Decimal Fractions

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Decimal fractions allow us to be more accurate with our calculations and measurements.

Because most of us have ten fingers, it is thought that this is the reason the decimal fraction system is based around the number 10!

So we can think of decimal fractions as being fractions with powers of 10 in the denominator.

Write in this space EVERYTHING you already know about decimal fractions.

s a go! Give thi

Q To make dark-green coloured paint, you can mix yellow and blue together, using exactly 0.5 (half) as much yellow as you do blue.

How much dark-green paint will you make if you use all of the 12.5 mL of blue paint you have?

Work through the book for a great way to do this

Decimal Fractions H 6 1 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions

Place value of decimal fractions

Decimal fractions represent parts of a whole number or object.

thousandths thousands of • thousandths Millionths Tens ThousandsHundredsTens Ones TenthsHundredthsThousandthsTen HundredMillionthsTen W H O L E D E C I M A L 1 10 10 # 000 000 000 000 000

100 100 # ' 1000 1000 # '

# Decimal point 10 10 ' 000 000

# 100 ' 1

' 10

' ' st 1 1 decimal place: '10 ==# one tenth 10

nd 1 2 decimal place: '10 ==# one hundredth 100 Add ‘th’ to the name for decimal rd 1 3 decimal place: '10 ==# one thousandth place values 1000

th 1 4 decimal place: '10 ==# one ten thousandth etc... 10 000

Write the place value of each digit in the number 465.2703

4 6 5 . 2 7 0 3

Multiply by multiples of 10 Divide by multiples of 10

Expanded forms Place values

4...... 41# 00 = 400 = 4 hundred

6...... 61# 0 = 60 = 6 tens (or sixty) Integer parts

5...... 51# = 5 = 5 ones (or five)

1 2 2...... 21' 02# = = 2 tenths 1st decimal place `or 10 j 10 1 7 7...... 71' 00or 7 # = = 7 hundredths 2nd decimal place ` 100 j 100 1 0 00...... '1000or 0 # = = 0 thousandths 3rd decimal place ` 1000 j 1000 1 3 33...... '10 000or 3 # = = 3 ten thousandths 4th decimal place ` 10 000 j 10 000

2 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions

Place value of decimal fractions

1 Write the decimal fraction that represents these:

a 2 hundredths b 9 tenths c 1 ten thousandth 0.02 Always put a zero in front (called a leading zero) when there are no whole numbers d 3 thousandths e 6 hundred thousandths f 8 millionths

2 Write the fraction that represents these:

a 3 tenths b 7 thousandths c 1 hundredth

d 9 ten thousandths e 51 hundredths f 11 ten thousandths

3 Write the place value of the digit written in square brackets for each of these decimal fractions:

a 31.325 b 10.231 c 451.046 6 @ 6 @ 6 @

d 50.05043 e 60.79264 f 08.56309 6 @ 6 @ 6 @

4 Circle the digit found in the place value given in square brackets:

a [tenths] b [thousandths] c [hundred thousandths] 8.171615 4.321230 100.1001001

d [hundredths] e [ten thousandths] f [millionths] 9.12421 16.123210 3.120619

Decimal Fractions H 6 3 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions

Place value of decimal fractions

Each digit is multiplied by the place value and then added together when writing a number in expanded form.

Write the decimal fraction 23.401 in expanded form

1 1 1 23.4012=+##10 31++4 ##0 +1 # Multiply each digit by its place value 10 100 1000 1 1 =+21##0314++##1 Zero digits can be removed to simplify 10 1000

5 Write these decimal fractions in expanded form:

a 41. 9 =

b 29. 281 =

c 40. 2685 =

d 37. 4932 =

e 02. 306 =

f 0. 0085 = ALUE CE V OF LA DE P CI M A S L N 6 Simplify these numbers written in expanded form: O F I R

T A

C C

T A

I R

1 1 F

a ## # N L

11++4 6 = S

10 100 M

..../...../20... P

D C

O V

A E L 1 1 U b 41##09++10##+=7 10 100 1 1 1 c 51##00 ++2100##12++1 ##+=8 10 100 1000 1 1 1 1 1 d 61##++8 5 ##++0 2 ##+=9 10 100 1000 10 000 100 000

Psst: Remember to include a leading zero for these ones.

1 1 1 e 2 ##++0 3 # = 10 100 1000 1 1 1 1 f 6 ##++7 0 # +=1 # 100 1000 10 000 100 000 1 1 1 1 1 g 3 ##++4 1 ##++0 8 # = 10 100 1000 10 000 100 000

4 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Decimal Fractions

Approximations through rounding numbers

Look at these two statements made about a team of snowboarders: • They have attempted 4937 tricks since starting = Accurate statement • They have attempted nearly 5000 tricks since starting = Rounded offapproximation

Rounding off values is used when a great deal of accuracy is not needed. The next digit following the place value where a number is being rounded off to is the important part. Next digit

0 1 2 3 4 5 6 7 8 9

Closer to lower value, so round down Closer to higher value, so round up Leave the place value unchanged Add 1 to the place value

Here are some examples to see how we round off numbers.

