Decimal Fractions. Decimal Fractions. Curriculum Ready.

Transcription

1 Decimal Fractions Curriculum Ready

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3 Decimal fractions allow us to be more accurate with our calculations and measurements. Because most of us have ten finers, it is thouht that this is the reason the decimal fraction system is based around the number 0! So we can think of decimal fractions as bein fractions with powers of 0 in the denominator. Write in this space EVERYTHING you already know about decimal fractions. Give this a o! Q To make dark-reen coloured paint, you can mix yellow and blue toether, usin exactly 0.5 (half) as much yellow as you do blue. How much dark-reen paint will you make if you use all of the 2.5 ml of blue paint you have? Work throuh the book for a reat way to do this H SERIES TOPIC

4 How does it work? Place value of decimal fractions Decimal fractions represent parts of a whole number or object. # Tens of thousands # 000 # 00 Thousands Hundreds Tens Ones # 0 Decimal point st decimal place: ' 0 # 0 one tenth 2 nd decimal place: ' 0 # 00 one hundredth 3 rd decimal place: ' 0 # 000 one thousandth 4 th decimal place: ' 0 # one ten thousandth etc... # ' 0 ' 00 Tenths Hundredths Thousandths Ten thousandths W H O L E D E C I M A ' 000 ' ' Hundred thousandths Millionths ' L ' Add th to the name for decimal place values Ten Millionths Write the place value of each diit in the number Multiply by multiples of 0 Divide by multiples of 0 Expanded forms Place values # hundred... # 0 0 tens (or sixty) Inteer parts # 5 5 ones (or five) ' 0 or 2 ` # 0 j ' 00 or 7 ` # 00 j ' 000 or 0 ` # 000 j ' or 3 ` # j tenths 7 hundredths 0 thousandths 3 ten thousandths st decimal place 2 nd decimal place 3 rd decimal place 4 th decimal place 2 H SERIES TOPIC

5 How does it work? Your Turn Place value of decimal fractions Write the decimal fraction that represents these: a d 2 hundredths b 9 tenths c ten thousandth 0.02 Always put a zero in front (called a leadin zero) when there are no whole numbers 3 thousandths e hundred thousandths f 8 millionths 2 Write the fraction that represents these: a 3 tenths b 7 thousandths c hundredth d 9 ten thousandths e 5 hundredths f ten thousandths 3 Write the place value of the diit written in square brackets for each of these decimal fractions: a 3@ d 5@ Circle the diit found in the place value iven in square brackets: a [tenths] b [thousandths] c [hundred thousandths] d [hundredths] e [ten thousandths] f [millionths] H SERIES TOPIC 3

6 How does it work? Your Turn Place value of decimal fractions Each diit is multiplied by the place value and then added toether when writin a number in expanded form. Write the decimal fraction in expanded form # + # + # + # + # # + # + # + # Multiply each diit by its place value Zero diits can be removed to simplify 5 Write these decimal fractions in expanded form: a 49. b c d e f Simplify these numbers written in expanded form: a b 4 # + # + # # + # + # + # /.../20... LACE VALUE OF DECIMAL FRACTIONS PPLACE VALUE OF DECIMAL FRACTIONS c d # + # + # + # + # + # # + # + # + # + 2 # + 9 # Psst: Remember to include a leadin zero for these ones. e f # # # 7 0 # + # + # + # # # # # # 4 H SERIES TOPIC

7 How does it work? Approximations throuh roundin numbers Look at these two statements made about a team of snowboarders: They have attempted 4937 tricks since startin Accurate statement They have attempted nearly 5000 tricks since startin Rounded off approximation Roundin off values is used when a reat deal of accuracy is not needed. The next diit followin the place value where a number is bein rounded off to is the important part. Next diit Closer to lower value, so round down Leave the place value unchaned Closer to hiher value, so round up Add to the place value Here are some examples to see how we round off numbers. Round these numbers (i) 242 to the nearest hundred The diit 4 is in the hundreds position The next diit is a, so round up by addin to 4 Chane the other smaller place value diits to 0 s ` rounded to the nearest hundred (ii) to one decimal place (or to the nearest tenth) The diit 3 is in the first decimal place The next diit is a, so round down Write decimal fraction with one decimal place only ` rounded to one decimal place (iii) to four decimal places (or to the nearest ten thousandth) The diit is in the fourth decimal place The next diit is a 9, so round up by addin to Write decimal fraction with four decimal places only ` rounded to four decimal places H SERIES TOPIC 5

