Decimal numbers. Chapter

Chapter 6 Deimal numbers Contents: A B C D E F G H I Plae value Ordering deimal numbers Adding and subtrating deimal numbers Multiplying and dividing by powers of 0 Multiplying deimal numbers Dividing deimal numbers Terminating and reurring deimals Deimal approximations Comparing sizes

6 DECIMAL NUMBERS (Chapter 6) Deimal numbers are widely used in everyday life. We see them frequently in money and in measurements of length, area, weight and so on. HISTORICAL NOTE ² The earliest deimal system was probably invented by Elamites of Iran in the period 300-200 BC. ² The deimal system was developed in Anient India and Arabia. ² The deimal point we use in this ourse was probably invented by Bartholomaeus Pitisus in 62. ² A omma is used instead of a deimal point in some European ountries. OPENING PROBLEM If we divide by 3 using a alulator, the deimal number 0:333 333 3::::::: is formed. We all this number zero point 3 reurring as the line of 3s is endless. It is said that all reurring deimals are rational numbers whih result from one whole number being divided by another. If this is so, what fration is equal to zero point 9 reurring or 0:999 999 99::::::? A PLACE VALUE We have seen that the number system we use today is a plae value system using base 0. In this hapter we extend the plae value system to inlude parts of a whole. In Chapter 4 we introdued ommon frations so that we ould write numbers less than one. If we restrit ourselves to frations where the denominator is a power of 0, we an use the plae value system to represent both whole and frational numbers. We introdue a mark alled a deimal point to separate the whole number part from the frational part. For example: 73:24 represents 700 + 30 + + 2 0 + 4 + 24:09 represents 20 + 4 + + 9 When written as a sum like this, we say the number is in expanded form. The plae value table for 73:24 and 24:09 is: hundreds tens units tenths hundredths thousandths 73:24 7 3. 2 4 24:09 2 4. 0 9

DECIMAL NUMBERS (Chapter 6) 7 When a deimal number does not ontain any whole number part, we write a zero in the ones plae. This gives more emphasis to the deimal point. For example, we write 0:7 instead of :7. Zeros are also very important within deimal numbers. For example, 6:702 = 6 + 7 0 + 2 whereas 6:72 = 6 + 7 0 + 2 : Example Write in expanded form: 7:802 7:802 = 7 + 8 0 + 2 Example 2 Write in deimal form: 6 6 = 60 + = 6 + =0:06 EXERCISE 6A. Express the following in expanded form: a 3:6 b 8:07 0:23 d 2:06 e 3:007 f 0:000 4 g 3:08 h 0:0632 i 3:707 j 0:006 07 2 Write the following in deimal form: 2 a 0 b 3 24 d 83 e 0 + 3 37 f 3 g + 7 6 h 0 + + 9 3 i + 7 0 000 9 j + 9 k 0 + 6 7 l 0 + + + 7 0 000 Example 3 State the value of the digit 3 in 0:632 0:632 0 6 3 3 ) the 3 stands for. 3 State the value of the digit in the following: a 23 b 3:8 3:07 d 87:062 e 0:02 f 3 077 g 8:94 h 389:64

8 DECIMAL NUMBERS (Chapter 6) Example 4 You should be able Express 4 37 in deimal form. 4 37 =4+ + 30 + 7 =4+ 0 + 3 + 7 =4:37 to see how to do this in one step. 4 Express in deimal form: a 8 8 0 b 2 7 3 8 d 46 e 7 4 f 3 7 g 6 0 000 h 39 648 666 i j k 68 706 0 l CONVERTING DECIMALS TO FRACTIONS Our understanding of deimals and the plae value system allows us to onvert deimals to frations. Some frations an be anelled down to their simplest form by dividing both the numerator and denominator by their highest ommon fator (HCF). For example, 0:36 = 36 = 9 2 when all ommon fators have been anelled. Example Write as a fration in simplest form: a 0:8 b 3:88 0:37 a 0:8 = 8 0 = 4 b 3:88 =3+ 88 =3 22 2 0:37 = 37 = 3 8 EXERCISE 6A.2 Write as a fration in simplest form: a 0:3 b 0:9 :2 d 2: e :7 f 3:2 g 0: h 0:6 i 0:02 j 0:07 k 0:04 l 0:2 2 Write as a fration in simplest form: a 0:27 b 0:84 0:004 d 0:0 e 0:0004 f 0:27 g 0:82 h 0:002 i 0:62 j 0:000 0 k 4:08 l 0:07

