D e c i m a l s DECIMALS.

Transcription

1 D e i m l s DECIMALS

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3 Deimls DECIMALS A deiml numer is sed on ple vlue hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed nd deiml numer needs to e pproximted. Sometimes deimls go on forever like Answer these questions, efore working through the hpter. I used to think: Is 23.4 nerer to 23 or 24? Wht does it men, if the rtio of penuts to shew nuts in ke is 4:1? Wht do the dots in o o men? Answer these questions, fter working through the hpter. But now I think: Is 23.4 nerer to 23 or 24? Wht does it men, if the rtio of penuts to shew nuts in ke is 4:1? Wht do the dots in o o men? Wht do I know now tht I didn't know efore? 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 1

4 Deimls Bsis Deimls nd Rounding Is 21.4 loser to 21 or 22? 'Rounding' or 'rounding off' is method of pproximting numer's vlue. For exmple we my need numer urte to the nerest unit, or nerest 10 or nerest thousndth, depending on different situtions. If the digit in the deiml ple fter the required ury is 4 or less, then round down. If it is 5 or more, then round up. Rounding exmples Round 3.4 to the nerest unit. Chek this digit The digit fter the unit is 4 so round down to the nerest unit. The nerest unit to 3.4 is 3 Round 3.7 to the nerest unit. Chek this digit The digit fter the unit is 7 so round up to the nerest unit. The nerest unit to 3.7 is 4 Write to the nerest hundred. Chek this digit The digit fter the hundred digit (3) is 2 so round down to the nerest hundred. The nerest hundred to is 300 d Write to the nerest hundredth. Chek this digit The digit fter the hundredth digit (7) is 6 so round up to the nerest hundredth. The nerest hundredth is e Write orret to two deiml ples. Chek this digit The digit fter the seond deiml ple is n 8 so round up. The numer orret to two deiml ples is 0.37 f Write orret to one deiml ple. Chek this digit The digit fter the first deiml ple is 3 so round down. The numer orret to one deiml ple is 0.6 g Write orret to four deiml ples. Chek this digit The digit fter the fourth deiml ple is 1 so round down. The numer orret to four deiml ples is % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

5 Deimls Questions Bsis 1. Round the following numers to the nerest unit: d Round s sked: Write orret to 2 deiml ples. Write orret to 1 deiml ple. Write to the nerest hundred. d Write orret to the nerest unit. e Write orret to 1 deiml ple. f Write orret to 5 deiml ples. 3. The length of room is 5.52 m nd the redth is 3.41 m. Find the re orret to 2 deiml ples. (only round t the end) 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 3

6 Deimls Knowing More Signifint Figures In deiml numers, some of the digits re 'signifint' nd others re not. This depends on the ury of mesurement. Eh numer hs first signifint figure nd lst signifint figure hs first signifint figure 9 nd lst signifint figure 3. (The words 'digit' nd 'figure' men the sme thing here) How to find the first signifint figure The first signifint figure is the leftmost nonzero digit The length of the penil is 10 m whih ould e written s 0.10 m or km. In eh mesurement of the length of the penil, the first signifint figure is 1 euse it is the first nonzero digit. For exmple, hs first signifint digit 2 nd hs first signifint digit 8. Finding the lst signifint figure n e it triky. There re three types of numers we need to look out for when identifying the lst signifint digit. Type 1: The lst digit is nonzero. If the lst digit is nonzero, then the lst digit is lso the lst signifint digit. For exmple: Lst digit is nonzero (8) Lst digit is nonzero (7) Both these numers hve first signifint digit 5 nd lst signifint digit 8. Type 2: The lst digit is 0 nd is fter the deiml point The lst digit is zero fter the deiml point The lst digit is zero fter the deiml point Both these numers hve first signifint digit 1 nd lst signifint digit 0. Type 3: A whole numer ending with 0 (no deiml point) This is where it gets triky. It depends on the ury of the mesurement. Look t the numer If it ws rounded to the nerest hundred (for exmple, from 225 or 183) then only the 2 is urte nd it hs only one signifint figure (the 2). If it ws rounded to the nerest ten (for exmple from 199 or 202) then the first two digits re urte nd so there re two signifint figures (the 2 nd the first 0). If it ws rounded to the nerest unit then ll three digits re signifint % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