8 How does it work? Your Turn APPROXIMATION THROUGH ROUNDING NUMBERS. Approximations throuh roundin numbers Round these whole numbers to the place value iven in square brackets /.../20... a [nearest ten] b [nearest hundred] c [nearest thousand] (i) 53. (i) (i) (ii) 854. (ii) 474. (ii) (iii) (iii) (iii) Round these decimal fractions to the decimal places iven in the square brackets. a [nearest tenth] b [nearest hundredth] c [nearest thousandth] (i) (i) (i) (ii) (ii) (ii) (iii). 85. (iii) (iii) Approximate the followin distance measurements: a A roup of people form an 8.82 m lon line when they stand toether. (i) How lon is this line to the nearest 0 cm (i.e. decimal place)?. (ii) What is the approximate lenth of this line to the nearest 0 metres?. b Under a microscope the lenth of a dust mite was m (i) Approximate the lenth of this dust mite to the nearest ten thousandth of a metre.. (ii) Approximate the lenth of this dust mite to the nearest hundredth of a metre.. c If Lichen City is m away from Moss City: (i) What is this distance approximated to the nearest km? (i.e. nearest thousand). (ii) What is the approximate distance between the cities to the nearest 00 km?. (iii) Are the diits 2, 3 or even 5 important to include when describin the total distance between the two cities? Briefly explain here why/why not. H SERIES TOPIC

9 How does it work? Your Turn Approximations throuh roundin numbers Roundin up can affect more than one diit when the number 9 is involved. Round 0.95 to one decimal place The diit 9 is in the tenths position 9 rounds up to 0, so the 9 becomes 0 and is added to the diit in front The next diit is a 5, so round up by addin to 9 Chane the other smaller place value diits to 0s ` rounded to one decimal place 4 Round off these numbers accordin to the square brackets. a [one decimal place] 98.. d [nearest ones] [nearest thousand] b [nearest ten] 398. e [three decimal places] h [nearest ones] c [two decimal places] f [three decimal places] i [four decimal places] Approximate these values: a A call centre receives an averae of calls each day durin one month. (i) Approximate the number of calls received to the nearest hundreds. (ii) Approximately how many thousands of calls did they receive? (iii) Estimate the number of calls received daily throuhout the month.... b A swimmin pool had a slow leak, causin it to empty L in one week. (i) How much water was lost to the nearest 0 litres? (ii) How much water was lost to the nearest ml if ml 000 L?.. (iii) Is the diit important when approximatin to the nearest whole litre? Briefly explain here why/why not. H SERIES TOPIC 7

10 How does it work? Decimal fractions on the number line The smallest place value in a decimal fraction is used to position points accurately on a number line. Decimal fractions are based on the number 0, so there are always ten divisions between values E: Here is the value 3. on a number line: Six tenths of the way from 3.0 to 4.0 The major intervals on the number line are marked accordin to the second last decimal place value 8 So its eiht thousandths of the way from.240 to Here are some more examples involvin number lines: (i) What value do the plotted points represent on the number lines below? a) Point is four steps from 0. towards 0.2, so the plotted point is: 0.4 b) Point is nine steps from 0.0 towards 0.07, so the plotted point is: 0.09 (ii) Round the value of the plotted points below to the nearest hundredth. a) Point is three steps from 2.4 towards 2.5, so the plotted point is 2.43 ` the value of the plotted point to the nearest hundredth is: 2.4 b) Point is five steps from 8.79 towards 8.80, so the plotted point is ` the value of the plotted point to the nearest hundredth is: H SERIES TOPIC

