUJGZ UIF HSFBUFS MFOHUI JO FBDI PG UIFTF QBJST PG MFOHUIT ćFO DBMDVMBUF UIF EJČFSFODF
CFUXFFO UIF UXP MFOHUIT (JWF ZPVS BOTXFS JO UIF NPTU BQQSPQSJBUF VOJUT

(a) LN N (b) N DN
(c) DN NN (d) DN N
(e) N LN (f) LN DN

5 $POWFSU UIF GPMMPXJOH BSFB NFBTVSFNFOUT UP UIF VOJUT HJWFO

(a) DN2 = NN2 (b) DN = NN2
(c) DN = NN2 (d) LN2 = N2
(e) N2 = (f) N = NN2
LN2

Cubic centimetres or cm3 is 6 $POWFSU UIF GPMMPXJOH WPMVNF NFBTVSFNFOUT UP UIF VOJUT HJWFO
sometimes written as cc. For
example, a scooter may have a (a) DN3 = NN3 (b) DN3 = NN3
50 cc engine. That means the (c) DN3 = NN3 (d) N3 = DN3
total volume of all cylinders in the (e) NN3 = DN3 (f) NN3 = N3
engine is 50 cm3.

7 /BFFN MJWFT LN GSPN TDIPPM BOE 4BEJRB MJWFT N GSPN TDIPPM )PX NVDI DMPTFS UP
UIF TDIPPM EPFT 4BEJRB MJWF

8 " DPJO IBT B EJBNFUFS PG NN *G ZPV QMBDFE DPJOT JO B SPX
IPX MPOH XPVME
UIF SPX CF JO DN

9 " TRVBSF PG GBCSJD IBT BO BSFB PG NN2 8IBU BSF UIF MFOHUIT PG UIF TJEFT PG
UIF TRVBSF JO DN

10 )PX NBOZ DVCPJE TIBQFE CPYFT
FBDI XJUI EJNFOTJPOT DN × DN × DN
DBO ZPV
ĕU JOUP B WPMVNF PG N3

13.2 Time
t 5JNF JT OPU EFDJNBM I NFBOT POF IPVS BOE 1650 PS 14
PG BO IPVS
OPU I
t 0OF IPVS BOE NJOVUFT JT XSJUUFO BT
t 5JNF DBO CF XSJUUFO VTJOH B N BOE Q N OPUBUJPO PS BT B IPVS UJNF VTJOH UIF OVNCFST GSPN UP UP

HJWF UIF UJNFT GSPN NJEOJHIU PO POF EBZ I
UP POF TFDPOE CFGPSF NJEOJHIU

&WFO JO UIF IPVS DMPDL TZTUFN
UJNF JT OPU EFDJNBM ćF UJNF POF NJOVUF BęFS JT



Tip

:PV DBO FYQSFTT QBSUT PG BO IPVS BT B EFDJNBM %JWJEF UIF OVNCFS PG NJOVUFT CZ

'PS FYBNQMF NJOVUFT = 12 = 1 = 0 2 IPVST ćJT DBO NBLF ZPVS DBMDVMBUJPOT FBTJFS
60 5

82 Unit 4: Number

13 Understanding measurement

Exercise 13.2

1 'JWF QFPQMF SFDPSE UIF UJNF UIFZ TUBSU XPSL
UIF UJNF UIFZ ĕOJTI BOE UIF MFOHUI PG UIFJS
MVODI CSFBL

(a) $ PQZ BOE DPNQMFUF UIJT UBCMF UP TIPX IPX NVDI UJNF FBDI QFSTPO TQFOU BU XPSL PO
UIJT QBSUJDVMBS EBZ

Name Time in Time out Lunch Hours worked
%BXPPU 1 QBTU )BMG QBTU ĕWF
3 IPVS
/BEJSB 4 Q N
+PIO 4
3PCZO B N
.BSJ Q N 1 IPVS
2
B N
NJO

IPVS

NJO

(b) $BMDVMBUF FBDI QFSTPO T EBJMZ FBSOJOHT UP UIF OFBSFTU XIPMF DFOU JG UIFZ BSF QBJE
QFS IPVS

2 0O B QBSUJDVMBS EBZ
UIF MPX UJEF JO )POH ,POH IBSCPVS JT BU ćF IJHI UJEF JT BU
UIF TBNF EBZ )PX NVDI UJNF QBTTFT CFUXFFO MPX UJEF BOE IJHI UJEF

