# # 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

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1 How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0 # 000 on thousnth th iml pl: ' 0 # on tn thousnth t # ' 0 ' 00 Tnths Hunrths Thousnths Tn thousnths W H O L E D E C I M A ' 000 ' ' Hunr thousnths Millionths ' L ' A th to th nm or iml pl vlus Tn Millionths Writ th pl vlu o h igit in th numr Multiply y multipls o 0 Divi y multipls o 0 Expn orms Pl vlus # hunr # 0 0 tns (or sixty) Intgr prts # ons (or iv) ' 0 or ` # 0 j 7 7' 00 or 7 ` # 00 j 0 0' 000 or 0 ` # 000 j ' or ` # j tnths 7 hunrths 0 thousnths tn thousnths st iml pl n iml pl r iml pl th iml pl H Mthltis Pssport P Lrning

2 How os it work? Your Turn Pl vlu o imls Writ th iml tht rprsnts ths: hunrths 9 tnths tn thousnth 00 Alwys put zro in ront (ll ling zro) whn thr r no whol numrs thousnths hunr thousnths 8 millionths Writ th rtion tht rprsnts ths: tnths 7 thousnths hunrth 9 tn thousnths hunrths tn thousnths Writ th pl vlu o th igit writtn in squr rkts or h o Cirl th igit oun in th pl vlu givn in squr rkts: [tnths] [thousnths] [hunr thousnths] [hunrths] [tn thousnths] [millionths] Mthltis Pssport P Lrning H SERIES TOPIC

3 How os it work? Your Turn Pl vlu o imls Eh igit is multipli y th pl vlu n thn togthr whn writing numr in xpn orm Writ th iml 0 in xpn orm # + # + # + # + # # + # + # + # Multiply h igit y its pl vlu Zro igits n rmov to simpliy Writ ths imls in xpn orm: Simpliy ths numrs writtn in xpn orm: # + # + # # + # + # + # 0 00 //0 LACE VALUE OF DECIMALS PPLACE VALUE OF DECIMALS # + # + # + # + # + # # + # + # + # + # + 9 # Psst: Rmmr to inlu ling zro or ths ons g # # # 7 0 # + # + # + # # # # # # H Mthltis Pssport P Lrning

4 How os it work? Approximtions through rouning numrs Look t ths two sttmnts m out tm o snoworrs: Thy hv ttmpt 97 triks sin strting Aurt sttmnt Thy hv ttmpt nrly 000 triks sin strting Roun o pproximtion Rouning o vlus is us whn grt l o ury is not n Th nxt igit ollowing th pl vlu whr numr is ing roun o to is th importnt prt Nxt igit Closr to lowr vlu, so roun own Lv th pl vlu unhng Closr to highr vlu, so roun up A to th pl vlu Hr r som xmpls to s how w roun o numrs Roun ths numrs (i) to th nrst hunr 0 0 Th igit is in th hunrs position Th nxt igit is, so roun up y ing to Chng th othr smllr pl vlu igits to 0 s ` 00 roun to th nrst hunr (ii) 0 to on iml pl (or to th nrst tnth) Th igit is in th irst iml pl Th nxt igit is, so roun own Writ iml with on iml pl only ` 0 0 roun to on iml pl (iii) 89 to our iml pls (or to th nrst tn thousnth) Th igit is in th ourth iml pl Th nxt igit is 9, so roun up y ing to Writ iml with our iml pls only ` 89 8 roun to our iml pls Mthltis Pssport P Lrning H SERIES TOPIC

5 How os it work? Your Turn APPROXIMATION THROUGH ROUNDING NUMBERS Approximtions through rouning numrs Roun ths whol numrs to th pl vlu givn in squr rkts //0 [nrst tn] [nrst hunr] [nrst thousn] (i) (i) 0 (i) 98 (ii) 8 (ii) 7 (ii) 8 0 (iii) 90 (iii) (iii) 00 Roun ths imls to th iml pls givn in th squr rkts [nrst tnth] [nrst hunrth] [nrst thousnth] (i) 07 (i) 0 (i) 0 7 (ii) 7 (ii) (ii) 08 (iii) 8 (iii) 00 (iii) 0080 Approximt th ollowing istn msurmnts: A group o popl orm n 88 m long lin whn thy stn togthr (i) How long is this lin to th nrst 0 m (i iml pl)? (ii) Wht is th pproximt lngth o this lin to th nrst 0 mtrs? Unr mirosop th lngth o ust mit ws m (i) Approximt th lngth o this ust mit to th nrst tn thousnth o mtr (ii) Approximt th lngth o this ust mit to th nrst hunrth o mtr I Lihn City is 8 m wy rom Moss City: (i) Wht is this istn pproximt to th nrst km? (i nrst thousn) (ii) Wht is th pproximt istn twn th itis to th nrst 00 km? (iii) Ar th igits, or vn importnt to inlu whn sriing th totl istn twn th two itis? Brily xplin hr why/why not H Mthltis Pssport P Lrning