11 How does it work? Your Turn 4 Decimal fractions on the number line Display these decimal fractions on the number lines below: a 0.7 b DECIMAL FRACTIONS ON THE NUMBER LINE.../.../20... c 0.3 d e 2.34 f Label these number lines and then display the iven decimal fraction on them: a. b 4.2 c 0.94 d 7.07 e f Round the value of the plotted points below to the nearest place value iven in square brackets. a [tenth] b [hundredth] ` the value. ` the value. c [tenth] d [hundredth] ` the value. ` the value. e [thousandth] f [thousandth] ` the value. ` the value. [thousandth] h [thousandth] ` the value. ` the value. H SERIES TOPIC 9

12 How does it work? Multiplyin and dividin by powers of ten Move the decimal point dependin on the number of zeros decimal point moves riht, decimal point moves left Calculate these multiplication and division questions involvin powers of 0: (i) 5 # 000 We can simply add the same number of zeros to the end of the whole number 5 # # The whole number in decimal fraction form Fill the empty bounces with 0s If the decimal point is on the left after dividin, an extra 0 is placed in front. (ii) 8 00 ' 8' ' 00 The whole number in decimal fraction form Remember to include the leadin zero '00 has 2 zeros, so move decimal point 2 spaces left 0.08 Fill the empty bounces with 0s and put a zero in front (iii) # # Move decimal point 4 spaces riht No empty bounces to fill, so this is the answer (iv) ' ' Move decimal point 5 spaces left Fill empty bounces with 0s and put a zero in front (v) 205. # # 205. ' # is the same as ' Move decimal point 3 spaces left Place a leadin zero in front of the decimal point 0 H SERIES TOPIC

13 How does it work? Your Turn Multiplyin and dividin by powers of ten Calculate these multiplications. Remember, multiply means move decimal point to the riht: a 8 00 # b 3.4 # 0 c 29 # 000 d # e 0.52 # 00 f # Calculate these divisions. Remember, divide means move decimal point to the left: a 2 ' 00 b 4590 ' 000 c. 004 ' 0 d ' e ' f ' Here are some of the powers of 0 in exponent form. The power the number of zeros Calculate these mixed problems written in exponent form: a # b 2400 ' 0 5 c # 0 d # e ' f ' 0 7 H SERIES TOPIC

14 How does it work? Your Turn Multiplyin and dividin by powers of ten MULTIPLYING AND DIVIDING BY POWERS OF TEN 4 For these calculations: (i) Show where our character needs to spray paint a new decimal point, and (ii) write down the two numbers the new decimal point is between to solve the puzzle.../.../20... a # I 9 and 2 b ' N c # A d ' O e # X f # T # R h ' I i # D j # P This is another mathematical name for a decimal point: I 0 and 9 8 and 9 8 and 7 9 and 2 0 and 7 3 and 9 8 and 2 0 and 8 3 and 8 and 7 2 H SERIES TOPIC

15 How does it work? Terminatin decimal fractions to fractions These have decimal fraction parts which stop (or terminate) at a particular place value. The place value of the last diit on the riht helps us to write it as a fraction. Write 0.3 as a fraction: Decimal fraction 0.3 Fraction 3 0 Decimal fraction diits in the numerator Last diit is in tenths position Inteers in front of the decimal fraction values are simply written in front of the fraction. Write.07 as a fraction: Decimal fraction diits in the numerator Last diit is in hundredths position 07 is just 7 Always simplify the fraction parts if possible. These two examples show you how. Write each of these decimal fraction as an equivalent (equal) fraction in simplest form (i) Equivalent, un-simplified fraction 00 Last diit is in hundredths position 25 ' ' 25 Divide numerator and denominator by HCF 4 Equivalent fraction in simplest form (ii) Equivalent, un-simplified mixed number Last diit is in thousandths position 2 05 ' ' 5 Divide numerator and denominator by HCF Equivalent mixed number in simplest form H SERIES TOPIC 3