3 4BSBI T QMBOF XBT EVF UP MBOE BU Q N )PXFWFS
JU XBT EFMBZFE BOE JU MBOEFE BU )PX
NVDI MBUFS EJE UIF QMBOF BSSJWF UIBO JU XBT NFBOU UP

4 )PX NVDI UJNF QBTTFT CFUXFFO

(a) Q N BOE Q N PO UIF TBNF EBZ
(b) B N BOE Q N PO UIF TBNF EBZ
(c) Q N BOE B N UIF OFYU EBZ
(d) B N BOE PO UIF TBNF EBZ

5 6TF UIJT TFDUJPO GSPN B CVT UJNFUBCMF UP BOTXFS UIF RVFTUJPOT UIBU GPMMPX

Chavez Street
Castro Avenue
Peron Place
Marquez Lane

(a) 8IBU JT UIF FBSMJFTU CVT GSPN $IBWF[ 4USFFU
(b) )PX MPOH EPFT UIF KPVSOFZ GSPN $IBWF[ 4USFFU UP .BSRVF[ -BOF UBLF
(c) " CVT BSSJWFT BU 1FSPO 1MBDF BU RVBSUFS QBTU UFO ćF CVT JT NJOVUFT MBUF "U XIBU UJNF

EJE JU MFBWF $IBWF[ 4USFFU
(d) 4BODIF[ NJTTFT UIF CVT GSPN $BTUSP "WFOVF )PX MPOH XJMM IBWF UP XBJU CFGPSF UIF

OFYU TDIFEVMFE CVT BSSJWFT
(e) ćF CVT GSPN $IBWF[ 4USFFU JT EFMBZFE JO SPBEXPSLT CFUXFFO $BTUSP "WFOVF BOE

1FSPO 1MBDF GPS NJOVUFT )PX XJMM UIJT BČFDU UIF SFTU PG UIF UJNFUBCMF

Unit 4: Number 83

13 Understanding measurement

13.3 Upper and lower bounds
t "MM NFBTVSFNFOUT XF NBLF BSF SPVOEFE UP TPNF EFHSFF PG BDDVSBDZ ćF EFHSFF PG BDDVSBDZ GPS FYBNQMF UIF OFBSFTU

NFUSF PS UP UXP EFDJNBM QMBDFT
BMMPXT ZPV UP XPSL PVU UIF IJHIFTU BOE MPXFTU QPTTJCMF WBMVF PG UIF NFBTVSFNFOUT

ćF IJHIFTU QPTTJCMF WBMVF JT DBMMFE UIF VQQFS CPVOE BOE UIF MPXFTU QPTTJCMF WBMVF JT DBMMFE UIF MPXFS CPVOE

t 8IFO ZPV XPSL XJUI NPSF UIBO POF SPVOEFE WBMVF ZPV OFFE UP VTF UIF VQQFS BOE MPXFS CPVOET PG FBDI

dp means decimal places Exercise 13.3
sf means significant figures
1 &BDI PG UIF OVNCFST CFMPX IBT CFFO SPVOEFE UP UIF EFHSFF PG BDDVSBDZ TIPXO JO UIF
CSBDLFUT 'JOE UIF VQQFS BOE MPXFS CPVOET JO FBDI DBTF

(a) OFBSFTU XIPMF OVNCFS

(b) OFBSFTU XIPMF OVNCFS

(c) TG

(d) EQ

(e) EQ

(f) UP OFBSFTU UFOUI

(g) OFBSFTU UFO

(h) TG


2 " CVJMEJOH JT N UBMM NFBTVSFE UP UIF OFBSFTU NFUSF

(a) 8IBU BSF UIF VQQFS BOE MPXFS CPVOET PG UIF CVJMEJOH T IFJHIU
(b) * T NFUSFT B QPTTJCMF IFJHIU GPS UIF CVJMEJOH

&YQMBJO XIZ PS XIZ OPU

3 ćF EJNFOTJPOT PG B SFDUBOHVMBS QJFDF PG MBOE BSF N CZ N ćF NFBTVSFNFOUT BSF FBDI
DPSSFDU UP POF EFDJNBM QMBDF

(a) 'JOE UIF BSFB PG UIF QJFDF PG MBOE
(b) $BMDVMBUF UIF VQQFS BOE MPXFS CPVOET PG UIF BSFB PG UIF MBOE

4 6TBJO #PMU IPMET UIF XPSME SFDPSET GPS UIF N BOE N TQSJOUT
)F XBT
BMTP B NFNCFS PG UIF +BNBJDBO GPVS CZ N SFMBZ UFBN UIBU TFU B OFX XPSME SFDPSE PG
TFDPOET JO "VHVTU