DECIMAL NUMBERS (Chapter 6) 9 B ORDERING DECIMAL NUMBERS Just like whole numbers, deimal numbers may be shown on a number line. To do this we generally divide eah segment of the number line into ten equal parts. Example 6 Plae the values 2:, 2:4 and 2:8 on a number line. Divide a number line from 2 to 3 into ten equal parts. 2 2. 3 2: is plaed 0 of the way from 2 to 3. 2:4 lies half-way between 2:4 and 2:. 2:8 is plaed 8 0 of the way from 2 to 3. Example 7 Write down the values of A and B on the number line: 0.3 A B 0.4 The segment between 0:3 and 0:4 is divided into 0 equal parts, so the number line shows 0:30, 0:3, 0:32,..., 0:39, 0:40. A lies at 0:32 B lies half-way between 0:38 and 0:39, so B is 0:38. EXERCISE 6B Plae the following deimal numbers on separate number lines: a :2, :3, :9 b 4:3, 4:7, 4:8 68:7, 68:2, 69: d :8, 6:9, 6:2 e 0:22, 0:26, 0:29 f :8, :8, :88 2 Write down the values of A and B on the following number lines: a b 2 A B 3 0. B A 0.6 e 0.2 A B 0.2 0.9 A B 0.92 d f 2.2 A B 2.3 6.73 A B 6.74

20 DECIMAL NUMBERS (Chapter 6) Example 8 > means Insert >, < or = between the numbers :302 and :3. Both numbers have wholes and three tenths. :302 has zero hundredths whereas :3 has one hundredth. So, :302 < :3 is greater than. < means is less than. 3 Insert <, > or = between these pairs of numbers: a 3:63, 3:6 b 7:07, 7:7 0:008 76, 0:0786 d 0:229, 0:292 e 0:47, 0: f 2:0, 2:0 67 g 0:746, 0:467 h 0:076, i 0:306, 0:603 0 j, 0: k 7:, 7:00 l 0:7, 70 Example 9 Write the following deimal numbers in asending order (from smallest to largest): 7:3, 7:28, 7:09 To help ompare the numbers we write them with the same number of plaes after the deimal: 7:30, 7:280, 7:09 The numbers eah have the same whole number part: 7:30, 7:280, 7:09 but different values in the tenths plae: So, 7:30, 7:280, 7:09 7:09, 7:28, 7:3 are in asending order. We an write zeros at the end of deimal numbers without hanging the plae value of the other digits. 4 Write in asending order: a 2:36, 2:3, 2:036 b 9:43, 9:34, 9:3, 9:04 0:, 0:49, 0:02 d 8:7, 8:7, 8:6, 9: e 8:0, 7:99, 8: f 7:209, 7:092, 7:902, 7:29 g 3:, 3:09, 3:2, 3:009 h 0:9, 0:09, 0:99, 0:099 Matthew s best four times for an 80 m sprint are 9:9 seonds, 9:09 seonds, 9:99 seonds and 9:89 seonds. Plae these times in order from fastest to slowest. 6 On Monday the US dollar ould be exhanged for 0:72 euros. This means that eah US dollar was worth a little more than 72 euro ents. For the rest of the week the exhange figures were: Tuesday 0:722, Wednesday 0:720, Thursday 0:702, and Friday 0:722. Plae the exhange rates in order from highest to lowest.