16 How does it work? Your Turn Terminatin decimal fractions to fractions Write each of these decimal fractions as equivalent fractions: a 0. b. c 009. d f h. e i 0.29 j. k l Write each of these decimal fractions as equivalent fractions and then simplify: a 0.5 b 0. c 0.02 Simplest form Simplest form Simplest form d 0.08 e f Simplest form Simplest form Simplest form 0.2 h 0.25 i Simplest form Simplest form Simplest form j k l Simplest form Simplest form Simplest form 4 H SERIES TOPIC

17 TERMINATING DECIMAL FRACTIONS TO FRACTIONS * Where does it work? Your Turn Terminatin decimal fractions to fractions 3 Write each of these decimal fractions as equivalent mixed numbers: /.../20... a 23. b. c e f.00 d Write each of these decimal fractions as equivalent mixed numbers and then simplify: a 2.8 b.4 c.0 4 Simplest form Simplest form Simplest form 3 5 e 275. f d.0 Simplest form Simplest form Simplest form. 004 h i. 344 Simplest form Simplest form Simplest form H SERIES TOPIC 5

18 How does it work? Fractions to terminatin decimal fractions Where possible, just write as an equivalent fraction with a power of 0 in the denominator first. numerator denominator 3 3 # # 2 0 ` 0. Multiply numerator and denominator by the same value Equivalent fraction with a power of 0 in the denominator Three fifths six tenths zero point six Sometimes it is easier to first simplify the fraction before chanin to a decimal fraction. Write these as an equivalent decimal fraction (i) ' 3 2 ' 3 4 Simplify fraction 4 4 # # Equivalent fraction with a power of 0 in the denominator ` 025. Three twelfths one quarter twenty five hundredths zero point two five (ii) ' 3 5 ' Simplify fraction part 2 # 2 5 # Equivalent fraction with a power of 0 in the denominator Two and three fifteenths two and one fifth two and two tenths two point two H SERIES TOPIC

19 How does it work? Your Turn Fractions to terminatin decimal fractions include a leadin zero Write each of these fractions as equivalent decimal fractions. a 9 b 3 c d Write each of these as equivalent fractions with a power of 0 in the denominator. a c 2 b d 20 9 e 8 f 3 h i 4 j k (i) Write each of these as equivalent fractions with a power of 0 in the denominator. (ii) Chane to equivalent decimal fractions. a 5 b 4 c 25 d 4 25 e 200 f h 200 i H SERIES TOPIC 7

20 How does it work? Your Turn Fractions to terminatin decimal fractions 4 Chane each of these fractions to equivalent decimal fractions after first simplifyin. Show all your workin. a 2 20 b c d /.../20... G DECIMAL FRACTIONS NFRACTIONS TO TERMINATI e 9 75 f h H SERIES TOPIC

21 How does it work? Your Turn Fractions to terminatin decimal fractions When chanin the denominator to a power of 0 is not easy, you can write the numerator as a decimal fraction and then divide it by the denominator. Write this fraction as an equivalent decimal fraction ' 8 Write numerator as a decimal fraction and divide by the denominator If you need more decimal place 0s, you can add them in later! Complete division, keepin the decimal point in the same place ` Five eihths zero point six two five 5 Complete these divisions to find the equivalent decimal fraction: a ' 5 b ' 4 3 c ' ' ' ' d e f H SERIES TOPIC 9

22 How does it work? Your Turn Fractions to terminatin decimal fractions Simplify these fractions and then write as an equivalent decimal fraction usin the division method. Show all your workin. a 2 5 b 9 2 c 49 d e 8 24 f 2 20 H SERIES TOPIC

23 Where does it work? Addin and subtractin decimal fractions Just add or subtract the diits in the same place value. To do this, line up the decimal points and matchin place values vertically first. Add 2.45 to.3 (i.e ) Decimal points lined up vertically Add matchin place values toether Subtract 5.8 from.89 (i.e ) Decimal points lined up vertically Subtract matchin place values Calculate each of these further additions and subtractions (i) Add matchin place values toether Any place value spaces are treated as 0s ` Roundin decimal fractions before addin is sometimes used to quickly approximate the size of the answer. (ii) Round each value in question (i) to the nearest whole number before addin. Decimal points lined up vertically ` Values rounded to nearest ones Approximate value for addition Note: Roundin values before addin/subtractin is not as accurate as roundin after addin/subtractin. (iii) Fill place value spaces in the top number with a 0 when subtractin ` Decimal points lined up vertically Subtract matchin place values H SERIES TOPIC 2