Round these numbers

(i) 2462 to the nearest hundred 2462 The digit ‘4’ is in the hundreds position

2462 The next digit is a 6, so round up by adding 1 to 4

2500 Change the other smaller place value digits to 0’s

` 2462. 2500 roundedo t the nearest hundred

(ii) 0.3145 to one decimal place (or to the nearest tenth)

0.3145 The digit ‘3’ is in the first decimal place

0.3145 The next digit is a 1, so round down

0.3 Write decimal fraction with one decimal place only

` 03..1450. 3 rounded to one decimal place

(iii) 26.35819 to four decimal places (or to the nearest ten thousandth)

26.35819 The digit ‘1’ is in the fourth decimal place

26.35819 The next digit is a 9, so round up by adding 1 to 1

26.3582 Write decimal fraction with four decimal places only

` 26.3 58192. 63. 582 rounded to four decimal places

Decimal Fractions H 6 5 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions

APPROXIM S AT R . I E 5 O B 4 N

Approximations through rounding numbers M 4 U T

4 U

G D H

3 U R 1 Round these whole numbers to the place value given in square brackets. O ..../...../20...

a [nearest ten] b [nearest hundred] c [nearest thousand]

(i) 536 . (i) 14 302 . (i) 98 542 .

(ii) 8514 . (ii) 4764 . (ii) 18 401 .

(iii) 93025 . (iii) 80 048 . (iii) 120510 .

2 Round these decimal fractions to the decimal places given in the square brackets.

a [nearest tenth] b [nearest hundredth] c [nearest thousandth]

(i) 07. 3 . (i) 24. 06 . (i) 10.4762 .

(ii) 34. 7 . (ii) .0007 . (ii) 03. 856 .

(iii) 11.85 . (iii) 1.003 . (iii) 00. 48640 .

3 Approximate the following distance measurements: a A group of people form an 8.82 m long line when they stand together. (i) How long is this line to the nearest 10 cm (i.e. 1 decimal place)? .

(ii) What is the approximate length of this line to the nearest 10 metres? .

b Under a microscope the length of a dust mite was 0.000194 m (i) Approximate the length of this dust mite to the nearest ten thousandth . of a metre. (ii) Approximate the length of this dust mite to the nearest hundredth of a metre. .

c If Lichen City is 3 458 532 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 100 km? . (iii) Are the digits 2, 3 or even 5 important to include when describing the total distance between the two cities? Briefly explain here why/why not.

6 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions

Approximations through rounding numbers

Rounding up can affect more than one digit when the number 9 is involved.

Round 0.95 to one decimal place 0.95 The digit ‘9’ is in the tenths position

0.95 The next digit is a 5, so round up by adding 1 to 9 9 rounds up to 10, so the 9 becomes 0 and 1 1.0 Change the other smaller place value digits to 0s is added to the digit in front. ` 09..51. 0 rounded to one decimal place

4 Round off these numbers according to the square brackets.

a [one decimal place] b [nearest ten] c [two decimal places] 19. 8 . 398 . 11.899 .

d [nearest ones] e [three decimal places] f [three decimal places] .799 . 01. 398 . 21. 995 .

g [nearest thousand] h [nearest ones] i [four decimal places] 49798 . 1999. . .989999 .

5 Approximate these values:

a A call centre receives an average of 2495.9 calls each day during 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 throughout the month. .

b A swimming pool had a slow leak, causing it to empty 9599.5896 L in one week.

(i) How much water was lost to the nearest 10 litres? .

(ii) How much water was lost to the nearest mL if 1mL = 1 L? . 1000

(iii) Is the digit 6 important when approximating to the nearest whole litre? Briefly explain here why/why not.

Decimal Fractions H 6 7 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions

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 10, so there are always ten divisions between values Eg: Here is the value 3.6 on a number line:

6 Six tenths of the way from 3.0 3.6 4.0 3.0 to 4.0

• The major intervals on the number line are marked according to the second last decimal place value

1.240 1.248 1.250

So its eight thousandths of the way from 1.240 to 1.250

Here are some more examples involving number lines:

(i) What value do the plotted points represent on the number lines below?

4 a)

0.1 0.2 Point is four steps from 0.1 towards 0.2, so the plotted point is: 0.14

9 b) 10.06 10.07 Point is nine steps from 10.06 towards 10.07, so the plotted point is: 10.069

(ii) Round the value of the plotted points below to the nearest hundredth.