DECIMAL NUMBERS (Chapter 6) 2 C ADDING AND SUBTRACTING DECIMAL NUMBERS When adding or subtrating deimal numbers, we write the numbers under one another so the deimal points are diretly underneath eah other. When this is done, the digits in eah plae value will also lie under one another. We then add or subtrat as for whole numbers. Example 0 Find: :76 + 0:96 : 760 + 0 : 9 6 2 : 72 Notie that the deimal points are plaed diretly underneath eah other. Example Find: a 4:632 :07 b 8 0:706 a 4 : 6 2 3 2 2 : 07 b 3 : 2 7 8 : 0 9 0 9 0 0 0 : 706 7 : 294 Plae the deimal points diretly under one another and subtrat as for whole numbers. We insert :000 after the 8 so we have the same number of deimal plaes in both numbers. EXERCISE 6C Find: a 0:3 +0:6 b 0:8 + 0:23 0:7 + :36 d 0:2 + 0:9 + 2 e 0:076 + 0:6 f :6 + 8:072 g 0:07 + 0:477 h 2:66 + :302 i 0:0037 + 0:628 j 0:02 + 0:979 k 3:69 + 8:09 l 0:6 + 2:09 + 0:89 2 Find: a : 0:8 b 2:6 :7 0:3 d 3 0:72 e 3:2 0:6 f 0:99 g 0:9999 h :6 0:9 i :2 3:6 j 0:083 0:009 k :2 0:6 l 0:6 + 0:093 0:3

22 DECIMAL NUMBERS (Chapter 6) 3 Add: a 38:76, 32:8 and 9:072 b 8:6, 236:9 and 072:4 9:04, 360:8 and 0:034 d 0:76, 0:6, 08:77 and 0:862 4 Subtrat: a 8:6 from 9:837 b 4:86 from 28:79 :603 from 20 d 9:674 from 68:3 A 20 m length of rope is ut into 4 piees. Three of the piees have lengths :62 m, 8:0 m, and 2:6 m. Find the length of the fourth piee. 6 A weightlifter snathes 3:8 kg, 42:9 kg, and 3:7 kg in onseutive lifts. Find the total mass lifted. 7 How muh hange would you expet from a $20 note if you purhased artiles osting $8:63, $:09 and $4:73? 8 Rosemary s Visa Card Statement is given alongside: What would be the total at the bottom of the statement? Tesos $30:80 Sports World $288:00 The Red Lion Inn $46:42 B&Q $387:9 Morrisons $9:46 Sainsburys $4:9 9 Taxation E07:90 Private Health Cover E9:20 Superannuation E9:62 Union Fees E4:82 Eah fortnight Alex is paid E700 less the dedutions given in the table alongside. What is Alex s atual take home pay eah fortnight? 0 Continue the number sequenes by writing the next three terms: a 0:, 0:2, 0:3,... b 0:2, 0:4, 0:6,... 0:03, 0:0, 0:07,... d 0:0, 0:06, 0:07,... e :2, :, :0,... f 0:, 0:22, 0:33,... g 7:4, 7:2, 7:0,... h 3:2, 2:8, 2:4,... i 0:6, 0:, 0:,... D MULTIPLYING AND DIVIDING BY POWERS OF 0 MULTIPLICATION Consider multiplying 2:36 ² by : 2:36 = 236 = 236 ² by : 2:36 = 236 = 236 0 = 2360