(a) 6TBJO #PMU JT DN UBMM
DPSSFDU UP UIF OFBSFTU DFOUJNFUSF
BOE IJT NBTT JT LH
DPSSFDU
UP UIF OFBSFTU LJMPHSBN 'JOE UIF VQQFS BOE MPXFS CPVOET PG IJT IFJHIU BOE NBTT

(b) ćF +BNBJDBO DPBDI TBZT IJT UFBN DBO SVO UIF N SFMBZ JO TFDPOET #PUI UIFTF
NFBTVSFNFOUT BSF HJWFO UP UXP TJHOJĕDBOU ĕHVSFT 8IBU JT UIF NBYJNVN TQFFE JO
NFUSFT QFS TFDPOE
BU XIJDI UIFZ DBO SVO UIF SFMBZ (JWF ZPVS BOTXFS DPSSFDU UP UXP
EFDJNBM QMBDFT

5 ćF UXP TIPSU TJEFT PG B SJHIU BOHMFE USJBOHMF BSF DN UP UIF OFBSFTU NN
BOE DN UP
UIF OFBSFTU NN
$BMDVMBUF VQQFS BOE MPXFS CPVOET GPS

(a) UIF BSFB PG UIF SFDUBOHMF
(b) UIF MFOHUI PG UIF IZQPUFOVTF
(JWF ZPVS BOTXFST JO DFOUJNFUSFT UP GPVS EFDJNBM QMBDFT

84 Unit 4: Number

13 Understanding measurement

13.4 Conversion graphs
t $POWFSTJPO HSBQIT BMMPX ZPV UP DPOWFSU GSPN POF VOJU PG NFBTVSF UP BOPUIFS CZ QSPWJEJOH UIF WBMVFT PG CPUI VOJUT

PO EJČFSFOU BYFT 5P ĕOE POF WBMVF x
XIFO UIF PUIFS y
JT HJWFO
ZPV OFFE UP ĕOE UIF y WBMVF BHBJOTU UIF HSBQI BOE

UIFO SFBE PČ UIF DPSSFTQPOEJOH WBMVF PO UIF PUIFS BYJT

Tip Exercise 13.4Rupiah (in thousands)

.BLF TVSF ZPV SFBE UIF 1 4IFJMB
BO "VTUSBMJBO
JT HPJOH PO IPMJEBZ UP UIF JTMBOE PG #BMJ JO *OEPOFTJB 4IF ĕOET UIJT
MBCFMT PO UIF BYJT TP UIBU DPOWFSTJPO HSBQI UP TIPX UIF WBMVF PG SVQJBI UIF DVSSFODZ PG *OEPOFTJB
BHBJOTU UIF
ZPV BSF SFBEJOH PČ UIF "VTUSBMJBO EPMMBS
DPSSFDU WBMVFT
Exchange rate
Tip 600

ćF 64" NBJOMZ TUJMM 500
VTFT UIF 'BISFOIFJU
TDBMF GPS UFNQFSBUVSF 400
"QQMJBODFT
TVDI BT TUPWFT

NBZ IBWF UFNQFSBUVSFT 300
JO 'BISFOIFJU PO UIFN

QBSUJDVMBSMZ JG UIFZ BSF BO 200
"NFSJDBO CSBOE
100

0
0 10 20 30 40 50 60 70 80 90 100
Australian dollars

(a) 8IBU JT UIF TDBMF PO UIF WFSUJDBM BYJT
(b) )PX NBOZ SVQJBI XJMM 4IFJMB HFU GPS

(i) "VT (ii) "VT (iii) "VT
(c) ćF IPUFM TIF QMBOT UP TUBZ BU DIBSHFT SVQJBI B OJHIU

(i) 8IBU JT UIJT BNPVOU JO "VTUSBMJBO EPMMBST
(ii) )PX NVDI XJMM 4IFJMB QBZ JO "VTUSBMJBO EPMMBST GPS BO FJHIU OJHIU TUBZ

