UNCORRECTED. 6E Multiplication of decimals 6F Division of decimals 6G Decimals and fractions 6H Decimals and percentages

Transcription

1 Chpter 6Decimls Wht you will lern 6A Decimls nd plce vlue (Consolidting) 6B Rounding decimls 6C Addition nd sutrction of decimls (Consolidting) 6D Multiplying nd dividing y powers of 0 Victorin curriculum NUMBER AND ALGEBRA Rel numers Multiply nd divide frctions nd decimls using efficient written strtegies nd digitl technologies (VCMNA44) Round decimls to specified numer of deciml plces (VCMNA46) Connect frctions, decimls nd percentges nd crry out simple conversions (VCMNA47) 6E Multipliction of decimls 6F Division of decimls 6G Decimls nd frctions 6H Decimls nd percentges Find percentges of quntities nd express one quntity s percentge of nother, with nd without digitl technologies. (VCMNA48) Money nd finncil mthemtics Investigte nd clculte est uys, with nd without digitl technologies (VCMNA0)

2 Decimls t the MotoGP The clcultion of lp times nd finishing times of the Austrlin MotoGP t Phillip Islnd is the responsiility of group of timing officils, who, with the id of the ltest time keeping technology from Tissot, clculte the time for ech rider correct to the nerest millisecond. They do this y plcing high tech rdio module underneth the front firing of ech ike, which communictes with the digitl sensors plced long the trck, including the finish line. This timing technology, long with high-speed cmers, is used to ensure tht single winner cn e crowned. Online resources Chpter pre-test Video demonstrtions of ll worked exmples Interctive widgets Interctive wlkthroughs Downlodle HOTsheets Access to ll HOTmths Austrlin Curriculum courses Access to the HOTmths gmes lirry In 0 the rce ws won y 0.49 seconds y Mrc Mrquez with rce time of (40 minutes nd.849 seconds). His grid time for the rce ws He hd pole position nd et the 0 world chmpion Jorge Lorenzo, who cme in second. On the 4448 m Phillip Islnd circuit, Mrc s fstest lp time of 8.08 is record for the course. This time ws first recorded in 0, rce won y Jorge Lorenzo, in time of on the reduced 9 lps in rce full of drm, including ike swps nd tyre issues nd rce in which Mrc ws disqulified.

3 06 Chpter 6 Decimls 6A Decimls nd plce vlue CONSOLIDATING Key ides Some quntities chnge y whole numer mounts, such s the numer of people in room, ut there re mny quntities tht increse or decrese continuously, such s your height, weight nd ge. We often tlk out ge s whole numer (e.g. Mike is yers old) ut, in relity, our ge is n ever-incresing (continuous) quntity. For exmple, if Mike is yers, 4 months, weeks, dys, hours, 6 minutes nd seconds old, then Mike is ctully yers old! The price of chocolte r is often expressed using decimls. e.g $.8 There re mny numers in tody s society tht re not whole numers. For exmple, it is unusul to uy n item in supermrket tht is simply whole numer of dollrs. The price of lmost ll shopping items involves oth dollrs nd cents. A chocolte r my cost $.9, which is n exmple of deciml numer. Let s strt: Split-second timing Orgnise students into pirs nd use digitl stopwtch. Mny students wtches will hve suitle stopwtch function. Try to stop the stopwtch on exctly 0 seconds. Hve two ttempts ech. Were you le to stop it exctly on 0.00 seconds? Wht ws the closest time? Try these dditionl chllenges with your prtner. c Stop the wtch exctly on: i iii seconds seconds How quickly cn you strt nd stop the stopwtch? How ccurtely cn you time minute without looking t the stopwtch? ii iv 8.7 seconds 4. seconds A deciml point is used to seprte the whole numer from the deciml or frction prt. When deling with deciml numers, the plce vlue tle must e extended to involve tenths, hundredths, thousndths etc. The numer 48.7 mens: Hundreds Tens Units. Tenths Hundredths Thousndths

4 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 07 Exmple Understnding deciml plce vlue Wht is the vlue of the digit 8 in the following numers? SOLUTION The vlue of 8 is The vlue of 8 is 0 Exmple Chnging to decimls EXPLANATION The 8 is in the first column fter the deciml point, which is the tenths column. The 8 is in the third column fter the deciml point, which is the thousndths column. Express ech of the following proper frctions nd mixed numerls s decimls. 7 0 SOLUTION c 7 EXPLANATION 7 0 = mens seven-tenths, so put the 7 in the 0 tenths column. = 0.0 mens five-hundredths, so put the in the hundredths column. c 7 =.7 7 mens units nd 7 one-hundredths. 7 hundredths is one-tenth nd seven-hundredths.

5 08 Chpter 6 Decimls Exmple Exmple, Exmple Arrnging deciml numers in order Arrnge the following deciml numers in scending order (i.e. smllest to lrgest)..7, 7.,.7,.7, 7.,.7 SOLUTION EXPLANATION.7,.7,.7,.7, 7., 7. The units column hs higher vlue thn the tenths column, nd the tenths column hs higher vlue thn the hundredths column..7 is igger thn.7 ecuse it hs seven-tenths, which is igger thn three-tenths. Exercise 6A For the numer 8.7, give the vlue of the following digits. c 7 A stopwtch is stopped t 6.7 seconds. Wht is the digit displyed in the tenths column? Wht is the digit displyed in the units column? c Wht is the digit displyed in the hundredths column? d Is this numer closer to 6 or 7 seconds? Wht is the vlue of the digit 6 in the following numers? c d 0.69 e 6.4 f 8.68 g.64 h.96 4 Stte whether ech of the following sttements is true or flse. 7.4 < 7.8. < 0.89 c 4.6 > 4.7 d 8.09 > 8.4 e.8 Ä 8. f.8 Ä.800 g 7.9 Å 8.4 h. Å.09 i 0 = 0 j 7 0 = 70 Express ech of the following proper frctions s deciml. e f 8 0 k 0 l 0 0 c g 4 7(½) 0 4 7(½) d h (½) UNDERSTANDING FLUENCY

