decimals - grade 5 aswto one decimal place (or to the nearest tenth) (iii) 26.35819 to four decimal...

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PASSPORT

Deci

mal

s DECIMALSDECIMALSDECIMALS

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6HSERIES TOPIC

1Decimals

Mathletics Passport © 3P Learning

��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

Give this a go!Give this a go!

Decimals�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�system�is�based�around�the�number�10!

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

Write�in�this�space�EVERYTHING�you�already�know�about�decimals.

Q

2 Decimals


6HSERIES TOPIC

DecimalsHow does it work?

1st�decimal�place:� 10101

#' = = �one�tenth

2nd�decimal�place:� 101001

#' = = �one�hundredth

3rd�decimal�place:� 1010001

#' = = �one�thousandth

4th�decimal�place:� 1010000

1#' = = �one�ten�thousandth etc...

Decimal�point

Add�‘th’�to�the�name�for�decimal�place�values

or

or

or

or

......

......

......

......

2 10 2101

7 100 71001

0 0 1000 010001

3 3 10 000 310000

1

2

7

#

#

#

#

'

'

'

'

`

`

`

`

j

j

j

j

......

......

......

4 100

6 10

5 1

4

6

5

#

#

#

102

1007

10000

100003

=

=

=

=

400

60

5

=

=

=

Multiply�by�multiples�of 10 Divide�by�multiples�of 10

1st�decimal�place

2nd�decimal�place

3rd�decimal�place

4th�decimal�place

= 2 tenths

= 7 hundredths

= 0 thousandths

= 3 ten�thousandths

= 4�hundred

= 6�tens�(or�sixty)

= 5�ones�(or�five)

# 1

0 0

00

# 1

000

# 1

00

# 1

0

# 1

' 1

0

' 1

00

' 1

000

' 1

0 0

00

' 1

00 0

00

' 1

000 0

00

' 1

0 0

00 0

00

•

Tens�of�thousands

Tenths

Hundredths

Thousandths

Ten�thousandths

Hundred�thousandths

Millionths

Ten�M

illionths

Thousands

Hundreds

Tens

Ones

Place value of decimals

Decimals�represent�parts�of�a�whole�number�or�object.

W H O L E D E C I AM L

Write�the�place�value�of�each�digit�in�the�number�465.2703

4 6 5 . 2 7 0 3

Expanded forms Place values

Integer�parts

6HSERIES TOPIC

3Decimals


How does it work? Your Turn Decimals


Write�the�decimal�that�represents�these:

b ca

e fd

Write�the�fraction�that�represents�these:

Write�the�place�value�of�the�digit�written�in�square�brackets�for�each�of�these�decimals:

2�hundredths 9�tenths 1�ten�thousandth

3�thousandths 6�hundred�thousandths

b ca

e fd

3�tenths 7�thousandths 1�hundredth

9�ten�thousandths 51�hundredths 11�ten�thousandths

b ca

e fd

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

.3 1 3256 @ .1 0 2316 @ 1 .4 5 0466 @

.5 0 050436 @ .6 0 792646 @ .0 8 563096 @

b ca

e fd

[tenths]

8 . 1 7 1 6 1 5

[thousandths]

4 . 3 2 1 2 3 0

[millionths]

3 . 1 2 0 6 1 9

[hundredths]

9 . 1 2 4 2 1

[ten�thousandths]

1 6 . 1 2 3 2 1 0

[hundred�thousandths]

1 0 0 . 1 0 0 1 0 0 1

Always�put�a�zero�in�front�(called�a�leading zero)�when�there�are�no�whole�numbers

0.02

8�millionths

2

3

4

1

4 Decimals


6HSERIES TOPIC


Write�the�decimal�23.401�in�expanded�form


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

PLACE VALUE OF DECIMALS PLACE VALUE

OF DECIMALS

..../...../20...

a

b

c

d

e

f

.

.

.

.

.

.

4 19

29 281

40 2685

3 74932

0 2306

0 0085

=

=

=

=

=

=

5

.23 401 2 10 3 1 4101 0

1001 1

10001

2 10 3 1 4101 1

10001

# # # # #

# # # #

= + + + +

= + + +

Multiply�each�digit�by�its�place�value

Zero�digits�can�be�removed�to�simplify

6

Write�these�decimals�in�expanded�form:

Simplify�these�numbers�written�in�expanded�form:

1 1 4101 6

1001

4 10 9 1 0101 7

1001

5 100 2 10 0 1 2101 1

1001 8

10001

6 1 8101 5

1001 0

10001 2

100001 9

1000001

# # #

# # # #

# # # # # #

# # # # # #

+ + =

+ + + =

+ + + + + =

+ + + + + =

a

b

c

d

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

e

f

g

2101 0

1001 3

10001

61001 7

10001 0

10 0001 1

1000001

3101 4

1001 1

10001 0

100001 8

1000001

# # #

# # #

# # # # #

#

+ + =

+ + + =

+ + + + =

6HSERIES TOPIC

5Decimals


How does it work? Decimals

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 off approximation

Closer�to�lower�value,�so�round down

Leave�the�place�value�unchanged

Closer�to�higher�value,�so�round up

Add�1�to�the�place�value

Round�these�numbers

The�digit�‘4’�is�in�the�hundreds�position��

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

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

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

The�next�digit�is�a�1,�so�round down

Write�decimal�with�one�decimal�place�only

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

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

Write�decimal�with�four�decimal�places�only�

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

(i)� 2462��to�the�nearest�hundred

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

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

2462 2500` . rounded�to�the�nearest�hundred

. .0 3145 0 3` . rounded�to�one�decimal�place

.3 .26 5819 26 3582` . rounded�to�four�decimal�places

2 6 . 3 5 8 1 9

2 6 . 3 5 8 1 9

2 6 . 3 5 8 2

0 . 3 1 4 5

0 . 3 1 4 5

0 . 3

2 4 6 2

2 4 6 2

2 5 0 0

0 1 2 3 4 5 6 7 8 9

Next�digit

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.

6 Decimals


6HSERIES TOPIC



1 � Round�these�whole�numbers�to�the�place�value�given�in�square�brackets.

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

Approximate�the�following�distance�measurements:��3

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?�

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.

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.

4

544

3

APPROXIMATION THROUGH ROUNDING

NUMBE

RS.

..../...../2

0...

a

a

[nearest�ten]

[nearest�tenth]

b

b

c

c

[nearest�hundred]

[nearest�hundredth]

[nearest�thousand]

[nearest�thousandth]

(i)

(i)

(ii)

(ii)

(iii)

(iii)

536 .

8514 .

93025 .

(i)

(i)

(ii)

(ii)

(iii)

(iii)

14302 .

4764 .

80048 .

(i)

(i)

(ii)

(ii)

(iii)

(iii)

98542 .

18401 .

120510 .

.0 73 .

.3 47 .

.11 85 .

.2 406 .

.0 007 .

.1 003 .

.10 4762 .

.0 3856 .

.0 048640 .

.

.

.

.

.

.

a

b

c

6HSERIES TOPIC

7Decimals




4 � �Round�off�these�numbers�according�to�the�square�brackets.��

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

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

The�next�digit�is�a�5,�so�round up�by�adding�1�to�9

Change�the�other�smaller�place�value�digits�to�0s�

. .0 95 1 0` . rounded�to�one�decimal�place

9�rounds�up�to�10,�so�the 9�becomes�0�and�1 is�added�to�the�digit�in�front.��

a � [one�decimal�place]

.1 98 .

d � [nearest�ones]

.79 9 .

g � [nearest�thousand]

49798 .

b � [nearest�ten]

398 .

e � [three�decimal�places]

.0 1398 .

h � [nearest�ones]

.199 9 .

c � [two�decimal�places]

.11 899 .

f � [three�decimal�places]

.2 1995 .

i � [four�decimal�places]

.9 89999 .

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 = 10001 L?

� (iii)� �Is�the�digit�6�important�when�approximating�to�the�nearest�whole�litre?

Briefly�explain�here�why/why�not.

.

.

.

.

.

0 . 9 5

0 . 9 5

1 . 0

8 Decimals


6HSERIES TOPIC


Decimals on the number line

The�smallest�place�value�in�a�decimal�is�used�to�position�points�accurately�on�a�number�line.

•� �Decimals�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:

So�its�eight�thousandths�of�the�way�from�1.240�to�1.250

Six�tenths�of�the�way�from�3.0�to�4.0

Here�are�some�more�examples�involving�number�lines:

1.240 1.248 1.250

8

3.0 3.6 4.0

6

0.1 0.2

4a)

2.14 2.15

a)

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

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

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

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

Point�is�nine�steps�from�10.06�towards�10.07,�so�the�plotted�point�is:�10.069

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

10.06 10.07

9b)

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

8.79 8.80

b)

3

5

6HSERIES TOPIC

9Decimals




1 � �Display�these�decimals�on�the�number�lines�below:��

2 � Label�these�number�lines�and�then�display�the�given�decimal�on�them:

3 � Round�the�value�of�the�plotted�points�below�to�the�nearest�place�value�given�in�square�brackets.

` the�value��.








4

DECIMA

LS ON THE NUMBER LINE

DECIMALS ON THE NUMBER

LINE

..../...../2

0...