24 ADDING AND SUBTRACTING DECIMAL FRACTIONS Where does it work? Your Turn Addin and subtractin decimal fractions Complete these additions and subtractions: a b. 8 + c d e f h Calculate these additions and subtractions, showin all workin: a Add 8.75 to.24 b Subtract 3.5 from /.../20... c Add 0.93 to 0.85 d Add 2.9, 5. and 0.3 e Subtract from 8.02 f Add 0.20, 4.4 and H SERIES TOPIC

25 Where does it work? Your Turn Addin and subtractin decimal fractions 3 a Approximate these calculations by roundin each value to the nearest whole number first. (i) (ii) (iii) (iv) (v) (vi) b Calculate parts (v) and (vi) aain, this time roundin after addin the numbers to et a more accurate approximate value. (i) (ii) Calculate these subtractions, showin all your workin: a - b c H SERIES TOPIC 23

26 Where does it work? Multiplyin with decimal fractions Just write the terms as whole numbers and multiply. Put the decimal point back in when finished. The number of decimal places in the answer the number of decimal places in the question! Calculate 4 # # Multiply both terms as whole numbers 48 decimal place in question decimal place in answer ` 4 # Calculate 0.02 # # Multiply both terms as whole numbers ` 0.02 # decimal places in question 4 decimal places in answer How does this work when multiplyin with decimal fractions? Excellent question! Very lad you asked! Let s do the second one aain but this time chane the decimal fractions to equivalent fractions first 002. # # # 45 00# Chanin the decimal fractions to fractions Multiply numerators and denominators toether Number of zeros in denominator total of decimal places in question 290' Dividin by moves decimal point four places to the left ` 4 decimal places in question 4 decimal places in answer Try this method for yourself on the first example above, rememberin that 4 4 as a fraction. 24 H SERIES TOPIC

27 MMULTIPLYING WITH DECIMAL FRACTIONS Where does it work? Your Turn Multiplyin with decimal fractions Calculate these whole number and decimal fraction multiplications, showin all you workin: a # b 5 5. c # 0.4 # 0.2 # 4 3 # # 2 d e f 2 Calculate these decimal fraction multiplications, showin all your workin: 3.8 # # # 25. a b c.../.../20... ULTIPLYING WITH DECIMAL FRACTIONS 7. # # # 93. d e f H SERIES TOPIC 25

28 Where does it work? Dividin with decimal fractions Opposite to multiplyin, we move the decimal point before dividin if needed. To find the quotient involvin decimal fractions, the question must be chaned so the divisor is a whole number. dividend ' divisor quotient Calculate 428. ' Divisor already a whole number so no chane needed ` 4.28 ' 4.07 Calculate ' ' ' 000 Move both decimal points riht until divisor is a whole number 45. ' Quotient 2 Dividend if divisor ` ' Drop off any 0s at the front of the answer Here s another example showin how to treat remainders Calculate 2. ' ' ' 0.8 Move both decimal points riht until divisor is a whole number 2. ' Add 0s on the end of the dividend for each new remainder ` 2. ' Drop off any 0s at the front 2 H SERIES TOPIC

29 Where does it work? Your Turn Dividin with decimal fractions Calculate these decimal fraction and whole number divisions: 3.' ' 5 a b c. 2' 9.../.../20... DDIVIDING WITH DECIMAL FRACTIONS IVIDING WITH DECIMAL FRACTIONS ` 3.' 4 ` 7.5 ' 5 `.2 ' ' ' ' 7 d e f ` 0.3 ' 3 ` 0.489' 5 ` 0.97 ' 7 2 Calculate these decimal fraction divisions, showin all your workin: 52. ' ' ' 08. a b c ` 5.2' 0.4 ` 9.' 0. ` 0.5 ' ' ' ' d e f `.58 ' 0.4 ` ' 0.05 ` ' 0.00 H SERIES TOPIC 27