6 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 09 Exmple c Exmple 6 Express ech of the following mixed numerls s deciml c d e 4 8 f 7 g 7 h Write the following numer phrses s decimls. seven nd six-tenths c thirty-three nd four-hundredths d e eight nd forty-two hundredths f twelve nd nine-tenths twenty-six nd fifteen-hundredths ninety-nine nd twelve-thousndths 8 Find the difference etween ech of the following deciml numers nd their nerest whole numer c 8.98 d 6. e f 4.99 g 0.8 h , 9 9, 0 Arrnge these groups of numers in scending order (i.e. smllest to lrgest)..,.0,.,. 0.6,.06,.6, 0. c 7.8,.78,.87,.87 d 6.9, 9.6, 9.6,.96,.69 0 The tting verges for five retired Austrlin Cricket test cptins re: Adm Gilchrist 47.60, Steve Wugh.06, Mrk Tylor 4.49, Alln Border 0.6 nd Kim Hughes 7.4. List the five plyers in descending order of tting verges (i.e. lrgest to smllest). Ricky Ponting s test tting verge is 6.7. Where does this rnk him in terms of the retired Austrlin test cptins listed ove? 9 FLUENCY PROBLEM-SOLVING 6A

7 0 Chpter 6 Decimls 6A The depth of river t 9:00 m on six consecutive dys ws: Dy :. m Dy :.8 m Dy :.49 m Dy 4:.47 m Dy :. m Dy 6:.6 m On which dy ws the river level highest? On which dy ws the river level lowest? c On which dys ws the river level higher thn the previous dy?, nd c re digits nd > > c. Write these numers from smllest to lrgest. Note tht the dot represents the deciml point..,.c,.c, c.c, c.,..c,.c,., c., c.c,.c,.,.,.c Write s decimls, if is digit. 0 Different deciml comintions c 0 + d, Write s mny different deciml numers s you cn nd plce them in scending order using: i the digits 0, nd deciml point. Ech digit cn e used only once. ii the digits 0,, nd deciml point. Ech digit cn e used only once. iii the digits 0,,, nd deciml point. Ech digit cn e used only once. Clculte the numer of different deciml numers tht could e produced using the digits 0,,,, 4 nd deciml point. 4 PROBLEM-SOLVING REASONING ENRICHMENT

8 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 6B Rounding decimls Deciml numers sometimes contin more deciml plces thn we need. It is importnt tht we re le to round deciml numers when working with money, mesuring quntities, including time nd distnce, or writing nswers to some division clcultions. For exmple, the distnce round the school ovl might e km, which rounded to deciml plce is 0.4 km or 400 m. The rounded figure, lthough not precise, is ccurte enough for most pplictions. Running events re electroniclly mesured nd rounded to deciml plces. Usin Bolt hs repetedly roken his own world records. In August 009 he set new world record of 9.8 seconds over m t the World Chmpionships in Germny, which ws -hundredths (0.) of second fster thn his Beijing Olympic Gmes (August 008) record of 9.69 seconds. Let s strt: Rounding rinstorm Usin Bolt s -metre sprint world record is mesured to deciml plces. In group of four, rinstorm occsions when it my e useful to round or estimte deciml numers. Aim to get more thn 0 common pplictions. In pirs one person sttes deciml numer nd the prtner needs to stte nother deciml numer tht would llow the two numers to dd up to whole numer. Use mentl rithmetic only. Strt with deciml plce nd try to uild up to or 4 deciml plces. Rounding involves pproximting deciml numer to fewer deciml plces. To round deciml: Cut the numer fter the required deciml plce; e.g. round to deciml plces. To determine whether you should round your nswer up or down, consider only the digit immeditely to the right of the specified plce. For rounding purposes this cn e referred to s the criticl digit..6 7 cut is the criticl digit in this exmple Key ides

9 Chpter 6 Decimls Key ides If the criticl digit is less thn (i.e. 0,,, or 4), then you round down. This mens write the originl numer to the plce required, leving off ll other digits. This cn e referred to s simply leving the numer s it is. If the criticl digit is or more (i.e., 6, 7, 8 or 9), then you round up. This mens write the originl numer to the plce required, ut increse this digit y. Leve off ll other digits. Exmple 4 Determining the criticl digit The following deciml numers need to e rounded to deciml plces. Drw line where the numer must e cut nd then circle the criticl digit SOLUTION EXPLANATION. 9 8 A line is drwn directly fter the specified numer of deciml plces, in this cse, The criticl digit is lwys the numer stright fter the specified numer of deciml plces. Exmple Rounding decimls to deciml plce Round ech of the following to deciml plce SOLUTION EXPLANATION.7 The criticl digit is 8 nd therefore the tenths column must e rounded up from 6 to 7.. The criticl digit is 4 nd therefore the tenths column remins the sme, in effect rounding the originl numer down to.. Exmple 6 Rounding decimls to different deciml plces Round ech of the following to the specified numer of deciml plces. Round to deciml plces. Round to deciml plces. c Round to deciml plces.

10 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Exmple 4 Exmple SOLUTION EXPLANATION 8.47 The criticl digit is, therefore round down The criticl digit is 9, therefore round up. c The criticl digit is 6, therefore round up. Rounding up hs resulted in digits eing crried over. Rememer to show the stted numer of deciml plces; hence, the zeros must e displyed. Exercise 6B For ech of the following, select the closer lterntive. Is.79 closer to.7 or.8? Is.4 closer to or? c Is 8 closer to 80 or 90? d Is closer to 6.77 or 6.78? The following decimls need to e rounded, correct to deciml plces. Drw line where the numer must e cut nd then circle the criticl digit tht must e checked s to whether to round up or down c 7.8 d e f 0.79 g h 7.99 To round correctly to specified numer of plces, you must know which digit is the criticl digit. Rememer: The criticl digit is lwys the digit immeditely to the right of the specified numer of plces. Stte the criticl digit in ech of the following numers. i.874 rounded to deciml plce. Criticl digit = ii.874 rounded to deciml plces. Criticl digit = iii.874 rounded to deciml plce. Criticl digit = iv.874 rounded to the nerest whole numer. Criticl digit = Stte the correct rounded numers for the numers in prts i to iv ove (½) Round ech of the following to deciml plce c.6 d 0.87 e 6.8 f 9.94 g. h (½) 9(½) UNDERSTANDING FLUENCY

