Yr 9 New Century Maths Chapter 01- Exploring Numbers

Number

Exploring numbers

1

Working with numbers is an important part of our lives. We use numbers when we make calculations or when we shop. Numbers such as fractions are used in recipes and for labelling shoe sizes. Numbers are also used to record temperatures below and above zero. Numbers are used to store and pass on information, to solve problems and to help us in our daily lives.

01_NC_Maths_9_Stages_5.2/5.3 Page 2 Friday, February 6, 2004 2:07 PM

■ define a rational number■ convert between fractions, decimals and percentages and order them■ identify significant figures and round numbers to a specified number of

decimal places and significant figures■ estimate answers and use the language of estimation■ write recurring decimals in fraction form■ find a fraction or percentage of a quantity■ increase and decrease a quantity by a given percentage■ express a quantity as a fraction or percentage of another■ apply the unitary method to fraction and percentage problems

■ integer An integer is any positive or negative whole number or zero. Any of the whole numbers …, −3, −2, −1, 0, 1, 2, 3, … are integers.

■ fraction A fraction is a part of something. The fraction means ‘twoparts out of 7’. ‘Fraction’ comes from Latin (a language used by the ancient Romans). The Latin word fractus means ‘break’.

■ percentage A fraction out of a hundred. It has the symbol %. Percentages are used in many different applications, such as interest rates, unemployment rates, and the increase or decrease in the value of shares on the stockmarket.

■ unitary method This method involves finding the value of one part, or one item, when given the value of several parts or items.

■ rational number A rational number is any number that can be written in the form , where a and b are integers. So , 0.6 and 37% are all examples of rational numbers.

A capacity crowd of 110 000 people, an Australian television audience of over 10 000 000 and 2.5 billion viewers around the world watched the Sydney 2000 Olympic Opening Ceremony. How precise are these figures?

In this chapter you will:

Wordbank

27---

ab-- 3

5---

Think!

EXP LOR ING NUMBERS 3

CHAPTER 1


4

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

The magic of numbers

Numbers have always fascinated people. Numbers are used in everyday life, they are used to pass on information and ideas, and they are also used in puzzles and problems. There are odd numbers, even numbers, square numbers, triangular numbers, palindromic numbers, and many more types of numbers.

1 Answer the following:a 6 + 4 × 3 b 12 ÷ 4 + 5 × 3 c 24 ÷ 6 − 18 ÷ 3d 60 ÷ 5 ÷ 2 e 48 ÷ 3 × 4 f 44 − 3 × 7

2 Find the answers to the following, rounded to the nearest 5 cents:a the cost of 42 litres of petrol at $1.07 per litreb the cost of 18 kg of potatoes at 99 cents per kilogramc the cost of 23 pencils at 35 cents each.

3 Simplify each of the following fractions:

a b c d

4 Express each of these as a percentage:

a b 0.7 c 0.08 d 1 e 0.355

5 Express each of these as a decimal:

a b 15% c 1 d e 2.5%

6 Find:a 10% of $84 b 2% of 800 c 15% of 120

d 4% of 300 e 5% of 560 f 2 % of 400

35100--------- 16

40------ 135

120--------- 18

15------

15--- 1

2---

310------ 3

4--- 13

20------

12---

Start up

Worksheet 1-01

Brainstarters 1

Example 1

Change 347 into a palindromic number.

SolutionTo change any number into a palindromic number:• reverse the digits• add the new number to the original number• repeat the previous steps until a palindromic number is formed.

Reverse the digits: 743Add the new number to the original number: + 347

= 1090

Reverse the digits: 0901Add this to 1090: + 1090

= 1991

The number 347 has been changed into the palindromic number 1991.


EXP LOR ING NUMBERS

5

CHAPTER 1

1 a Copy and complete:2 + 4 =

2 + 4 + 6 =2 + 4 + 6 + 8 =

2 + 4 + 6 + 8 + 10 =2 + 4 + 6 + 8 + 10 + 12 =

b Find a quick way to obtain the totals you found in part a. Explain your quick method.c Find the sum of the first:

i 8 even numbers ii 10 even numbersiii 18 even numbers iv 50 even numbers

2 a Copy and complete this pattern:1 + 3 =

1 + 3 + 5 =1 + 3 + 5 + 7 =

�

1 + 3 + … + 15 =b What is the pattern for adding the odd numbers?c Find the sum of the first:

i 10 odd numbers ii 15 odd numbersiii 40 odd numbers iv 100 odd numbers.