0.0 1.0

0.2 0.3

0.8 0.9

1.994 1.995

2.902 2.903

0.1 0.2

2.3 2.4

a

a

c

e

g

a

c

c

e

e

0.7

[tenth]

[tenth]

[thousandth]

[thousandth]

[hundredth]

[hundredth]

[thousandth]

[thousandth]

1.6

0.13

0.94

2.34

2.053

b

d

f

4.2

7.07

9.538

2.0 3.0

1.03 1.04

0.08 0.09

8.103 8.104

0.989 0.990

9.1 9.2

5.21 5.22

b

b

d

f

h

d

f

2.1

9.15

5.212

10 Decimals


6HSERIES TOPIC


Multiplying and dividing by powers of ten

Move�the�decimal�point�depending�on�the�number�of�zeros

(i)� 5 1000#

(ii)� 8 100'

(iii)� . 10001 25893 0#

(iv)� . 10 0024 905 0 0'

(v)� . 1260 151000

#

� Calculate�these�multiplication�and�division�questions�involving�powers�of�10:

=��decimal�point�moves�right��,�� =��decimal�point�moves�left��

.

. .

8 100 8 0 100

8 0

' '=

=

0.08=

. . .

.

1 25893 10000 1 258 9 3

1258 9 3

# =

=

. . .

.

24 905 100000 24 905

0 00024905

' =

=

. .

. .

260 1510001 260 15 1000

2 60 15

# '=

=

The�whole�number�in�decimal�form�

'100�has�2�zeros,�so�move�decimal�point�2�spaces�left

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

Move�decimal�point�4�spaces�right

No�empty�bounces�to�fill,�so�this�is�the�answer�

Move�decimal�point�5�spaces�left

Fill�empty�bounces�with�0s�and�put�a�zero�in�front

1000

1# is�the�same�as�' 1000

Move�decimal�point�3�spaces�left

Place�a�leading�zero�in�front�of�the�decimal�point

.

..

5 1000 5 0 1000

5 0

# #=

=

The�whole�number�in�decimal�form�

Fill�the�empty�bounces�with�0s�

We�can�simply�add�the�same�number�of�zeros�to�the�end�of�the�whole�number�

0.2 6015=

Remember�to�include�the�leading�zero�

5000=

1 2 3

2 1

1 2 3 4

5 4 3 2 1

3 2 1

If�the�decimal�point�is�on�the�left�after�dividing,�an�extra�0�is�placed�in�front.

6HSERIES TOPIC

11Decimals




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

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

3 � Calculate�these�mixed�problems�written�in�index�form:

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

10 10

10 10000

1

4

=

=

10 100

10 100000

2

5

=

=

10 1000

10 1000000

3

6

=

=

a

a

a

d

d

d

b

b

b

c

c

c

e

e

e

f

f

f

8 100# 29 1000#3.4 10#

12.45 10000# 0.512 100# 0.0000469 1000000#

1002 ' 4590 1000' .0 014 10'

70. 0 100008 ' .1367 512 1000' 421900 100000000'

31 102# 2400 105

' 0.0027 106#

90.008 104# .3 45 103

' 2159 951 107'

12 Decimals


6HSERIES TOPIC



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�

2 8 3 0 3 9 2 0

2 3 8 5 7

0 4 7 6 3 8 9 2

3 8 2 9 6 2

1 9 2 3 8 0 7

8 9 2 3 6 7 0 1

2 0 9 1 7 9 8 3

8 3 9 1 7

9 0 2 8 7 3 2 0 1

0 0 8 3 9 0

I 9 and 2

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

MU

LTIPLYING AND DIVIDING BY POWERS

OF TEN

..../...../20.

..

a

b

c

d

e

f

g

h

i

j

2830.3920 100#

23857 1000'

0.4763892 105#

382 961 10000'

19238.07 101#

8.9236701 10000#

20 917 9831000000

1#

83917 105'

902873.02110

12

#

0.08390 103#

N

A

O

X

T

R

I

D

P

I

6HSERIES TOPIC

13Decimals



MU


OF TEN

..../...../20.

..

(i)�0.25

07�is�just 7

� Write�each�of�these�decimals�as�an�equivalent�(equal)�fraction�in�simplest�form

Write�1.07�as�a�fraction:��

�Last�digit�is�in�hundredths�position

Decimal�digits�in�the�numerator

0.2510025= Equivalent,�un-simplified�fraction

Divide�numerator�and�denominator�by�HCF

Equivalent�fraction�in�simplest�form

Equivalent,�un-simplified�mixed�numeral

Divide�numerator�and�denominator�by�HCF

Equivalent�mixed�numeral�in�simplest�form

10025

2525

''=

41=

2.105 21000105=

100105

552

0 ''=

220021=

1.07 11007=

103=Write�0.3�as�a�fraction:��

�Last�digit�is�in�tenths�position

Decimal�digits�in�the�numerator0.3

Decimal Fraction

�Last�digit�is�in�hundredths�position

Terminating decimals to fractions

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

The�place�value�of�the�last digit on the right�helps�us�to�write�it�as�a�fraction.

