Number Decimals - Wikispacesmrgrenfellsmaths.wikispaces.com/file/view/chapter07.pdfrevise place value and order for decimals write a (terminating) decimal as a common fraction add

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Number

Decimals

7

You should feel comfortable working with decimal fractions because they are used everywhere. That mountain bike you want costs $259.95, or you need a desk 0.75 metres wide to fit between your bed and your wardrobe. The building, business and banking world could not function without the use of decimals.

■ revise place value and order for decimals■ write a (terminating) decimal as a common fraction■ add and subtract decimals■ multiply and divide a decimal by a power of 10, a whole number or

another decimal■ use estimation and the calculator to find decimal answers■ convert common fractions to terminating or recurring decimals■ use the notation for recurring decimals■ round decimals to a given number of places■ solve a variety of real-life problems involving decimals and money.

■ decimal A number that includes a decimal point and place value to indicate parts of a whole.

■ number of decimal places The number of digits after the decimal point.■ estimate An educated guess concerning the answer to a calculation.■ recurring decimal A decimal fraction that has one or more digits that

repeat forever.■ terminating decimal A decimal fraction that is not recurring, but comes

to an end.■ round To write a number to a given number of decimal places.

Why do you think people now find decimals more convenient to use than common fractions when describing parts of a whole number?

■ How is writing 0.5 better than writing ?

■ How is writing 0.2 easier than writing ?

■ How is 0.75 better than ?

In this chapter you will:

Wordbank

Think!

12---

15---

34---

DECIMALS 199

CHAPTER 7

200

NEW CENTURY MATHS 7

Place value

You are familiar with numbers that have decimal points, for example: • money $3.52• measurements 6.2 m.

The digits after the decimal point indicate a part of a whole.

The position of a digit in a number shows its size. This is known as

place value

.

1 Arrange the numbers in each of these sets in ascending order:a 21, 32, 34, 17, 8 b 23, 26, 54, 66, 33c 101, 100, 110, 99, 102, 111 d 251, 247, 256, 254, 244

2 Arrange the numbers in each of these sets in descending order:a 44, 39, 42, 45, 38, 40 b 55, 56, 53, 50, 49, 52c 1001, 1000, 1100, 1010, 1111 d 301, 310, 299, 318, 295

3 a Write 27 to the nearest ten. b Write 752 to the nearest hundred.c Write 9079 to the nearest thousand. d Write 16 837 to the nearest thousand.e Write 29 537 to the nearest hundred. f Write 40 219 to the nearest ten.

4 Evaluate:a 123 + 32 + 456 b 432 + 45 + 2341 + 7c 6 + 78 + 32 + 9 + 199 d 47 852 + 365 + 1458 + 81e 234 − 45 f 1138 − 445g 894 − 35 h 114 782 − 36 989i 58 × 3 j 126 × 8k 589 × 13 l 789 × 26m 2920 ÷ 8 n 2163 ÷ 7o 10 836 ÷ 21 p 625 ÷ 25

5 Evaluate:a 32 × 10 b 578 × 100c 325 × 1000 d 400 × 10e 400 ÷ 10 f 1200 ÷ 100g 1 000 000 ÷ 1000 h 81 000 ÷ 100

Start up

Skillsheet 7-02

Decimal fractions

Worksheet 7-01

Brainstarters 7

Skillsheet 7-01

Rounding whole numbers

Example 1

What does the number 532.81 mean?

Solution5 3 2 . 8 1

hundreds tens units tenths hundredthsdecimal

point

So 532.81 is 5 × 100 + 3 × 10 + 2 × 1 + 8 × + 1 × .110------ 1

100---------

DECIMALS 201 CHAPTER 7

Example 2

Write each of these decimals in expanded form:a 604.75 b 29.04

Solutiona 604.75 = 6 × 100 + 0 × 10 + 4 × 1 + 7 × + 5 × .

b 29.04 = 2 × 10 + 9 × 1 + 0 × + 4 × .

What is the place value of the digit 8 in the number 612.87?

SolutionThe 8 in 612.87 is in the tenths column, so it has a value of 8 tenths (or ).

Write in decimal form:

a 6 + + b 2 × 10 + 4 ×

Solutiona 6.18 b 20.04

110------ 1

100---------

110------ 1

100---------

Example 3

810------

Example 4

110------ 8

100--------- 1

100---------

1 Copy this place-value table. Put the 12 given numbers in the table, with the digits in their correct columns.

a 14.82 b 6.014 c 931.02d 70.8 e 297.86 f 11.14g 503.92 h 8.3 i 0.375j 200.047 k 4.025 l 0.81

2 Write each of the following (a to p) as a decimal:a five and four-tenths b six and fifteen-hundredthsc eleven and thirty-eight hundredths d eight and three tenthse fourteen and six-hundredthsf four hundred and two and three-thousandthsg 6 + 2 tenths + 3 hundredths h 3 + 7 tenthsi nineteen and nine-tenths j 14 + 3 tenths + 9 hundredthsk 2 tens + 4 + 0 tenths + 9 hundredths l 2 + 6 hundredthsm 8 + 7 tenths + 5 hundredths n seventy-three hundredthso nine-thousandths p 8 tenths + 5 hundredths + 9 thousandths

hundreds tens units tenths hundredths thousandths

Exercise 7-01Example 1

202 NEW CENTURY MATHS 7

Understanding the point

3 Write in expanded form:a 1.234 b 102.34 c 30.12 d 0.751 e 2.09f 12.71 g 8.003 h 4.509 i 0.04 j 0.386

4 What is the place value of the digit 4 in each of these numbers?a 431.70 b 31.047 c 761.04 d 114.37e 3.734 f 907.431 g 0.064 75 h 42 376i 72.314 j 26.74 k 100.0407 l 94.071

5 What is the place value of the digit 7 in each number listed in Question 4?

6 Write in decimal form:

a 4 + b 3 +

c 4 + + d 12 + +

e 5 + + + f 0 + + +

g 19 + h 2 + +

i 11 + + j 3 × 1 + 2 × + 7 ×

k 2 × 100 + 7 × 10 + 6 × + 1 × l 7 × 10 + 6 × 1 + 5 ×

m 3 × + 4 × + 1 × n 9 × + 7 ×

o 2 × 10 + 7 + 2 × + 3 × p 4 × 10 + 9 ×

q 9 + 4 × + 1 × r 5 × 100 + 4 × + 3 ×

110------ 8

10------

310------ 5

100--------- 9

10------ 3

100---------

210------ 7

100--------- 9

1000------------ 4

100--------- 6

1000------------ 1

10 000----------------

91000------------ 7

10------ 3

1000------------

610------ 2

10 000---------------- 1

10------ 1

100---------

110------ 1

100--------- 1

100---------

110------ 1

100--------- 1

1000------------ 1

10------ 1

1000------------

110------ 1

100--------- 1

100---------

1100--------- 1

1000------------ 1

10------ 1

1000------------

Example 4

Example 3

Example 2

Example 5

Where would you place the decimal point in this weather report?The day was fine and warm, with a maximum temperature of 245˚C.

SolutionSince a warm day has a temperature in the mid-twenties, we would place the decimal point after the 4 so it reads 24.5˚C.

Read the following story carefully. List the numbers appearing in parts a to q and place a decimal point in each so that the story makes sense. a Maria filled her Holden car, which was nearly empty, with 434 litres of petrolb at 849 cents per litrec and handed the attendant a $5000 note.d After she received her change of $1315e she picked up her friend Fred, a tall man of 188 metres in height.



Ordering decimalsThe number of digits after the decimal point tells us the number of decimal places in the decimal number.

f Into the car jumped Charlie the dog, weighing 144 kilograms,g happily wagging his 150 centimetres of tail.h Maria and Fred had each packed a 200 kg backpack.i They stopped for a snack at McDougall’s and bought two beefburgers each, so that they

could have their daily intake of 250 kg of meat.j After they had driven for several hours, averaging 850 kilometres per hour, they were

within sight of Mt Kosciuszko, the highest mountain in Australia,k about 23000 m above sea level.l They enjoyed seeing the small eucalyptus trees about 105 m high.m They also saw a kangaroo, which jumped about 80 m,n 110 times the Olympic record for the long jump that was set in Mexico City.o After a pleasant walk of 134 km they arrived back at their campsite.p They slept soundly and, next morning, drove the 2500 km homeq in about 300 hours.