3 a Write down the first ten triangular numbers.b Write down the first ten square numbers.c Write down the first ten cube numbers.d Which number is both a triangular and a square number?e Which number is both a square and a cube number?f What is the result if you add any two consecutive triangular numbers?g What is the result if you halve the product of any two consecutive whole numbers?

4 The first five Fibonacci numbers are 1, 1, 2, 3 and 5.a How are the Fibonacci numbers formed?b Write down the next 10 Fibonacci numbers, ending with 610.c Every third Fibonacci number is divisible by 2. Which Fibonacci numbers are divisible by:

i 3? ii 5? iii 8? iv 13? v 21?

5 a Divide the sum of the first ten consecutive Fibonacci numbers by 11. What is the remainder?

b Divide the sum of the ten consecutive Fibonacci numbers after 3 by 11. What is the remainder?

c Divide the sum of any ten consecutive Fibonacci numbers by 11. What is the remainder?d What do the results from parts a, b and c show?

6 Which Fibonacci number is also:a a square number? b a cube number?

7 The German mathematician Christian Goldbach claimed in 1742 that:i Every even number greater than 2 can be writen as the sum of two prime numbers.ii Every odd number greater than 7 can be written as the sum of three prime numbers.

a Write down the first 24 prime numbers, ending with 97.b Test the first claim on the even numbers 10, 24, 90 and 122.c Test the second claim on the odd numbers 33, 45, 57 and 95.

Exercise 1-01Worksheet

1-02Find this number!


6 NEW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

8 Two primes that differ by 2, such as 11 and 13, are called twin primes. Find all eight pairs of twin primes that are less than 100.

9 A perfect number is a number that is the sum of its factors (excluding itself). For example, 6 is a perfect number because its factors are 1, 2, 3 and 6, and 1 + 2 + 3 = 6.a There is another perfect number below 50. Find it.b Verify that 496 is also a perfect number.

10 Palindromic numbers are numbers that read the same backwards as forwards. For example 3443 and 23 345 154 332 are palindromic numbers.a What is the smallest palindromic number larger than:

i 1000? ii 1000 000?b What is the largest palindromic number less than:

i 1000? ii 1000 000?c What is the only palindromic year this century?

11 Construct a palindromic number from each of these:a 36 b 127 c 79

12 Figurate numbers (or polygonal numbers) are whole numbers named after geometric figures.a Draw a diagram showing the triangular numbers

1, 3, 6, 10, 15 and 21.b Draw a diagram showing the square numbers

1, 4, 9, 16 and 25.c The diagram on the right shows the pentagonal

numbers 1, 5, 12 and 22. What are the next two pentagonal numbers?

d Write the first five hexagonal numbers (draw a diagram if necessary).

Example 1

Spreadsheet 1-01

Palindromes

The geniusKarl Friedrich Gauss, the son of poor parents, was born in Brunswick, Germany, on 30 April 1777. Evidence of Gauss’ genius appeared before he was three years old when, watching his father making out the payroll, he told his father that the figures were wrong.

Gauss started school at seven, but it was another two years before he showed his extraordinary ability in mathematics. He was admitted to a class in arithmetic. As none of the students had heard of arithmetic progressions, the teacher gave them a long problem that would take

some time to complete. The problem was of the following type: 1 + 2 + 3 + … + 100.

The teacher had barely finished giving the question when Gauss laid his slate on the table, indicating that he already had the answer. At the end of the lesson, the teacher looked at the answers and found that Gauss, the youngest student, was the only one with the correct answer. This proved to be very favourable for Gauss, and by the age of 14 he had come to the attention of the Duke of Brunswick. The Duke was very impressed by Gauss on meeting him, and gave him an assurance that his education would be continued.

What is the answer to 1 + 2 + 3 + … + 98 + 99 + 100?

Just for the record


EXP LOR ING NUMBERS 7 CHAPTER 1

Rational numbersWhen we add, subtract or multiply whole numbers, the answer is always a whole number. However, dividing any two whole numbers will not always result in an answer that is a whole number. For example, 15 ÷ 4 will not give a whole number answer. This leads us to the notion of rational numbers.