6 How os it work? Your Turn Approximtions through rouning numrs Rouning up n t mor thn on igit whn th numr 9 is involv Roun 09 to on iml pl 0 9 Th igit 9 is in th tnths position 9 rouns up to 0, so th 9 oms 0 n is to th igit in ront Th nxt igit is, so roun up y ing to 9 Chng th othr smllr pl vlu igits to 0s ` 09 0 roun to on iml pl Roun o ths numrs oring to th squr rkts [on iml pl] 98 [nrst ons] 79 9 g [nrst thousn] 9798 [nrst tn] 98 [thr iml pls] 098 h [nrst ons] 999 [two iml pls] 899 [thr iml pls] 99 i [our iml pls] Approximt ths vlus: A ll ntr rivs n vrg o 99 lls h y uring on month (i) Approximt th numr o lls riv to th nrst hunrs (ii) Approximtly how mny thousns o lls i thy riv? (iii) Estimt th numr o lls riv ily throughout th month A swimming pool h slow lk, using it to mpty L in on wk (i) How muh wtr ws lost to th nrst 0 litrs? (ii) How muh wtr ws lost to th nrst ml i ml 000 L? (iii) Is th igit importnt whn pproximting to th nrst whol litr? Brily xplin hr why/why not Mthltis Pssport P Lrning H SERIES TOPIC 7

7 How os it work? on th numr lin Th smllst pl vlu in iml is us to position points urtly on numr lin r s on th numr 0, so thr r lwys tn ivisions twn vlus Eg: Hr is th vlu on numr lin: 0 0 Six tnths o th wy rom 0 to 0 Th mjor intrvls on th numr lin r mrk oring to th son lst iml pl vlu 8 So its ight thousnths o th wy rom 0 to Hr r som mor xmpls involving numr lins: (i) Wht vlu o th plott points rprsnt on th numr lins low? ) 0 0 Point is our stps rom 0 towrs 0, so th plott point is: 0 ) Point is nin stps rom 00 towrs 007, so th plott point is: 009 (ii) Roun th vlu o th plott points low to th nrst hunrth ) Point is thr stps rom towrs, so th plott point is ` th vlu o th plott point to th nrst hunrth is: ) Point is iv stps rom 879 towrs 880, so th plott point is 879 ` th vlu o th plott point to th nrst hunrth is: H Mthltis Pssport P Lrning

8 How os it work? Your Turn on th numr lin Disply ths imls on th numr lins low: 07 ECIMALS ON THE NUMBER LINE //0 DDECIMALS ON THE NUMBER LINE Ll ths numr lins n thn isply th givn iml on thm: Roun th vlu o th plott points low to th nrst pl vlu givn in squr rkts [tnth] [hunrth] ` th vlu ` th vlu [tnth] [hunrth] ` th vlu ` th vlu [thousnth] [thousnth] ` th vlu ` th vlu g [thousnth] h [thousnth] ` th vlu ` th vlu Mthltis Pssport P Lrning H SERIES TOPIC 9

9 How os it work? Multiplying n iviing y powrs o tn Mov th iml point pning on th numr o zros iml point movs right, iml point movs lt Clult ths multiplition n ivision qustions involving powrs o 0: (i) # 000 W n simply th sm numr o zros to th n o th whol numr # # Th whol numr in iml orm Fill th mpty ouns with 0s I th iml point is on th lt tr iviing, n xtr 0 is pl in ront (ii) 8 00 ' 8' ' 00 Th whol numr in iml orm Rmmr to inlu th ling zro 80 '00 hs zros, so mov iml point sps lt 008 Fill th mpty ouns with 0s n put zro in ront (iii) 89 # # Mov iml point sps right 89 No mpty ouns to ill, so this is th nswr (iv) 90' ' Mov iml point sps lt Fill mpty ouns with 0s n put zro in ront (v) 0 # # 0 ' # is th sm s ' Mov iml point sps lt 00 Pl ling zro in ront o th iml point 0 H Mthltis Pssport P Lrning

10 How os it work? Your Turn Multiplying n iviing y powrs o tn Clult ths multiplitions Rmmr, multiply mns mov iml point to th right: 8 00 # # 0 9 # # 0 # # Clult ths ivisions Rmmr, ivi mns mov iml point to th lt: ' ' ' ' ' 900 ' Hr r som o th powrs o 0 in inx orm Th powr th numr o zros Clult ths mix prolms writtn in inx orm: 0 # 00 ' # # 0 ' 9 9 ' 0 7 Mthltis Pssport P Lrning H SERIES TOPIC