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

Always�simplify�the�fraction�parts�if�possible.�These�two�examples�show�you�how.

(ii)�2.105

�Last�digit�is�in�thousandths�position

14 Decimals


6HSERIES TOPIC



1 Write�each�of�these�decimals�as�equivalent�fractions:

c da

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

Simplest�form

.0 1 = .0 09 = .0 03 =

b ca 0.5 = = 0.6 = = 0.02 = =

b .0 7 =

g he 0. 100 = .0 013 = .0 049 =f 0.007 =

k li 0.129 = .0 1007 = .0 0601 =j .0 081 =

e fd 0.08 = = 0.004 = = 0.005 = =

h ig 0.12 = = 0.25 = =

k lj 0.045 = = 0.0028 = = 0.0605 = =

Simplest�form Simplest�form

Simplest�formSimplest�form Simplest�form



0.022 = =

6HSERIES TOPIC

15Decimals


Where does it work? Your Turn Decimals


3 � Write�each�of�these�decimals�as�equivalent�mixed�numerals:�

b ca .2 3 = .1 1 = .03 7 =

e fd .01 3 = .4 001 = .002 9 =

b ca 2.8 = 1.4 = .04 6 =

e fd .03 5 = .2 75 = 5.005 =

h ig .1 004 = .0252 = .3 144 =

4 � Write�each�of�these�decimals�as�equivalent�mixed�numerals�and�then�simplify:

..../...

../20...

0. 5 =

1

2TE

RMINAT

ING DECIMALS TO FRACTIO

NS *

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

Simplest�form

=

16 Decimals


6HSERIES TOPIC


Fractions to terminating decimals

Write�these�as�an�equivalent�decimal

(i)��123

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

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

.

123

41

41

41

10025

0 25

33

2525

#

#

`

'' =

=

=

=

.

2153 2

51

251 2 2

102

2 2

33

2#

#

'' =

=

=

.

53

53

106

0 6

22

#

#

`

=

=

=

numeratordenominator

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

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

Three�fifths��= �six�tenths��= �zero�point�six

Multiply�numerator�and�denominator�by�the�same�value

Equivalent�fraction�with�a�power�of�10�in�the�denominator

(ii)� 2153

Simplify�fraction


Simplify�fraction�part


6HSERIES TOPIC

17Decimals



include�a� leading�zero


1 � Write�each�of�these�fractions�as�equivalent�decimals.�

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

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

c da 12

=43 =

209 =b

52 =

g he258 =

20011 =

1252 =f

2503 =

c da109 =

10011 =

10007 =b

1003 =

i kj

b ca51 =

=

e fd

h ig

154 = 3

251 = 6

207 =

41 =

=

2511 =

=

254 =

=

2001 =

=

1256 =

=

2259 =

=

12001 =

=

8507 =

=

18 Decimals


6HSERIES TOPIC



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

a 2012

c 2418

e 759

g 1 60036

b 2520

d 4022

f 34012

h 215012

FRACT

IONS TO TERMINATING DECIMAL

S

..../...../20...

1 = 0.52

6HSERIES TOPIC

19Decimals




When�changing�the�denominator�to�a�power�of�10�is�not�easy,�you�can�write�the�numerator�as�a�decimal�and�then�divide�it�by�the�denominator.

5 Complete�these�divisions�to�find�the�equivalent�decimal:�

a b.52 2 000 5'=

d e f

Write�numerator�as�a�decimal�and�divide�by�the�denominator

Complete�division,�keeping�the�decimal�point�in�the�same�place

Five�eighths�= �zero�point�six�two�five

If�you�need�more�decimal�place�0s,�you�can�add�them�in�later!

.00041 1 4'= c .000

83 3 8'=

.000 558 8 '= 1.000

811 1 8'= .000

427 27 4'=

.

.

.

85 5 000 8

8 5 0 0 0

0 6 2 5

'

`

=

=

=

g

.

.

0 6 2 5

8 5 0 0 05 2 4= g

Write�this�fraction�as�an�equivalent�decimal

.5 2 0 0 0=

=

g

.5 8 0 0 0=

=

g .8 1 1 0 0 0=

=

g .4 2 7 0 0 0=

=

g

.4 1 0 0 0=

=

g .8 3 0 0 0=

=

g

20 Decimals


6HSERIES TOPIC



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

a b

c

1512

129

5649 d

e f2481

818

1626

6HSERIES TOPIC

21Decimals


Decimals

(i)� . . .24 105 11 06 6 5902+ +

•� Add�2.45�to�6.31�(i.e.�2.45 + 6.31)

Decimal�points�lined�up�vertically


Rounding�decimal�values�before�adding�is�sometimes�used�to�quickly�approximate�the�size�of�the�answer.