A rational number is any number that can be expressed in the form , where a and b are integers, and b ≠ 0.

So , 24, −6, 1 , , 0.2 and 27% are all rational numbers, since they can all be written in the

form , b ≠ 0.

Fractions, decimals, percentages and integers are all examples of rational numbers.

FractionsThe rational number is also a fraction , where a is the numerator and b is the denominator.

So and are proper fractions, since their numerators are less than their denominators.

So and are improper fractions.

Multiplying by multiples of 101 Examine these examples:

a 64 × 10 = 640b 37 × 20= 37 × 2 × 10 c 18 × 200= 18 × 2 × 100

= 74 × 10 = 36 × 100= 740 = 3600

d 300 × 32= 32 × 3 × 100 e 60 × 400= 6 × 10 × 4 × 100= 96 × 100 = 6 × 4 × 10 × 100= 9600 = 24 × 10 × 100

= 24 000

2 Now write answers for these:a 36 × 10 b 57 × 100 c 942 × 10 d 862 × 100e 16 × 20 f 40 × 80 g 900 × 20 h 800 × 20i 15 × 200 j 48 × 200 k 300 × 15 l 8 × 50m 11 × 400 n 500 × 12 o 75 × 20 p 500 × 40q 300 × 400 r 70 × 50 s 70 × 2000 t 3000 × 15

Skillbank 1A SkillTest 1-01

Multiplying by multiples of 10

Skillsheet 1-01

Order of operations

Skillsheet 1-02

Integers using diagrams

ab---

13--- 3

4--- 6

-7-----

ab---

ab---

Fractions in which the numerator is less than the denominator are called proper fractions.

35--- 15

16------

Fractions in which the numerator is greater than the denominator are called improper fractions.

157------ 4

3---



So 1 and 5 are mixed numerals.

So , and are equivalent fractions.

An equivalent fraction can be obtained by multiplying or dividing the numerator and the denominator by the same number.

A mixed numeral is made up of a whole number and a fraction.

34--- 3

8---

Fractions that have equal value are called equivalent fractions.

34--- 6

8--- 12

16------

Example 2

What number could be placed in the box to make a fraction between 3 and 4?

SolutionMethod A (trial and error):

Try 13: = 13 ÷ 5

= 2 which is smaller than 3.

Try 14: = 2 which is also smaller than 3.

Try 15: = 3 which is still not big enough.

Try 16: = 16 ÷ 5

= 3 which is greater than 3 and less than 4.

∴ The missing number could be 16.

(By trial and error, we find that the numbers 17, 18 and 19 could also be placed in the box.)

Method B:

The number of fifths in 3= 3 × 5

= 15

The number of fifths in 4= 4 × 5

= 20

∴ The number in the box must be between 15 and 20,so the number in the box could be 16, 17, 18 or 19.

5------

135------

35---

145------ 4

5---

155------

165------

15---

1 Change the following to improper fractions.

a 1 b 3 c 2 d 5 e 7 f 2

2 Change the following to mixed numerals.

a b c d e f

12--- 4

5--- 2

3--- 7

8--- 3

4--- 3

8---

72--- 9

5--- 9

4--- 22

3------ 69

11------ 47

6------

Exercise 1-02



Comparing fractionsWe can compare the sizes of fractions if they have a common (the same) denominator.

� (since the denominators are the same and 4 � 1)

A common denominator can be found by multiplying the denominators of the fractions together. To rewrite two fractions with a common denominator, multiply each fraction by the other fraction’s denominator.

3 Complete these pairs of fractions:

a = b = c = d = e =

f = g = h = i = j =

4 Simplify:

a b − c d e − f −

5 What number could be placed in the box so that:

a has a value between 6 and 7? b has a value between 3 and 4?

c has a value between 2 and 3? d has a value between 10 and 12?

6 a If has a value between 5 and 6, the number in the box could be:

A 28 B 25 C 15 D 27

b If has a value between 3 and 4, the number in the box could be:

A 30 B 36 C 32 D 33

c If has a value between 4 and 5, the number in the box could be:

A 14 B 22 C 30 D 38

d If has a value between 2 and 3, the number in the box could be:

A 4 B 10 C 6 D 8

e If has a value between 5 and 6, the number in the box could be:

A 5 B 3 C 4 D 2

f If has a value between 2 and 3, the number in the box could be:

A 10 B 16 C 14 D 12

7 What number could be placed in the box so that:

a has a value between 3 and 4? b has a value between 2 and 3?

c has a value between 4 and 5? d has a value between 3 and 5?