11 MULTIPLYING AND DIVIDING BY POWERS OF TEN How os it work? Your Turn Multiplying n iviing y powrs o tn For ths lultions: (i) Show whr our hrtr ns to spry pint nw iml point, n (ii) writ own th two numrs th nw iml point is twn to solv th puzzl // # I 9 n ' 8 7 N 0789 # A ' 8 9 O # X 8970# T g # R h ' I i # D j # P This is nothr mthmtil nm or iml point: I 0 n 9 8 n 9 8 n 7 9 n 0 n 7 n 9 8 n 0 n 8 n 8 n 7 H Mthltis Pssport P Lrning

12 How os it work? Trminting imls to rtions Ths hv iml prts whih stop (or trmint) t prtiulr pl vlu Th pl vlu o th lst igit on th right hlps us to writ it s rtion Writ 0 s rtion: Diml 0 Frtion 0 Diml igits in th numrtor Lst igit is in tnths position Intgrs in ront o th iml vlus r simply writtn in ront o th rtion Writ 07 s rtion: Diml igits in th numrtor Lst igit is in hunrths position 07 is just 7 Alwys simpliy th rtion prts i possil Ths two xmpls show you how Writ h o ths imls s n quivlnt (qul) rtion in simplst orm (i) 0 0 Equivlnt, un-simplii rtion 00 Lst igit is in hunrths position ' 00 ' Divi numrtor n nomintor y HCF Equivlnt rtion in simplst orm (ii) Equivlnt, un-simplii mix numrl Lst igit is in thousnths position 0 ' 000 ' Divi numrtor n nomintor y HCF 00 Equivlnt mix numrl in simplst orm Mthltis Pssport P Lrning H SERIES TOPIC

13 How os it work? Your Turn Trminting imls to rtions Writ h o ths imls s quivlnt rtions: g 00 h i 09 j k 0007 l Writ h o ths imls s quivlnt rtions n thn simpliy: g 0 h 0 i 00 j 00 k 0008 l 000 H Mthltis Pssport P Lrning

14 TERMINATING DECIMALS TO FRACTIONS * Whr os it work? Your Turn Trminting imls to rtions Writ h o ths imls s quivlnt mix numrls: 0 // Writ h o ths imls s quivlnt mix numrls n thn simpliy: g 00 h 0 i Mthltis Pssport P Lrning H SERIES TOPIC

15 How os it work? Frtions to trminting imls Whr possil, just writ s n quivlnt rtion with powr o 0 in th nomintor irst numrtor nomintor # # 0 ` 0 Multiply numrtor n nomintor y th sm vlu Equivlnt rtion with powr o 0 in th nomintor Thr iths six tnths zro point six Somtims it is sir to irst simpliy th rtion or hnging to iml Writ ths s n quivlnt iml (i) ' ' Simpliy rtion # # 00 Equivlnt rtion with powr o 0 in th nomintor ` 0 Thr twlths on qurtr twnty iv hunrths zro point two iv (ii) ' ' Simpliy rtion prt # # 0 Equivlnt rtion with powr o 0 in th nomintor Two n thr itnths two n on ith two n two tnths two point two H Mthltis Pssport P Lrning

16 How os it work? Your Turn Frtions to trminting imls inlu ling zro Writ h o ths rtions s quivlnt imls Writ h o ths s quivlnt rtions with powr o 0 in th nomintor g h 0 00 i j k 7 0 (i) Writ h o ths s quivlnt rtions with powr o 0 in th nomintor (ii) Chng to quivlnt imls 00 g 9 h 00 i Mthltis Pssport P Lrning H SERIES TOPIC 7

17 How os it work? Your Turn Frtions to trminting imls Chng h o ths rtions to quivlnt imls tr irst simpliying Show ll your working FRACTIONS TO TERMINATING DECIMALS 0 // g 00 h 0 8 H Mthltis Pssport P Lrning

18 How os it work? Your Turn Frtions to trminting imls Whn hnging th nomintor to powr o 0 is not sy, you n writ th numrtor s iml n thn ivi it y th nomintor Writ this rtion s n quivlnt iml ' 8 Writ numrtor s iml n ivi y th nomintor I you n mor iml pl 0s, you n thm in ltr! g g Complt ivision, kping th iml point in th sm pl ` 0 Fiv ighths zro point six two iv Complt ths ivisions to in th quivlnt iml: 000 ' 000 ' 000 ' 8 8 g 000 g g ' 000 ' ' 8 g g g Mthltis Pssport P Lrning H SERIES TOPIC 9

19 How os it work? Your Turn Frtions to trminting imls Simpliy ths rtions n thn writ s n quivlnt iml using th ivision mtho Show ll your working H Mthltis Pssport P Lrning

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