30 Where does it work? Recurrin decimal fractions Non-terminatin decimal fractions have decimal parts that do not stop. They keep oin on and on Three dots means it keeps oin If the decimal parts have a repeatin number pattern, they are called recurrin decimal fractions Here are some examples involvin recurrin decimal fractions A dot above the start and end diit of the repeatin pattern is used to show it is a recurrin decimal fraction. (i) Write these recurrin decimal fractions usin the dot notation The pattern 2 keeps repeatin in the decimal parts a) b) c) o o A bar over the whole pattern can also be used instead of dots (ii) Calculate ' Start End 0. 8 o o Start End o o Start and End 047. o or.047r 0. ' 0. ' Identify the start and end of the repeatin pattern Dot above start and end of the repeatin pattern Identify the start and end of the repeatin pattern Dot above start and end of the repeatin pattern Identify the start and end of the repeatin pattern Dot above start and end of the repeatin pattern ' Write as a decimal fraction with a few 0s ` ' o Repeats the same remainder when dividin Recurrin decimal fraction in simplest notation 28 H SERIES TOPIC

31 Where does it work? Your Turn Recurrin decimal fractions What is the name of the horizontal line above the repeated numbers in a recurrin decimal fraction? Hihliht the boxes that match the recurrin decimal fractions in each row with the correct simplified notation in each column to find the answer. Not all of the matches form part of the answer! r. 4. o o o o o C z F h N d W c D b A a U n P t L f O m Y n A m R f T t K z E h R d I c U b S a L a D b A m I h M t B f S c A d U z Q n R h Z d A n E z A c N t 0 a M b A h G f A f T z P c H d T a Y n A t A h C m A b I d Y t A b U n H m I z E f S m I t T a A b L a D t E f A d N c L m E z O d N h W c J f B d A a X h M m A b U n A A z P m V c E a F b A n B d T Y f E c I t H t A n A m A m U f A b A h A a D d R c c z h m n a f b 2 Calculate these divisions which have recurrin decimal fractions as a result. Write answers usin dot notation. ' 3 4' 9 5' a b c ` ' 3 ` 4' 9 ` 5'.' 25. ' ' 3 d e f `. ' ` 25. ' 9 ` 034. ' 3 H SERIES TOPIC 29

32 Where does it work? Your Turn Recurrin decimal fractions 3 (i) Complete the followin divisions to five decimal places. (ii) Determine whether the answer is a recurrin decimal fraction or not. 2' 3 ' ' 7 a b c.../.../20... ECURRING DECIMAL FRACTIONS... RRECURRING DECIMAL FRACTIONS... ` 2' 3 ` ' ` ' 7 Recurrin decimal fraction? Recurrin decimal fraction? Recurrin decimal fraction? Yes No Yes No Yes No. ' ' ' 08. d e f `. ' 7 ` 29. ' 3 ` 0.33 ' 0.8 Recurrin decimal fraction? Recurrin decimal fraction? Recurrin decimal fraction? Yes No Yes No Yes No 0.8' ' ' h i ` 0.8 ' 0.3 ` 0.09' 0.0 ` ' Recurrin decimal fraction? Recurrin decimal fraction? Recurrin decimal fraction? Yes No Yes No Yes No 30 H SERIES TOPIC

33 What else can you do? Simple recurrin decimal fractions into sinle fractions Only recurrin, non-terminatin decimal fractions can be written in fraction form. Here is a quick way for simple decimal fractions with the pattern startin riht after the decimal point o 9 One diit in repeatin pattern, so that diit over o o ' 3 99 ' Two diits in repeatin pattern, so those two diits over 99 Always simplify fractions o o Three diits in repeatin pattern, so those three diits over 999 Here are some other examples includin mixed numbers. Write each of these recurrin decimal fractions as mixed numbers in simplest form (i) o One diit in repeatin pattern, so that diit over Diits in front of decimal point form the whole number part (ii) o o Three diits in repeatin pattern, so those diits over Diits in front of decimal point form the whole number 345 ' ' Simplify the fraction part H SERIES TOPIC 3