7 Deimls Knowing More Here re some exmples. Find the first nd lst signifint figures of the following The first signifint figure is 1. The first signifint figure is 3. The lst signifint figure is 8 (Type 1). The lst signifint figure is 0 (Type 2) (fter roudning to nerest 10) (fter rounding to the nerest 1000) The first signifint figure is 3. The first signifint figure is 2. The lst signifint figure is 8 (Type 3). The lst signifint figure is 5 (Type 3). Counting Signifint Figures To find the numer of signifint figures, ount the digits from the leftmost nd rightmost signifint numers. Find the first nd lst signifint figures of the following The first signifint figure is 5. The first signifint figure is 2. The lst signifint figure is 8 (Type 1). The lst signifint figure is the seond 0 (Type 2). There re 3 signifint figures. There re 5 signifint figures d The first signifint figure is 5. The first signifint figure is 4. The lst signifint figure is 1 (Type 1). The lst signifint figure is the third 0 (Type 2). There re 4 signifint figures. There re 7 signifint figures. e 3200 f 3200 (if this hs een rounded to nerest hundred) (if tht hs een rounded to the nerest 10) The first signifint figure is 3. The first signifint figure is 3. The lst signifint figure is 2 (Type 3). The lst signifint figure is the first 0 (Type 3). There re 2 signifint figures. There re 3 signifint figures. g 3200 h (if this hs een rounded to nerest unit) The first signifint figure is 3. The first signifint figure is 3. The lst signifint figure is the third 0 (Type 2). The lst signifint figure is the seond 0 (Type 3). There re 5 signifint figures. There re 4 signifint figures. 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 5

8 Deimls Knowing More Rounding to Signifint Figures Sometimes rounding is done to signifint figures insted of deiml ples. The sme method pplies, the only differene is insted of heking the next digit ording to deiml ple, hek the next digit ording to signifint figures. If the digit fter the required ury is 4 or less, then round down. If it is 5 or more, then round up. Round the following numers Round 3.77 to 2 signifint figures to 2 signifint figures is 3.8. Not signifint Write orret to 2 signifint figures to 2 signifint figures is Write orret to 3 signifint figures to 3 signifint figures is d Write orret to 4 signifint figures orret to 4 signifint figures. The zero on the end is neessry to mke 4 signifint figures 6 100% Deimls SERIES TOPIC Mthletis 100% 3P Lerning

9 Deimls Questions Knowing More 1. How mny signifint figures re in eh of these numers? d e f g h Round these numers: Write orret to 3 signifint figures. Write orret to 4 signifint figures. Write orret to 3 signifint figures. d Write orret to 3 signifint figures. 3. Find 2 possile numers whih n e written orret to 3 signifint figures s: % Deimls Mthletis 100% 3P Lerning SERIES TOPIC 7

10 Deimls Using Our Knowledge Rtios Rtios re esier to understnd from n exmple thn from their definition t first. So let's strt with n exmple In lss of oys nd girls, there re 12 oys How mny girls re in the lss if the rtio of oys to girls is 1:2? This mens tht for every 1 oy there re 2 girls. So if there re 12 oys then there re 12 # 2 = 24 girls. How mny girls re in the lss if the rtio of oys to girls is 2:1? This mens tht for every 2 oys there is 1 girl. So in lss with 12 oys there re 12 ' 2 = 6 girls. If the lss hs 18 girls, then wht is the rtio of oys to girls? There re 12 oys nd 18 girls so the rtio of oys to girls is 12:18 whih n e simplified to 2:3. d If the rtio of oys to girls is 3:2 then how mny students in totl re in the lss? For every 3 oys there re 2 girls. This rtio n lso e written s 12:8 (y multiplying oth sides y 4). So if there re 12 oys it mens there re 8 girls. The totl numer of students is = 20. Sometimes rtios n ompre more thn 2 things In upord the rtio of green soks to red soks to lue soks is 2:4:7. If there re 4 green soks, how mny red nd lue soks re there? For every 2 green soks there re 4 red soks. If there re 4 (or 2# 2) green soks there must e 2# 4 = 8 red soks. For every 2 green soks there re 7 red soks. If there re 4 (or 2# 2) green soks there must e 2# 7 = 14 lue soks. If there re 12 red soks, how mny green nd lue soks re there? For every 4 red soks there re 2 green soks. If there re 12 (or 3# 4) green soks there must e 3# 2 = 6red soks. For every 4 red soks there re 7 green soks. If there re 12 (or 3# 4) green soks there must e 3# 7 = 21lue soks. If there re 39 soks ltogether, then how mny green soks re in the upord? Aording to the rtio for every 13 (or ) soks, 2 of them re green. This mens 13 2 of the soks re green. ` green soks = 2 13 # totlsoks = 2 # 39 = 6 13 Rtios n e defined s ompring the numer of like quntities reltive to eh other % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

11 Deimls Using Our Knowledge Rtes Rtes ompre unlike quntities. The most ommon is speed whih ompres distne to time using km/h or m/s. For exmple, if r is driving t 20 km/h it mens tht fter every hour it will trvel distne of 20 km. While you're on hike you notie you wlk 2 metres eh seond. How long will it tke you to wlk kilometre? A kilometre is 1000 metres. If you wlk t 2 m/s, to wlk 1000 m it will tke 1000 m 2 m/s = 500 s So, it will tke 500 seonds for you to wlk kilometre whih is 8 minutes nd 20 seonds. Sometimes it's importnt to know how to onvert etween the units of rtes, like hnging n mount from m/s to km/h. A pipe fills tnk with wter t onstnt rte of 10.8 L/h (Litres per hour). Convert this to ml/s (millilitres per seond). Step 1: Write this s frtion: 10.8 L 1h Step 2: Convert the numertor nd denomintor s neessry: ^10. 8# 1000h ml = ^1# 3600h s Step 3: Simplify: = ml 3600 s = 3 ml/s How mny kilometres does driver driving t onstnt speed of 30 m/s trvel in 5 hours? Step 1: Convert the speed to km/h: 30 m 1 s ^30 ' 1000h km = = km # 3600 ^1' 3600h h 1h 1 = 108 km/h Step 2: Use the new rte to find the numer of metres trvelled in 5 hours: kilometres trvelled = 108km/h # 5h = 540 km 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 9