DECIMAL NUMBERS (Chapter 6) 23 When we multiply by, the deimal point of 2:36 shifts 2 plaes to the right. 2:36 beomes 236. When we multiply by, the deimal point shifts 3 plaes to the right. 2:360 beomes 2360. When multiplying by 0 n we shift the deimal point n plaes to the right. The number beomes 0 n times larger than it was originally. Remember 0 =0 0 2 = 0 3 = 0 4 = 0 000 The index or power indiates the number of zeros.. Example 2 Find: a 9:8 0 b 0:07 3:026 0 000 a 9:8 0 =9:8 0 =98 b 0:07 =0:07 0 2 =7: 3:026 0 000 =3:0260 0 4 = 30 260 f0 = 0, so shift the deimal point plae right.g f = 0 2, so shift the deimal point 2 plaes right.g f0 000 = 0 4, so shift the deimal point 4 plaes right.g EXERCISE 6D. a Multiply 8:7 by: i 0 ii iii iv 0 b Multiply 0:073 by: i 0 ii iii 0 4 iv 0 6 2 Find: a 38 0 b 9 3:2 0 d 0:8 0 e 0:7 f 2:8 g 0:6 h 0:83 i :89 0 4 j 0:03 0 3 k 0:083 l 0:87 000

24 DECIMAL NUMBERS (Chapter 6) DIVISION Consider dividing 3:7 by : 3:7 = 37 0 = 37 0 = 37 =0:037 and by : 3:7 = 37 0 = 37 0 000 =0:0037 When we divide by, the deimal point in 3:7 shifts 2 plaes to the left. 003:7 beomes 0:037. When we divide by, the deimal point in 3:7 shifts 3 plaes to the left. 0003:7 beomes 0:0037. When dividing by 0 n we shift the deimal point n plaes to the left. The number beomes 0 n times smaller than it was originally. Example 3 Find: a 0:4 0 b 0:8 a 0:4 0 =0:4 0 =0:04 b 0:8 = 000:8 0 3 = 0:000 8 f0 = 0, so shift the deimal point plae leftg f = 0 3, so shift the deimal point 3 plaes leftg EXERCISE 6D.2 a Divide 0:9 by: i 0 ii iii 0 4 b Divide 70:6 by: i ii 0 000 iii 0 8 2 Find: a 7 0 b 89 0 463 0 d 463 e 463 f 463 0 000 g 0:8 0 h 0:8 i 0:73 j 0:07 0 k 0:07 l 0:083 m 0:0023 n 0:0028 0 000 o 0:000 0 000

DECIMAL NUMBERS (Chapter 6) 2 E MULTIPLYING DECIMAL NUMBERS We an explain how deimal numbers are multiplied by first onverting the deimals into frations. For example, onsider the produt 4 0:03. If we first onvert to frations, we have 4 0:03 = 4 3 fonvert to frationsg = 2 fmultiply frationsg =0:2: fonvert bak to deimalg =0:2 From this example we see that: ² we multiply the whole numbers 4 and 3 ² then divide by a power of 0, in this ase. Now onsider finding 0:4 0:0. If we first onvert to frations, we have 0:4 0:0 = 4 0 fonvert to frationsg = 20 fmultiply frationsg =0:020: fonvert bak to deimalg =0:02 One again we an see that ² we multiply the whole numbers 4 and ² then divide by a power of 0, in this ase. With pratie we do not need to onvert the deimals to frations first. We multiply the deimal numbers as though they were whole numbers, then divide by the appropriate power of 0. Example 4 Find: 0:3 0:07 0:3 0:07 We multiply 0:3 by 0 and 0:07 by, =(3 7) then balane by dividing by 0 =. =2 = 0:02 fshifting the deimal point 3 plaes leftg EXERCISE 6E Find the value of: a 0:3 0:2 b 0: 0:07 0:02 0:4 d (0:4) 2 e (0:06) 2 f 0:03 0:004 g 0:004 40 h 60 0:8 i 600 0:07 j 4000 0:6 k 0:04 40 000 l 0:3 0: 0:7