3 a)

2.14 2.15 Point is three steps from 2.14 towards 2.15, so the plotted point is 2.143 ` the value of the plotted point to the nearest hundredth is: 2.14

5 b) 8.79 8.80 Point is five steps from 8.79 towards 8.80, so the plotted point is 8.795 ` the value of the plotted point to the nearest hundredth is: 8.80

8 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions

L FRACT MA I I 4 O C N S E

Decimal fractions on the number line D

Display these decimal fractions on the number lines below: N

U R M E

..../...../20...B a 0.7 b 2.1

0.0 1.0 2.0 3.0 c 0.13 d 9.15

0.1 0.2 9.1 9.2 e 2.34 f 5.212

2.3 2.4 5.21 5.22

2 Label these number lines and then display the given decimal fraction on them: a 1.6 b 4.2

c 0.94 d 7.07

e 2.053 f 9.538

3 e Round th value of the plotted points below to the nearest place value given in square brackets. a [tenth] b [hundredth]

0.2 0.3 1.03 1.04 ` the value . ` the value . c [tenth] d [hundredth]

0.8 0.9 0.08 0.09 ` the value . ` the value . e [thousandth] f [thousandth]

1.994 1.995 8.103 8.104 ` the value . ` the value . g [thousandth] h [thousandth]

2.902 2.903 0.989 0.990 ` the value . ` the value .

Decimal Fractions H 6 9 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions

Multiplying and dividing by powers of ten

Move the decimal point depending on the number of zeros

= decimal point moves right , = decimal point moves left

Calculate these multiplication and division questions involving powers of 10:

(i) 51# 000 51##000 = 50. 1000 The whole number in decimal fraction form 1 2 3 = 50. . We can simply add the same number of zeros to the end Fill the empty bounces with 0s of the whole number = 5000

If the decimal point is on the left after dividing, an extra 0 is placed in front. (ii) 81' 00 81''00 = 80. 100 The whole number in decimal fraction form

2 1 Remember to include '100 has 2 zeros, so move decimal point 2 spaces left the leading zero  = . 80.

= 0.08 Fill the empty bounces with 0s and put a zero in front

(iii) 12.15893# 000 0 1 2 3 4 12..5893 # 10 000 = 125893. Move decimal point 4 spaces right

= 125893. No empty bounces to fill, so this is the answer

(iv) 24.19050' 0000 5 4 3 2 1 24..9051' 00 000 = 24. 905 Move decimal point 5 spaces left

= 0. 00024905 Fill empty bounces with 0s and put a zero in front 1 (v) 2601. 5 # 1000 1 1 2601..5 # = 260151' 000 # is the same as ' 1000 1000 1000 3 2 1 = ..26015 Move decimal point 3 spaces left

= 0.26015 Place a leading zero in front of the decimal point

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Multiplying and dividing by powers of ten

1 Calculate these multiplications. Remember, multiply means move decimal point to the right:

a 81# 00 b 3.41# 0 c 29 # 1000

d 12.451# 0 000 e 0.5121# 00 f 0.0000469# 1000 000

2 Calculate these divisions. Remember, divide means move decimal point to the left:

a '1002 b 4590' 1000 c 00. 14' 10

d 70.0'10 0008 e 1367.5121' 000 f 421900 '100 000 000

Here are some of the powers of 10 in exponent form. The power = the number of zeros.

101 = 10 102 = 100 103 = 1000

104 = 10 000 105 = 100 000 106 = 1000 000

3 Calculate these mixed problems written in exponent form:

2 5 6 a 31 # 10 b 2400 '10 c 0.00271# 0

4 3 7 d 90.008 # 10 e 34. 51' 0 f 2159 951 '10

Decimal Fractions H 6 11 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions

PLYING TI A L N U D M Multiplying and dividing by powers of ten D

T I

4 For these calculations: F

E P W w (i) Sho where our character needs to spray paint a new decimal point, and O e (ii) writ down the two numbers the new decimal point is between to solve the puzzle ..../...../20...

a 2830.3920 # 100 28303920 I 9 and 2

b 23 857' 1000 23857 N

5 c 0.4763892 # 10 04763892 A

d 382 961 '10 000 382962 O

1 e 19 238.07 # 10 1923807 X

f 8.92367011# 0 000 89236701 T

1 g 20 917 983 # 20917983 R 1000 000

h 83 9171' 05 83917 I

1 i 902873.021 # 902873201 D 102

3 j 0.083901# 0 008390 P

This is another mathematical name for a decimal point:

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 6 and 7

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PLYING TI A L N U D M D Terminating decimal fractions to fractions

T I

N These have decimal fraction parts which stop (or terminate) at a particular place value.

Y R

E P W

..../...../20...O The place value of the last digit on the right helps us to write it as a fraction. Decimal fraction Fraction Decimal fraction digits in the numerator Write 0.3 as a fraction: 0.3 = 3 10

Last digit is in tenths position

Integers in front of the decimal fraction values are simply written in front of the fraction.