2 4UVEZ UIF DPOWFSTJPO HSBQI BOE BOTXFS UIF RVFTUJPOT

Conversion graph,
Celsius to Fahrenheit
250

200

150
Temperature

in °F 100

50

–20 0 20 40 60 80 100
Temperature
in °C

(a) 8IBU JT TIPXO PO UIF HSBQI

(b) 8IBU JT UIF UFNQFSBUVSF JO 'BISFOIFJU XIFO JU JT

(i) °$ (ii) °$ (iii) °$

Unit 4: Number 85

13 Understanding measurement

(c) 4 BSBI ĕOET B SFDJQF GPS DIPDPMBUF CSPXOJFT UIBU TBZT TIF OFFET UP DPPL UIF NJYUVSF BU
°$ GPS POF IPVS "ęFS BO IPVS TIF ĕOET UIBU JU IBT IBSEMZ DPPLFE BU BMM 8IBU DPVME
UIF QSPCMFN CF

(d) + FTT JT "NFSJDBO 8IFO TIF DBMMT IFS GSJFOE /JDL JO &OHMBOE TIF TBZT
A*U T SFBMMZ DPME IFSF

NVTU CF BCPVU EFHSFFT PVU 8IBU UFNQFSBUVSF TDBMF JT TIF VTJOH )PX EP ZPV LOPX UIJT

3 ćJT HSBQI TIPXT UIF DPOWFSTJPO GBDUPS GPS QPVOET JNQFSJBM NFBTVSFNFOU PG NBTT

BOE LJMPHSBNT

Conversion graph, pounds to kilograms
60

40
Kilograms

20

0 40 80 120 160
Pounds

(a) /FUUJF TBZT TIF OFFET UP MPTF BCPVU QPVOET )PX NVDI JT UIJT JO LJMPHSBNT
(b) +PIO TBZT IF T B XFBLMJOH )F XFJHIT QPVOET )PX NVDI EPFT IF XFJHI JO LJMPHSBNT
(c) 8IJDI JT UIF HSFBUFS NBTT JO FBDI PG UIFTF DBTFT

(i) QPVOET PS LJMPHSBNT
(ii) LJMPHSBNT PS QPVOET
(iii) LJMPHSBNT PS QPVOET

13.5 More money
t 8IFO ZPV DIBOHF NPOFZ GSPN POF DVSSFODZ UP BOPUIFS ZPV EP TP BU B HJWFO SBUF PG FYDIBOHF $IBOHJOH

UP BOPUIFS DVSSFODZ JT DBMMFE CVZJOH GPSFJHO DVSSFODZ

t &YDIBOHF SBUFT DBO CF XPSLFE PVU VTJOH DPOWFSTJPO HSBQIT BT JO

CVU NPSF PęFO
UIFZ BSF XPSLFE
PVU CZ EPJOH DBMDVMBUJPOT

t %PJOH DBMDVMBUJPOT XJUI NPOFZ JT KVTU MJLF EPJOH DBMDVMBUJPOT XJUI EFDJNBMT CVU ZPV OFFE UP SFNFNCFS UP JODMVEF
UIF DVSSFODZ TZNCPMT JO ZPVS BOTXFST

Tip Exercise 13.5

$VSSFODZ SBUFT DIBOHF BMM 6TF UIF FYDIBOHF SBUF UBCMF CFMPX GPS UIFTF RVFTUJPOT
UIF UJNF ćFTF SBUFT XFSF Currency exchange rates – October 2011
DPSSFDU JO
CVU UIFZ
NBZ CF WFSZ EJČFSFOU UPEBZ Currency US Euro UK Indian Aus Can SA NZ Yen
$ (€) £ rupee $ $ rand $ (¥)
The inverse rows show the 1 US $
exchange rate of one unit of the JOWFSTF
currency in the column to the 1 Euro
currency above the word inverse. JOWFSTF
For example, using the inverse row 1 UK £
below US $ and the euro column, JOWFSTF
€1 will buy $1.39.