26 DECIMAL NUMBERS (Chapter 6) 2 Given that 87 23 = 8 3, evaluate: a 8:7 23 b 8:7 2:3 8:7 2:3 d 87 0:23 e 0:87 0:23 f 8:7 0:23 g 0:87 2:3 h 870 0:23 i 8:7 0:002 3 3 Evaluate: a 0:3 6 b 0: 4:0 0:03 7 d 0:03 700 e 2:8 f 0:6 0:8 g 0:8 0:0 h 0:0 0:4 i (0:2) 2 j (2:) 2 k 0:4 0: l (0:03) 2 m (0:2) 3 n (0:3) 3 o 2 0:0004 p +0:2 0:3 q 0:08 0:08 0:2 r (0:3 ) 2 4 a Find the ost of 72 books at E:7 eah. b Find the ost of 8: 6 m of plasti sheeting at $4: 62 per metre. In order to bake akes for the shool fair, I buy 80 kg of flour at $0:84 per kg and 2 kg of sugar at $:7 per kg. How muh money have I spent? 6 I load 40 bags of salt onto my lorry, eah having mass 0: kg. Find the total mass of all bags. 7 House briks have a mass of 4:3 kg eah and I buy 200 of them to build a wall around my ourtyard. a Find the total mass of the briks. b If my truk an arry only 2 tonnes at a time, how many truk loads are neessary to transport the briks? F INVESTIGATION What to do: DIVIDING DECIMAL NUMBERS DIVISION OF DECIMALS Copy and omplete the following divisions. Look for patterns to use when the divisions involve deimals. a 800 200 =, 80 20 =, 8 2=, 0:8 0:2 = b 800 20 =, 80 2=, 8 0:2 =, 0:8 0:02 = 80 200 =, 8 20 =, 0:8 2=, 0:08 0:2 = 2 In eah set of divisions, what did you notie about the answers? 3 Did you find that in eah set the division by the smallest whole number was the easiest?

DECIMAL NUMBERS (Chapter 6) 27 From the Investigation you should have observed that multiplying or dividing both numbers in a division by the same fator does not hange the result. This observation leads to the following rules of division: When dividing a deimal number by a whole number, arry out the division as normal, writing deimal points under eah other. Example Make sure you write Find: a 32: b 0:47 3 deimal points under one another. a 32: 2 6 : Answer: 6: b 3 0 : 4 2 7 0 : 3 9 Answer: 0:39 When dividing a deimal number by another deimal number, write the division as a fration. Multiply top and bottom by the same power of 0 so the denominator beomes a whole number. Then perform the division. Example 6 Find: a 8 0:06 b 0:02 :4 a 8 0:06 = 8 0:06 = 800 6 = 300 b 0:02 :4 = 0:02 0 :4 0 = 0:2 4 =0:0 0. 0 4 0. 2 0 4 7 0 7 0 0 EXERCISE 6F Evaluate: a 8:4 2 b :6 3 20:4 4 d 0: e 0:26 9 f :6 7 g 49:8 6 h 3:04 4 i 0:66 j 0:040 3 k 3:92 8 l 0:392 7 2 Calulate: a 0:9 0:3 b 4:9 0:7 0: d 0:36 0:6 e 0:8 0:6 f 0:2 0:0 g 3:2 0:08 h 2:7 0:003 i 0:84 0:2 j 0:7 0:7 k 0:2 0:03 l 2:88 0:023 3 Evaluate: a 0:36 4 b 3 3:2 : d 0:08 4 e 0:08 0:4 f 0:08 0:004 g :2 h :2 00