Decimal fraction digits in the numerator 1.07 = 1 7 Write 1.07 as a fraction: 100 Last digit 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) 0.25 25 0.25 = Equivalent, un-simplified fraction 100

Last digit is in hundredths position = 25 ' 25 Divide numerator and denominator by HCF 100 ' 25

1 = Equivalent fraction in simplest form 4

(ii) 2.105

2.105 = 2 105 Equivalent, un-simplified mixed number 1000

Last digit is in thousandths position 105 5 = 2 ' Divide numerator and denominator by HCF 1000 ' 5

= 2 21 Equivalent mixed number in simplest form 200

Decimal Fractions H 6 13 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions

Terminating decimal fractions to fractions

1 Write each of these decimal fractions as equivalent fractions:

a 01. = b 07. = c 00. 9 = d 00. 3 =

e 0. 100 = f 0.007 = g 00. 13 = h 00. 49 =

i 0.129 = j 00. 81 = k 01. 007 = l 00. 601 =

2 Write each of these decimal fractions as equivalent fractions and then simplify:

a 0.5 == b 0.6 == c 0.02 ==

Simplest form Simplest form Simplest form

d 0.08 == e 0.004 == f 0.005 ==

Simplest form Simplest form Simplest form

g 0.12 == h 0.25 == i 0.022 ==

Simplest form Simplest form Simplest form

j 0.045 == k 0.0028 == l 0.0605 ==

Simplest form Simplest form Simplest form

14 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning Where does it work? Your Turn Decimal Fractions

ING DEC AT IM IN A M 1 L R

E F

T 2 R

Terminating decimal fractions to fractions A

* C

S 5 =

I .

T T

C O A

R 3 Write each of these decimal fractions as equivalent mixed numbers: 0 F ..../...../20...

a 23. = b 11. = c 37.0 =

d 13.0 = e .4001 = f 29.00 =

4 Write each of these decimal fractions as equivalent mixed numbers and then simplify:

a 2.8 = b 1.4 = c 46.0 =

= = =

Simplest form Simplest form Simplest form

d 35.0 = e 27. 5 = f 5.005 =

= = =

Simplest form Simplest form Simplest form

g .1004 = h .0252 = i 31. 44 =

= = =

Simplest form Simplest form Simplest form

Decimal Fractions H 6 15 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions

Fractions to terminating decimal fractions

Where possible, just write as an equivalent fraction with a power of 10 in the denominator first.

numerator denominator

3 = 3 # 2 Multiply numerator and denominator by the same value 5 5 # 2 6 = Equivalent fraction with a power of 10 in the denominator 10 ` = 06.

Three fifths = six tenths = zero point six

Sometimes it is easier to first simplify the fraction before changing to a decimal fraction.

Write these as an equivalent decimal fraction

(i) 3 12 3 3 1 ' = Simplify fraction 12 ' 3 4

1 = 1 # 25 4 4 # 25 25 = Equivalent fraction with a power of 10 in the denominator 100

` = 02. 5

Three twelfths = one quarter = twenty five hundredths = zero point two five

(ii) 2 3 15 3 3 1 2 ' = 2 Simplify fraction part 15 ' 3 5

2 12# = 2 2 5 # 2 10

= 22. Equivalent fraction with a power of 10 in the denominator

Two and three fifteenths = two and one fifth = two and two tenths = two point two

16 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions

include a leading zero Fractions to terminating decimal fractions

1 Write each of these fractions as equivalent decimal fractions.

a 9 = b 3 = c 11 = d 7 = 10 100 100 1000

2 Write each of these as equivalent fractions with a power of 10 in the denominator.

2 3 9 a 1 = b = c = d = 2 5 4 20

8 3 e = f = g 11 = h 2 = 25 250 200 125

4 1 7 i 1 = j 3 = k 6 = 5 25 20

3 (i) Write each of these as equivalent fractions with a power of 10 in the denominator. (ii) Change to equivalent decimal fractions.

1 1 11 a = b = c = 5 4 25

= = =

4 1 6 d = e = f = 25 200 125

= = =

9 1 7 g 2 = h 1 = i 8 = 25 200 50

= = =

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Fractions to terminating decimal fractions

4 Change each of these fractions to equivalent decimal fractions after first simplifying. Show all your working.

12 20 a b 20 25

TIONS T AC O R T F E R

M S 5 I N

1 N

= A

I T

T I

2 N A G

18 22 D

E L C A I c d M 24 40 ..../...../20...

9 12 e f 3 75 40

12 g 1 36 h 2 600 150

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Fractions to terminating decimal fractions

When changing the denominator to a power of 10 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

5 = 5. 000' 8 Write numerator as a decimal fraction and divide by the denominator 8

= 85. 000 g If you need more 06. 25 decimal place 0s, you = 85. 520040 Complete division, keeping the decimal point in the same place can add them in later! g

` = 06. 25

Five eighths = zero point six two five

5 Complete these divisions to find the equivalent decimal fraction: 2 1 3 a = 2. 000' 5 b = 14.000 ' c = 38.000 ' 5 4 8

= 52. 000 = 41. 000 = 83. 000 g g g

= = =

8 11 27 d = 8.000' 5 e = 181.000 ' f = 27.000 ' 4 5 8 4

= 58. 000 = 8110. 00 = 4270. 00 g g g

= = =

Decimal Fractions H 6 19 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions

Fractions to terminating decimal fractions

6 Simplify these fractions and then write as an equivalent decimal fraction using the division method. Show all your working.

a 12 b 9 15 12

c 49 d 18 56 8

e 81 f 26 24 16

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Adding and subtracting decimal fractions

Just add or subtract the digits in the same place value. To do this, line up the decimal points and matching place values vertically first.