11 4 Chpter 6 Decimls 6B Exmple 6, Exmple 6c Write ech of the following correct to deciml plces c.96 d 0.9 e.46 f g.0 h i 6.89 j k 9.99 l Round ech of the following to the specified numer of deciml plces, given s the numer in the rckets..9 () 7.89 () c.6 (0) d 0. (0) e.86 () f 9.9 () g 9.9 () h () Round ech of the following to the specified numer of deciml plces..98 () () c 6.94 (0) d () 8 Round ech of the following to the nerest whole numer c.99 d.7 e.6 f 7. g.6 h Round ech of the following mounts to the nerest dollr. $.8 $0.0 c $7.0 d $66.80 e $0.4 f $9. g $.9 h $ , 0 Some wise shoppers hve the hit of rounding ll items to the nerest dollr s they plce them in their shopping sket. They cn then keep running totl nd hve close pproximtion s to how much their finl ill will cost. Use this technique to estimte the cost of the following. Jenette purchses 0 items: $., $0.8, $4.6, $8.99, $.0, $7.0, $.90, $.9, $4.8, $.99 Adm purchses items: $0., $.00, $.40, $8.90, $6.90, $.9, $.0, $.0, $., $0.0, $4.9, $. c Jenette s ctul shopping totl is $0.8 nd Adm s is $44.7. How ccurte were Jenette s nd Adm s estimtions? 0, FLUENCY PROBLEM-SOLVING

12 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Electronic timing pds re stndrd in Ntionl Swimming competitions. In recent Ntionl Under -m freestyle rce, Edwin receives rounded time of.8 seconds nd Jsmine time of.7 seconds. If the timing pds cn clculte times only to the nerest second, wht will e the time difference etween the two swimmers? If the timing pds cn clculte times only to the nerest tenth of second, wht will e the time difference etween the two swimmers? c Wht is the time difference etween the two swimmers, correct to deciml plces? d If the timing pds cn mesure to deciml plces, wht would e the quickest time tht Edwin could hve swum the rce in? Without using clcultor, evlute , correct to deciml plces. Wht is the lest numer of deciml plces you need to find in the quotient to ensure tht you hve rounded correctly to deciml plces? Smr elieves should e rounded up to 0., ut Cssndr elieves it should e rounded down to 0.4. Mke n rgument to support ech of their sttements, ut then show the flw in one girl s logic nd clerly indicte which girl you think is correct. Rounding with technology 4 Most clcultors re le to round numers correct to specified numer of plces. Find out how to do this on your clcultor nd check your nswers to Questions nd 6. Spredsheet softwre pckges cn lso round numers correct to specified numer of plces. Find out the correct syntx for rounding cells in spredsheet progrm, such s Microsoft Excel, nd then check your nswers to Questions 7 nd 8., 4, PROBLEM-SOLVING REASONING ENRICHMENT 6B

13 6 Chpter 6 Decimls 6C Addition nd sutrction of decimls CONSOLIDATING Key ides Addition nd sutrction of decimls follows the sme procedures s those for whole numers. To dd or sutrct whole numers you must line up the units, tens, hundreds nd so on, nd then you dd or sutrct ech column. When deling with the ddition or sutrction of decimls the routine is the sme. Consider how similr the following two sums re: Let s strt: Wht s the totl? Ech student thinks of three coins (gold or silver) nd writes their totl vlue on sheet of pper. Ech student in the clss then estimtes the totl vlue of the mounts written down in the clssroom. Record ech student s estimted totl. Ech student then writes the vlue of the three coins they thought of on the ord (e.g. $.70, $0.80 etc.). Students copy down the vlues into their workooks nd dd the Accounting involves the ddition nd sutrction of deciml numers. deciml numers to determine the totl vlue of the coins in the clssroom. Which student hs the closest estimted totl? When dding or sutrcting decimls, the deciml points nd ech of the deciml plces must e ligned under one nother. The loction of the deciml point in the nswer is directly in line with the loction of ech of the deciml points in the question. Once the numers re correctly ligned, proceed s if completing whole numer ddition or sutrction. If the numers of deciml plces in the numers eing dded or sutrcted re different, it cn e helpful to plce dditionl zeros in the shorter numer(s) to prevent clcultion errors.

14 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 7 Exmple 7 Adding decimls Find: SOLUTION Exmple 8 Sutrcting decimls EXPLANATION Find: SOLUTION Mke sure ll deciml points nd deciml plces re correctly ligned directly under one nother. Align deciml points directly under one nother. Fill in missing deciml plces with dditionl zeros. Crry out ddition, following the sme procedure s tht for ddition of whole numers. EXPLANATION Mke sure ll deciml points nd deciml plces re correctly ligned directly under one nother. Align deciml points directly under one nother nd fill in missing deciml plces with dditionl zeros. Crry out sutrction, following the sme procedure s tht for sutrction of whole numers.

15 8 Chpter 6 Decimls Exmple 7 Exmple 8 Exercise 6C 7., 8. nd.0 must e dded together. Which of the following is the est wy to prepre these numers redy for ddition? A 7. B 7. C 7.0 D Which of the following is the correct wy to present nd solve the sutrction prolem ? A 77.8 B 77.8 C D Find ech of the following. 4 7., c 0.0. d Find ech of the following c. +. d e f Find: c.94 d (½) Find: c d e f (½) 4 6(½) UNDERSTANDING FLUENCY