24.105 11.06 6.5902 41.7552` + + =

•� Subtract�5.18�from�11.89�(i.e.�11.89 - 5.18)

(ii)�Round�each�value�in�question�(i)�to�the�nearest�whole�number�before�adding.��

�(iii)� . .80 09 72 6081-

. . .24 105 11 06 6 5902 24 11 7

42

` .

.

+ + + +

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

Any�place�value�spaces�are�treated�as�0s

Fill�place�value�spaces�in�the�top�number�with�a�‘0’�when�subtracting�

80.09 72.6081 7.4819` - =

Where does it work?

2 4 . 1 0 5 +

1 1 . 0 6

1 6 . 5

1 9 0 2

4 1 . 7 5 5 2


Add�matching�place�values�together

Values�rounded�to�nearest�ones

Approximate�value�for�addition


Subtract�matching�place�values

2 . 4 5 +

6 . 3 1

8 . 7 6

1 1 . 8 9 -

5 . 1 8

6 . 7 1

Add�matching�place�values�together

Subtract�matching�place�values

Adding and subtracting decimals

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.�

Calculate�each�of�these�further�additions�and�subtractions

8 10 .

10 9

10

10 -

71 21 . 6

01 81 1

7 . 4 8 1 9

22 Decimals


6HSERIES TOPIC



1 � Complete�these�additions�and�subtractions:��

2 Calculate�these�additions�and�subtractions,�showing�all�working:

c da b

g he f

Add�8.75�to�1.24a Subtract�3.15�from�4.79b

Add�0.936�to�0.865c

Subtract�0.9356�from�8.6012e

Add�2.19, 5.6�and�0.13d

Add�10.206, 4.64�and�8.0159f

ADDING AND SUBTRACTING DECIMALS +

- +

- .

..../.....

/20...

0 . 1 4 +

0 . 7 3

0 . 9 9 -

0 . 2 6

1 . 6 8 +

5 . 3 0

0 . 2 4 6 +

0 . 8 3 2

5 . 2 4 -

0 . 8 3

5 . 0 7 4 -

1 . 0 6 4

1 2 . 1 9 4 +

9 . 0 5 7

2 4 . 1 5 8 -

1 3 . 6 9 4

6HSERIES TOPIC

23Decimals




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

b

c

4

a b

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

(i)� �(ii)��. . .2 71 3 80 1 92+ +. . .8 34 1 61 0 54+ +

Calculate�these�subtractions,�showing�all�your�working:��

7.8 2.56- . .13 09 8 4621- . .0 52 0 12532-

5.7 + 6.2 .

.

+

8.3 - 1.9 .

.

-

8.34 + 1.61 + 0.54 .

.

2.71 + 3.80 + 1.92 .

.

+ ++ +

11.3 - 0.2 .

.

-

0.9 + 9.4 .

.

+(i)

(iii)

(v) (vi)

(iv)

(ii)

24 Decimals


6HSERIES TOPIC

Where does it work? Decimals

Multiplying with decimals

How�does�this�work�when�multiplying�with�decimals?�Excellent�question!�Very�glad�you�asked!�

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��

2 Calculate

4 1.2#

0.02 1.45#

4 12 4 8# =

. .

4 8

4 1 2 4 8#` =

1

Multiply�both�terms�as�whole�numbers

1�decimal�place�in�question��= 1�decimal�place�in�answer

Multiply�both�terms�as�whole�numbers

4�decimal�places�in�question��= 4�decimal�places�in�answer

2 145 2 9 0# =

0.02 1.45 0 . 0 2 9 0` # =

2 9 0

4 3 2 1

Let’s�do�the�second�one�again�but�this�time�change�the�decimals�to�equivalent�fractions�first

Changing�the�decimals�to�fractions

Multiply�numerators�and�denominators�together

Number�of�zeros�in�denominator�=�total�of�decimal�places�in�question

Dividing�by�10 000�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,�remembering�that�414= �as�a�fraction.�

. .

.

.

0 02 1 451002

100145

100 1002 145

10 000290

290 10 000

0 2 9 0

0 0290

# #

##

'

=

=

=

=

=

=

4 3 2 1

6HSERIES TOPIC

25Decimals



Multiplying with decimals

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

a b c0.8 2# .5 1 5# 0.14 6#

d e f0.62 4# 3 .0 032# 1.134 2#

2 � Calculate�these�decimal�multiplications,�showing�all�your�working:��

a b c.8 .23 0# . .1 09 0 08# . .2 7 2 5#

d e f. .47 1 1# 3. .21 2 1# . .17 2 9 3#

MULTIPLY

ING WITH DECIMALS MULTIPLYING WITH DE

CIMALS

..../...../2

0...

26 Decimals


6HSERIES TOPIC


Calculate� . .1 26 0 8'

•� Calculate� .4 28 4'

Divisor�already�a�whole�number�so�no�change�needed

•� Calculate� . .0 0456 0 006'

0.0456 0.006 .7 6` ' =

. . . .