• Add 2.45 to 6.31 (i.e. 2.45 + 6.31) 2 . 4 5 + Decimal points lined up vertically 6 . 3 1

8 . 7 6 Add matching place values together

• Subtract 5.18 from 11.89 (i.e. 11.89 - 5.18) 1 1 . 8 9 - Decimal points lined up vertically 5 . 1 8

6 . 7 1 Subtract matching place values

Calculate each of these further additions and subtractions

(i) 24..1051++1066.5902 2 4 . 1 0 5 + Decimal points lined up vertically 1 1 . 0 6

1 6 . 51 9 0 2 4 1 . 7 5 5 2 Add matching place values together Any place value spaces are treated as 0s ` 24.105++11.066.59024= 1.7552

Rounding decimal fractions before adding is sometimes used to quickly approximate the size of the answer. (ii) Round each value in question (i) to the nearest whole number before adding.

` 24..1051++1066.59022. 41++17 Values rounded to nearest ones

. 42 Approximate value for addition

Note: Rounding values before adding/subtracting is not as accurate as rounding after adding/subtracting.

(iii) 80.. 09- 72 6081 1 1 1 1 8 0 . 0 9 0 0 - Decimal points lined up vertically

71 21 . 6 01 81 1 Subtract matching place values 7 . 4 8 1 9 Fill place value spaces in the top number with ` 80.097-=2.6081 7.4819 a ‘0’ when subtracting

Decimal Fractions H 6 21 Mathletics Passport © 3P Learning SERIES TOPIC Where does it work? Your Turn Decimal Fractions

Adding and subtracting decimal fractions

1 Complete these additions and subtractions:

a 0 . 1 4 + b 1 . 6 8 + c 0 . 2 4 6 + d 1 2 . 1 9 4 + 0 . 7 3 5 . 3 0 0 . 8 3 2 9 . 0 5 7

e 0 . 9 9 - f 5 . 0 7 4 - g 5 . 2 4 - h 2 4 . 1 5 8 - 0 . 2 6 1 . 0 6 4 0 . 8 3 1 3 . 6 9 4

AND SUBT NG RA I C D T 2 Calculate these additions and subtractions, showing all working: D I A N

. G

E -

a Add 8.75 to 1.24 b Subtract 3.15 from 4.79 +

S F

N R O A I C

T ..../...../20...

c Add 0.936 to 0.865 d Add 2.19, 5.6 and 0.13

e Subtract 0.9356 from 8.6012 f Add 10.206, 4.64 and 8.0159

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Adding and subtracting decimal fractions

3 a Approximate these calculations by rounding each value to the nearest whole number first.

(i) 5.7 + 6.2 . + (ii) 0.9 + 9.4 . +

. .

(iii) 8.3 - 1.9 . - (iv) 11.3 - 0.2 . -

. .

(v) 8.34 + 1.61 + 0.54 . + + (vi) 2.71 + 3.80 + 1.92 . + +

. .

b Calculate parts (v) and (vi) again, this time rounding after adding the numbers to get a more accurate approximate value.

(i) 83..41++61 05. 4 (ii) 27 .. 13++80 19. 2

4 Calculate these subtractions, showing all your working:

a 7.82- .56 b 13..09 -84621 c 05..20- 12532

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Multiplying 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!

1 Calculate 41# .2 41# 24= 8 Multiply both terms as whole numbers 1 48 1 decimal place in question = 1 decimal place in answer

` 41# ..24= 8

2 Calculate 0.02 # 1.45 21# 45 = 290 Multiply both terms as whole numbers 4 3 2 1 290

` 0.02 # 1.45 = 0.0290 4 decimal places in question = 4 decimal places in answer

How does this work when multiplying with decimal fractions? Excellent question! Very glad you asked! Let’s do the second one again but this time change the decimal fractions to equivalent fractions first

2 145 00..21##45 = Changing the decimal fractions to fractions 100 100

2145 = # Multiply numerators and denominators together 1001# 00

290 = Number of zeros in denominator = total of decimal places 10 000 in question

= 2901' 0 000

4 3 2 1 = 02. 90 Dividing by 10 000 moves decimal point four places to the left

= 00. 290 ` 4 decimal places in question = 4 decimal places in answer

Try this method for yourself on the first example above, remembering that 4 = 4 as a fraction. 1

24 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning Where does it work? Your Turn Decimal Fractions