Worksheet7-02

Dewey decimals

Example 6

How many decimal places are there in:a 3.6567? b 15.801?

Solutiona 3.6567 has four decimal places. b 15.801 has three decimal places.

Arrange these numbers in ascending order (smallest to largest):67.41 67.14 6.714 67.04

SolutionTo compare decimals more easily, insert zeros at the end to give all the numbers the same amount of decimal places.

67.41 67.14 6.714 67.04become: 67.410 67.140 6.714 67.040In ascending order: 6.714, 67.04, 67.14, 67.41

Arrange these numbers in descending order (largest to smallest):0.5 0.08 1.7 0.85

Solution0.5 0.08 1.7 0.85

become: 0.50 0.08 1.70 0.85In descending order: 1.7, 0.85, 0.5, 0.08

1234 123

Example 7

Example 8


1 How many decimal places does each of these numbers have?a 1.65 b 3.881 c 15.3062 d 0.005 e 7.045 73 f 814.3g 9.100 001 h 203.222 22 i 0.040 400 4

2 Arrange these numbers in ascending order (smallest first):a 43.89, 56.324, 9.998, 80.879, 400, 23.89, 56.314b 0.568, 0.684, 0.099, 1.002, 0.586, 5.608, 0.0586c 1.23, 0.891, 1.814, 0.222, 7.007, 0.89d 0.5, 0.05, 0.005e 3.441, 3.404, 3.4, 3.44, 3.004, 3.044f 0.2, 0.202, 0.22, 0.022g 1.01, 1.002, 1.012, 1.21h 0.07, 0.67, 0.71, 0.007, 7

3 Arrange these numbers in descending order (largest first):a 570.25, 125.63, 0.9899, 4000.99, 1256.3, 400.099b 5.37, 6.539, 5.639, 5.367, 3.659, 3.66, 5.369c 1.6, 1.61, 1.599, 1.601, 1.509d 6, 0.06, 0.6, 6.6e 0.7, 0.07, 0.707, 0.77, 0.007, 7.07f 0.4004, 0.044, 0.404, 0.44g 0.1, 0.08, 0.65, 0.029h 0.92, 0.921, 0.09, 0.099

4 Insert � or � to make each of these true:a 0.2 0.25 b 0.731 0.73c 0.035 0.305 d 0.007 0.070d 1.59 1.059 f 0.099 0.99g 44.44 4.444 h 0.7932 0.7239

5 Copy each of those number lines carefully and fill in the values of the points marked with purple dots:a

b

c

d

e

f

g

1.6 1.9 2.3

4.07 4.09 4.11

0.65 0.67 0.71

8.0 8.16 8.28.1

0.07 0.23

2.0 2.6

4.3 4.5 4.74.4 4.6


Example 8

Example 7


6 Look at the following diagram. a Find a path from the START circle to the TOP circle. You can make your first move in

either direction, but then you can only move to a circle with a larger number.

b Now write out the number sequence for:i the longest path ii a path requiring seven moves

iii the route requiring nine movesc There are several paths that use the smallest number of moves.

i Find and write the number sequence for all the shortest paths you can find.ii How many moves are in the shortest path?

TOP

0.121

0.05

START

0.170.12

0.029 0.2

0.09 0.071 0.11 0.14

0.30.081

0.005 0.015

Ordering decimalsA spreadsheet can be used for ordering a group of numbers, from smallest to largest or vice versa.

InstructionsStep 1: Type your numbers into the green column, like the green box below.

(You can overtype Spreadsheet 7-01.)Step 2: Hold down the left mouse key and drag, to highlight the numbers that you typed.Step 3: Click on the Data menu, and choose ‘Sort’.Step 4: You must choose this column to sort, but you can decide whether to choose to sort

in ascending order or descending order.Step 5: When you press ‘OK’, the spreadsheet will automatically order the data for you.

Example: 0.2

0.23

0.232

0.233

0.3

0.32

0.322

Using technology

Spreadsheet 7-01

Ordering decimals


Decimals are special fractionsA decimal number usually includes a part of a whole unit. The unit is divided into 10 parts (tenths), 100 parts (hundredths), 1000 parts (thousandths), or any ‘power of 10’ parts.

This shape has been divided into 10 equal parts. Nine out of the 10 parts are coloured blue.

Writing this as a common fraction, or nine-tenths of the shape is coloured.

Writing it as a decimal fraction, 0.9 or ‘zero-point-nine’ of the shape is coloured.

A decimal number has a decimal point to mark where the whole part is separated from the fraction part of the number.

Worksheet 7-03

Decimals squaresaw 1

910------

Example 9

Write each of these as a decimal:

a b 3 c eighty-five thousandths

Solution

a = 0.22

b 3 = 3.7

c eighty-five thousandths =

= 0.085

22100--------- 7

10------

22100---------

710------

851000------------

Note:• tenths have one decimal place• hundredths have two decimal places• thousandths have three decimal places, and so on.

Example 10

Write each of these as a common fraction:a 0.09 b 0.273 c 8.1

Solutiona 0.09 = (two decimal places means hundredths)

b 0.273 = (three decimal places means thousandths)

c 8.1 = 8 + = 8 (one decimal place means tenths)

Note that the number of zeros in the denominator (bottom number) of the common fraction is the same as the number of decimal places in the decimal.

9100---------

2731000------------

110------ 1

10------


1 What part of each of these shapes has been shaded? (Give your answers as decimals.)a

b c

d

2 Write each of these as a decimal:

a b c d

e four-tenths f g h

i j eighty-seven hundredths k

l m n o

p q r s


a b c

d e f

g h thirty-two hundredths i five-thousandths

j k l

m n o seven-millionths


a 1 b 4 c 23

d 6 e forty-five and twenty-three thousandths

f 2 g 6 h 89

i 42 j 5 k 4

5 Write each of these as a common fraction:a 0.7 b 0.4 c 0.39d 0.572 e 0.003 f 0.05g 0.11 h 0.309 i 0.9j 0.999 k 0.013 l 0.0004m 0.0471 n 0.3333 o 0.5001p 0.91 q 0.087 r 1.9s 27.33 t 2.007 u 10.349v 7.41 w 101.3 x 6.0102

910------ 15

100--------- 79

100--------- 60

100---------

23100--------- 6

10------ 411

1000------------

7041000------------ 7

100---------

14100--------- 8

10------ 235

1000------------ 247

1000------------

17100--------- 368

1000------------ 9345

10 000---------------- 493

1000------------

67100--------- 3

10------ 4

100---------

111000------------ 8

100--------- 6

1000------------

1100---------

1710 000---------------- 2

100--------- 57

1000------------

3310 000---------------- 46

100 000-------------------

67100--------- 47

100--------- 9

10------

810------

310------ 11

1000------------ 143

1000------------

010------ 0

100--------- 6

1000------------

Exercise 7-04

Example 10

Example 9


Applying strategies and reflecting: Decimal distancesEquipment: A tape measure and some coloured chalk.

Step 1: Form a group of three or four students and find a flat area such as a path, basketball court, or the classroom floor.

Step 2: Mark a starting point on your flat area.

Step 3: Choose a distance between 1 and 4 metres, for example 3.25 m.

Step 4: Using chalk, take turns to mark your estimate of 3.25 m from the starting point. Measure the exact length and mark it with an X.

Step 5: Give points to each person. Score 5 points for best estimate, 3 points for next best, 2 points for next and 1 point for next.

Take turns to choose distances, and play the game until someone reaches 20 points and wins.

Sample score sheet for one person:

Trial Distance Guess Points

1 3.25 2.57

2 4.50 3.16

3 3.50 3.41

4 1.75 1.82

5 2.20 2.38

Total

Working mathematically

Estimating answersA quick way of estimating the answer to a calculation is to round each number in the calculation.