34 SIMPLE RECURRING DECIMAL FRACTIONS INTO SINGLE FRACTIONS What else can you do? Your Turn Simple recurrin decimal fractions into sinle fractions Use the shortcut method to write each of these recurrin decimal fractions as a fraction in simplest form: 04. o 08r. 0. o a b c 0. oo 027. o o 057. o o d e f 2 Use the shortcut method to write each of these recurrin decimal fractions as mixed numbers in simplest form. 5. o 27r. 43r. a b c 3r o o d. e f o o h i 3 (i) Write 09. o as a fraction in simplest form. (Ii) Does anythin unusual seem to be happenin with your answer? Explain /.../ H SERIES TOPIC

35 What else can you do? Combinin decimal fraction techniques to solve problems All the techniques in this booklet can be used to solve problems. These examples show different ways decimal fractions pop up in every-day life (i) These rainfall measurements were taken durin three days of rain from a small weather aue: 3.8 mm 3. mm 27. mm What was the total rainfall for the three days, to the nearest whole mm? mm Add the decimal fraction values toether Round to nearest whole mm ` The total rainfall over the three days was approximately 78 mm Answer with a statement (ii) The results for five runners in a 00 m race were plotted on the number line below seconds a) What was the fastest time run (to the nearest thousandth of a second)? Fastest time left-most plotted point.22 seconds b) What time did two runners finish the race toether on? Two runners with the same time two dots at the same point.223 seconds c) What was the averae time ran by all runners in this race? Averae time The sum of all the times ran divided by the number of runners ( ) ' ' The averae time ran by all the runners in the race. 2242seconds Read off all the times Add, then divide by 5 Answer with a statement H SERIES TOPIC 33

36 COMBINING DECIMAL FRACTION TECHNIQUES TO SOLVE PROBLEMS What else can you do? Your Turn Combinin decimal fraction techniques to solve problems To make dark-reen coloured paint, you can mix yellow and blue toether, usin exactly 0.5 (half) as much yellow as you do blue. a Use multiplication to show how much yellow paint you will need if you use all of the 2.4 ml of blue paint you have. Remember me? b How many millilitres of dark-reen paint can you make with 8.45 ml of yellow paint in the mix? Round your answer to the nearest tenth of a ml. 2 Derek types his essays at an averae speed of words every minute. How many words does he type in five minutes (to the nearest whole word)? 3 Nine people were tryin out for a speed roller skatin team around an oval flat track. The shortest time to complete six full laps of the track for each person were recorded on the number line below:.../.../ seconds a What was the slowest time recorded to 3 decimal places? b To make the team, a skater had to complete the six laps in less than seconds. How many skaters made it into the team? c How many skaters missed out makin the team by less than 0.0 seconds? 34 H SERIES TOPIC

37 What else can you do? Your Turn Combinin decimal fraction techniques to solve problems 4 The wireless transmitter in Laura s house reduces in sinal strenth by for every metre of distance she moves her computer away from the transmitters antenna. Her computer displays sinal strenth usin bars as shown below: 4 bars 08. to.0 sinal strenth 3 bars 0. to 0.8 sinal strenth 2 bars 0. 4 to 0. sinal strenth bar 0. 2 to 0.4 sinal strenth 0 bars 0.2 or below sinal strenth How many bars of sinal strenth would Laura have if usin her computer.25m away from the antenna? 5 Ruofan is puttin toether a video of a recent karaoke party with her friends. She will be usin five of her favourite music tracks for the video. The lenth of time each of the tracks play for is: 3.55 min, 5.4 min, 2.27 min, 3.8 min and 4.8 min If she uses the entire lenth of the tracks with a 0.5 min break in each of the four aps between sons, how lon will her video run for (to the nearest whole minute)? Show all your workin. H SERIES TOPIC 35