.

0 0456 0 006 0 045 6 0 006

45 6 6

' '

'

=

=

dividend�' �divisor�= �quotient

. 2

.

4 4 8

1 0 72g

4.28 4 1.07` ' =

Move�both�decimal�points�right�until�divisor�is�a�whole�number

4 .

.

6 5 6

0 7 64 3g

Drop�off�any�0s�at�the�front�of�the�answer

Quotient�2 �Dividendif�divisor�1 1

Here’s�another�example�showing�how�to�treat�remainders

1.26 0.8 1.2 6 0.8

.12 6 8

' '

'

=

=

. . .1 26 0 8 1 575` ' =

8 .

.

1 2 6 0 0

0 1 5 7 51 4 6 4

= g

Move�both�decimal�points�right�until�divisor�is�a�whole�number

Add�0s�on�the�end�of�the�dividend�for�each�new�remainder

Drop�off�any�0s�at�the�front

Dividing with decimals

Opposite�to�multiplying,�we�move�the�decimal�point�before�dividing�if�needed.��

To�find�the�quotient�involving�decimals,�the�question�must�be�changed�so�the�divisor�is�a�whole�number.

6HSERIES TOPIC

27Decimals




1 � Calculate�these�decimal�and�whole�number�divisions:

a b c3.6 4' 17.5 5' .16 2 9'

d e f0.63 3' .0 489 5' .10 976 7'

3.6 4` ' = 17.5 5` ' = 16.2 9` ' =

0.63 3` ' = 0.489 5` ' = 10.976 7` ' =

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

a b c. .45 2 0' . .9 6 0 6' . .0 56 0 8'

d e f. .1 58 0 4' 0. .8125 0 05' . .5 3682 0 006'

5.2 0.4` ' = 9.6 0.6` ' = 0.56 0.8` ' =

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

DIVIDING WITH DECIMALS DIVIDING

WITH DECIMAL

S

..../...../20...÷

g

g

g

g

g

g

g

g

g

g

g

g

= = =

28 Decimals


6HSERIES TOPIC


Recurring decimals

Identify�the�start�and�end�of�the�repeating�pattern

Dot�above�start�and�end�of�the�repeating�pattern





Write�1�as�a�decimal�with�a�few�0s

Repeats�the�same�remainder�when�dividing

Recurring�decimal�in�simplest�notation

If�the�decimal�parts�have�a�repeating�number�pattern,�they�are�called�recurring�decimals.

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

. ...

.

0 2052052

0 205= o o

. ...0 3582942049

. ...5 212121

A�dot�above�the�start�and�end�digit�of�the�repeating�pattern�is�used�to�show�it�is�a�recurring�decimal.

(i)� Write�these�recurring�decimals�using�the�dot�notation

. ...

.

10 81818

10 81= o o

Three�dots�means�it�keeps�going

Start��End

Start��End

or

.

.. ...

1 0000 6

6 1 0 0 0 00 16 6 6

4 4 4 4

'=

g

...

.

1047777

1 047= o

Start�and�End

1.047= r

0.1 0.6 1 6' '=

(ii)� Calculate�0.1 0.6'

1 6 0.1666 ... 0.16` ' = = o

A�bar�over�the�whole�pattern�can�also�be�used�instead�of�dots

The�pattern�21�keeps�repeating�in�the�decimal�parts

Here�are�some�examples�involving�recurring�decimals�

a)

b)

c)

10.81818...

0.2052052...

1.047777...

. .0 205 0 205=o o

6HSERIES TOPIC

29Decimals



Recurring decimals

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

� �Highlight�the�boxes�that�match�the�recurring�decimals�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!

2 � �Calculate�these�divisions�which�have�recurring�decimals�as�a�result.�� Write�answers�using�dot�notation.

a b c1 3' 4 9' 5 6'

d e f1.6 6' .2 5 9' .0 34 3'

1 3` ' = 4 9` ' = 5 6` ' =

.1 6 6` ' = .2 5 9` ' = .0 34 3` ' =

c z h m n a f b

g

g

g

g

g

g

.0 14o .0 4r .41 1o .0 144 0.141o o .0 41o .414 .0 401o o .4 1o .0 41o o

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

30 Decimals


6HSERIES TOPIC


Recurring decimals

3 � (i)� Complete�the�following�divisions�to�five�decimal�places.�� (ii)� Determine�whether�the�answer�is�a�recurring�decimal�or�not.

a b c2 3' 1 6' 1 7'

d e f.1 6 7' .2 9 3' . .0 33 0 8'

2 3` ' = 1 6` ' = 1 7` ' =

.1 6 7` ' = .2 9 3` ' =

Recurring�decimal?Yes� ��No









g h i0.6 .38 0' 0. .019 0 06' 0. 0.00644 002'

0.68 0.3` ' 0.019 0.06` ' 0.00644 0.002` '

RECU

RRING DECIMALS... RECURRING DECIMALS... RECURRIN

G DECIMALS...