1 Examine these examples:

a 631 + 280 + 51 + 43 + 96 ≈ 600 + 300 + 50 + 40 + 100= (600 + 300 + 100) + (50 + 40)= 1000 + 90= 1090 (Actual answer = 1101)

b 55 + 132 − 34 + 17 − 78 ≈ 60 + 130 − 30 + 20 − 80= (60 + 20 − 80) + (130 − 30)= 0 + 100= 100 (Actual answer = 92)

c 67 × 13 ≈ 70 × 12= 840 (Actual answer = 871)

d 78 × 7 ≈ 80 × 7= 560 (Actual answer = 546)

Skillbank 7

SkillTest 7-01

Estimation


Adding and subtracting decimalsCareful setting out is essential when adding and subtracting decimal numbers.

When adding or subtracting whole numbers, we must write them under one another in their correct place-value columns:

67 Not: 67486 486

9 9302 302

+ 59 + 59

923 ???

The same is true when adding and subtracting numbers that have decimal points.

e 929 ÷ 5 ≈ 1000 ÷ 5= 200 (Actual answer = 185.8)

f 510 ÷ 24 ≈ 500 ÷ 20≈ 50 ÷ 2= 25 (Actual answer = 21.25)

2 Now estimate the answers to these:

a 27 + 11 + 87 + 142 + 64 b 55 + 34 − 22 − 46 + 136

c 684 + 903 d 35 + 81 + 110 + 22 + 7

e 517 − 96 f 210 − 38 − 71 + 151 − 49

g 766 − 353 h 367 × 2

i 83 × 81 j 984 × 16

k 828 ÷ 3 l 507 ÷ 7

Example 11

Add 16.27, 10.92, 4.03, 0.89, 32, 0.6

SolutionEstimate: 16 + 11 + 4 + 1 + 32 + 1 = 65.

The answer should be about 65.16.2710.924.030.89

32.0 Remember that 32 is the same as 32.0 or 32.00.+ 0.6

64.71

When adding or subtracting decimals, keep decimal points under one another.


To really understand decimals, we need to be able to count up and down by decimal fraction steps. Use the link to go to an activity enabling you to practise this skill.

Example 12

1 Subtract 8.914 from 46.029.

SolutionEstimate: 46 − 9 = 37.The answer should be about 37. 46.029

− 8.914

37.115

2 Find the answer to 4.31 − 2.183.

SolutionEstimate: 4 − 2 = 2. 4.310 Hint: Fill the spaces with zeros.

− 2.183

2.127

Spreadsheet 7-02

Counting with decimals

1 Copy and complete these addition calculations:a 5.3 b 4.723 c 43.5 d 12.755

6.2 0.01 116.29 1.35+ 0.5 + 12.2 7.3 18.675

+ 0.227 + 49

e 0.0076 f 37 g 6.83 h 0.0571.23 3.45 0.5 0.1324

+ 0.9 1.98 12.387 5.6731+ 548.7 + 5.001 + 38.009

2 Copy and complete these subtraction calculations:a 8.936 b 57.703 c 6.1 d 23.57

− 3.513 − 16.21 − 0.2 − 16.88

e 22.6 f 30.001 g 6.03 h 48.6− 13.54 − 1.73 − 4.091 − 9.951

3 Find the answers to these:a 103.67 + 9.81 + 0.24 + 3.7 b 4.6 + 2.9 + 15 + 0.16 + 32.32c 98.64 − 41.09 d 7.41 − 3.95e 38.624 + 1.109 + 23.7 + 0.65 f 81.91 + 3.254 + 681.7 + 458g 2.91 + 32.5 − 18.9 h 1.57 − 0.89 + 2.34i 58.94 − 2.31 − 46.13 j 14.62 − 0.195 + 6.37

4 An electrician needed these lengths of cable to complete a wiring job: 12.3 m, 4.8 m, 18.7 m, 7.98 m, 13.65 m and 23.6 m.a How many metres of cable did the electrician use?b If the full spool of cable was 100 m long, how many metres of cable were left in the

spool after the electrician completed the job.


Example 12


5 Add each set of prices. Calculate the exact change from the amount shown in brackets.a $7.01 $0.34 $2.19 ($10)b $2.14 $1.34 39c $0.45 ($5)c $5.26 $10.09 $14.31 $11.36 ($50)d $0.85 $4.34 $1.17 $8.79 ($20)e $11.34 $9.15 $3.95 $7.92 $2.36 ($50)f $1.46 $13.74 $42.99 $7.46 $29.05 ($100)

6 To keep fit, Angela runs each day. Last week she ran 3.8 km, 4.1 km, 2.3 km, 2.6 km, 3.0 km, 0.9 km and 1.8 km. How far did she run last week?

7 A truck carrying sand had a total mass of 13 248.56 kg. If the truck alone had a mass of 5210.8kg, what is the mass of the sand?

8 Five runners in the school’s 100 m race recorded the following times: 13.5 s, 13.81 s, 12.7 s, 14.62 s, 12.45 sa Place these times in order, from fastest to slowest.b What is the time difference between the first runner to finish and the last?c If the runner in second place had run 0.3 seconds faster, would she have won the race?

Explain your answer.

9 Krysten’s expenses for one week are shown in the table on the right.a How much did she spend?b How much did she have left out of $620.80?

10 A block of wood 11.27 cm thick has 0.34 cm shaved off one side and 0.55 cm shaved off the other side. How thick is the block of wood then?

11 Wendy was making teddy bears. She needed these amounts of material for five bears: 2.6 m, 0.8 m, 1.2 m, 0.75 m and 0.88 m. How much material did she need altogether?

12 The odometer in a car measures the total distance the car has travelled. The odometer below reads 21 456.9 km. The blue digit shows tenths of a kilometre.

Find the distance travelled during a holiday if the odometers below give the readings at the start of the holiday and at the end of the holiday.

2 1 4 5 6 9

2 1 5 7 6 4

2 2 3 1 5 3

Item Cost ($)

Food 128.80

Clothing 88.45

Car 58.35

Rent 185.00

Entertainment 78.95

Savings 66.00


Applying strategies: Comparing heights1 Use the clues below to find each girl’s name and height.

• Mandy is taller than Sarah.• Kelly is taller than Sarah but shorter than Mandy.• Sarah is shorter than Yin.• Mandy is not the tallest.• The heights of the girls are 168.5 cm, 166.3 cm, 164.2 cm and 160.7 cm.

2 Use the clues below to find each boy’s name and height.• Steve is 164.7 cm tall. • Mike is 14.7 cm taller than Milof.• Steve is 3.9 cm shorter than Milof. • Ganesh is 1.6 cm taller than Mike.

3 Use the clues below to find each student’s name and height.• Yollande is 15.1 cm taller than Peter. • Jade is 13.7 cm shorter than Yollande.• Karl is 20.6 cm taller than Jade. • Yollande is 6.9 cm shorter than Karl.• Peter is 163 cm tall.

4 Devise your own problem using four to six class members. Estimate their heights and height differences, and write a set of clues. (Don’t forget to change their names!)

C DA B

C DA B

C DA B



Multiplying and dividing by powers of 10This place-value chart shows what happens when you multiply and divide a decimal by powers of 10. Check it using your calculator.

Note how the decimal point moves to the right when multiplying and to the left when dividing. The number of places moved is the same as the number of zeros in the power of 10 you are multiplying or dividing by.