16 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 9 7 Find the missing numers in the following sums c. d How much greter is 6. thn 76.?. 9 9 Sturt wnts to rise $ for the Rinow Clu chrity. He lredy hs three dontions of $0.0, $0.0 nd $.00. How much does Sturt still need to rise? 0 Dily rinflls for 4 dys over Ester were. mm,. mm, 0.6 mm nd.76 mm. Wht ws the totl rinfll over the 4-dy Ester holidy? Complete the ddition tle elow , Michelle erned $78.6 working prt-time over -yer period. However, she ws required to py her prents $0 per week for ord for weeks. Michelle lso spent $4.6 on clothing nd $6.90 on presents for her fmily nd friends during the yer. She plced the rest of her money in the nk. How much did Michelle nk for the yer? If =.8, =. nd c =.98, find: + + c + - c c c + - d c - ( + ) 4 Write down three numers etween nd 0, ech with deciml plces, tht would dd to.6. Cn you find solution to prt tht uses ech digit from to 9 exctly once ech? Money, money, money Investigte the following procedures nd shre your findings with friend. Choose n mount of money tht is less thn $0.00 (e.g. $.). Reverse the order of the digits nd sutrct the smller numer from the lrger numer (e.g. $. - $. = $.98). c Reverse the order of the digits in your new nswer nd now dd this numer to your most recent totl (e.g. $.98 + $8.9 = $0.89). Did you lso get $0.89? Repet the procedure using different strting vlues. Try to discover pttern or rule. Justify your findings. 0, 4 PROBLEM-SOLVING REASONING ENRICHMENT 6C

17 0 Chpter 6 Decimls 6D Multiplying nd dividing y powers of 0 Key ides Powers of 0 include 0, 0, 0, 0 4,..., which correspond to the numers 0,, 0, 0 000,... Note tht the numer of zeros in the numer is the sme s the power of 0 for tht numer. For exmple, 0 4 = 0 000, the numer ten thousnd hs four zeros nd it is equl to ten to the power of four. Let s strt: Dynmic lep frog A set of lrge numer crds, enough for one crd per student in the clss, is required. The set of crds should include the following digits, numers nd symols: 0,,,, 4,, 6, 7, 8, 9,.,,,, 0,, 0, 0 000, 000, The deciml plce crd is vitl! Crds should e ig enough to e red from the ck of the clssroom. Any of the digits cn e douled up to increse the totl numer of crds. Ech student receives one crd. Four students with one of the 0 to 9 digit crds stnd up t the front nd mke 4-digit numer. The student with the deciml plce crd then positions themselves somewhere within this numer or on either end. Now student with the or opertion comes up the front. Finlly, student with power of 0 crd comes up nd performs the ctul clcultion y gently moving the deciml plce! Repet numer of times with students swpping crds on severl occsions. Every numer contins deciml point ut it is usully not shown in integers. For exmple: 4 is 4.0 nd 00 is 00.0 Extr zeros cn e dded in the column to the right of the deciml point without chnging the vlue of the deciml. For exmple:. =.0 =.00 =.000 etc. When multiplying y powers of 0: Move the deciml point to the right the sme numer of plces s there re zeros in the multiplier. For exmple, if multiplying y 0, move the deciml point plces to the right = 78.9 Note: The deciml point ctully stys still nd ll the digits move three plces to the left, ut this is hrder to visulise.

18 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger When dividing y powers of 0: Move the deciml point to the left the sme numer of plces s there re zeros in the multiplier. For exmple, if dividing y, move the deciml point plces to the left = 9.76 Note: The deciml point ctully stys still nd ll the digits move two plces to the right, ut this is hrder to visulise. Exmple 9 Multiplying y powers of 0 Evlute: SOLUTION EXPLANATION 6. = 6. hs two zeros, therefore deciml point ppers to move plces to the right = 4 Deciml point ppers to move 4 plces to the right nd dditionl zeros re inserted s Exmple 0 Dividing y powers of 0 necessry. 4. Evlute: SOLUTION EXPLANATION = hs one zero, therefore deciml point is moved plce to the left = Deciml point is moved plces to the left nd dditionl zeros re inserted s necessry Key ides

19 Chpter 6 Decimls Exmple Working with the missing deciml point Evlute: SOLUTION EXPLANATION = If no deciml point is shown in the question, it must e t the very end of the numer. Four dditionl zeros must e inserted to move the invisile deciml point 4 plces to the right = 0.0 Deciml point is moved plces to Exmple Evluting using order of opertions Clculte this expression, using the order of opertions: ( ) SOLUTION ( ) = (4. -.6) = = =.6 Exercise 6D the left. 0.0 EXPLANATION First, we must clculte the rckets. The division y cn lso e done in the first step. 0(4. -.6) mens 0 (4. -.6). Fill in the correct numer of zeros in the multiplier to mke the following product sttements correct. The first one hs een done for you = = c =.7 d 00 = UNDERSTANDING

20 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Exmple 9 Exmple 0 Exmple Exmple Fill in the correct numer of zeros in the divisor to mke the following division sttements correct. The first one hs een done for you = = 7.84 c = d 890 = How mny plces nd in wht direction does the deciml point in the numer pper to move if the following opertions occur? i ii 0 iii iv v 0 vi 0 vii 0 viii If ll of the opertions ove hd tken plce on numer, one fter the other, wht would e the finl position of the deciml plce reltive to its strting position? Clculte: c d 4.04 e f.9 6 g.7 0 h 4. 0 i j. 0 k l Clculte: c d 689. e.8 0 f g.9 h i j 6.7 k l Clculte: c d. e f (½), 7(½) 4 7(½) Clculte: c 0 d 6 0 e 4 f 4 g 7 0 h i (½), 9 8(½), 9, 0 Clculte the following, using the order of opertions c + 0(4 + 8) d 0( ) e f g h (64 + ) 4 7(½) 0 UNDERSTANDING FLUENCY PROBLEM-SOLVING 6D

21 4 Chpter 6 Decimls 6D 9 A service sttion chrges $.7 per litre of petrol. How much will it cost Tnish to fill her cr with litres of petrol? 0 A lrge ee frm produces 00 litres of honey per dy. If there re 0 millilitres in litre, how mny millilitres of honey cn the frm s ees produce in one dy? The frm s honey is sold in millilitre jrs. How mny jrs of honey cn the frm s ees fill in one dy? Wendy is on moile phone pln tht chrges her cents per text messge. On verge, Wendy sends 0 text messges per dy. Wht will it cost Wendy for dys of sending text messges t this rte? Give your nswer in cents nd then convert your nswer to dollrs. Drren wishes to purchse shres t $. per shre. Given tht there is lso n dditionl $00 rokerge fee, how much will it cost Drren to purchse the shres? The weight of mtchstick is kg. Find the weight of oxes of mtches, with ech ox contining mtches. The weight of one empty mtch ox is kg. 4 Complete the tle elow, listing t lest one possile comintion of opertions tht would produce the stted nswer from the given strting numer. Strting numer Answer Possile two-step opertions.7.7 0, c.dfg.cdfg, 0 d.swkk fwy dswkk f.wy, 4 The numer undergoes series of multipliction nd division opertions y different powers of 0. The first four opertions re: 0,, nd 0. Wht is the fifth nd finl opertion if the finl numer is ? 4, PROBLEM-SOLVING REASONING

