Chapter. Contents: A Constructing decimal numbers


 Colin Peters
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1 Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting frtions to deimls
2 166 DECIMALS (Chpter 9) OPENING PROBLEM Arrnge these mounts of money in sending order: $56:65, $56:05, $50:65, $55:50, $56:50 Whih is lrgest? DECIMAL NUMBERS ARE EVERYWHERE ALL THESE ITEMS AT ONE LOW PRICE.99 9 e Retil gints rep $37.4n A CONSTRUCTING DECIMAL NUMBERS We hve seen previously how whole numers re onstruted y pling digits in different ple vlues. For exmple, 384 hs ple vlue tle sine 384= hundreds tens units Numers like 0:37 nd 4:569 re lled deiml numers. We use them to represent numers etween the whole numers. The deiml point or dot seprtes the whole numer prt to the left of the dot, from the frtionl prt to the right of the dot.
3 DECIMALS (Chpter 9) 167 If the whole numer prt is zero, we write zero in front of the deiml point. So, we write 0:37 insted of : :37 is short wy of writing , nd 4:569 is relly So, the ple vlue tle for 0: 37 nd 4: 569 is: Deiml numer units tenths hundredths thousndths Expnded form 0:37 0 : :569 4 : Exmple 1 Express in written or orl form: 0:9 3:06 11:407 0:9 is zero point nine. 3:06 is three point zero six. 11:407 is eleven point four zero seven. Orl form mens how you would sy it. Exmple 2 Write in ple vlue tle: 7 hundredths tens units tenths hundredths thousndths Numer Written Numerl 7 hundredths : 0 7 0: : :409 EXERCISE 9A 1 Express the following in written or orl form: 0:6 0:45 0:908 d 8:3 e 6:08 f 96:02 g 5:864 h 34:003 i 7:581 j 60:264 2 Convert into deiml form: eight point three seven twenty one point zero five nine point zero zero four d thirty eight point two zero six
4 168 DECIMALS (Chpter 9) 3 Write in ple vlue tle nd then s deiml numer: d + 6 e f Numer thousnds hundreds tens units 4 Write in ple vlue tle nd then s deiml numer: 8 tenths 3 thousndths 7 tens nd 8 tenths d 9 thousnds nd 2 thousndths e 2 hundreds, 9 units nd 4 hundredths f 8 thousnds, 4 tenths nd 2 thousndths g 5 thousnds, 20 units nd 3 tenths h 9 hundreds, 8 tens nd 34 thousndths i 6 tens, 8 tenths nd 9 hundredths j 36 units nd 42 hundredths : tenths hundredths thousndths PRINTABLE WORKSHEET Written Numerl If word for digit ends in ths then the numer follows the deiml point. Exmple 3 Express 5:706 in expnded form: 5:706 = = Express the following in expnded form: 5:4 14:9 2:03 d 32:86 e 2:264 f 1:308 g 3:002 h 0:952 i 4:024 j 2:973 k 20:816 l 7:777 m 9:008 n 154:451 o 808:808 p 0:064 6 Write the following in deiml form: d e 7 f g h i j k l m n o
5 Exmple 4 Stte the vlue of the digit 6 in the following: 0: :3964 So, the 6 stnds for. DECIMALS (Chpter 9) s 1 s 1 s 7 Stte the vlue of the digit 3 in the following: 4325:9 6:374 32:098 d 150:953 e 43:4444 f 82:7384 g 24:8403 h 3874:941 8 Stte the vlue of the digit 5 in the following: 18: :08 4:5972 d 94:8573 e :264 f 275:183 g :843 h 0:0005 Exmple 5 Write 39 in deiml form. 39 = = =0:03 + 0:009 =0:039 9 Write in deiml form: d 117 e 469 f 703 g 600 h 540 i j Convert the following to deiml form: d e f seventeen nd four hundred nd sixty five thousndths twelve nd ninety six thousndths three nd six hundred nd ninety four thousndths four nd twenty two hundredths 9 hundreds, 8 tens nd 34 thousndths 36 units nd 42 hundredths 11 Stte the vlue of the digit 2 in the following: d e f g h 5 652
6 170 DECIMALS (Chpter 9) B REPRESENTING DECIMAL NUMBERS DECIMAL GRIDS Deimls n lso e represented on 2dimensionl grids. Suppose this grid represents one whole unit. This shded prt is or of the whole unit. This is 0:4 or 0:40. This shded prt is 27 or 0:27 of the whole unit. Exmple 6 Wht deiml numer is represented y: 0 : 3 6 units tenths hundredths MULTI ATTRIBUTE BLOCKS Multi Attriute Bloks or MABs re prtil 3dimensionl wy to represent deimls. represents unit or whole mount. represents tenth or 0:1 of the whole. represents hundredth or 0:01 of the whole. nd represents thousndth of 0:001 of the whole. The smller the deiml numer, the more zeros there re fter the deiml point.