28 DECIMAL NUMBERS (Chapter 6) i :2 0:0 j 0:2 000 k 0:2 0 l 0:2 0: m 0:02 0:000 n 9 4 o 3 8 p 3: 2 q 0:03 2: r 0:049 0:07 4 Use your alulator to solve the following problems: a b d e f How many :6 m lengths of rope an be ut from a roll :2 m long? If E400:80 is distributed equally amongst 7 people, how muh does eah get? 4 gold nuggets have mass 0:7 kg, 0:369 kg, 0:836 kg and 2:93 kg respetively. Find: i the total mass in kg ii the average mass of the 4 nuggets. How many $:2 pakets of almonds an be bought for $2? Determine the number of 2:4 m lengths of piping required to onstrut a 720 m drain. How many tins of preserved fruit eah osting $2: an be purhased with $8:6? Example 7 Use your alulator to simplify: 3:4 + 0: 0:03 0:0 We must divide the whole of the numerator by the whole of the denominator, so we use the braket keys: Key in ( 3:4 + 0: ) ( 0:03 0:0 ) = Answer: 2670 Use your alulator to simplify: a 2:7 8 b 2+0:7 0:3 : 0:2 3:2 0:4 0:06 d e 0:02 0:0 0:76 f 2:6+:3 0:0 :6 0:2 0:6 4 0:0 0:6 G TERMINATING AND RECURRING DECIMALS Every rational number an be written as either a terminating or a reurring deimal. TERMINATING DECIMALS Terminating deimals result when the rational number has a denominator whih has no prime fators other than 2 or. For example, 8 =0:2 where the only prime fator of 8 is 2 3 20 =0: where the prime fators of 20 are 2 and.

DECIMAL NUMBERS (Chapter 6) 29 Example 8 Use division to write the following frations as deimals: a b 7 40 is really. a =0:2 : 0 0 : 2 b 7 40 = :7 4 fdividing top and bottom by 0g =0:42 4 : 7 0 2 0 0 : 4 2 Another method of onverting frations to terminating deimals is to write the denominator as a power of 0. For example: ² for halves multiply by to onvert to tenths ² for fifths multiply by 2 2 to onvert to tenths ² for quarters multiply by 2 2 to onvert to hundredths. Example 9 Write the following in deimal form, without arrying out a division: a 3 b 7 2 8 a 3 = 3 2 2 = 6 0 =0:6 b 7 2 = 7 4 2 4 = 28 =0:28 8 = 2 8 2 = 62 =0:62 EXERCISE 6G. Use division to write the following frations as terminating deimals: 9 a 0 b 4 4 d 3 e 8 3 7 f 40 g 20 h i 3 4 j 8 3 0 2 What would you multiply the following by to onvert the denominators to powers of 0? a 9 20 b 0 8 d 9 40 e 200 3 Write the following as deimals without arrying out a division: a 3 4 b 7 7 0 d 93 3 200 e 2 8 f 00 g 4 h 3 2 i 9 2 j 2 7 20

30 DECIMAL NUMBERS (Chapter 6) RECURRING DECIMALS Reurring deimals repeat the same sequene of numbers without stopping. Reurring deimals result when the denominator of a rational number has one or more prime fators other than 2 or. 3 For example, 7 =0:428 7 428 7 428 7 :::::: We indiate a reurring deimal by writing the full sequene one with a line over the repeated setion. We an also indiate it using dots. For example, 3 =0:3 or 0: _3 and 3 7 =0:428 7 or 0: _428 7_ Example 20 Write as deimals: a 7 9 b Some deimals take a long time to reur. For example, 7 a 9 =0:7777:::::: =0:7 b =0:444:::::: =0:4 9 7 : 0 7 0 7 0 7 0 :::::: 0 : 7 7 7 7 :::::: : 0 6 0 0 6 0 0 :::::: 0 : 4 4 4 :::::: 7 =0:88232947647 EXERCISE 6G.2 Write as reurring deimals: a 2 3 b 3 9 7 d 7 e 6 2 Use your alulator to write as reurring deimals: 2 a b 9 4 3 d 23 4 e 23 4 INVESTIGATION 2 CONVERTING RECURRING DECIMALS TO FRACTIONS Although it is quite easy to onvert frations into reurring deimal form, it is not always as easy to do the reverse proess. What to do: 2 Write down deimal expansions for 9, 9 and 7 9. a From your results, predit the fration equal to: i 0:4 ii 0:8 b Copy and omplete: 0:aaaaaa:::::: = 0:a = : 23 2 Use your alulator to write down deimal expansions for 99, 47 99 and 83 99 : a From your results, predit the fration equal to: i 0:7 ii 0:3 b Copy and omplete: 0:abababab:::::: = 0:ab = :