36.7 3 6 7

36.7 × 10 3 6 7

36.7 × 100 3 6 7 0

36.7 × 1000 3 6 7 0 0

2.35 2 3 5

2.35 × 10 2 3 5

2.35 × 100 2 3 5

2.35 × 1000 2 3 5 0

36.7 ÷ 10 3 6 7

36.7 ÷ 100 0 3 6 7

36.7 ÷ 1000 0 0 3 6 7

2.35 ÷ 10 0 2 3 5

2.35 ÷ 100 0 0 2 3 5

Applying strategies and reasoning: Contents of envelopesThe objects listed below are typical of what we put in envelopes. • pin 0.06 g • stamp 0.13 g• lined notepaper 3.69 g • staple 0.07 g• cheque 0.97 g • paper clip 0.22 g

The total mass (in grams) is printed on each envelope on the right. If a standard envelope has a mass of 2.23 g, identify the contents of each of these envelopes from its total mass. Each envelope can contain more than one of each object listed above.a Contents: 1 and 1

b Contents: 1 and 1

c Contents: 2 , 1 and 5

d Contents: 1 , 1 , 1 and 3

e Contents: 2 , 1 and 1

f Contents: 3 , 2 , 1 , 1 and 1

3.42 g

8.16 g

15.66 g

6.14 g

7.32 g

9.9 g

c

a b

d

e

f


Thousa

nds

thou

sandth

sTen

thou

sands

Hundreds

Tens

Units

tenth

s

hundredth

s

thou

sandth

s

ten

Calculat

ion


Multiplying decimalsEstimation and ‘fixing the point’If you know the result of a whole number multiplication, you can use estimation to work out the answer to similar decimal multiplications and correctly position the decimal point.

1 Copy and complete each of these statements:a To multiply by 10, move the decimal point place to the .b To multiply by 100, move the decimal point places to the .c To multiply by 1000, move the decimal point places to the .d To divide by 10, move the decimal point places to the .e To divide by 100, move the decimal point places to the .f To divide by 1000, move the decimal point places to the .

2 Copy and complete:a 46.3 ÷ 10 = b 507 ÷ 100 = c 1203 ÷ 1000 =d 36.4 ÷ 100 = e 705 ÷ 1000 = f 3102 ÷ 10 =g 6.43 ÷ 1000 = h 64.3 ÷ 1000 = i 4.28 ÷ 1000j 66 ÷ 100 = k 0.31 ÷ 1000 = l 0.02 ÷ 10m 24.9 × 100 = n 0.81 × 10 = o 37.42 × 1000 =p 0.416 × 100 = q 0.81 × 100 = r 2.192 × 1000 =s 60 451 × 100 = t 6.02 × 100 = u 0.031 × 10 =

3 Evaluate, using the rules you have found:a 45.213 × 100 b 10.64 × 1000 c 6304 ÷ 100d 5.98 ÷ 1000 e 847.612 × 100 f 0.0592 × 10g 36.28 ÷ 100 h 519.4 × 1000 i 40.075 ÷ 10j 81.348 ÷ 1000 k 502 × 100 l 0.61 ÷ 100m 0.4 ÷ 1000 n 17.01 × 100 o 12.3 × 10 000p 66 ÷ 10 000 q 5.2 × 10 ÷ 100 r 14.71 ÷ 100 × 10

4 Evaluate:a 469 187 ÷ 100 000 b 437.421 ÷ 1000 c 27.43 ÷ 1 milliond 1 200 000 ÷ 1 000 000 e 235 000 137 ÷ 10 000 f 137.429 × 1000 ÷ 10 000

Exercise 7-06

Worksheet 7-04

Where’s the point?

Example 13

1 Given that 17 × 12 = 204, find:a 1.7 × 12 b 1.7 × 120

Solution

Multiplication Estimate Answer

17 × 12 204 (given)

a 1.7 × 12 2 × 10 = 20 20.4 use the digits 204 to make a number near 20

b 1.7 × 120 2 × 100 = 200 204 use the digits 204 to make a number near 200


2 Given that 23 × 47 = 1081, find:a 2.3 × 47 b 230 × 4.7 c 23 × 0.47

Solution

Multiplication Estimate Answer

23 × 47 1081 (given)

a 2.3 × 4.7 2 × 5 = 10 10.81 use the digits 1081 to make a number near 10

b 230 × 4.7 200 × 5 = 1000 1081 use the digits 1081 to make a number near 1000

c 23 × 0.47 20 × 0.5 = 10 10.81 use the digits 1081 to make a number near 10

1 Copy and complete each of the following tables:

2 Given that 63 × 34 = 2142, use estimates to find:

a 630 × 340 b 0.63 × 3.4 c 0.63 × 3400

d 3.4 × 630 e 6.3 × 34 000 f 63 × 340

a Multiplication Estimate Answer

69 × 18 1242

690 × 180

6.9 × 180

6.9 × 1.8

690 × 1.8

b Multiplication Estimate Answer

104 × 42 4368

10.4 × 42

1.04 × 4.2

104 × 4.2

0.104 × 4.2

c Multiplication Estimate Answer

38 × 92 3496

3.8 × 92

38 × 920

38 × 0.92

380 × 0.92



3 Given that 1.7 × 1.2 = 2.04, use estimates to find:a 1.7 × 12 b 17 × 12 c 0.17 × 1.2d 120 × 170 e 0.12 × 17 f 17 000 × 0.12

4 Given that 7.2 × 3.4 = 24.48, use estimates to find:a 7.2 × 34 b 72 × 3.4 c 0.72 × 3.4d 72 × 34 e 7.2 × 0.34 f 720 × 340

5 Given that 1.26 × 6 = 7.56, use estimates to find:a 12.6 × 6 b 126 × 6 c 1.26 × 0.6d 0.126 × 6 e 0.126 × 0.6 f 126 000 × 600

6 Use the fact that 0.3 × 0.24 = 0.072 to find:a 3 × 0.24 b 0.3 × 2.4 c 0.3 × 24d 30 × 0.24 e 300 × 2.4 f 0.03 × 0.24

Reflecting and reasoning: Decimal places in multiplication answers

1 What happens when you multiply by a number less than one? As a group activity, consider the product of 12 × 0.8.a Is the answer more or less than 12? Why?b Estimate the answer to 12 × 0.8.c How many decimal places have 12 and 0.8?d Use a calculator to evaluate 12 × 0.8. How many decimal places has the answer?

2 a Is the answer to 0.7 × 0.3 more or less than 0.7? Why?b Estimate the answer to 0.7 × 0.3.c How many decimal places have 0.7 and 0.3?d Use a calculator to evaluate 0.7 × 0.3. How many decimal places has the answer?

3 a Is the answer to 2.5 × 4.1 more or less than 2.5? Why?b Estimate the answer to 2.5 × 4.1.c How many decimal places have 2.5 and 4.1?d Use a calculator to evaluate 2.5 × 4.1. How many decimal places has the answer?e What is the relationship between the number of decimal places in the question

and the number of decimal places in the answer?

4 a If 82 × 6 = 492, what do you think is the answer to 82 × 0.6? Why?b If 4 × 17 = 68, what do you think is the answer to 0.4 × 1.7? Why?c If 367 × 51 = 18 717, what do you think is the answer to 3.67 × 5.1? Why?d What is the answer to 0.5 × 0.9?

5 Make up a question about multiplying decimals. Swap questions with all members of the group. If you disagree about the correct answers, check with your teacher.

When multiplying decimals, the number of decimal places in the answer equals the total number of decimal places in the question.



Using the multiplication rule

Example 14

1 Evaluate 3.05 × 4.8.

SolutionStep 1: Do the multiplication without decimal points.

305× 48

244012 200

14 640Step 2: Decide where to place the decimal point. There are two decimal places in 3.05

and one decimal place in 4.8. So there are a total of three decimal places in the question. We write the answer, 14 640, with three decimal places: 14.640 = 14.64.

2 Evaluate 0.6 × 4.1.

Solution41

× 6

246Because there is a total of two decimal places in the question, we need to write the answer, 246, with two decimal places. The answer is 2.46.