38 What else can you do? Your Turn Combinin decimal fraction techniques to solve problems After a recent study by a city council, the averae number of people in each household was determined.to be 3.4. Explain how this is possible if a household cannot actually have 0.4 of a person? psst: Check example on pae 33 to see how averae calculations are made. 7 A Mexican chef has split up a mystery inredient Sal-X into four exactly identical quantities in separate jars. He then distributes 382. o ml of the secret inredient Sa-Y amonst the four jars, producin in total 839. o ml of the special sauce SalSa-XY. How much of the mystery inredient Sal-X is there in each jar (to the nearest ml)? Show all your workin. 8 After completely flat water conditions (waves with a heiht of 0.0m), the heiht of the waves at a local beach start increasin by 0.2 m every 03. o hours. If the waves need to be at least.4 metres hih before surfers will ride them at this beach, how lon will it be until people start surfin there to the nearest minute? Show all your workin. psst:.0 hours 0 minutes 3 H SERIES TOPIC

39 What else can you do? Your Turn Reflection Time Reflectin on the work covered within this booklet: What useful skills have you ained by learnin about decimal fractions? 2 Write about one or two ways you think you could apply decimal fractions to a real life situation. 3 If you discovered or learnt about any shortcuts to help with decimal fractions or some other cool facts, jot them down here: H SERIES TOPIC 37

40 Cheat Sheet Here is a summary of the thins you need to remember for decimal fractions Place value of decimal fractions # # 000 # 00 Tens of thousands Thousands Hundreds Tens Ones W H O L E # 0 # ' 0 Tenths ' 00 ' 000 Hundredths Thousandths Ten thousandths D E C I M A ' ' Hundred thousandths Millionths ' L ' Ten Millionths Approximations throuh roundin numbers The next diit followin the place value where a number is bein rounded off to is the important part. Next diit Closer to lower value, so round down Leave the place value unchaned Closer to hiher value, so round up Add to the place value Decimal fractions on the number line The smallest place value in a decimal fraction is used to position points accurately on a number line Six tenths of the way from 3.0 to 4.0 Eiht thousandths of the way from.240 to.250 Multiplyin and dividin by powers of ten Move the decimal point dependin on the number of zeros decimal point moves riht, decimal point moves left 5 # # 000 8' ' H SERIES TOPIC

41 Cheat Sheet Terminatin decimal fractions to fractions The place value of the last diit on the riht helps us to write it as a fraction. Write 0.3 as a fraction: Last diit is in tenths position Decimal fraction 0.3 Fraction Decimal fraction Fraction 3 Write.07 as a fraction: Last diit is in hundredths position Fractions to terminatin decimal fractions Where possible, just write as an equivalent fraction with a power of 0 in the denominator first. E: 3 3 # 2 Multiply numerator and denominator by the same value 5 5 # 2 Equivalent fraction with a power of 0 in the denominator 0 ` 0. Three fifths six tenths zero point six When this method is not easy, write the numerator as a decimal fraction and then divide it by the denominator. Addin and subtractin decimal fractions Line up the decimal points and matchin place values vertically before addin or subtractin. Multiplyin and dividin decimal fractions Write the terms as whole numbers and multiply. Put the decimal point back in when finished. The number of decimal places in the answer the number of decimal places in the question! E: : 4 # : 0.02 # Dividin with decimal fractions The question must be chaned so the divisor is a whole number first. E: : 3.5 ' ' 4 : ' ' 3 dividend ' divisor quotient Recurrin decimal fractions These have decimal parts with a repeatin number pattern. E: : o o 5.2 : o o Start End Simple recurrin decimal fractions into sinle fractions Start Only recurrin, non-terminatin decimal fractions can be written in fraction form. This is the method for simple decimal fractions with the pattern startin riht after the decimal point. : : o o One diit in repeatin Two diits in repeatin pattern, pattern, so that diit over 9 so those two diits over End Always simplify fractions : o o Three diits in repeatin pattern, so those three diits over 999, Keep whole number out the front. H SERIES TOPIC 39

42 Notes 40 H SERIES TOPIC

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