86 Unit 4: Number

13 Understanding measurement

(a) 8IBU JT UIF FYDIBOHF SBUF GPS

(i) 64 UP ZFO (ii) 6,b UP /;

(iii) FVSP UP *OEJBO SVQFF (iv) $BOBEJBO EPMMBS UP FVSP

(v) ZFO UP QPVOE (vi) 4PVUI "GSJDBO SBOE UP 64

(b) )PX NBOZ *OEJBO SVQFFT XJMM ZPV HFU GPS

(i) 64 (ii) FVSPT (iii) b

(c) )PX NBOZ ZFO XJMM ZPV OFFE UP CVZ

(i) 64 (ii) FVSPT (iii) b

Mixed exercise 1 $POWFSU UIF GPMMPXJOH NFBTVSFNFOUT UP UIF VOJUT HJWFO

(a) LN UP NFUSFT (b) DN UP NN (c) UPOOFT UP LJMPHSBNT

(d) HSBNT UP LJMPHSBNT (e) HSBNT UP NJMMJHSBNT (f) MJUSFT UP NJMMJMJUSFT

(g) NM UP MJUSFT (h) DN2 UP NN (i) N UP LN2

(j) N3 UP DN (k) LN3 UP N (l) NN3 UP DN3

2 ćF BWFSBHF UJNF UBLFO UP XBML BSPVOE B USBDL JT POF NJOVUF BOE TFDPOET )PX MPOH XJMM
JU UBLF ZPV UP XBML BSPVOE UIF USBDL UJNFT BU UIJT SBUF

3 " KPVSOFZ UPPL I NJO BOE T UP DPNQMFUF 0G UIJT
I NJO BOE T XBT TQFOU IBWJOH
MVODI PS TUPQT GPS PUIFS SFBTPOT ćF SFTU PG UIF UJNF XBT TQFOU USBWFMMJOH )PX NVDI UJNF
XBT BDUVBMMZ TQFOU USBWFMMJOH

4 5BZP T IFJHIU JT N
DPSSFDU UP UIF OFBSFTU DN $BMDVMBUF UIF MFBTU QPTTJCMF BOE HSFBUFTU
QPTTJCMF IFJHIU UIBU IF DPVME CF

5 ćF OVNCFS PG QFPQMF XIP BUUFOEFE B NFFUJOH XBT HJWFO BT
DPSSFDU UP UIF OFBSFTU

(a) *T JU QPTTJCMF UIBU QFPQMF BUUFOEFE &YQMBJO XIZ PS XIZ OPU
(b) *T JU QPTTJCMF UIBU QFPQMF BUUFOEFE &YQMBJO XIZ PS XIZ OPU

6 ćF EJNFOTJPOT PG B SFDUBOHMF BSF DN BOE DN
FBDI DPSSFDU UP UISFF TJHOJĕDBOU
ĕHVSFT

(a) 8SJUF EPXO UIF SBOHF PG QPTTJCMF WBMVFT PG FBDI EJNFOTJPO
(b) 'JOE UIF MPXFS BOE VQQFS CPVOET PG UIF BSFB PG UIF SFDUBOHMF
(c) 8SJUF EPXO UIF MPXFS BOE VQQFS CPVOET PG UIF BSFB DPSSFDU UP UISFF TJHOJĕDBOU ĕHVSFT

Unit 4: Number 87

13 Understanding measurement

Litres Conversion graph for 7 4UVEZ UIF HSBQI BOE BOTXFS UIF RVFTUJPOT
imperial gallons to litres (a) 8IBU EPFT UIF HSBQI TIPX
45 (b) $POWFSU UP MJUSFT
40 (i) HBMMPOT (ii) HBMMPOT
35 (c) $POWFSU UP HBMMPOT
30 (i) MJUSFT (ii) MJUSFT
25 (d) /BSFTI TBZT IF HFUT NQH JO UIF DJUZ BOE NQH PO UIF IJHIXBZ JO IJT DBS
20 (i) $POWFSU FBDI SBUF UP LN QFS HBMMPO
15 (ii) (JWFO UIBU POF HBMMPO JT FRVJWBMFOU UP MJUSFT
DPOWFSU CPUI SBUFT
10 UP LJMPNFUSFT QFS MJUSF
5
0 6TF UIF FYDIBOHF SBUF UBCMF CFMPX SFQFBUFE GSPN &YFSDJTF
UP BOTXFS UIF GPMMPXJOH
0 2 4 6 8 10 RVFTUJPOT