Animation 7-01

Multiplying decimals

1 Use the method shown above to calculate:a 3.05 × 4 b 1.02 × 7 c 2.001 × 9d 17.1 × 2 e 10 × 2.25 f 3 × 4.20g 6.95 × 5 h 1.0004 × 8 i 0.18 × 5j 0.4 × 12 k 6 × 0.002 l 0.25 × 11m 10.2 × 4 n 3 × 4.2 o 0.5 × 10p 0.71 × 2 q 7 × 2.193 r 10.941 × 3s 0.63 × 5 t 492 × 0.12 u 0.11 × 365

2 Calculate:a 0.4 × 0.8 b 3.9 × 0.5 c 0.8 × 0.6d 0.3 × 0.24 e 0.08 × 0.04 f 3.1 × 0.4g 12.6 × 0.06 h 0.28 × 3 i 0.39 × 9j 2.93 × 0.3 k 6.80 × 4 l 0.54 × 20m 0.9 × 0.75 n 8.06 × 4.1 o 0.11 × 1.01p 2.45 × 0.1 q 0.02 × 3.5 r 6.3 × 0.04

3 Find:a 47.9 × 23 b 6.43 × 7.2 c 83.4 × 6.3d 2.95 × 5.13 e 2.45 × 3.9 f 94.6 × 5.1g 0.237 × 1.2 h 9.2 × 7.9 i 11.1 × 0.621j 0.023 × 0.042 k 0.7521 × 3.6 l 321.2 × 8.1


Worksheet7-05

Which decimals?


Applying strategies and reflecting: Multiplication estimation gameThis is either a game for two players, or a class team game.

Players should copy this table into their books or print it from the CD-ROM, but should not commence calculating.

RulesStep 1: You each have a two-minute time limit in which to estimate all the answers for

the 10 multiplication questions in the table, correct to one decimal place.

Step 2: On the instruction ‘Go’ given by your partner, you begin to write the estimates for the multiplication questions in your table.

Step 3: After two minutes, stop. Then the other person completes Step 2.

Step 4: Use a calculator to find the exact answers, and score yourself for each multiplication.• 5 points for each estimate within 0.1 of the correct answer.• 4 points for each estimate within 0.2 of the correct answer.• 3 points for each estimate within 0.3 of the correct answer.• 2 points for each estimate within 0.4 of the correct answer.• 1 point for each estimate within 0.5 of the correct answer.• 0 points if the estimate is more than 0.5 from the correct answer.

Step 5: Total your points, and compare your total with that of your partner.

Investigate further

1 Did you tend to overestimate or underestimate?

2 Now make up and try your own set of 10 decimal multiplications. See if you perform better with practice.

3 Use this link to try an Excel version of this game.

Multiplication Estimation Correct answerfrom calculator Points

a 42 × 3.5

b 5.1 × 2.4

c 1.6 × 7.3

d 3.8 × 6.2

e 2.2 × 8.8

f 7.5 × 3.2

g 5.6 × 5.7

h 2.6 × 5.1

i 9.3 × 3.9

j 4.8 × 3.3

Total


Spreadsheet 7-03

Multiplication game


Calculating changeExample 15

Find the change from $50 if the following items are bought: • 3 kg of butter at $2.50 per kilogram • 500 g of cheese at $12.90 per kilogram• 5 kg of meat at $5.75 per kilogram • 2 dozen eggs at $1.77 per dozen.

Round the total to the nearest 5 cents.

Solution3 × 2.50 = 7.50

0.5 × 12.90 = 6.455 × 5.75 = 28.752 × 1.77 = + 3.54

Total 46.24

Rounded to the nearest 5 cents, this total is $46.25.Change from $50: 50.00

− 46.25

3.75

Exact change will be $3.75.

MILK 1LTBUTTERBREAD

RND TOTALCASHCHANGE

SALESPERSON 31

10:52 12/12/2004

1.131.861.64

4.655.000.35

Worksheet

Which decimals?7-05

1 Calculate the change from $50 for each of the following purchases. Round totals to the nearest 5 cents.a • meat $25.36

• 2 dozen eggs at $1.36 per dozen• 5.5 kg of apples at $1.49 per kilogram

b • 3 kg of tomatoes at $2.45 per kilogram• 10 kg of potatoes at $1.29 per kilogram• 4 L of milk at $1.37 per litre• 2 kg of butter at $3.08 per kilogram

c • 3.5 kg of chops at $3.89 per kilogram• 2.5 kg of T-bone steak at $8.10

per kilogram• 5 L of soft drink at $1.26 per litre

d • toothpaste $2.56• jam $1.13• cheese $3.08• 4 kg of oranges at $1.74 per kilogram• 3 kg of sausages at $1.85 per kilogram

e • 2 kg of nails at $2.51 per kilogram• 2 m of wood at $13.48 per metre• 2.5 L of glue at $6.74 per litre


SkillBuilder 1-09

Rounding money

Worksheet7-06

Shopping and change


f • 7 pens at $2.15 each• 12 glue sticks at $1.99 each• 4 erasers at $0.35 each

g • 2 tubes of toothpaste at $3.15 each• 2 bottles of drink at $1.37 each• 4 boxes of tissues at $3.29 each• 1.5 kg of tomatoes at $2.65 per kilogram

h • 46.5 litres of petrol at 93.6 cents per litre• 0.5 litres of premium oil at $6.60 per litre

2 Find as many words as you can beginning with ‘deci’, meaning ‘ten’.

Calculating changeYour task is to solve all of the parts of Question 1 in Exercise 7-09 using a spreadsheet similar to the one below.

1 What formula will you use to calculate the cost?

2 What formula will you use to calculate the total?

3 What formula will you use to calculate the change?

A B C D E F

6 Item Quantity Price/Unit Cost7

8 Meat 1 $ 25.36

9 Eggs 2 $ 1.36

10 Apples 5.5 $ 1.49

11

12 TOTAL:13

14 Amount given: $ 50.00

15 CHANGE:

Using technology

Spreadsheet 7-04

Calculating change

Applying strategies and reflecting: Multiplication maze1 Can you work out how to get through the maze on the facing page? Start with 1 in

your calculator display and, as you travel along each path, multiply the number in the display by the number you pass.You may travel along each path only once, but it can be in any direction. You must finish with a 5 in the display.

2 How many steps are needed to complete the maze?

3 Is there only one solution to this maze?



Dividing decimalsDividing decimals by whole numbers

Always check by estimation that your answers sound reasonable.

START 51

× 2.5

× 1.5 × 4 × 2 × 0.4

× 5

× 2× 10

× 0.2× 0.5

× 0.25

× 0.1

FINISH

When dividing by a whole number, keep decimal points under one another.

Example 16

Evaluate each of the following:a 10 ÷ 4 b 13.62 ÷ 3 c 0.018 ÷ 6 d 2.63 ÷ 4

Solution

a b c d10.042.5

13.6234.54

60.0030.018 2.63004

0.6575

1 Evaluate each of the following:a 4.8 ÷ 2 b 18.6 ÷ 3 c 20.8 ÷ 5d 32.8 ÷ 8 e 29.3 ÷ 2 f 8.79 ÷ 4g 0.056 ÷ 7 h 10.71 ÷ 4 i 195.6 ÷ 8j 7.35 ÷ 2 k 4.15 ÷ 8 l 19.23 ÷ 5m 13.42 ÷ 8 n 22.47 ÷ 7 o 0.318 ÷ 3

2 Evaluate each of the following:a 12 ÷ 5 b 13.56 ÷ 12 c 23 ÷ 4d 88.88 ÷ 11 e 107.1 ÷ 9 f 82.5 ÷ 6g 18.48 ÷ 15 h 177 ÷ 12 i 0.044 ÷ 11j 205 ÷ 50 k 165.2 ÷ 20 l 2075.6 ÷ 8m 58 ÷ 8 n 0.732 ÷ 6 o 6.23 ÷ 50



Dividing decimals by decimalsLook at this pattern:

18 ÷ 3 = 6180 ÷ 30 = 61800 ÷ 300 = 6

When dividing two numbers, if we multiply both numbers by the same power of 10 first, the answer stays the same. We can use this property to help us divide decimals. For example:

9.8 ÷ 0.08 = 980 ÷ 8 (multiplying both numbers by 100)= 122.5

The answer to 980 ÷ 8 is easier to find because it involves division by a whole number.

This works because in Steps 1 and 2 we multiply both decimals by the same power of 10.

Always check by estimation that your answers sound reasonable.