22 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Stndrd form 6 Extremely lrge numers nd extremely smll numers re often written in more prcticl wy, known s stndrd form or scientific nottion. For exmple, the distnce from the Erth to the Sun is kilometres! The distnce of 0 million kilometres cn e written in stndrd form s. 0 8 kilometres. On clcultor, cn e represented s.e nd.e8 represent the sme lrge numer nd indicte tht the deciml plce needs to e moved 8 plces to the right. c d e f.e8 =. 0 8 = = Represent these numers in stndrd form. i ii iii Use clcultor to evlute the following. i ii The distnce from the Erth to the Sun is stted ove s 0 million kilometres. The more precise figure is kilometres. Reserch how stronomers cn clculte the distnce so ccurtely. Hint: It is linked to the speed of light. Crry out further reserch on very lrge numers. Crete list of 0 very lrge numers (e.g. distnce from Erth to Pluto, the numer of grins in kg of snd, the numer of strs in the glxy, the numer of memory ytes in teryte... ). Rnk your 0 lrge numers in scending order. How re very smll numers, such s , represented in stndrd form? Represent the following numers in stndrd form. i ii iii ENRICHMENT 6D

23 6 Chpter 6 Decimls 6E Key ides Multipliction of decimls There re countless rel-life pplictions tht involve the multipliction of deciml numers. For exmple, finding the re of lock of lnd tht is 4. m long nd. m wide, or pricing 4.-hour jo t rte of $.7 per hour. In generl, the procedure for multiplying deciml numers is the sme s multiplying whole numers. There is, however, one extr finl step, which involves plcing the deciml point in the correct position in the nswer. Let s strt: Multipliction musings Finding the re of lock of lnd usully involves multiplying decimls. Consider the following questions within your group. Wht hppens when you multiply y numer tht is less thn? Consider the product of 0.7. Will the nswer e more or less thn? Why? Estimte n nswer to 0.7. Wht is the totl numer of deciml plces in the numers nd 0.7? Clculte 0.7. How mny deciml plces re there in the nswer? When multiplying decimls, strt y ignoring ny deciml points nd perform the multipliction s you would normlly. On rriving t your nswer, plce the deciml point in the correct position. The correct position of the deciml point in the nswer is found y following the rule tht the totl numer of deciml plces in the question must equl the numer of deciml plces in the nswer. For exmple:.4. deciml plces in the question 4 deciml points 068 ignored here = deciml plces in the nswer It is lwys worthwhile estimting your nswer. This llows you to check tht your deciml point is in the correct plce nd tht your nswer mkes sense. When multiplying y multiples of 0, initilly ignore the zeros in the multiplier nd ny deciml points nd perform routine multipliction. On rriving t your nswer, position your deciml point, rememering to move your deciml point ccording to the rules of multiplying y powers of 0.

24 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 7 Exmple Multiplying decimls Clculte: SOLUTION EXPLANATION Perform multipliction, ignoring deciml point. There re deciml plces in the question, so there will e deciml plces in the nswer.. 7 = 86.7 Estimtion is less thn ( 7 = 84) Ignore oth deciml points. Perform routine multipliction. Totl of deciml plces in the question, so there must e deciml plces in the nswer =.047 Estimtion is less thn 8 ( 4 7 = 8). Exmple 4 Multiplying decimls y multiples of 0 Clculte: SOLUTION = = = 9. 6 = EXPLANATION Ignore the deciml point nd zeros. Multiply = 9. Position the deciml plce in your nswer. There re deciml plces in the question, so must hve deciml plces in the nswer. Move the deciml point 4 plces to the right. Ignore the deciml point nd zeros. Multiply 6. Position deciml plce in the nswer. Shift deciml plce plces to the right ecuse there re two zeros in the question.

25 8 Chpter 6 Decimls Exmple Exmple 4 Exercise 6E Work out the totl numer of deciml plces in ech of the following product sttements c 4. d 8.7. e f.7 0. g h 0.00 i Insert the deciml point into ech of the following nswers so tht the multipliction is true. 6.4 = = 9 c = 9 Why is it worthwhile to estimte n nswer to multipliction question involving decimls? 4 Wht is the difference etween deciml point nd deciml plce? How mny deciml points nd how mny deciml plces re in the numer 4.67? Copy nd complete the rule for multiplying deciml numers (see the Key ides in this section). The totl numer of deciml plces must equl the numer of in the nswer. 6 7 Clculte: c.9 8 d 4 7. e 7.8 f.9 g h.4. i j k l Clculte: c d e f g h i Clculte nd then round your nswer to the nerest dollr. $6.0 $7. c 4 $8.70 d $.4 e $ f 7. $00 g 4. $.60 h 0.06 $70.00 i 0.08 $.0 6 8(½) 6 8(½) 9, (½) 9 Anit requires 4. m of mteril for ech dress she is mking. She is plnning to mke totl of seven dresses. How much mteril does she need? 0 The net weight of cn of spghetti is 0.44 kg. Find the net weight of eight cns of spghetti. Jimo rn. km ech dy for the month of Mrch. How mny kilometres did he run for the month? UNDERSTANDING FLUENCY PROBLEM-SOLVING