Multiplying with decimal fractions

1 Calculate these whole number and decimal fraction multiplications, showing all you working:

a 0.82# b 51# .5 c 0.14 # 6

d 0.62# 4 e 3.# 0 032 f 1.1342#

WITH DE ING CIM LY AL IP F T R L A U C M T I

2 M

Calculate these decimal fraction multiplications, showing all your working: T

F P

G C

E W D I

T a 30.8# .2 b 10..90# 08 c 27..# 25 H ..../...../20...

d 71..# 1 4 e 3.# 21 .21 f 17..29# 3

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Dividing with decimal fractions

Opposite to multiplying, we move the decimal point before dividing if needed. To find the quotient involving decimal fractions, the question must be changed so the divisor is a whole number.

dividend ' divisor = quotient

• Calculate 42. 84' 10. 7 44.228 Divisor already a whole number so no change needed g ` 4.28 ' 41= .07

• Calculate 00..4560' 006

00..4560''006 = 0045.60006 Move both decimal points right until divisor is a whole number

= 45.66' 07. 6 4 3 Quotient 2 Dividend 654.6 if divisor 1 1 g ` 0.0456' 0.006= 76 . Drop off any 0s at the front of the answer

Here’s another example showing how to treat remainders

Calculate 12..60' 8

1.26 ''0.81= .2 60.8 Move both decimal points right until divisor is a whole number

= 12.68'

01. 575 = 8.121 466040 Add 0s on the end of the dividend for each new remainder g `12..60' 81= .575 Drop off any 0s at the front

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DIVIDIN G W S IT ON H I T D C E Dividing with decimal fractions C A I R M F

1 A

Calculate these decimal fraction and whole number divisions: C C

..../...../20...T

÷ W

G D N I I V D

a 3.64' b 17.5 ' 5 c 16.29' I

g g g ` 3.64' = `17.5 ' 5= `16.2 ' 9 =

d 0.63' 3 e 04. 89' 5 f 10.9767'

g g g ` 0.63' 3= ` 0.4895' = `10.976 ' 7 =

2 Calculate these decimal fraction divisions, showing all your working:

a 52..' 0 4 b 96..' 06 c 05..60' 8

g g g ` 5.20' .4 = ` 9.60' .6 = ` 0.56' 0.8 =

d 15..80' 4 e 0.8125 ' 00 . 5 f 53..682' 0 006

g g g `1.58 ' 0.4 ` 0.8125' 0.05 ` 5.3682' 0.006

= = =

Recurring decimal fractions

Non-terminating decimal fractions have decimal parts that do not stop. They keep going on and on.

03. 582942049 ... Three dots means it keeps going

If the decimal parts have a repeating number pattern, they are called recurring decimal fractions. 52. 12121... The pattern 21 keeps repeating in the decimal parts Here are some examples involving recurring decimal fractions

A dot above the start and end digit of the repeating pattern is used to show it is a recurring decimal fraction. (i) Write these recurring decimal fractions using the dot notation

a) 10.81818... 10.81818 ... Identify the start and end of the repeating pattern Start End

= 10.81o o Dot above start and end of the repeating pattern

b) 0.2052052... 02. 052052 ... Identify the start and end of the repeating pattern Start End

= 02. o 05o Dot above start and end of the repeating pattern

c) 1.047777... 1047777 ... Identify the start and end of the repeating pattern Start and End 02..o 05o =0205 = 10. 47o Dot above start and end of the repeating pattern A bar over the whole or pattern can also be used instead of dots = 1.047r (ii) Calculate 0.10' .6

0.10''.6 = 16

= 1. 0000' 6 Write 1 as a decimal fraction with a few 0s

01. 666... 61.44440000 Repeats the same remainder when dividing g ` 16' ==0.1666 ... 0.16o Recurring decimal fraction in simplest notation

Recurring decimal fractions

1 e What is th name of the horizontal line above the repeated numbers in a recurring decimal fraction?

Highlight the boxes that match the recurring 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!

01. 4o 04.r .411o 01. 44 0.141oo 04. 1o 41. 4 04. o 01o 41. o 04. o 1o 4.1414 ... C z F h N d W c D b A a U n P t L f O m 0.144144 ... Y n A m R f T t K z E h R d I c U b S a 0.1444 ... L a D b A m I h M t B f S c A d U z Q n 0.401401 ... R h Z d A n E z A c N t 0 a M b A h G f 4.111 ... A f T z P c H d T a Y n A t A h C m A b 0.4111... I d Y t A b U n H m I z E f S m I t T a 0.4141 ... A b L a D t E f A d N c L m E z O d N h 41.111 ... W c J f B d A a X h M m A b U n A A z 0.444 ... P m V c E a F b A n B d T Y f E c I t 0.1411411 ... 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 recurring decimal fractions as a result. Write answers using dot notation.