7 DECIMALS (Chpter 9) 171 Exmple 7 Write the deiml vlue represented y the MABs if the lrgest lok represents one unit. 1 : units tenths hundredths thousnds 2 : units tenths hundredths thousnds EXERCISE 9B 1 Write the deiml tht represents the shded re: d There re no hundredths shown. We must write tht with zero, 0. 2 Write the deiml vlue represented y the MABs if the lrgest lok represents one unit:
8 172 DECIMALS (Chpter 9) d e C DECIMAL CURRENCY Deiml urreny is one of the most prtil wys to ring mening to deimls. When tlking out nd using money we re lso using deiml numers. The deiml point seprtes whole numers from the frtions. For exmple: E27:35 is E27 plus 35 of one E. Suppose ountry hs the following oins nd nknotes: $1 $2 $5 $10 $20 $50 $ The urreny is lled deiml euse it uses the se 10 system. 1 1 is or 0:01 of $1 2 is 2 or 0:02 of $1 5 5 is or 0:05 of $1 10 is 10 or 0:10 of $1 20 is 20 or 0:20 of $1 50 is 50 or 0:50 of $1
9 DECIMALS (Chpter 9) 173 Exmple 8 How muh money is shown? $20 $ $2 5 2 We hve =27 whole dollrs nd = 77 ents So, we hve $27:77 ltogether: Exmple 9 Using one euro (E) s the unit, hnge to deiml vlue: seven euros, 45 euro ents 275 euro ents E7: euro ents = 200 euro ents + 75 euro ents = E2 +E0:75 = E2:75 EXERCISE 9C 1 Chnge these urreny vlues to deimls of one dollr: $5 5 $20 50 $2 20 $2 $2 $50 $1 d $ 5 $10 10 $ $5 e $10 50 f $ 50 $2 20 $
10 174 DECIMALS (Chpter 9) 2 Write eh mount s dollrs using deiml point: 4 dollrs 47 ents 15 dollrs 97 ents seven dollrs fifty five ents d 36 dollrs e 150 dollrs f thirty two dollrs eighty ents g 85 dollrs 5 ents h 30 dollrs 3 ents 3 Chnge these mounts to deimls using the euro s the unit: i 35 ents ii 5 ents iii 405 ents iv 3000 ents v 487 ents vi 295 ents vii 3875 ents viii ents D Strting with the top row, wht is the sum of eh row ove in euros? Wht is the sum of eh olumn ove in euros? USING A NUMBER LINE Just s whole numers n e mrked on numer line, we n do the sme with deiml numers. Consider the following numer line where eh whole numer shown hs ten equl divisions. Eh division on this numer line represents 1 10 or 0:1 Exmple 10 Mke sure the mounts hve their deiml point extly elow the other Find the deiml vlues of A, B, C nd D mrked on the numer line shown A 1 B C D Eh division on the numer line represents 0:1 So, A is 0:7, B is 1:3, Cis 2:1 nd D is 3:2 We n divide our numer line into smller prts thn tenths. 1 Suppose we divide eh of the prts whih represent 10 into 10 equl prts. Eh unit is now 1 divided into equl prts nd eh division is or 0:01 of the unit