..../...../20.

..

g

g

g

g

g

g

g

g

g

= = =

0.33 .80` ' =

6HSERIES TOPIC

31Decimals


DecimalsWhat else can you do?

Simple recurring decimals into single fractions

Only�recurring,�non-terminating�decimals�can�be�written�in�fraction�form.��

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

(i)� . ...3 777

Three�digits�in�repeating�pattern,�so�those�three�digits�over�999

One�digit�in�repeating�pattern,�so�that�digit�over�9

Two�digits�in�repeating�pattern,�so�those�two�digits�over�99

One�digit�in�repeating�decimal�pattern,�so�that�digit�over�9

Digits�in�front�of�decimal�point�form�the�whole�number�part

Three�digits�in�repeating�decimal�pattern,�so�those�digits�over�999

Digits�in�front�of�decimal�point�form�the�whole�number

Simplify�the�fraction�part

. ... .3 7777 3 7

397

=

=

o

9912

334

33

''=

=

�(ii)�� . ...16 345345

0.111... 0.1

. ... .

91

0 1212 0 129912

= =

= =

o

o o

0.301301... 0.301999301= =o o

Here�are�some�other�examples�including�mixed�numerals.

. ... .16 345345 16 345

16999345

16999345

16333115

33

''

=

=

=

=

o o

Always�simplify fractions

Write�each�of�these�recurring�decimals�as�mixed�numerals�in�simplest�form

32 Decimals


6HSERIES TOPIC

What else can you do? Your Turn Decimals

Simple recurring decimals into single fractions

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

a b c.0 4o .0 8r .0 6o

d e f.0 11o o .0 27o o .0 57o o

g h i0.162 5.1485 0.4896o o

Use�the�shortcut�method�to�write�each�of�these�recurring�decimals�as�mixed�numerals�in�simplest�form.

2

a b c.1 5o .2 7r .4 3r

d e f.3 6r 5.12 0.117o o

3 � (i)� Write� .0 9o �as�a�fraction�in�simplest�form.

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

.../20...

= 9

= 0. ...

0. SIMPLE R

ECURRING DECIMALS INTO SINGLE FRACTIO

NS

6HSERIES TOPIC

33Decimals


What else can you do? Decimals

Combining decimal techniques to solve problems

All�the�techniques�in�this�booklet�can�be�used�to�solve�problems.

(i)� These�rainfall�measurements�were�taken�during�three�days�of�rain�from�a�small�weather�gauge:

Add�the�decimal�values�together

(ii)�� The�results�for�five�runners�in�a�100�m�race�were�plotted�on�the�number�line�below.

13.8

36.1

27.6

77.5

+

78. mm

Read�off�all�the�times

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

The�average�time�ran�by�all�the�runners�in�the�race�� .11 2242= seconds

( . . . . . )

.

11 221 11 223 11 223 11 226 11 228 5

56 121 5

'

'

= + + + +

=

5 5 6. 1 2 1 0

.11 2 2 4 21 11 2

= g

11.22 11.23seconds

These�examples�show�different�ways�decimals�pop�up�in�every-day�life

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

13.8 mm 36.1 mm 27.6 mm

Round�to�nearest�whole�mm

Answer�with�a�statement

Add,�then�divide�by�5

Answer�with�a�statement`��The�total�rainfall�over�the�three�days�was�approximately�78 mm

34 Decimals


6HSERIES TOPIC


Remember me?


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 Usemultiplicationtoshowhowmuchyellowpaintyouwillneedifyouuseall of the 12.46 mL of blue paint you have.

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 fiveminutes(tothenearestwholeword)?

3 Ninepeopleweretryingoutforaspeedrollerskatingteamaroundanovalflattrack. Theshortesttimetocompletesixfulllapsofthetrackforeachpersonwererecordedon the number line below:

a Whatwastheslowesttimerecordedto3 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?

..../...../20...

COMBIN

ING DECIMAL TECHNIQUES TO SOLVE PRO

BLEMS

126.22 126.23 126.24 126.25 126.26 126.27seconds

6HSERIES TOPIC

35Decimals




� �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� .0 81= �to�1.0�signal�strength3�bars�� 0. 16= �to�0.8�signal�strength2�bars�� 0. 14= �to�0.6�signal�strength1�bar�� 0. 12= �to�0.4�signal�strength0�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?�

� �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.

4

5

36 Decimals


6HSERIES TOPIC



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� .138 2o mL�of�the�secret�ingredient�“Sa-Y”�amongst�the�four�jars,�producing�in�total� .863 9o 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� .0 3o �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

6HSERIES TOPIC

37Decimals



Reflection Time

Reflecting�on�the�work�covered�within�this�booklet:

� �What�useful�skills�have�you�gained�by�learning�about�decimals?