78---

16------ 2

5---

10------ 2

3--- 10------ 27------ 3

7---

36------ 11

12------

10------ 72

80------ 48

56------

7------

7------ 30

35------ 90

100--------- 9------ 7------ 21

27------

912------ 30

50------ 36

44------ 100

300--------- 45

55------ 96

144---------

5------

3------

8------

4------

4------

9------

7------

15------

21------

28------

17------ 23------

49------ 28------

Example 2

45--- 1

5---



Example 3

Arrange the fractions , , in ascending order.

SolutionA common denominator is 60.

Ascending order: , , .

23--- 3

4--- 2

5---

∴ = , = and =

× 20 × 15 × 12

23--- 40

60------ 3

4--- 45

60------ 2

5--- 24

60------

× 20 × 15 × 1225--- 2

3--- 3

4---

1 Write �, � or = between the fractions in each of these pairs to make each of the following a true statement.

a b c d

e f g h

2 Arrange each of the following sets of fractions in ascending order.

a , , b , , c , , ,

3 Arrange each of the following sets of fractions in descending order.

a , , b , , c , , ,

35--- 7

10------ 7

9--- 7

11------ 11

12------ 7

8--- 4

7--- 2

3---

12--- 4

7--- 5

8--- 3

5--- 6

7--- 3

4--- 5

6--- 3

4---

56--- 4

5--- 2

3--- 4

5--- 3

4--- 5

7--- 1

2--- 1

3--- 3

5--- 4

7---

35--- 2

3--- 3

4--- 5

8--- 2

3--- 5

7--- 2

3--- 4

5--- 1

2--- 5

11------

Exercise 1-03

Example 3

Negative numbers in today’s worldNegative numbers are used in many ways in today’s world. For example, negative numbers are used by the stock exchange to indicate a downward change in the value of shares. So, if the value of shares drops from $9.00 to $5.03, the fall is shown in the newspaper as −3.97 and the percentage change is recorded as −44%. Similarly, if the share price increases from $2.18 to $4.25, the rise is shown as +2.07 and there is a percentage change of +95%.

Negative numbers are also used to record temperatures. If the temperature is 10°C below zero, we write −10°C.1 What is the lowest recorded

temperature?2 What is the highest recorded

temperature?3 What is meant by absolute zero

when recording temperature?

Just for the record

Share prices are displayed on this electronic board at the Australian Stock Exchange.



Operations with fractionsSimplifying fractionsWhen the numerator and denominator of a fraction cannot be divided exactly by the same whole number, the fraction is in simplest form or lowest form.

Adding and subtracting fractionsWhen adding or subtracting fractions, the denominators must be the same.

If the denominators are not the same, then the fractions need to be converted to equivalent fractions that do have the same denominator. A quick way to find a common denominator of two or more fractions is to multiply their denominators together.

Example 4

Simplify .

Solution=

=

1520------

1520------ 15 5÷

20 5÷---------------

35---

Example 5

a Find + . b Find − .

Solutiona + = + b − = −

= =

= 1

a Find 2 + 3 . b Find 4 − 2 . c Find 5 − 2 .

Solutiona 2 + 3 = 5 + + b 4 − 2 = 4 − 2 c 5 − 2 = 5 − 2

= 5 + + = 2 = 3 −

= 5 + = 2 −

= 5 + 1 = 2

= 6

25--- 3

4--- 7

8--- 2

3---

25--- 3

4--- 8

20------ 15

20------ 7

8--- 2

3--- 21

24------ 16

24------

2320------ 5

24------

320------

Example 6

34--- 1

3--- 3

5--- 1

3--- 1

2--- 3

5---

34--- 1

3--- 3

4--- 1

3--- 3

5--- 1

3--- 9

15------ 5

15------ 1

2--- 3

5--- 5

10------ 6

10------

912------ 4

12------ 4

15------ 1

10------

1312------ 10

10------ 1

10------

112------ 9

10------

112------



Multiplying fractionsWhen multiplying fractions, multiply the numerators and multiply the denominators. When multiplying mixed numerals, first change them to improper fractions.