12 Deimls Questions Using Our Knowledge 1. Simplify these rtios s muh s possile: 16:4 25:5 9:21 d 8:30 2. In group of people there re 4 oys nd 16 girls Wht is the rtio of oys to girls in simplest form? Wht is the rtio of girls to oys in simplest form? 3. In ke reipe the rtio for the ingredients honey, flour nd wter is 3:10:6 If ke ontins 30 prts of honey, how mny prts of flour nd wter does it ontin? If there re 57 prts of ll these ingredients together, how mny prts re flour? If 12 prts of wter re used, how mny prts in totl mke up ll the ingredients of the ke? % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

13 Deimls Questions Using Our Knowledge 4. $3600 is to e divided etween Meliss nd Grthe. Find how muh money eh gets if the rtio of Meliss s money to Grthe s money is 5:1 Find how muh money eh gets if the rtio of Meliss s money to Grthe s money is 2:7 5. Let s sy you red 434 pges in week. Convert this to dily rte? How long would it tke you to red series of ooks totlling 3255 pges? 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 11

14 Deimls Questions Using Our Knowledge 6. Convert 18 km/h to m/s. 7. Convert 345 m/s to km/h. 8. A tnk fills up t rte of 300 ml eh seond t onstnt rte: Write this s rte nd then onvert this rte to L/min. Convert the rte to L/h. How muh liquid will e in the tnk fter 2 hours? % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

15 Deimls Questions Using Our Knowledge 9. The sle of mp sys 1:7500. How fr re two points in rel life, if on the pge they re 10.2 m prt? 10. Petrol osts 0.13 ents per millilitre. Convert this to rte of dollrs per litre. If the r uses 6.5 L for 100 km, how muh petrol is needed to drive 300 km. How muh will it ost to drive 300 km. 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 13

16 Deimls Thinking More Reurring Deimls A reurring deiml is deiml numer whose deiml digits repet sequene without ever stopping. For exmple or or They're lso lled 'repeting deimls'. To write these esily, dots re pled over the digits tht repet to sve writing relly long deimls. Here re some exmples = 03. o (1 repeting digit) = o o (2 repeting digits) = o o (4 repeting digits) Eh reurring deiml n e written s frtion where nd re oth integers. How is this frtion found? Here re some exmples Let x = Let x = Multiply oth sides y 10 Multiply oth sides y 100 ` 10x = ` 10x- x = ` 9x = 8 ` x = 8 9 ` = 8 9 Chek these vlues on lultor ` 100x = ` 100x- x = ` 99x = 53 ` x = ` = In the seond step of oth sides we multiplied y 10, nd in the seond step of oth sides were multiplied y 100. When hnging reurring deimls to frtions multiply oth sides y 10 n where n is the mount of repeting digits. There is little extr trik when not ll the digits re repeting digits. Convert o o to frtion Let x = o o ` 100x = ` 100x- x = ` 99x = ` 990x = 463 ` x = ` o o... = Not ll the deiml digits re repeting. There will e n extr trik There re 2 repeting digits (6 nd 7) so multiply oth sides y 10 2 = 100 Extr trik multiply oth sides to remove the deiml % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

17 Deimls Questions Thinking More 1. Write these reurring deimls using dots on top of the digits: d Write these in repeting form: 04. o 032. o o o d o o 3. Convert these to frtions in their simplest form: 06. o 035. o o o o d 278. o o 100% Deimls Mthletis 100% 3P Lerning SERIES TOPIC 15

18 Deimls Questions Thinking More 4. Prove tht 0.9o = 1. (Hint: Convert 0.9 o into frtion) 5. Convert these deimls to frtions: (Hint: There is n extr trik in these questions) 0.45 o o o o d o o % Deimls SERIES TOPIC Mthletis 100% 3P Lerning

19 Deimls Answers Notes Bsis: Using Our Knowledge: pges per dy 52.5 dys 1 d m/s 2. d e f km/h m L/min 1080 L/h 2160 L 1. Knowing More: km d 4 e 5 f $1.30 perl 19.5 L g 8 h 4 $ Thinking More: d o 0.86 oo or o o d o o or Using Our Knowledge: 41 : 5:1 d : d 4: : : d prts of flour nd 60 prts of wter prts re flour 38 prts 4. Meliss gets $3000 nd Grthe gets $600 Meliss gets $800 nd Grthe gets $ d % Deimls Mthletis 100% 3P Lerning SERIES TOPIC 17

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