3 Find:a 0.651 ÷ 3 b 37 ÷ 8 c 0.078 ÷ 6d 675 ÷ 12 e 89.341 ÷ 7 f 4.275 ÷ 5g 2.75 ÷ 4 h 0.093 ÷ 11 i 117.44 ÷ 16j 79.86 ÷ 33 k 31.98 ÷ 13 l 130.76 ÷ 28m 0.471 ÷ 3 n 256.84 ÷ 4 o 0.696 ÷ 12

To divide a decimal by a decimal:Step 1: Make the second decimal a whole number by moving the decimal point the

appropriate number of places to the right.

Step 2: Move the decimal point in the first decimal the same number of places to the right.

Step 3: Divide the new first number by the new second number.

Example 17

Evaluate each of the following:a 0.4 ÷ 0.2 b 1.75 ÷ 0.5 c 122.4 ÷ 7.65

Solution

a 0.4 ÷ 0.2Move the decimal points one place to the right (multiplying both decimals by 10).= 4 ÷ 2 = 2

b 1.75 ÷ 0.5Move the decimal points one place to the right (multiplying both decimals by 10).= 17.5 ÷ 5 = 3.5

c 122.4 ÷ 7.65Move the decimal points two places to the right (multiplying both decimals by 100).= 12 240 ÷ 765 = 16

Animation 7-02

Dividing decimals


1 What happens when you divide by a number less than 1? a 18 ÷ 0.5 means ‘how many times does 0.5 go into 18’. Is the answer more or less

than 18? Why?b Estimate the answer to 18 ÷ 0.5.c Use the method from Example 17 to evaluate 18 ÷ 0.5.

2 a 20.4 ÷ 0.3 means ‘how many times does 0.3 go into 20.4’. Is the answer more or less than 20.4?

b Estimate the answer to 20.4 ÷ 0.3.c Find the exact answer to 20.4 ÷ 0.3.

3 Rewrite each of the following divisions so that the second decimal is a whole number.a 508.8 ÷ 1.2 b 17.82 ÷ 0.11 c 333 ÷ 4.5d 1.725 ÷ 2.5 e 129.2 ÷ 0.38 f 49.5 ÷ 1.5g 168 ÷ 0.75 h 14.823 ÷ 0.61 i 0.66 ÷ 0.022

4 Evaluate each of these, and check that your answers seem reasonable by estimating:a 3.48 ÷ 0.4 b 7.32 ÷ 0.2 c 2.94 ÷ 0.6d 16.28 ÷ 2.2 e 27 ÷ 0.25 f 10.08 ÷ 2.4g 10.4 ÷ 0.05 h 5.6 ÷ 0.07 i 1.71 ÷ 0.3j 40.82 ÷ 5.2 k 532 ÷ 3.5 l 78.12 ÷ 6.2m 0.272 ÷ 0.08 n 98.4 ÷ 0.08 o 465 ÷ 7.5


Reflecting and applying strategies: Practising your skillsThe cards for this set of questions have been printed with the decimal points in the wrong places. Insert the decimal points so that the numbers on the cards fit the clues.

1 The product of these two numbers is 134.64. The sum of the numbers is 58.5.

2 The difference between the numbers is 178.8. When you divide the greater number by the smaller number, the quotient is between 43 and 44.

3 The sum of the three numbers is 5.36. The product of the numbers is 1.2096.

4 The sum of the three numbers is 4.61. The product of the numbers is 0.9135.

5 The sum of the four numbers is 2.55. The product of two numbers is 0.196. The product of the other two numbers is 0.217.

561

183

2.4

42

8

15

42

29

36

21

14 147 31



Decimals at work

1 Find the cost of 352 units of electricity at 12.3 cents per unit.

2 A farmer wants to fence a rectangular paddock. The paddock is 35.6 metres long and 20.85 metres wide. How many metres of fencing will be needed?

3 Mark buys golf balls for $4.85 each and sells them for $5.15 each. How much money does he make if he sells 30 golf balls?

4 A drink bottle holds 0.75 litres. How many drink bottles can be filled from a tub that holds 4.5 litres?

5 A car travels 220.6 kilometres on 14 litres of petrol. How many kilometres would the car travel on one litre of petrol? (Give the answer to one decimal place.)

6 Pamela runs 3.8 kilometres each day of the week. How far does she run in one week?

7 A long distance train is made up of a diesel engine, two dining cars and 15 passenger carriages. The engine has a mass of 20.2 tonnes, each dining car has a mass of 14.35 tonnes and each passenger carriage has a mass of 13.96 tonnes. How heavy is the entire train?

8 George is cutting shelves from a board which is 4.6 metres long. Each shelf needs to be 1.25 metres long. How many shelves can be cut?

9 Samantha ran 100 metres in 15.21 seconds. How long would Samantha take to run 400 metres at this pace?

10 Henry has a faulty calculator; it does not show the decimal point. For each of the calculations in the table shown on the right, write the correct answer.

11 Mick walks to work and back each day. He works six days a week and, in one week, walks 16.8 kilometres. How far is Mick’s apartment from work?

12 The following are calculator displays for amounts in dollars and cents. Give the answers in dollars and cents.

a b c d

13 A sheet of paper is 0.01 cm thick. How many sheets would be in a stack 4 cm high?

Calculation Answer

3.42 × 12 4104

4.145 × 0.2 829

37.3 × 8.8 32824

0.03 × 157.64 47292

8.3902 × 0.3 251706

11115555....222233336666 6666....8888444411112222 444488887777....777755559999333311118888 1111222233334444....000044447777111122226666

Exercise 7-12


Converting common fractions to decimalsWe often use fractions in our everyday conversations:

Are we half-way there yet?Nearly two-thirds of the crowd barracked for NSW.

It is useful to recognise fractions in their decimal form.

Example 18

Write each of these as a decimal:a b

Solutiona means 3 ÷ 5 b means 5 ÷ 8

= 0.6 = 0.625

35--- 5

8---

35--- 5

8---

3.050.6

5.0008

0.6252 4

35--- 5

8---

1 Copy and complete this table. Use a calculator if you need to.

2 Use your completed table from Question 1 to help you find decimals equal to the following common fractions:

a b c d

e f g h

i j k l

3 Explain why some of the fractions in Question 2 have the same decimal value.

Common fraction Meaning as division Decimal Equivalent common fraction

a 3 ÷ 5 0.6

b 1 ÷ 2

c 0.25

d

e

f 3 ÷ 4 0.75

g 1 ÷ 5

h 1 ÷ 8

35--- 6

10------

12--- 5

10------

14---

45---

25---

25--- 3

8--- 3

4--- 2

2---

24--- 6

8--- 3

5--- 2

8---

58--- 7

8--- 4

8--- 5

5---



Recurring decimalsSometimes when a common fraction is converted to a decimal, we get a repeating or recurring decimal. One or more of the digits in the decimal repeat forever.

4 Write as decimals:

a 4 b 23 c 12 d 6

e 57 f 19 g 110 h 80

i 6 j 13 k 5 l 6

5 You can use Excel to change decimals into common fractions and vice versa. Use this link to go to an activity that will show you how.

810------ 3

4--- 5

8--- 3

5---

25--- 1

8--- 7

8--- 1

4---

15--- 1

2--- 3

8--- 2

4---

Spreadsheet 7-05

Common fractions to decimals

Worksheet 7-07

Fraction families

Example 19

1 Change to a decimal.

Solution means 1 ÷ 3

= 0.333… = 0. or 0.



= 0.8333… = 0.8 or 0.8



= 0.181 818… = 0. or 0.

Notice the repeating or recurring pattern in the numbers following the decimal point. To show that the pattern goes on forever, we place a dot or a line above the recurring digits.The dot or line shows the repeating section: 0.259 259 259 2… = 0. or 0.Non-repeating decimals are called terminating decimals. ‘Terminate’ means ‘to stop’.