a 13' b 49' c 56'

g g g `13' = ` 49' = ` 56' =

d 1.66' e 25. ' 9 f 03. 43'

g g g `16. ' 6 = ` 25. ' 9 = ` 03. 43' =

G DECIMAL IN F RR R U A C C E T Recurring decimal fractions I R O . . N

. S

3 I

(i) Complete the following divisions to five decimal places. E

(ii) Determine whether the answer is a recurring decimal fraction or not. I

G A

M D I E

..../...../20...C a 23' b 16' c 17'

g g g ` 23' = `16' = `17' =

Recurring Recurring Recurring decimal fraction? decimal fraction? decimal fraction? Yes No Yes No Yes No

d 16. ' 7 e 29. ' 3 f 03..30' 8

g g g `16. ' 7 = ` 29. ' 3 = ` 0.33 ' 0.8 =

Recurring Recurring Recurring decimal fraction? decimal fraction? decimal fraction? Yes No Yes No Yes No

g 0.6.80' 3 h 0.90' .01 06 i 0.' 0.00644002

g g g ` 0.68' 0.3 ` 0.0190' .06 ` 0.00644' 0.002

= = =

Recurring Recurring Recurring decimal fraction? decimal fraction? decimal fraction? Yes No Yes No Yes No

Simple recurring decimal fractions into single fractions

Only recurring, non-terminating decimal fractions can be written in fraction form.

Here is a quick way for simple decimal fractions with the pattern starting right after the decimal point.

0.111... ==0.1o 1 One digit in repeating pattern, so that digit over 9 9

01. 2120... ==.12o o 12 Two digits in repeating pattern, so those two digits over 99 99 Always simplify = 12 ' 3 fractions 99 ' 3

= 4 33

301 0.301301...== 0.301o o Three digits in repeating pattern, so those three digits over 999 999

Here are some other examples including mixed numbers. Write each of these recurring decimal fractions as mixed numbers in simplest form

(i) 3. 777... 3. 7777... = 37. o One digit in repeating pattern, so that digit over 9

7 = 3 Digits in front of decimal point form the whole number part 9

(ii) 16.345345... 16.345345... = 16.345o o Three digits in repeating pattern, so those digits over 999

345 = 16 Digits in front of decimal point form the whole number 999

= 16 345 ' 3 Simplify the fraction part 999 ' 3

= 16 115 333

Decimal Fractions H 6 31 Mathletics Passport © 3P Learning SERIES TOPIC What else can you do? Your Turn Decimal Fractions

Simple recurring decimal fractions into single fractions

1 Use the shortcut method to write each of these recurring decimal fractions as a fraction in simplest form:

a 04. o b 08.r c 06. o

d 01. oo1 e 02. o 7o f 05. o 7o

2 Use the shortcut method to write each of these recurring decimal fractions as mixed numbers in simplest form.

a 15. o b 27.r c 43.r

d 36.r e 5.12 f 0.117o o

g 0.162 h 5.1485 i 0.4896o o

RRING ECU DE R CI E M L A P L M I F 3 o S = 9 R

(i) Write 09. as a fraction in simplest form. A C

0. T I

= O

... S

0.I I

O R

I E N L

..../...../20...G (Ii) Does anything unusual seem to be happening with your answer? Explain.

Combining 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 during three days of rain from a small weather gauge:

13.8 mm 36.1 mm 27.6 mm

What was the total rainfall for the three days, to the nearest whole mm?

13.8 + Add the decimal fraction values together 36.1 27.6 77.5

. 78mm 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 100 m race were plotted on the number line below.

seconds 11.22 11.23 a) What was the fastest time run (to the nearest thousandth of a second)? Fastest time = left-most plotted point = 11.221 seconds

b) What time did two runners finish the race together on? Two runners with the same time = two dots at the same point = 11.223 seconds

c) What was the average time ran by all runners in this race? Average time = The sum of all the times ran divided by the number of runners

=+(.11 221112..23++11 223 11..226 +11 228) ' 5 Read off all the times

= 56.1215' Add, then divide by 5 11. 2242 = 556. 11121210 g The average time ran by all the runners in the race = 11.2242seconds Answer with a statement

Decimal Fractions H 6 33 Mathletics Passport © 3P Learning SERIES TOPIC What else can you do? Your Turn Decimal Fractions

Combining decimal fraction techniques to solve problems

1 To make dark-green coloured paint, you can mix yellow and blue together, using 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 12.46 mL of blue paint you have.

er me? Rememb

b How many millilitres of dark-green paint can you make with 18.45 mL of yellow paint in the mix? Round your answer to the nearest tenth of a mL.

2 Derek types his essays at an average speed of 93.45 words every minute. How many words does he type in five minutes (to the nearest whole word)?

DECIMAL NG F I RA N C I T B I M O

O N C

H S

N M

3 Nine people were trying out for a speed roller skating team around an oval flat track. I

U B

The shortest time to complete six full laps of the track for each person were recorded on R

O E

V S L

the number line below: ..../...../20...O

seconds 126.22 126.23 126.24 126.25 126.26 126.27

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 126.245 seconds. How many skaters made it into the team?

c How many skaters missed out making the team by less than 0.01 seconds?