1 Simplify each of the following:

a b c d e f


a + b + c + d + e +

3 Simplify:

a − b − c − d − e −

4 Answer the following:

a 3 + 2 b 2 + 2 c 13 − 2 d 5 − 1 e 7 − 1

f 2 + 1 g 2 − 1 h 6 + 4 i 15 − 4 j 4 − 2

5 a Jenny buys a roll of contact to cover her books. She uses of it to cover her project book

and to cover her science book. What fraction of the roll of contact is left?

b The sum of two numbers is 4 . If the first number is 1 , what is the second number?

c If you buy 8 m of material and then use 6 m for curtains, how much material is left?

d Sok spent of the day working, of the day sleeping, of the day eating, and the rest

of the day relaxing. What fraction of the day did he spend relaxing?

e Clare recorded two TV programs. The first program used of the video tape and the

second program used of it. What fraction of the tape was not used?

1830------ 24

36------ 9

27------ 50

60------ 48

56------ 5

10------

34--- 2

3--- 4

7--- 1

2--- 4

5--- 2

3--- 7

8--- 2

3--- 3

8--- 7

10------

12--- 5

8--- 7

10------ 1

3--- 2

3--- 4

5--- 3

4--- 5

12------ 3

5--- 2

3---

13--- 2

5--- 3

5--- 3

4--- 7

10------ 13

20------ 3

5--- 1

4--- 2

3--- 1

2---

23--- 4

5--- 1

2--- 4

5--- 3

4--- 1

10------ 2

7--- 1

2--- 3

4---

12---

13---

34--- 2

3---

34--- 5

8---

13--- 4

8--- 1

12------

13---

25---

Exercise 1-04Example 4

Example 5

Example 6

Skillsheet 1-03

Simplifying fractions

Worksheet 1-06

Magic squares

Skillsheet 1-04

Improper fractions and

mixed numerals

SkillBuilder 2-15–2-18

Mixed fractions

Example 7

Find:

Solution

Find:

a 1 × b 2 × 1

a × b 5 × c of 12 L

a × = b 5 × = × =

= 7

c of 12 L= × 12 L= × L

= L

= 7 L

35--- 2

7--- 3

2--- 3

5---

35--- 2

7--- 6

35------ 3

2--- 5

1--- 3

2--- 15

2------

12---

35--- 3

5--- 3

5--- 12

1------

365------

15---

Example 8

23--- 5

8--- 2

3--- 3

5---



Reciprocals

The reciprocal of 2 is since 2 × = 1.

The reciprocal of is since × = = 1.

To obtain the reciprocal of a number, write the number as a fraction and turn it upside down.

Solutiona 1 × = × b 2 × 1 = ×

= =

= 1 = 4

23--- 5

8--- 5

3--- 5

8--- 2

3--- 3

5--- 8

3--- 8

5---

2524------ 64

15------

124------ 4

15------


a × b × c × d × e ×

2 Find:

a × 8 b × 6 c 7 × d × 24 e 10 ×

3 Find:

a of 60 b of 5 m c of $76 d of 18 kg

e of $85 f of 20 km g of $15 h of 33 m

4 Simplify:

a 1 × b 4 × 2 c 1 × 2 d 5 × 1 e

f 5 × 5 g 3 × 2 h 1 × i 3 × 2 j 2 × 1

5 a At a football match, 12 players are each given three-quarters of an orange to eat. How many oranges are eaten?

b Ken worked 5 hours yesterday and of that today. How long did Ken work today?

c The cost of a car insurance policy for 6 months is of the annual premium.

If the annual premium is $500, find the cost for 6 months.

35--- 2

5--- 1

3--- 3

4--- 5

8--- 7

10------ 4

5--- 3

11------ 5

6--- 12

20------

35--- 3

4--- 2

3--- 5

8--- 3

8---

45--- 3

4--- 1

8--- 2

3---

710------ 5

8--- 3

10------ 5

6---

35--- 2

3--- 3

8--- 1

2--- 5

6--- 2

3--- 3

4--- 13

4---( )

2

57--- 5

16------ 2

3--- 3

7--- 3

10------ 5

7--- 5

8--- 1

7---

12--- 3

4---

58---


SkillBuilder 2-19

Multiplying fractions

SkillBuilder 2-20

Multiplying and simplifying fractions

Example 8

SkillBuilder 2-24–2-25Multiplying

mixed fractions

The product of a number and its reciprocal is 1.