Gallons (imperial) Currency exchange rates – October 2011

1 mile = 1.61 km $VSSFODZ 64 &VSP 6, b *OEJBO "VT $BO 4" /; :FO d

ħ
SVQFF SBOE
The US gallon is different from the 64
imperial gallon with a conversion JOWFSTF
factor of 1 US gallon to 3.785 litres. &VSP
*OWFSTF
6, b
JOWFSTF

8 +BO MJWFT JO 4PVUI "GSJDB BOE JT HPJOH PO IPMJEBZ UP *UBMZ )F IBT 3 UP FYDIBOHF GPS
FVSPT )PX NBOZ &VSPT XJMM IF HFU

9 1FUF JT BO "NFSJDBO XIP JT USBWFMMJOH UP *OEJB GPS CVTJOFTT )F OFFET UP FYDIBOHF
GPS SVQFFT
(a) 8IBU JT UIF FYDIBOHF SBUF
(b) )PX NBOZ SVQFFT XJMM IF HFU BU UIJT SBUF
(c) " U UIF FOE PG UIF USJQ IF IBT SVQFFT MFę PWFS 8IBU XJMM IF HFU JG IF DIBOHFT UIFTF
CBDL UP EPMMBST BU UIF HJWFO SBUF

10 +JNNZ JT #SJUJTI BOE IF JT HPJOH UP 4QBJO PO B QBDLBHF IPMJEBZ ćF DPTU PG UIF IPMJEBZ JT
FVSPT 8IBU JT UIJT BNPVOU JO 6, QPVOET

88 Unit 4: Number

14 Further solving of equations and
inequalities

14.1 Simultaneous linear equations
t 4JNVMUBOFPVT NFBOT ABU UIF TBNF UJNF
t ćFSF BSF UXP NFUIPET GPS TPMWJOH TJNVMUBOFPVT FRVBUJPOT HSBQIJDBMMZ BOE BMHFCSBJDBMMZ
t ćF HSBQIJDBM TPMVUJPO JT UIF QPJOU XIFSF UIF UXP MJOFT PG UIF FRVBUJPOT JOUFSTFDU ćJT QPJOU IBT BO x BOE B

y DPPSEJOBUF

t ćFSF BSF UXP BMHFCSBJD NFUIPET CZ TVCTUJUVUJPO BOE CZ FMJNJOBUJPO

o 4PNFUJNFT ZPV OFFE UP NBOJQVMBUF PS SFBSSBOHF POF PS CPUI PG UIF FRVBUJPOT CFGPSF ZPV DBO TPMWF UIFN
BMHFCSBJDBMMZ

o 'PS UIF TVCTUJUVUJPO NFUIPE
POF FRVBUJPO JT TVCTUJUVUFE JOUP UIF PUIFS

o 'PS UIF FMJNJOBUJPO NFUIPE ZPV OFFE FJUIFS UIF TBNF DPFďDJFOU PG x PS UIF TBNF DPFďDJFOU PG y JO CPUI
FRVBUJPOT

o *G UIF WBSJBCMF XJUI UIF TBNF DPFďDJFOU IBT UIF TBNF TJHO JO CPUI FRVBUJPOT
ZPV TIPVME UIFO TVCUSBDU
POF FRVBUJPO GSPN UIF PUIFS *G UIF TJHOT BSF EJČFSFOU UIFO ZPV TIPVME BEE UIF UXP FRVBUJPOT

o *G BO FRVBUJPO DPOUBJOT GSBDUJPOT
ZPV DBO NBLF FWFSZUIJOH NVDI FBTJFS CZ AHFUUJOH SJE PG UIF GSBDUJPOT
.VMUJQMZ FBDI UFSN CZ B TVJUBCMF OVNCFS B DPNNPO EFOPNJOBUPS
BOE ADMFBS UIF EFOPNJOBUPST PG UIF GSBDUJPOT

Exercise 14.1 A

1 %SBX UIF HSBQIT GPS FBDI QBJS PG FRVBUJPOT HJWFO ćFO VTF UIF QPJOU PG JOUFSTFDUJPO UP ĕOE
UIF TJNVMUBOFPVT TPMVUJPO ćF MJNJUT PG UIF x BYJT UIBU ZPV TIPVME VTF BSF HJWFO JO FBDI DBTF