DECIMAL NUMBERS (Chapter 6) 3 3 Predit frations equal to a 0:7 b 0:3628 0:234 4 Sometimes the reurring part of a deimal ours after one or more plaes from the deimal point. For example, 0:73 =0:7333333:::::: Find the fration equal to: a 0:73 b 0:92 0:76 d 0:ab H DECIMAL APPROXIMATIONS We are often given measurements as deimal numbers. In suh ases we approximate the deimal by rounding off to the required auray. We have previously seen how to round off whole numbers. For example: 3628 ¼ 3630 (to the nearest 0) ¼ 3600 (to the nearest ) ¼ 4000 (to the nearest ) We round off deimal numbers in the same way. For example: 0:3872 ¼ 0:387 ¼ 0:39 ¼ 0:4 (to 3 deimal plaes) (to 2 deimal plaes) (to deimal plaes) RULES FOR ROUNDING OFF DECIMAL NUMBERS ² If the digit after the one being rounded is less than, i.e., 0,, 2, 3 or 4, then we round down. ² If the digit after the one being rounded is or more, i.e.,, 6, 7, 8 or 9, then we round up. Example 2 Find 36 7 orret to 3 deimal plaes. Rather than do long division, we an use a alulator. Pressing 36 7 =, the result is 2:7 647 09 We divide to the fourth deimal plae and then round to 3 deimal plaes. 36 So, 7 ¼ 2:8 (to 3 deimal plaes)

32 DECIMAL NUMBERS (Chapter 6) ROUNDING DECIMAL NUMBERS USING A CALCULATOR Most alulators have a FIX mode for rounding numbers to a ertain number of deimal plaes. To ativate the FIX mode, press MODE, selet the FIX option, and then speify the number of deimal plaes required. While you remain in FIX mode, all answers will be given to this number of deimal plaes. EXERCISE 6H a Write 0:7690 orret to: i deimal plae ii 2 deimal plaes iii 3 deimal plaes. b Write 0:0749 orret to: i deimal plae ii 2 deimal plaes iii 4 deimal plaes. 2 Find deimal approximations for: a 8:7 to the nearest integer b :63 to the nearest tenth 0:63 to 2 deimal plaes d 0:46 to 3 deimal plaes e 0:7 to 4 deimal plaes f 0:8 to deimal plaes. 3 Use a alulator to evaluate orret to the number of deimal plaes shown in the square brakets: a 37 7 [] b 2:4 3:79 [] (0:8) 2 [] d (0:72) 2 8 [2] e 23 [2] f 0:3 :7 [2] g (0:043) 2 [3] h 2:3 4 0:6 [3] i 3 [3] 4 Darren s batting average is alulated as 3:683. Round this to one deimal plae. Romandy Gold makes an annual profit of $36:748 million. Round this figure to one deimal plae. 6 Jordan shoots 32:87 points per basketball game. Round this to two deimal plaes. 7 Tye alulates the interest due on her savings aount to be $78:983. Round this to the nearest ent. 8 Use your alulator to round to the number of deimal plaes in brakets: a 4:67 [2] b :387 [3] 8:6604 [2] d 98:99 [] e :9962 [2] f 2:09 [] g :784 96 [4] h 7:499 [0] i 0:006 2 [] j 39 7 [4] k 6:9 :7 [3] l (0:367) 2 [3] m (8:39) 3 [3] n 0:637 (0:2) 2 [4] o 2:62 (8:37) 2 [3]