12--- 1

2---

23--- 3

2--- 2

3--- 3

2--- 6

6---



Dividing fractionsTo divide by a fraction, multiply by its reciprocal.

When dividing mixed numerals, first change them to improper fractions.

Example 9

Find ÷ .

Solution

÷ = ×

=

Find:

a ÷ 1 b 5 ÷ 3 c 2 ÷ 1

Solution

a ÷ 1 = ÷

= ×

=

=

b 5 ÷ 3 = ÷ 3

= ×

=

= 1

= 1

c 2 ÷ 1 = ÷

= ×

=

= 1

38--- 2

3---

38--- 2

3--- 3

8--- 3

2---

916------

Example 10

34--- 1

2--- 1

4--- 1

2--- 3

5---

34--- 1

2--- 3

4--- 3

2---

34--- 2

3---

612------

12---

14--- 21

4------

214------ 1

3---

2112------

912------

34---

12--- 3

5--- 5

2--- 8

5---

52--- 5

8---

2516------

916------

1 Find the reciprocal of:

a b −7 c d 3 e −2


a ÷ b ÷ c ÷ d ÷ 3 e ÷

3 Simplify:

a 2 ÷ b 5 ÷ 1 c 3 ÷ 2 d 12 ÷ e 5 ÷ 2

f 3 ÷ 1 g 5 ÷ 1 h 4 ÷ 2 i 4 ÷ 5 j 2 ÷ 1

4 a It takes of a can of soft drink to fill a large glass. How many glasses can be filled from

20 cans of drink?

b A car is sold for 2 times its cost price. Find the cost price if the car was sold for $3000.

c Ilse’s stride, when walking, is metre. Her running stride is 1 metres. Find how many

more strides Ilse takes over a distance of 100 metres when she walks rather than runs.

34--- 12

5------ 3

8--- 1

2---

45--- 2

3--- 7

8--- 3

4--- 2

5--- 2

3--- 3

5--- 5

24------ 2

3---

34--- 1

3--- 1

2--- 3

5--- 1

2--- 1

4--- 1

2--- 1

10------

13--- 2

3--- 3

5--- 1

4--- 3

5--- 1

3--- 13

20------ 2

3--- 7

10------

45---

12---

23--- 1

4---

Exercise 1-06

Example 9

Example 10


Dividing fractions



Expressing quantities as fractionsQuantities can be compared by expressing one quantity as a fraction of another quantity.

d A weightlifter is able to lift a mass of 210 kg, which is 2 times the weightlifter’s mass.Find the mass of the weightlifter.

e A driver-education course requires 20 hours of classroom instruction. How many lessons

are required if each lesson lasts 1 hours?

12---

14---

Example 11

Solutiona What fraction is 8 of 50? b What fraction is 15 minutes of 2 hours?

a Fraction =

=

b 15 minutes out of 2 hours=

=

=

850------

425------

15 minutes2 hours

--------------------------

15120---------

18---

1 What fraction is:a 9 of 27? b 24 of 80? c 10 of 25? d 36 of 48?e 12 of 30? f 27 of 54? g 42 of 70? h 55 of 85?

2 What fraction is:a 5 min of 1 h? b 10 cm of 1 m? c $1.50 of $5?d 300 mL of 1 L? e 20 min of 1 h? f 200 m of 1 km?g 50 m of 1 km? h 300 mL of 2 L? i 40 min of 1 h?

3 a In a test consisting of 50 questions, Matthew made 15 errors. What fraction did Matthew have correct?


Worksheet1-05

Back-to-front problems



DecimalsDecimal numbers are based on powers of 10. The decimal point separates the whole number part from the fractional part of a decimal number.

Decimals are also rational numbers because they can be expressed as fractions with denominators of

10, 100, 1000, and so on. For example, 75.436 = 7 × 10 + 5 × 1 + 4 × + 3 × + 6 × .

Decimal placesThe term ‘decimal places’ refers to the number of figures (or digits) after the decimal point in a decimal number. The number 75.436 has three decimal places.

b Christine had $80 and spent $35. What fraction of the $80 did Christine spend?c In a class of 28 students, there are 12 boys. What fraction of the class are boys?d A football team won 15 of the 24 games it played. What fraction of games did the

team win?e Disha’s mobile phone bill for last month was $210, of which $90 was for SMS messages.