26 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 9 Bernrd is mking cuy house for his children. He needs lengths of timer, ech. m long. Wht is the totl length of timer needed to uild the cuy house? Wht is the cost of the timer if the price is $.9 per metre? A lwyer chrges $.00 per hour to ssist her client. How much does the lwyer chrge the client if she works on the jo for. hours? 4 4, 4 According to its mnufcturer, prticulr cr cn trvel 4. km on litre of petrol. How fr could the cr trvel on litres of petrol? The cr hs.4 litres of fuel in the tnk nd must complete journey of 0 km. Will it mke the journey without refuelling? c If the cr does mke the journey, how much petrol is left in the tnk t the end of the trip? If the cr doesn t mke the journey, how mny extr litres of fuel is needed? Write down two numers, ech with deciml plces, tht when multiplied y.8 will give n nswer etween 0.4 nd Write down one numer with 4 deciml plces tht when multiplied y 4.6 will give n nswer etween nd. 7 If 68 7 = 876, wht is the nswer to 6.8.7? Why? If = 76, wht is the nswer to.? Why? c If 0 00 = 7 000, wht is the nswer to. 0.? Why? d Wht is 7 6? Wht is the nswer to ? Why? Creting simple csh register 8 Using spredsheet progrm, such s Excel, design user-friendly csh register interfce. You must e le to enter up to 0 different items into your spredsheet. You will need quntity column nd cost per item column. 7 Using pproprite formuls, the totl cost of the ill should e displyed, nd there should then e room to enter the mount of money pid nd, if necessry, wht chnge should e given. When your spredsheet is set up, enter the following items. 4 chocolte $.8 ech $4.9 loves of $.9 ech kg $.99 per kg $.40 tomto $.0 litres of $.70 ech wshing $8.9 pckets of Tim $. ech. litres of soft $0.99 ech Money pid = $ PROBLEM-SOLVING REASONING ENRICHMENT 6E If your progrm is working correctly, the mount of chnge given should e $.0.

27 0 Chpter 6 Decimls 6F Key ides Division of decimls Similr to multipliction of deciml numers, there re countless rel-life pplictions tht involve the division of deciml numers. However, unlike multiplying deciml numers, where we siclly ignore the deciml points until the very end of the question, with division we try to mnipulte the question in such wy s to prevent dividing y deciml numer. Terminology reminders Exmple: 4 4 = 6 or 4 4 = 6 or ) is known s the dividend (the mount you hve, or the numer eing divided), 4 is known s the divisor (the numer doing the dividing) nd 6 is known s the quotient (or the nswer). Let s strt: Division decisions Consider the following questions within your group. Wht hppens when you divide y numer tht is less thn? Consider the nswer of Will the nswer e more or less thn 0? Why? Estimte n nswer to Clculte the nswer of. How does this compre to the nswer of 0 0.? Cn you think of n esier wy to clculte ? Division of deciml numers y whole numers Complete s you would normlly with ny other division question. The deciml point in the quotient (nswer) goes directly ove the deciml point in the dividend. For exmple: ) Division of deciml numers y other decimls Chnge the divisor into whole numer. Whtever chnge is mde to the divisor must lso e mde to the dividend. For exmple: = 46. When dividing y multiples of 0, initilly ignore the zeros in the divisor nd perform routine division. On rriving t your nswer, you must then re-position your deciml point ccording to the rules of dividing y powers of 0. For ech zero in the question tht you ignored initilly, the deciml point must move plce to the left.

28 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Exmple Dividing decimls y whole numers Clculte: SOLUTION 4.79 ) ) Exmple 6 Dividing decimls y decimls EXPLANATION Clculte: SOLUTION = 6.6 = 077. ) =. =.76 ) Crry out division, rememering tht the deciml point in the nswer is plced directly ove the deciml point in the dividend. Rememer to plce zeros in the nswer every time the divisor doesn t go. Agin, lign the deciml plce in the nswer directly ove the deciml plce in the question. An dditionl zero is required t the end of the dividend to terminte the deciml nswer. EXPLANATION Need to divide y whole numer Move ech deciml point plces to the right. Crry out the division question 6.6. Multiply divisor nd dividend y Move ech deciml point plces to the right. Crry out the division question..

29 Chpter 6 Decimls Exmple 7 Dividing decimls y multiples of 0 Clculte SOLUTION EXPLANATION ) 08.8 Ignore the three zeros in the Divide y = = Exmple 8 Evluting using order of opertions Clculte using the order of opertions: SOLUTION = = =.8 Exercise 6F EXPLANATION Now divide y 0, resulting in moving the deciml point plces to the left. First crry out nd, working from left to right. Then crry out + nd -, working from left to right. 4, 4 For the question = 9., which of the following options uses the correct terminology? A 6. is the divisor, 0.4 is the dividend nd 9. is the quotient. B 6. is the dividend, 0.4 is the divisor nd 9. is the quotient. C 6. is the quotient, 0.4 is the dividend nd 9. is the divisor. D 6. is the divisor, 0.4 is the quotient nd 9. is the dividend. Explin where you plce the deciml point in the quotient (i.e. nswer), when dividing deciml y whole numer. Clculte: c 0. d. 0.0 e Explin why these questions ll give the sme nswer. 4 For ech of the following pirs of numers, move the deciml points the sme numer of plces so tht the second numer ecomes whole numer..46, , 0. c , d 4., UNDERSTANDING

30 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger Exmple Exmple 6 Exmple 7 Exmple 8 6 Clculte: c 64.0 d.8 4 e 4.7 f.6 7 g h 49.6 i j 4. k.47 l Clculte: c d. 0.6 e f g h i j k l Design three deciml division questions for your prtner. Mke sure you clculte the nswer to ech question. Swp questions with your prtner. Go to work solving your prtner s questions. Pss your nswers ck for your prtner to correct. Discuss ny mistkes mde y either person. 8 6(½), 7, 8(½) 6(½), 8 9(½) Clculte: c d e f Clculte the following, rounding your nswers to deciml plces.. kg $. 7 c 8.6 m 0.6 m d 87 g. e 48. L 0. f $ Clculte the following, using the order of opertions ( ) c (8. -.) d e ( ) ( ) f ( ) Find the missing digits in these division questions. ) (½), 0(½), ) c ). d ) (½), 8 9(½), Chrlie pid $.7 to fill his ride-on lwnmower with 8 L of fuel. Wht ws the price per litre of the fuel tht he purchsed? Diden is picture frmer nd hs recently purchsed 4.6 m of timer. The verge-sized picture frme requires 90 cm (0.9 m) of timer. How mny verge picture frmes could Diden mke with his new timer? FLUENCY PROBLEM-SOLVING 6F