11 DECIMALS (Chpter 9) 175 Exmple 11 Find the deiml vlues of A, B, C nd D mrked on the numer line shown. 2.4 A 2.5 B C 2.6 D Eh division on the numer line represents 0:01. So, A is 2:43, B is 2:51, C is 2:57 nd D is 2:62 EXERCISE 9D 1 Write down the vlue of the numer t N on the following numer lines. 0 N 1 2 N 3 6 N 7 d 21 N 22 e 11 N 12 f 8 N 9 2 Copy the numer lines given nd mrk the following numers on them. A =1:6, B =2:5, C =2:9, D =4: E =13:7, F =14:2, G =15:3, H =16: Red the temperture on the thermometer shown C 4 Red the length of Christin s skirt from the tpe mesure. 5 Wht weight is shown on the sles? How muh milk is in the jug? kg 1 litre
12 176 DECIMALS (Chpter 9) 6 Write down the vlue of the numer t N on the following numer lines. N N N d N e N f N Copy the numer lines given nd mrk the following numers on them. A =4:61, B =4:78, C =4:83, D =4: E =10:35, F =10:46, G =10:62, H =10: E ORDERING DECIMALS We n use numer line to help ompre the sizes of deiml numers. For exmple, onsider the following numer line: As we go from left to right, the numers re inresing. So, 1:08 < 1:25 < 1:7 < 1:89 To ompre deiml numers without hving to onstrut numer line, we ple zeros on the end so eh numer hs the sme numer of deiml ples. We n do this euse dding zeros on the end does not ffet the ple vlues of the other digits. Exmple 12 Put the orret sign >, < or =, in the ox to mke the sttement true: 0:305 0:35 0:88 0:808 We strt y writing the numers with the sme numer of deiml ples. 0:305 0:350 So, 0:305 < 0:350 0:880 0:808 So, 0:880 > 0:808
13 DECIMALS (Chpter 9) 177 EXERCISE 9E 1 Write down the vlues of the numers A nd B on the following numer line, nd determine whether A > BorA< B: B 6 7 A B A A B d A B e f A B B A Insert the orret sign >, < or = to mke the sttement true: 0:7 0:8 0:06 0:05 0:2 0:19 d 4:01 4:1 e 0:81 0:803 f 2:5 2:50 g 0:304 0:34 h 0:03 0:2 i 6:05 60:50 j 0:29 0:290 k 5:01 5:016 l 1:15 1:035 m 21:021 21:210 n 8:09 8:090 o 0:904 0:94 3 Arrnge in sending order (lowest to highest): 0:8, 0:4, 0:6 0:4, 0:1, 0:9 0:14, 0:09, 0:06 d 0:46, 0:5, 0:51 e 1:06, 1:59, 1:61 f 2:6, 2:06, 0:206 g 0:095, 0:905, 0:0905 h 15:5, 15:05, 15:55 4 Arrnge in desending order (highest to lowest): 0:9, 0:4, 0:3, 0:8 0:51, 0:49, 0:5, 0:47 0:6, 0:596, 0:61, 0:609 d 0:02, 0:04, 0:42, 0:24 e 6:27, 6:271, 6:027, 6:277 f 0:31, 0:031, 0:301, 0:311 g 8:088, 8:008, 8:080, 8:880 h 7:61, 7:061, 7:01, 7:06 5 Continue the numer ptterns y writing the next three terms: 0:1, 0:2, 0:3,... 0:9, 0:8, 0:7,... 0:2, 0:4, 0:6,... d 0:05, 0:07, 0:09,... e 0:7, 0:65, 0:6,... f 2:17, 2:13, 2:09,... g 7:2, 6:4, 5:6,... h 0:25, 0:50, 0:75,... i 1:111, 1:123, 1:135,... j 0, 0:125, 0:250,...