(a) x 3 y − < x <

x y = 0

(b) x 2 y − < x <

x 2y = 3

(c) 2x + 2
2y 3x −1 0 − < x <


(d) x = 4 − < x <

y x − 7

(e) x 3y = 3 − < x <

y 2x + 3

Unit 4: Algebra 89

14 Further solving of equations and inequalities

2 ćF HSBQI CFMPX TIPXT MJOFT DPSSFTQPOEJOH UP TJY FRVBUJPOT

(a) 'JOE UIF FRVBUJPOT PG MJOFT A UP F

(b) 6TF UIF HSBQIT UP ĕOE UIF TPMVUJPOT UP UIF GPMMPXJOH QBJST PG TJNVMUBOFPVT FRVBUJPOT

(i) A BOE C (ii) D BOE F (iii) A BOE E

(c) /PX DIFDL ZPVS TPMVUJPOT BMHFCSBJDBMMZ

10 y D
AC

8B

6F

4

2 x
2 4 6 8 10
–10 –8 –6 –4 –2 0
–2 E
–4

–6

–8

–10

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WBMVFT JOUP POF PG UIF FRVBUJPOT

(a) y = 2 (b) y x = 3 (c) x + y = 4 (d) 2x y 7
x+y=6 y 3x = 5 2x + 3y = 12 3x y 8

4 4PMWF GPS x BOE y CZ VTJOH UIF FMJNJOBUJPO NFUIPE $IFDL FBDI TPMVUJPO

(a) x + y = 5 (b) 3 y 1 (c) 2x 3y = 12
x y=7 2x y 4 3x 3y = 30

(d) 2x 3y = 6 (e) 2x 5y = 11 (f) y 2x = 1
4x 6y = −4 3x 2y = 7 2y 3x = 5

Remember that you might 5 4PMWF TJNVMUBOFPVTMZ
need to rearrange one or both
equations before solving. 3x y 7 2x + y = 3 3x 2y = 4 x y
(a) 3x y 5 2 (c) 2x y 3 3 2
(b) (d) + = 2

x + y = 3 x + 4 y 11
2

5x + y = 10 3x + y = 4 7x 6y = 17 3x + y = 1
2 4 2 2 2
(e) (f ) (g) 2x 1 (h)
x 1 3y 3 + y = 6 3 5x 3 7y = 2
4 + y = 3 4 x + 2 = 7

2x 9y

(i) x = 6y 9
3

90 Unit 4: Algebra

14 Further solving of equations and inequalities

6 4BMMZ CPVHIU UXP DIPDPMBUF CBST BOE POF CPY PG HVNT GPS BOE &WBO QBJE GPS POF
DIPDPMBUF BOE UXP CPYFT PG HVNT *G UIFZ CPVHIU UIF TBNF CSBOET BOE TJ[FT PG QSPEVDUT
XIBU
JT UIF DPTU PG B DIPDPMBUF CBS BOE UIF DPTU PG B CPY PG HVNT

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CF NFNCFST PG UIF #PUBOJDBM 4PDJFUZ BOE JG ZPV BSF B NFNCFS
ZPV DBO WJTJU UIF HBSEFOT GPS GSFF
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GPS GSFF BOE JU DPTU UIF SFTU PG UIF HSPVQ UP HP JO )PX NBOZ TUVEFOUT XFSF JO UIF HSPVQ

8 &QISBIJN IBT DPJOT JO IJT QPDLFU
DPOTJTUJOH PG RVBSUFST BOE EJNFT POMZ *G IF IBT JO
IJT QPDLFU
IPX NBOZ PG FBDI DPJO EPFT IF IBWF

Exercise 14.1 B

1 'PSN UXP FRVBUJPOT VTJOH UIF JOGPSNBUJPO PO FBDI EJBHSBN BOE TPMWF UIFN TJNVMUBOFPVTMZ
UP ĕOE UIF WBMVFT PG x BOE y

(a) 20 (b) – 3x (c) 20
y –3y – 6
6y 2x + 4

2 x + 6 9y 2x
3

2y – 4x 4x – 2y

(d) (e) 2x + 7 (f )
x+y 5x – 2y
9–y

1 x 5 – 3y
2 2

3x 2x + 4

9 + 3y

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B TIPQ TFMMT B UPUBM PG EFTLT BOE DIBJST GPS B UPUBM BNPVOU PG ćF EFTLT
TFMM GPS FBDI BOE UIF DIBJST TFMM GPS FBDI
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MFU c FRVBM UIF OVNCFS PG DIBJST
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14.2 Linear inequalities
t ćF TPMVUJPO UP B MJOFBS JOFRVBMJUZ JT B SBOHF PG WBMVFT
t ćF TPMVUJPO DBO CF SFQSFTFOUFE PO B OVNCFS MJOF