What fraction of Disha’s bill was for SMS messages?

4 In a school there are 950 students, of which 400 are boys.a What fraction of the students are boys? b What fraction of the students are girls?

5 Angela went to a restaurant with a group of friends. The total bill was $450. Her friends contributed $360 to the bill. What fraction of the bill did Angela pay?

6 A total of 300 vehicles were observed to pass a checkpoint during a traffic survey. Of these, 55 were Fords and 50 were Holdens. What fraction of the vehicles were neither Fords nor Holdens?

110------ 1

100--------- 1

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

Worksheet 1-03

Where’s the point?

= ‘tenth’ = one decimal place (0._)

= ‘hundredth’ = two decimal places (0._ _)

= ‘thousandth’ = three decimal places (0._ _ _)

10------

100---------

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

Calculating by collecting like terms1 Examine these examples:

a 87 × 35 + 13 × 35 = ? Think: 87 × 35 = 87 lots of 3513 × 35 = 13 lots of 35

There are (87 + 13) lots of 35∴ 87 × 35 + 13 × 35 = (87 + 13) × 35

= 100 × 35= 3500

Skillbank 1B

SkillTest 1-02

Collecting like terms



ApproximationWhen counting the number of people in a room, for example, we normally give an exact answer. However, sometimes we need to use approximations, especially when measurement is involved.

For instance, if Stacey jumps 4.276 m in a long-jump event, we may approximate the distance as 4.3 m or 4.28 m (which are both very close to 4.276 m), depending on the level of accuracy required.

Rounding using decimal places

The function or mode on a calculator may be used to round an answer to a given number of decimal places.

b 7 × 84 + 3 × 84 = 7 × 84 + 3 × 84= (7 + 3) × 84= 10 × 84= 840

2 Find the answers for the following:a 79 × 56 + 21 × 56 b 92 × 117 + 8 × 117 c 6 × 39 + 4 × 39d 11 × 77 + 77 × 89 e 64 × 83 + 17 × 64 f 91 × 352 + 9 × 352g 86 × 13 + 14 × 13 h 75 × 93 + 25 × 93 i 8 × 793 + 793 × 2

3 Examine these examples:a 123 × 17 − 23 × 17 = 123 × 17 − 23 × 17

= (123 − 23) × 17= 100 × 17= 1700

b 17 × 87 − 7 × 87 = 17 × 87 − 7 × 87= (17 − 7) × 87= 10 × 87= 870

4 Find the answers for the following:a 12 × 63 − 2 × 63 b 142 × 39 − 42 × 39 c 1013 × 16 − 13 × 16d 172 × 38 − 72 × 38 e 13 × 157 − 3 × 157 f 27 × 102 − 27 × 2g 68 × 1019 − 68 × 19 h 84 × 14 − 4 × 84 i 378 × 116 − 378 × 16

Example 12

Round 54.3584 to:a three decimal places b two decimal places c one decimal place.

Solutiona 54.3584 = 54.358, rounded to three decimal placesb = 54.36, rounded to two decimal placesc = 54.4, rounded to one decimal place

FIX



Rounding using significant figuresAnother way of rounding is to give the most relevant or important digits of a number. For example, a crowd of 89 123 at Stadium Australia, Homebush, is usually written as 89 000, which is rounded to the nearest thousand, or correct to two significant figures.

1 Round each of the following to the number of decimal places indicated in the brackets.a 39.056 (2) b 100.534 (1) c 7.3777 (3)d 0.0051 (1) e 0.0515 (2) f 15.086 (0)g 245.67 (0) h 2.198 (1) i 42.9995 (3)

2 Use your calculator to evaluate each of the following. Give your answer to the number of decimal places indicated in the brackets in each case.

a 14.5 ÷ 4.2 + 9.75 (1) b (9.74 − 3.456) ÷ 5.1 (4)

c 6.78 × 4.32 + 8.96 ÷ 2.4 (2) d (3)

e 24.31 ÷ 6.2 (4) f 5.73 × 2.8 + 19.7 (1)

g 24.37 ÷ 8.9 ÷ 2.1 (3) h 33 ÷ (15 + 12) (2)

i (1) j (6.4)2 ÷ 3.7 (0)

k 19.8 − 42.3 ÷ 7 (4) l (1)

m 42.5 ÷ 0.18 × 5.6 (3) n (42.3 − 15.7) ÷ (18.7 + 2.9) (2)

45 37–97 24+------------------

94 37+25 18+------------------

64.39.2 15 3.1÷+---------------------------------


Spreadsheet 1-02

Rounding decimals



We use significant figures to indicate the size of a number by writing it to the nearest 1, 10, 100,

1000, …, or the nearest , , , … .