31 4 Chpter 6 Decimls 6F 4 A wter ottle cn hold 600 ml of wter. How mny wter ottles cn e filled from lrge drink continer tht cn hold 6 L? Six friends go out for dinner. At the end of the evening, the resturnt s ill is $98.0. As the ill is split eqully mong the six friends, how much does ech person py? Given tht they re hppy with the food nd the service, they decide to round the mount they ech py to $70. Wht is the witer s tip? 6 Clr purchses. kg of pples for $.90. Her friend Sophi uys 900 g of nns for $.79 t the sme shop. Find the cost per kilogrm of ech fruit. Which type of fruit is the est vlue in terms of price per kilogrm? 7 A police rdr gun mesures cr to e. m wy. At 0.6 seconds lter, the rdr gun mesures the sme cr to e 6.8 m wy. Determine the speed of the cr in metres per second (m/s). Multiply your nswer to prt y.6 to convert your nswer to km/h. c The cr is trvelling long n 80 km/h stretch of rod. Is the cr speeding? 6 6, 7 8 Given tht = 48.4, write down the vlue of ech of the following questions, without using clcultor c d e f Wht numer m I? 9 I m thinking of numer. Given the following clues for ech, find the numer. When I dd 4. nd then multiply y 6, the nswer is 0. When I divide it y nd then dd.9, the nswer is. c When I multiply it y nd then dd 9, the nswer is 0. d When I multiply it y nd then dd hlf, the nswer is 6. e When I sutrct 0.8, then divide y 0. nd then divide y 0., the nswer is 00. f Mke up three of your own numer puzzles to shre with the clss. 7, 8 9 PROBLEM-SOLVING REASONING ENRICHMENT

32 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 6A 6A 6A 6B 6C 6D 6E/F 6C/E 6C F Progress quiz Wht is the plce vlue of the digit 6 in the following numers? Express ech of the following frctions s deciml c 0 4 Arrnge the following deciml numers in scending order..4, 4., 0.4, 0.4, 0.04 Round ech of the following to the specified numer of deciml plces to deciml plces.4999 to deciml plces c to deciml plce Find: c d Evlute: Clculte: c d e f 0.06 g h i j Insert the deciml point in the nswer so tht ech mthemticl sentence is true = = 9744 It costs $9.8 for kilogrms of dog shmpoo. Find: the chnge from pying with one $0 note nd one $0 note the cost of the shmpoo per kilogrm c the numer of kilogrms tht cn e purchsed with $ note

33 6 Chpter 6 Decimls 6G Key ides Decimls nd frctions Decimls nd frctions re oth commonly used to represent numers tht re not simply whole numers. It is importnt tht we know how to convert deciml numer to frction, nd how to convert frction to deciml numer. In the photo opposite, we cn see tht or 0. of the cke hs een cut nd 4 or 0.7 of the cke remins. 4 Let s strt: Mtch my cll In pirs, nominte one student to e Frction kid nd the other to e Deciml expert. Frction kid strts nming some common frctions nd Deciml expert tries to give the equivlent deciml vlue. Strt with esy questions nd uild up to hrder ones. After 0 turns, swp round. This time Deciml expert will nme some deciml numers nd Frction kid will ttempt to cll out the equivlent frction. Discuss the following question in pirs: Which is esier, converting frctions to decimls or decimls to frctions? Converting decimls to frctions Using your knowledge of plce vlue, express the deciml plces s frction whose denomintor is power of 0. Rememer to simplify the frction whenever possile. Converting frctions to decimls When the denomintor is power of 0, we cn simply chnge the frction to deciml through knowledge of plce vlue. When the denomintor is not power of 0, try to find n equivlent frction whose denomintor hs power of 0 nd then convert to deciml. A method tht will lwys work for converting frctions to decimls is to divide the numertor y the denomintor. e.g. 0. = = 4 e.g. 7 = 0.7 e.g. = 4 0 = 0.4 Recurring decimls re decimls with repeted pttern. A dot, dots or r ove numer or numers indictes repeted pttern. e.g. = 0. = 0.. =.8 88 =.. 8. or.8 7 = = or.74 8 e.g = 8 ) = 0.6

34 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 7 Exmple 9 Converting decimls to frctions Convert the following decimls to frctions in their simplest form SOLUTION = Exmple 0 Converting frctions to decimls Convert the following frctions to decimls. 7 SOLUTION EXPLANATION 0.9 = 9 thousndths 0. = hundredths, which cn e simplified further y dividing the numertor nd denomintor y the highest common fctor of. c EXPLANATION 7 = hundredths = 6 0 = is n equivlent frction of, whose denomintor is power of 0. c 7 = or 0.8. )

35 8 Chpter 6 Decimls Exmple 9 Exmple 9 Exmple 0 Exmple 0 Exmple 0c Exercise 6G, Complete ech of these sttements, which convert common frctions to decimls. = 0 = 0. 4 = = 0. c 4 = = 0. d 4 = 0 = 0. Complete ech of these sttements, which convert decimls to frctions, in simplest form = = 0. = = 0 c 0.8 = 8 = d 0.64 = 64 = Stte whether ech of the following is true or flse. Use the exmples in the Key ides to help. 0. = = 0.. c. =.. d.77 =.7. e.6 66 =.. 6. f = 4.6 Convert the following decimls to frctions in their simplest form c 0. d 8. e. f 0.0 g 9.07 h.9 Convert ech of these frctions to decimls e f 9 c g 0 d h 79 Convert the following frctions to decimls, y first chnging the frction to n equivlent frction whose denomintor is power of e f 0 4 Convert the following frctions to decimls, y dividing the numertor y the denomintor. e f 6 8 c g c g 7(½) (½) d h d h (½) UNDERSTANDING FLUENCY