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Unit 4: Algebra 91

14 Further solving of equations and inequalities

Tip Exercise 14.2

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XIBU 1 %SBX B OVNCFS MJOF UP SFQSFTFOU UIF QPTTJCMF WBMVFT PG UIF WBSJBCMF JO FBDI DBTF
ZPV EP UP POF TJEF PG BO
JOFRVBMJUZ ZPV NVTU EP UP (a) x ȳ (b) y ≥ o (c) f o
UIF PUIFS #VU
XIFO ZPV (d) ȳ a ȳ (e) o ȴ n (f) –m
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TJEFT PG BO JOFRVBMJUZ CZ B
OFHBUJWF OVNCFS
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SFNFNCFS UP SFWFSTF UIF
EJSFDUJPO PG UIF JOFRVBMJUZ (a) x ȳ (b) o ȳ h ȳ (c) 2 a
(d) o ȳ s ȳ (e) − 1 ȳ e (f) ɀ b ȳ ɀ

3

3 4PMWF FBDI PG UIF GPMMPXJOH JOFRVBMJUJFT 4PNF PG UIF BOTXFST XJMM JOWPMWF GSBDUJPOT -FBWF

ZPVS BOTXFST BT GSBDUJPOT JO UIFJS TJNQMFTU GPSN XIFSF BQQSPQSJBUF

(a) x ȳ o (b) x ¦ (c) x ¦ ȴ (d) x ȳ
4

(e) x + 8 ȴ (f ) x − 5 (g) x ¦
(h) 2 e ȳ e
3 4
3

(i) r + 1 (j) x ¦
ȴ x
(k) g ¦ ¦ g
4 8

(l) y+4 ≥ y − 1 (m) 3 3 6 o (n) n ¦
¦ ¦ n
n

12 2

(o) 3 ⎛ 2x 1⎞ 2 (3x 33) ≥ 5 4 3+1
8 ⎝ 3⎠
9 3

14.3 Regions in a plane
t 8IFO UIF SFMBUJPOTIJQ CFUXFFO UXP WBSJBCMFT JT FYQSFTTFE BT BO JOFRVBMJUZ
UIBU SFMBUJPOTIJQ JT SFQSFTFOUFE HSBQIJDBMMZ

BT B SFHJPO PO UIF $BSUFTJBO QMBOF

t 8IFO UIF JOFRVBMJUZ JODMVEFT FRVBM UP ȳ PS ȴ

UIF CPVOEBSZ MJOF PO UIF HSBQI NVTU CF JODMVEFE BT B TPMJE MJOF PO
UIF HSBQI

t 8IFO UIF JOFRVBMJUZ EPFT OPU JODMVEF FRVBM UP PS

UIF CPVOEBSZ MJOF JT TIPXO BT B CSPLFO MJOF

Tip Exercise 14.3

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TIPX UIF SFHJPO UIBU SFQSFTFOUT UIF GPMMPXJOH JOFRVBMJUJFT CZ TIBEJOH UIF
JT JO UIF GPSN y mx D
VOXBOUFE SFHJPO
UIFO
(a) y x (b) y ȴ o x (c) y > –x
t UIF JOFRVBMJUZ
y mx c JT BCPWF UIF 2 4IBEF UIF VOXBOUFE SFHJPO UIBU SFQSFTFOUT FBDI JOFRVBMJUZ PO TFQBSBUF BYFT
line
(a) y ȳ o (b) x ¦ y (c) x ¦ y ȴ
t UIF JOFRVBMJUZ
y mx c JT CFMPX (d) x y ¦ (e) o y ȳ (f ) x y > 3
UIF MJOF 2
*G UIF FRVBUJPO JT OPU JO 4
UIF GPSN y mx c
ZPV
OFFE UP DIPPTF B QPJOU UP
POF TJEF PG UIF MJOF BOE
UFTU XIFUIFS JU JT JO PS OPU
JO UIF SFHJPO

92 Unit 4: Algebra