14 178 DECIMALS (Chpter 9) F We re often given mesurements s deiml numers. For exmple, my throom sles tell me I weigh 59:4 kg. In relity I do not weigh extly 59:4 kg, ut this is n pproximtion of my tul weight. Mesuring my weight to greter ury is not importnt. We round off deiml numers in the sme wy we do whole numers. We look t vlues on the numer line either side of our numer, nd work out whih is loser. For exmple, onsider 1: :23 is loser to 1:2 thn it is to 1:3, so we round down. 1:23 is pproximtely 1:2. ROUNDING DECIMAL NUMBERS Consider 5716 ¼ 5720 (to the nerest 10) ¼ 5700 (to the nerest ) ¼ 6000 (to the nerest ) Likewise, 0:5716 ¼ 0:572 ¼ 0:57 ¼ 0:6 (to 3 deiml ples) (to 2 deiml ples) (to 1 deiml ple) RULES FOR ROUNDING OFF DECIMAL NUMBERS ² If the digit fter the one eing rounded is less thn 5, i.e., 0, 1, 2, 3 or 4, then we round down. ² If the digit fter the one eing rounded is 5 or more, i.e., 5, 6, 7, 8 or 9, then we round up. Exmple 13 Round: 3:26 to 1 deiml ple 5:273 to 2 deiml ples 4:985 to 2 deiml ples 3:26 is loser to 3:3 thn to 3:2, so we round up. So, 3:26 ¼ 3:3. 5:273 is loser to 5:27 thn to 5:28, so we round down. So, 5:273 ¼ 5:27. 4:985 lies hlfwy etween 4:98 nd 4:99, so we round up. So, 4:985 ¼ 4:99 : EXERCISE 9F 1 Write these numers orret to 1 deiml ple: 2:43 3:57 4:92 d 6:38 e 4:275 2 Write these numers orret to 2 deiml ples: 4:236 2:731 5:625 d 4:377 e 6:5237
15 DECIMALS (Chpter 9) Write 0:486 orret to: 1 deiml ple 2 deiml ples. 4 Write 3:789 orret to: 1 deiml ple 2 deiml ples. 5 Write 0: orret to: 1 deiml ple 2 deiml ples 3 deiml ples d 4 deiml ples. 6 Find deiml pproximtions for: 3:87 to the nerest tenth 4:3 to the nerest integer 6:09 to one deiml ple d 0:4617 to 3 deiml ples e 2:946 to 2 deiml ples f 0: to 4 deiml ples. G CONVERTING DECIMALS TO FRACTIONS Deiml numers n e esily written s frtions with powers of 10 s their denomintors. Exmple 14 Write s frtion or s mixed numer: 0:7 0:79 2:013 0:7 = :79 = 79 2:013 =2+ 13 =2 13 We hve seen previously how some frtions n e onverted to simplest form or lowest terms y dividing oth the numertor nd denomintor y their highest ommon ftor. Exmple 15 Write s frtion in simplest form: 0:4 0:72 0:275 0:4 0:72 0:275 = 4 10 = = 2 5 = 72 = = = 275 = = 11 40
16 180 DECIMALS (Chpter 9) EXERCISE 9G 1 Write the following s frtions in simplest form: 0:1 0:7 1:5 d 2:2 e 3:9 f 4:6 g 0:19 h 1:25 i 0:18 j 0:65 k 0:05 l 0:07 m 2:75 n 1:025 o 0:04 p 2:375 2 Write the following s frtions in simplest form: 0:8 0:88 0:888 d 3:5 e 0:49 f 0:25 g 5:06 h 3:32 i 0:085 j 3:72 k 1:096 l 4:56 m 0:064 n 0:625 o 0:115 p 2:22 Exmple 16 Write s frtion: 0:45 kg 3:40 m 0:45 kg 3:40 m = 45 kg = kg = 9 20 kg =3m m =3m m =3 2 5 m 3 Write these mounts s frtions or mixed numers in simplest form: 0:20 kg 0:25 hours 0:85 kg d 1:50 km e 1:75 g f 2:74 m g 4:88 tonnes h 6:28 L i E1:25 j E1:76 k E3:65 l E4:21 m E8:40 n E5:125 o $3:08 p $4:11 q $18:88 r $52:25 H CONVERTING FRACTIONS TO DECIMALS We hve lredy seen tht it is esy to onvert frtions with denomintors 10,,, nd so on into deiml numers. Sometimes we n mke the denomintor power of 10 y multiplying the numertor nd denomintor y the sme numers. 3 For exmple, 5 = = 6 10 =0: = = 28 =0:28 We need to multiply the numertor nd denomintor y the sme mount so we do not hnge the vlue of the frtion.