2 � Write�about�one�or�two�ways�you�think�you�could�apply�decimals�to�a�real�life�situation.

3 � �If�you�discovered�or�learnt�about�any�shortcuts�to�help�with�decimals�or�some�other�cool�facts,�jot�them�down�here:

1

38 Decimals


6HSERIES TOPIC

Cheat Sheet Decimals

1.240 1.248 1.250

8

=��decimal�point�moves�right��,�� =��decimal�point�moves�left��

.

.

5 1000 5 0 1000

5 0

# #=

=

5000=

.

. .

8 100 8 0 100

8 0

' '=

=

.0 08=

1 2 3 2 1

Closer�to�lower�value,�so�round downLeave�the�place�value�unchanged

Closer�to�higher�value,�so�round upAdd�1�to�the�place�value

0 1 2 3 4 5 6 7 8 9

Next�digit

•

# 1

0 0

00

# 1

000

# 1

00

# 1

0

# 1

Tens�of�thousands

Thousands

Hundreds

Tens

Ones

W H O L E

' 1

0

' 1

00

' 1

000

' 1

0 0

00

' 1

00 0

00

' 1

000 0

00

' 1

0 0

00 0

00

Tenths

Hundredths

Thousandths

Ten�thousandths

Hundred�thousandths

Millionths

Ten�M

illionths

D E C I AM L

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



The�next�digit�following�the�place�value�where�a�number�is�being�rounded�off�to�is�the�important�part.�


The�smallest�place�value�in�a�decimal�is�used�to�position�points�accurately�on�a�number�line.

3.0 3.6 4.0

6

Six�tenths�of�the�way�

from�3.0�to�4.0

Eight�thousandths�of�the�

way�from�1.240�to�1.250


Move�the�decimal�point�depending�on�the�number�of�zeros

6HSERIES TOPIC

39Decimals


Cheat Sheet Decimals

.1 07 11007=

Eg:

.

53

53

106

0 6

22#

#

`

=

=

=

Multiplynumeratoranddenominatorbythesamevalue

Equivalentfractionwithapowerof10 in the denominator

Threefifths= six tenths = zero point six


The place value of the last digit on the righthelpsustowriteitasafraction.

.0 3103=Write 0.3asafraction:

Lastdigitisintenthsposition

Decimal DecimalFraction Fraction


Wherepossible,justwriteasanequivalentfractionwithapowerof10inthedenominatorfirst.

When this method is not easy, write the numerator as a decimal and then divide it by the denominator.


Lineupthedecimalpointsandmatchingplacevaluesverticallybeforeaddingorsubtracting.

Multiplying and dividing decimals

Writethetermsaswholenumbersandmultiply.Putthedecimalpointbackinwhenfinished. The number of decimal places in the answer = thenumberofdecimalplacesinthequestion!

4 1.2 4.8#: = 0.02 1.45 0.0290#: =Eg:

Eg:

Eg:

Recurring decimals

Thesehavedecimalpartswitharepeatingnumberpattern.


Thequestionmustbechangedsothedivisorisawholenumberfirst. dividend ' divisor = quotient

13.5 0.4 135 4: ' '= 89.25 0.003 89250 3: ' '=

AlwayssimplifyfractionsSimple recurring decimals into single fractions

Onlyrecurring,non-terminatingdecimalscanbewritteninfractionform.Thisisthemethodforsimpledecimalswiththepatternstartingrightafterthedecimalpoint.

0.111... 0.191: = =o 0.1212... 0.12

9912

334: = = =o o

Onedigitinrepeatingpattern,sothatdigitover9

Twodigitsinrepeatingpattern,so those two digits over 99

8.301301... 8.301 8999301: = =o o

Threedigitsinrepeatingpattern,sothose three digits over 999, Keep whole number out the front.

5.212121... 5.21 5.21: = =o o 0.3698698... 0.3698 0.3698: = =o o

Start StartEnd End

Write 1.07asafraction:

Lastdigitisinhundredthsposition

40 Decimals


6HSERIES TOPIC

Decimals Notes

4

DECIMA

LS ON

THE NUMBER LINE DECIMALS ON THE NUM

BE

R LINE

..../.....

/20...

MU


OF TEN

..../...../20.

..

MULTIPLY

ING WITH DECIMALS MULTIPLYING WITH DE

CIMALS

..../.....

/20...

DIVIDING WITH DECIMALS DIVIDING

WITH DECIMAL

S

..../...../20...÷

PLACE

VALUE OF DECIMALS PLACE VALUE O

F DECIMALS

..../...../20...

decimals - grade 5 aswto one decimal place (or to the nearest tenth) (iii) 26.35819 to four decimal...

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