36 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 9 8 Copy nd complete the following frction/deciml tles. The qurters tle (prt c) hs lredy een done for you. It s well worth trying to memorise these frctions nd their equivlent deciml vlues. hlves thirds c Frction Deciml qurters Frction Deciml Arrnge the following from smllest to lrgest., 0.7, 8, 0.4, 0.99, 4 d Frction Deciml fifths Frction Deciml 0 0 7, 0., 9, 0.8, 0.84, 4 0 Tn nd Lillin re trying to work out who is the etter chess plyer. They hve oth een plying chess gmes ginst their computers. Tn hs plyed 7 gmes nd eten the computer times. Lillin hs plyed only gmes nd hs eten the computer 6 times Using clcultor nd converting the pproprite frctions to decimls, determine who is the etter chess plyer. Lillin hs time to ply nother four gmes of chess ginst her computer. To e clssified s etter plyer thn Tn, how mny of these four gmes must she win? To estimte the thickness of one sheet of A4 pper, Christopher mesures rem of pper, which consists of 00 sheets of A4 pper. He determines tht the pile is mm thick. How thick is one sheet of A4 pper? Express your nswer s deciml numer nd lso s frction. 4 0 FLUENCY PROBLEM-SOLVING 6G

37 40 Chpter 6 Decimls 6G Copy nd complete the following frction/deciml tle. c Frction Deciml 4 Comment on the trend in the deciml vlues s the denomintor increses. Try to explin why this mkes sense. Copy nd complete the following deciml/frction tle. c Deciml Frction Comment on the trend in the frctions s the deciml vlue increses. Try to explin why this mkes sense. 6 4 Write three different frctions with different denomintors tht re etween the deciml vlue of.4 nd.. When 4 is expressed in deciml form, find the digit in the rd deciml plce. Give reson for 7 your nswer. Design deciml gme for the clss 6 Using the skill of converting decimls to frctions nd vice vers, design n pproprite gme tht students in your clss could ply. Ides my include vritions of Bingo, Memory, Dominoes etc. Try creting chllenging set of question crds REASONING ENRICHMENT

38 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 4 6H Decimls nd percentges Percentges re commonly used in tody s society. Per cent is derived from the Ltin words per centum, mening out of. Percentges give n ide of proportion. For exmple, if newspper sttes tht 000 people wnt council swimming pool We see percentges used round us in mny different situtions. constructed, then we know how mny wnt pool ut we don t know wht proportion of the community tht is. However, if there re 00 people in this community, the newspper cn stte tht 80% wnt swimming pool. This informs us tht mjority of the community (i.e. 80 out of every people) wnt swimming pool constructed. Let s strt: Cretive shding Drw squre of side length 0 cm nd shde exctly 0% or 0. of this figure. Drw squre of side length cm nd shde exctly 60% or 0.6 of this figure. Drw nother squre of side length 0 cm nd cretively shde n exct percentge of the figure. Ask your prtner to work out the percentge you shded. Wht percentge is shded? The symol, %, mens per cent. It comes from the Ltin words per centum, which trnsltes to out of. For exmple: % mens out of which equls = 0. To convert percentge to deciml, divide y. This is done y moving the deciml point plces to the left. For exmple: 4% = 4 = 0.4. To convert deciml to percentge, multiply y. This is done y moving the deciml point plces to the right. For exmple: 0.64 = 6.4. Therefore 0.64 = 6.4% Note: As in Section 6D, it is not ctully the deciml point tht moves; rther, it is the digits tht move round the sttionry deciml point. Key ides

39 4 Chpter 6 Decimls Exmple Converting percentges to decimls Express the following percentges s decimls. 0% 40% c.% d 0.4% SOLUTION EXPLANATION 0% = % =.4 40 =.4 c.% = 0. Deciml point moves plces to the left. d 0.4% = Deciml point moves plces to the left. Exmple Converting decimls to percentges Express the following decimls s percentges SOLUTION EXPLANATION 0.04 = 4.% Multiplying y moves the deciml point plces to the right. 7. = 70% Multiply 7. y. Exercise 6H Wht percentge of ech squre hs een shded? 6, 6 UNDERSTANDING

40 Essentil Mthemtics for the Victorin Curriculum Yer 7 Numer nd Alger 4 c 7.% is equivlent to which of the following decimls? A 7. B 7. C 0.7 D 7.0 4% is equivlent to which of the following decimls? A 0.4 B 4. C 4 00 D is equivlent to which of the following percentges? A 9% B.9% C 0.9% D 0.009% Prue nswered hlf the questions correctly for test mrked out of. Wht score did Prue get on the test? Wht percentge did Prue get on the test? c Find the score you would expect Prue to get if the test ws out of: i 0 ii 00 iii 40 iv d Find the percentge you would expect Prue to get if the test ws out of: i 0 ii 00 iii 40 iv 6 Fill in the empty oxes. 8% = 8 out of = 8 = 8 = 0 8 c % = out of = = =. 6% = 6 = = 6 =. d UNDERSTANDING 6H

41 44 Chpter 6 Decimls 6H Exmple, Exmple c, d Exmple 7 9(½) 7 9(½) 8 9(½) 7 Express the following percentges s decimls. % 7% c 68% d 4% e 6% f 9% g % h % i 8% j 4% k 7% l 99% 8 9 Express the following percentges s decimls..% 7.% c.% d 8.% e.% f 88.8% g 0% h 0% i 0.79% j 0.0% k.04% l 0.9% Express the following decimls s percentges c 0.4 d 0.7 e 0.46 f 0.7 g. h.4 i 0.0 j k.7 l Plce the following vlues in order from highest to lowest. 86%, 0.%, 0.6, 0., %, 7%, %, %,.4,.99%, 0.0%,.8, %, 7., 0 0,, At hockey mtch, 6% of the crowd supports the home tem. Wht percentge of the crowd supports the visiting tem? Lst Sturdy, Phil spent the 4 hours of the dy in the following wy: 0.4 of the time ws spent sleeping, 0. ws spent plying sport nd 0. ws spent eting. The only other ctivity Phil did for the dy ws wtch TV. Wht percentge of the dy did Phil spend wtching TV? Wht percentge of the dy did Phil spend either sitting down or lying down? Sugrlof Reservoir hs cpcity of 96 giglitres. However, s result of the drought it is only % full. How mny giglitres of wter re in the reservoir?, FLUENCY PROBLEM-SOLVING