DECIMAL NUMBERS (Chapter 6) 33 I COMPARING SIZES How muh bigger is Afria than Australia? To answer questions like this we need to obtain data from an atlas or textbook or from the internet. If we know the areas of the two ontinents, we an use: The number of times A is bigger than B = size of A size of B. Using the internet, we find that Afria is about 30:22 million km 2 and Australia is about 7:68 million km 2. The number of times Afria is bigger than Australia is 30:22 7:68 ¼ 3:94 So, Afria is almost four times bigger than Australia. EXERCISE 6I How many times higher is: a Mount Everest (8848 m) than Mount Hunter (444 m) in Alaska? b Mount Manaslu (863 m) in Nepal than Mount Yasus (300 m) in Afria? Mount Konkur (779 m) in China than Mount Logan (9 m) in Canada? 2 How many times is: a Asia (4:84 million km 2 ) than Antartia (4:00 million km 2 ) b Afria (30:22 million km 2 ) than South Ameria (7:84 million km 2 ) Asia (4:84 million km 2 ) than Australia (7:68 million km 2 )? 3 How many times longer is: a the river Nile (660 km) in Afria than the river Murray (220 km) in Australia b the river Amazon (6400 km) in South Ameria than the river Snake (670 km) in the USA the river Yangtze (6300 km) in China than the river Rhine (320 km) in Europe? 4 Jon s property is 4:63 times larger than Helen s. If Helen s property is 368:7 ha, find the area of Jon s property. KEY WORDS USED IN THIS CHAPTER ² deimal number ² deimal point ² expanded form ² highest ommon fator ² plae value ² rational number ² reurring deimal ² simplest form ² terminating deimal LEAP YEARS LINKS lik here Areas of interation: Human ingenuity, Environment

34 DECIMAL NUMBERS (Chapter 6) REVIEW SET 6A a Convert 0:3 to a fration. b State the value of the digit 6 in 7:3264 : Multiply 8:46 by. d Write 7+ 4 0 + 3 in deimal form. e Find 70:2 : f State the value of the digit 2 in 0:362: g Evaluate 0:02 : h Write 2:64 as a fration in simplest form. 2 Express 23:42 as a fration in simplest form. 3 Evaluate: a 0:62 + 2:3 b 0:28 0:43 4 Write: a 23:49 to 2 deimal plaes b 0:4723 to the nearest hundredth 0.4 to 2 deimal plaes. If you purhased artiles osting $:63, $3:72 and $2:40, how muh hange would you reeive from a $0 note? 6 A rae trak is 3:2 km long. How many laps are needed to omplete a 360 km rae? 7 If a man s height is :6 times that of his daughter, who is 2 m tall, determine the height of the man. 8 New Zealand has an area of 268:68 thousand km 2. It has a human population of 4:3 million, and is home to 40: million sheep. How many: a sheep are there per person b people are there per km 2? REVIEW SET 6B a Write 0:3826 orret to 3 deimal plaes. b State the value of the digit in 46:04. Express 4:02 as a fration in simplest form. d Given that 82 76 = 6232, evaluate 8:2 0:076. e Divide 0:42 by. f Write 6 as a reurring deimal. 2 Evaluate: a 0:37 + 2:4 b 2:8 2:4 3 Insert <, > or = between these pairs of numbers: a 2:0 and 2:0 b 0:966 and 0:696 4 Find: a 0:4398 to 3 deimal plaes b 3:2849 to the nearest hundredth Evaluate: a 3 2:6 0:3 b 3 0:3+2:6 7 6 Evaluate without a alulator: a 23:42 + 36:023 b 0:72 0:02 0: 0: 0:2 7 Solve the following problems: a By how muh does 32 exeed 30:876? b If E07:2 is shared equally between 3 people, how muh does eah reeive? In one day a truk delivers 48 tonnes of sand to a building site. The first 3 loads were :2 tonnes, 3:76 tonnes and 2:82 tonnes. How muh sand was delivered in the fourth and final load?