13---

13--- 1.0000…3

0.333…

13--- 3 3

56---

56--- 5.0000…6

0.8333…

56--- 3 3

211------

211------ 2.000000…11

0.18181…

211------ 18 18

259 259

Worksheet 7-08

Decimals squaresaw 2

1 Change these to repeating decimals:

a b c d e

f g h i j

k l m n o

19--- 1

3--- 1

6--- 1

7--- 2

3---

27--- 2

9--- 3

7--- 4

7--- 4

9---

57--- 5

9--- 6

7--- 7

9--- 8

9---



Rounding decimalsSometimes, to approximate an answer with many decimal places, we round to fewer decimal places. We need to be able to round when working with money, measuring quantities or writing answers to division calculations.

2 Copy and complete the following pattern:

Fraction:

Decimal: 0. 0.

3 Perform these calculations and write the answers as recurring decimals:a 11 ÷ 3 b 11.3 ÷ 7 c 58.43 ÷ 9d 1.9 ÷ 6 e 76 ÷ 9 f 0.67 ÷ 6g 2 ÷ 7 h 13.4 ÷ 6 i 149 ÷ 7j 13 ÷ 3 k 196 ÷ 11 l 12 ÷ 7m 7 ÷ 11 n 17 ÷ 12 o 18.1 ÷ 9

19--- 2

9--- 3

9--- 4

9--- 5

9--- 6

9--- 7

9--- 8

9---

1 2

Skillsheet 7-01

Rounding whole numbers

To round a decimal:• cut the number at the required decimal place• look at the digit immediately to the right of the specified place• if this digit is 0, 1, 2, 3 or 4, leave the number in the specified place unchanged• if the digit is 5, 6, 7, 8 or 9, add 1 to the number in the specified place.

Example 20

1 Round 86.246 correct to one decimal place.

Solution86.2 46

So 86.245 is 86.2 (correct to one decimal place).

2 Round 86.246 correct to two decimal places.

Solution86.24 6

So 86.246 is 86.25 (correct to two decimal places).

The number 0.087 1245 has seven decimal places. Round it to:a one decimal place b two decimal places c the nearest thousandth

Solutiona 0.1 b 0.09 c 0.087

cut the next digit is 4, so the number 2 does not change

cut the next digit is 6, so add 1 to 4 to give 5

Example 21


Rounding to:• the nearest tenth = one decimal place• the nearest hundredth = two decimal places• the nearest thousandth = three decimal places

1 Write each of the following correct to one decimal place:a 0.35 b 0.47 c 0.81 d 0.69e 2.55 f 0.32 g 0.90 h 2.88

2 Round each of the following to two decimal places:a 0.481 b 0.736 c 0.069 d 0.293e 0.309 f 0.655 g 2.096 h 3.995

3 Write each of the following correct to two decimal places:a 25.3456 b 341.7675 c 321.3333d 734.6541 e 27.757 575 f 1314.123 45

4 Copy this table into your book. Use a calculator to help you complete it.

5 Use a calculator to find the value of each of the following divisions. Give your answers to the nearest hundredths.a 2.89 ÷ 3 b 8.57 ÷ 6 c 0.812 ÷ 9d 7 ÷ 11 e 5.12 ÷ 6 f 11.71 ÷ 7g 8 ÷ 6 h 12 ÷ 7 i 18.87 ÷ 11j 12.62 ÷ 13 k 7 ÷ 12 l 9.38 ÷ 15

6 Round each of these numbers to four decimal places:a 10.33374 b 431.543 27 c 1.444 95d 3217.654 061 e 4.670 89 f 0.888 88

Question Calculator displayRounded to

one decimal placeRounded to

two decimal places

12.19 ÷ 3 4.063 333 3333 4.1 4.06

12.32 ÷ 6

19.82 ÷ 9

56.85 ÷ 11

17.13 ÷ 4

12.65 ÷ 12

4.875 ÷ 21

27.45 ÷ 8

17 ÷ 12

254.678 ÷ 32


Example 21


7 The answers to the following are whole numbers but, for particular reasons, some need to be rounded up and some need to be rounded down. Find the answers.a A box of chocolates with 33 chocolates is shared among a family of five people.

How many chocolates does each person receive?b A new bathroom requires 32 square metres of tiles. The tiles come in boxes

containing 1.5 square metres. How many boxes are needed to tile the bathroom?c A team of four golfers wins 27 new golf balls in a competition. How many does each

person receive?d Some timber comes in 1.8 m lengths. How many lengths are needed to build a chicken

house needing 23 m of timber?e Each blouse requires 1.3 m of material. How many blouses can be made from a 5 m

length of material.

8 Give a reason for:a rounding up b rounding down.

Questioning and reflecting: Rounding prices Since the use of 1c coins and 2c coins stopped in 1990, prices have been rounded to the nearest 5 cents. Find out:a when amounts are roundedb how amounts are rounded (check the major stores in your area)c what happens with bills such as electricity, phone, etc.d who decides how the rounding will work.

Working mathematically SkillBuilder 1-09

Rounding money

The salami technique

When banks started using computers to keep track of customers’ accounts, they left themselves open to a new type of crime: computer theft. One such crime employs the salami technique, where computer hackers steal a cent or a fraction of a cent from many bank accounts. They round down the decimal amount of an account balance (for example $234.6523 would become $234.65) and the stolen fraction of a cent ($0.0023) is deposited into the hacker’s account, with no one noticing it missing. When this is done to thousands of bank customers over a number of years, a considerable amount of money can be accumulated.

If the salami technique is applied to $1723.35631, $456.3277, $6701.2315 and $488.29891, how much will the computer criminal have in his or her account?Why do you think this type of crime is difficult to detect?

Why do you think it is called the ‘salami technique’?

Just for the record


More decimals at work

Example 22

Use the table below to find the cost of a 6-minute phone call made over a distance of 236 km on Friday at 11:00am.

SolutionThe distance is between 166 km and 745 km so we use the fourth column of costs. The time is a weekday during daytime, so we use the first row.

The call is charged at $0.43 a minute.Cost of call = 6 × $0.43

= $2.58

Cost of long-distance phone callsApproximate cost per minute (rounded to the nearest cent) with a minimum charge of 21c per call.

Distance Under50 km

50 km to 85 km

86 km to 165 km

166 km to 745 km

Over 745 km

Day8:00am–6:00pmMonday to Saturday

$0.14 $0.28 $0.35 $0.43 $0.60

Night6:00pm–10:00pmMonday to Friday

$0.09 $0.17 $0.26 $0.29 $0.41

Economy6:00pm Saturday to 8:00am Monday10:00pm–8:00ameveryday

$0.08 $0.13 $0.18 $0.21 $0.26

1 Long-distance phone callsUse the table in Example 22 to help you answer these questions:a Find the cost of a 5-minute call over a distance of 21 km made at 7:45pm on a Tuesday

night.b Find the cost of an 11-minute phone call from Manly to Penrith (60 km) on a Sunday

morning at 9:30am.c Janis called her boyfriend Mal, who lives 112 km away, on Saturday night at 6:10pm.

Find the cost of the call if they spoke for an hour-and-a-half.d A phone call from Sydney to Coffs Harbour (400 km) appeared on the phone bill as a

charge of $2.73. If the call was made on Monday at 10:30pm, work out how long the call took.

e Make up three questions about phone calls using the table of costs. Work out the answers and give a classmate the three questions to complete.



f Linh is an interstate truck driver. Her company gave her a $20 phone card before she left Melbourne on an interstate trip to Moree in New South Wales. After making deliveries at each of the towns marked on the strip map below, she telephoned her truck base in Bendigo. The phone calls were recorded in her log book.

Did Linh’s phone card cover the cost of her phone calls? Explain your answer.

2 FM radioIn the radio and television guide you will find a list of FM stations and their allocated frequencies measured in megahertz (MHz).

a Locate the stations on a number line.

b What is the frequency difference in megahertz between 2DAY and WS-FM?

c Find the smallest frequency difference between adjacent stations. (‘Adjacent’ means side-by-side.)

d What is the largest difference in frequency between adjacent stations?