17 DECIMALS (Chpter 9) 181 Exmple 17 Convert to deiml numers: = = 75 =0: = = 35 =0: = = 184 =0:184 EXERCISE 9H 1 By wht whole numer would you multiply the following, to otin power of 10? d 8 e 20 f 25 g 50 h 125 i 40 j 250 k 500 l Convert to deiml numers: d e f g h i j k 1 4 l m n o p q 3 8 r Copy nd omplete these onversions to deimls: 1 2 = :::::: 1 5 = ::::::, 2 5 = ::::::, 3 5 = ::::::, 4 5 = ::::::, 1 4 = ::::::, 2 4 = ::::::, 3 4 = :::::: d 1 8 = ::::::, 2 8 = ::::::, 3 8 = ::::::, 4 8 = ::::::, 5 8 = ::::::, 6 8 = ::::::, 7 8 = ::::::: You should rememer the deiml vlues of these frtions. KEY WORDS USED IN THIS CHAPTER ² deiml ² deiml urreny ² deiml point ² frtion ² highest ommon ftor ² hundredth ² mixed numers ² ple vlue ² round off ² simplest form ² tenth ² thousndth
18 182 DECIMALS (Chpter 9) REVIEW SET 9A 1 If the dollr represents the unit, wht re the deiml vlues of the following? $10 $5 50 $ $ $ $ 20 $ If eh grid represents one unit, wht deimls re represented y the following grids? 3 If represents one thousndth, write the deiml numers for: d 4 Write s deiml numer Write: 25 euros nd 35 euro ents s deiml numer $107 nd 85 ents s deiml numer five nd twenty nine thousndths in deiml form d e f 436 : in two different frtionl forms 2:049 in expnded form the mening of the 2 digit in 51: Write down the vlue of the numer t B on these numer lines: B B
19 DECIMALS (Chpter 9) Whih numer is greter, 3:2 or 3:1978? 8 Round 3:8551 to: one deiml ple two deiml ples. 9 Convert to frtion in simplest form: 0:8 0:75 0:375 d 0:68 10 Convert to deiml numer: d Write in sending order: 0:216, 0:621, 0:062, 0:206, 0: Continue this numer pttern for 3 more terms: 0:81, 0:78, 0:75, 0:72,... REVIEW SET 9B 1 If represents one thousndth, write the deiml vlues of: 2 If eh grid represents one unit, wht deiml numers re represented y: 3 Write s deiml numer: d Convert sixteen point five seven four to deiml form. Write 0:921 in two different nondeiml forms. Stte the vlue of the digit 3 in 41:039. d Write $12 nd 35 pene s deiml. 5 Write down the vlue of the numer t A on the following numer lines: A A Write the following deiml numers in desending order: 0:444, 4:04, 4:44, 4:044, 4:404 7 Continue the numer pttern y writing the next three terms: 0:3, 0:7, 1:1,...
20 184 DECIMALS (Chpter 9) 8 Whih numer is smller, 2:3275 or 2:3199? 9 Round 3:995 to: 1 deiml ple 2 deiml ples 10 Convert to frtion in simplest form: 0:62 0:45 0:875 d 10:4 11 Convert to deiml numer: d Copy nd omplete: 1 8 =0:125, 2 8 = ::::::, 3 8 = ::::::, 4 8 = ::::::, 5 8 = ::::::. ACTIVITY Wht to do: TANGRAMS 1 On piee of rd mrk out 20 m y 20 m squre. Then opy the following lines onto it nd ut long eh line. You should hve seven different piees. PRINTABLE TEMPLATE 2 Eh of the following shpes n e mde using ll seven piees of your tngrm. See how mny you n omplete. ridge puppy person running d t
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