34 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning What else can you do? Your Turn Decimal Fractions

Combining decimal fraction techniques to solve problems

4 The wireless transmitter in Laura’s house reduces in signal strength by 0.024 for every 1 metre of distance she moves her computer away from the transmitters antenna. Her computer displays signal strength using bars as shown below:

4 bars = 08. 1 to 1.0 signal strength 3 bars = 0. 16 to 0.8 signal strength 2 bars = 0. 14 to 0.6 signal strength 1 bar = 0. 12 to 0.4 signal strength 0 bars = 0.2 or below signal strength

How many bars of signal strength would Laura have if using her computer 16.25m away from the antenna?

5 Ruofan is putting together a video of a recent karaoke party with her friends. She will be using five of her favourite music tracks for the video. The length of time each of the tracks play for is:

3.55 min, 5.14 min, 2.27 min, 3.18 min and 4.86 min

If she uses the entire length of the tracks with a 0.15 min break in each of the four gaps between songs, how long will her video run for (to the nearest whole minute)? Show all your working.

Decimal Fractions H 6 35 Mathletics Passport © 3P Learning SERIES TOPIC What else can you do? Your Turn Decimal Fractions

Combining decimal fraction techniques to solve problems

6 After a recent study by a city council, the average 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 page 33 to see how average calculations are made.

7 A Mexican chef has split up a mystery ingredient “Sal-X” into four exactly identical quantities in separate jars. He then distributes 1382. o mL of the secret ingredient “Sa-Y” amongst the four jars, producing in total 39.86 o mL of the special sauce “SalSa-XY”. How much of the mystery ingredient “Sal-X” is there in each jar (to the nearest mL)? Show all your working.

8 After completely flat water conditions (waves with a height of 0.0m), the height of the waves at a local beach start increasing by 0.2 m every 03. o hours. If the waves need to be at least 1.4 metres high before surfers will ride them at this beach, how long will it be until people start surfing there to the nearest minute? Show all your working. psst: 1.0 hours = 60 minutes

36 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning What else can you do? Your Turn Decimal Fractions

Reflection Time

Reflecting on the work covered within this booklet:

1 What useful skills have you gained by learning 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:

Here is a summary of the things you need to remember for decimal fractions

Place value of decimal fractions

thousandths thousands of thousandths Millionths Tens ThousandsHundredsTens Ones TenthsHundredthsThousandthsTen HundredMillionthsTen W H O L E • D E C I M A L 1 10 10 # 000 000 000 000 000

100 100 # ' 1000 1000 # ' # 10 10 ' 000 000

# 100 ' 1

' 10

' '

Approximations through rounding numbers The next digit following the place value where a number is being rounded off to is the important part. Next digit

0 1 2 3 4 5 6 7 8 9

Closer to lower value, so round down Closer to higher value, so round up Leave the place value unchanged Add 1 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. 6 Six tenths of the way 3.0 3.6 4.0 from 3.0 to 4.0 8 Eight thousandths of the 1.240 1.248 1.250 way from 1.240 to 1.250

Multiplying and dividing by powers of ten Move the decimal point depending on the number of zeros

= decimal point moves right , = decimal point moves left

51##000 = 50. 1000 81''00 = 80. 100 1 2 3 2 1 = 50. = . 80. = 5000 = 00. 8

Terminating decimal fractions to fractions The place value of the last digit on the right helps us to write it as a fraction. Decimal fraction Fraction Decimal fraction Fraction 3 Write 0.3 as a fraction: 0.3 = Write 1.07 as a fraction: 10. 71= 7 10 100 Last digit is in tenths position Last digit is in hundredths position

Fractions to terminating decimal fractions Where possible, just write as an equivalent fraction with a power of 10 in the denominator first.

3 3 # 2 Eg: = Multiply numerator and denominator by the same value 5 5 # 2 6 = Equivalent fraction with a power of 10 in the denominator 10 ` = 06. 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.

Adding and subtracting decimal fractions Line up the decimal points and matching place values vertically before adding or subtracting.

Multiplying and dividing 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!

Eg: : 41# .2 = 4.8 : 0.02 # 1.45 = 0.0290

Dividing with decimal fractions dividend divisor quotient The question must be changed so the divisor is a whole number first. ' = Eg: : 13.5 ''0.41= 35 4 : 89.25'' 0.003= 89250 3

Recurring decimal fractions These have decimal parts with a repeating number pattern.

Eg: : 5.212121...5==.2o 15o .21 : 0.3698698...== 0.3698oo 0.3698 Start End Start End

Simple recurring decimal fractions into single fractions Always simplify fractions Only recurring, non-terminating decimal fractions can be written in fraction form. This is the method for simple decimal fractions with the pattern starting right after the decimal point.

: 0.111... ==0.1o 1 : 0.1212...0==.1o 2o 12 = 4 : 8.301301... ==8.30o 18o 301 9 99 33 999 One digit in repeating Two digits in repeating pattern, Three digits in repeating pattern, so pattern, so that digit over 9 so those two digits over 99 those three digits over 999, Keep whole number out the front.

Decimal Fractions