3 Decimal currencya Find out about the history of the decimal money system in Australia.

i When was it introduced?ii What system did it replace?

iii Why was the change made?iv When were the different coins and notes introduced?v What changes have occurred since?

b Design a set of notes ($5, $10, $20, $50, $100) that blind and sighted people could both use.

LOG

Call from Time Length of call

Echuca 7:30am 6 min

Narrandera 11:15am 9 min

Parkes 6:50pm 15 min

Dubbo 8:00am 4 min

Moree 4:15pm 22 min

80 km 230 km 230 km 110 km 340 km

Dubbo MoreeParkesNarranderaEchucaBendigo

StationFrequency

(MHz)

ABC 92.9

The Edge 96.1

STAR-FM 99.7

WS-FM 101.7

2MBS 102.5

FM 103 103.2

2DAY 104.1

MMM 104.9

JJJ 105.7

MIX 106.5

2SER 107.3

90 100 110


4 Decimal timea Investigate the idea of decimal time. How would

you measure time in this system?b Design a decimal time calendar.c How would decimal time affect your birthday

and age?d Do you think decimal time is possible? Explain.

1

2

345

67

89 10

1 Code breakerA long time ago, the warrior Wolf was trying to break a code that would open some dungeon doors. He had to free the prisoners before midnight so that they would not be turned into frogs by an evil spell. Each castle door was operated by a combination lock.a Use the following clues to match the various combinations with the doors to the

different rooms in the castle.Combinations Rooms

8.262 Queen’s chamber9.24 armoury9.96 throne room8.07 banquet hall8.16 kitchen8.79 dungeon

Clues• Combination 9.24 opens a door to a room that deals with food.• The combination to the armoury has a 6 in the hundredths place.• The combinations of the throne room and the banquet hall add to 18.03.• The Queen’s chamber has a combination that is bigger than 3.78 × 2.1 but smaller than

25.11 ÷ 3.1. (Hint: The doors in the dungeon and the throne room remained locked when Wolf tried 9.96 and 8.262.)

• The kitchen combination is one of the three largest combination numbers.b Invent another story that sets a code-breaking task involving decimals and ask your

classmates to solve it.

2 Calculate:a 0.02 × 0.02 b (0.02)3 c (1.1)3

d e f

3 Decide where the decimal point should be so that the numbers in the ovals fit the clues.a The product of the two numbers is 91.02.

The sum of the numbers is 28.3.b The sum of the three numbers is 14.

The product of the three numbers is 78.66.c The sum of the four numbers is 28.32.

The product of two of the numbers is 3.672. The product of the other two is 81.

0.04 0.36 0.0273

24637

5723 6

7251

45 18

Power plus


Topic overview• What parts of this topic were new to you? What parts did you already know?• Write any rules you have learnt about working with decimals.• What parts of this topic did you not understand? Be specific. Talk to a friend or your

teacher about them.• Give three examples of where decimals are used.• Copy this overview into your workbook and complete it. Check it with a friend or your

teacher, to be sure nothing is left out.

Language of mathsascending decimal decimal place decimal pointdescending estimate fraction hundredthplace value recurring decimal round terminating decimaltenth thousandth

1 Use the words ‘ascending’ and ‘descending’ in a non-mathematical way.

2 ‘The bushfire decimated the possum population of the forest.’ Look up the meaning of the word ‘decimate’.

3 What is a recurring dream or a recurring back pain? What does ‘recurring’ mean?

4 What happens when a train terminates at a station?

5 What is a ‘decimetre’?

6 Find examples of words beginning with ‘deci’ or ‘deca’ meaning ten of something.

Worksheet 7-09

Decimals crossword

DECIMALS

Operations

Problems

Recurringdecimals

Rounding

Place value

Converting common

fractions to decimals


Chapter 7 Review Topic testChapter 7

1 Write each of these in decimal form:

a 4 × 10 + 3 × + 6 × b 5 × 100 + 2 × 10 + 7 × + 3 ×

c twelve thousandths d 8 + 6 × + 9 ×

2 Read the story below and write the number in each part with a decimal point so that the story makes sense.a Angelo and Loren decided to go to the motorbike races. They caught the train at

12:45pm after walking 123 km to the station.b They gave the station master a $1000 notec and received $460 change.d The train trip took 15 hours.e Angelo said that each bike weighed 1400 kg.f There were 4500000 people at the race meetingg who paid a total of $50850000 in entrance costs.h The average price of a ticket was $113.i A mechanic told Loren that the price of race fuel was 968 cents per litre.

3 Draw a decimal number line for each of these sets of decimals and mark them in their correct locations.a 1.6 1.7 1.9 2.0 2.2 2.3 2.4b 4.00 4.02 4.03 4.06 4.08 4.11c 0.63 0.64 0.67 0.69 0.71d 8.0 8.02 8.06 8.1 8.12 8.16 8.2

4 Arrange these numbers in order, from largest to smallest:a 34.98 56.86 3.998 50.141 340 34.89 3.099b 0.136 0.86 0.652 0.662 0.23 1.006 0.086 8.6c 0.556 0.559 0.595 0.565d 1.015 1.293 1.1015 1.239e 16.409 16.087 16.009 16.49


a b c d e

6 Write each of these as a common fraction or mixed number:a 0.5 b 0.89 c 0.09 d 0.444 e 4.051

7 Find the value of each of these:a 12.35 + 4.53 + 0.56 + 3.125 + 24.7 + 20.09b 214.33 − 109.84c 0.568 + 23 + 4.027 − 16.28 + 0.65d 1600.8 − 562.9e 1453.6 + 1287.31 − 2344.4f 7 + 7.7 + 77.77 + 777.777 + 7777.7

Ex 7-01

110------ 1

100--------- 1

10------ 1

1000------------

1100--------- 1

1000------------

Ex 7-02

Ex 7-03

Ex 7-03

Ex 7-04

410------ 13

100--------- 7

100--------- 111

1000------------ 45

1000------------

Ex 7-04

Ex 7-05


g 204.9 + 23.24 + 0.015h 363.2 − 120.49i 0.14 + 0.375 + 0.0042j 16.95 + 28.2 − 12.4k 9.23 − 6.851l 16.51 + 9.003 + 125 + 0.9m 16.821 − 5.84

8 Find the answer to each of these:a 2.75 × 6 b 0.5 ×1.2 c 12.23 × 4d 6.1 × 1.2 e 0.92 × 10 f 100 × 6.7g 3.25 × 0.41 h 0.05 × 0.02 i 4.67 × 1.1

9 Find the answer to each of these:a 762.4 ÷ 2 b 12.5 ÷ 0.5 c 97.6 ÷ 10d 2.75 ÷ 100 e 12.72 ÷ 0.4 f 6.9 ÷ 0.03

10 The Liverpool Women’s Cricket Club was having a pizza night. They ordered 16 Super Supreme pizzas at $13.70 each and 10 Hawaiian pizzas at $12.10 each. How much will the club need to spend?

11 Maria saved $90 to go to a rock concert. Her return fare cost $5.60, her concert ticket cost $48.95, the program cost $11.00 and food cost $8.70. She did not have enough to buy the band’s latest compact disc (priced $24.00) after the concert. How much did she need to borrow from her friend Sam to buy the disc?

12 Rae bought 1200 bricks for $465.70. How much did one brick cost?

13 Change to decimals:

a b c d 6

14 Round each of these numbers to the place value shown:a 456.8 to the nearest ten b 125.84 to the nearest unitc 2345.876 to two decimal places d 3.8967 to the nearest tenthe 78 654.056 to the nearest thousand f 4821.623 to the nearest tenthg 678.439 to one decimal place h 102.007 to the nearest hundredi 102.007 to the nearest hundredth j 26.046 to two decimal places

Ex 7-08

Ex 7-11

Ex 7-12

Ex 7-12

Ex 7-12

Ex 7-1345--- 3

8--- 5

9--- 2

3---

Ex 7-15

Documents

Number Decimals - Wikispacesmrgrenfellsmaths.wikispaces.com/file/view/chapter07.pdfrevise place value and order for decimals write a (terminating) decimal as a common fraction add