So, in the example above, 89 123 (which has been written to five significant figures) has been rounded to 89 000 (which has two significant figures).

When rounding to significant figures:

the first significant figure in a number is the first non-zero digit.– The first significant figure in 3 756 000 is 3– The first significant figure in 0.000 217 is 2

zeros at the end of a whole number or at the beginning of a decimal are not significant. They are necessary place holders.– The red zeros in 1 340 000 and 0.007 032 are not significant.

zeros between non-zero digits or zeros at the end of a decimal are significant.– The red zeros in 3 608000 and 0.010 50 are significant.

110------ 1

100--------- 1

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

Example 13

Write each of the following numbers rounded to three significant figures.a 47.658 b 91.49 c 273 200

Solutiona 47.658 ≈ 47.7 b 91.49 ≈ 91.5 c 273200 ≈ 273 000

(The zeros are not significant but are placeholders necessary for showing the place values of the 2, 7 and 3.)

Write each of the following numbers correct to one significant figure.a 0.008 28 b 0.0035 c 0.998

Solutiona 0.00828 ≈ 0.008 b 0.0035 ≈ 0.004 c 0.998 ≈ 1

(The zeros between the decimal point and the first non-zero digit are not significant. They are placeholders.)

To how many significant figures has each of the following numbers been written?a 63.70 b 0.003 05 c 7600

Solutiona The zero after 7 is significant.

∴ 63.70 has been written to four significant figures.b The first significant figure is 3, and the zero between 3 and 5 is significant.

∴ 0.003 05 has three significant figures.c The zeros after 6 are not significant.

∴ 7600 has been written to two significant figures.

Example 14

Example 15



1 Round each of the following to the number of significant figures indicated in the brackets. a 37.6 (2) b 9430 (1) c 68.39 (3)d 2.813 (1) e 15.99 (3) f 63 500 (2)g 1769 000 (2) h 389 764 (4) i 189 371 (1)

2 Round each of the following to the number of significant figures indicated in the brackets. a 0.0637 (1) b 0.703 (2) c 0.8455 (2)d 0.000 017 (1) e 0.087 62 (3) f 0.038 71 (2)g 0.7995 (3) h 0.000 04 (4) i 0.95 (1)

3 To how many significant figures has each of the following numbers been written? a 457 b 0.23 c 15 000 d 4.0004e 0.0005 f 5000 g 0.002 07 h 89 072i 0.040 j 76 000 000 k 0.000 328 l 169.320

4 Round each of the following to the number of significant figures indicated in the brackets.a 7.478 (2) b 5712 (3) c 367 (1)d 0.007 66 (4) e 0.5067 (3) f 10 675 (2)g 1856.78 (3) h 0.000 78 (1) i 56 000 000 (1)

5 A company makes a profit of $27 846 521.78.a Approximate this amount to the nearest million and state the number of significant figures

in your answer.b Approximate the same amount in millions, to the nearest 10 million, and state the number

of significant figures in your answer.

6 a Australia’s population in 2001 was 19 387 000. To how many significant figures has this number been written?

b A total of 14 352 people attended a local football match. Express this number to three significant figures.

c The population of Sydney in 2001 was 4 085 400. Round this number to two significant figures.

7 Use your calculator to evaluate each of the following, giving your answer to the number of significant figures indicated in the brackets in each case.a 45.6 × 8.7 − 2.75 × 78.32 (2) b 15.5 − 9.87 ÷ 0.24 + 8.43 × 2.4 (1)

c (63.73 − 27.89) ÷ 5.82 (3) d (4)

e 95.34 × 4.7 − 2567.68 ÷ 5.78 (5) f (3)

g (5) h (4)

i (1) j 78.96 ÷ (23.6 + 94.7) (2)

k (3) l × 0.0715 (4)

8 Write 5002 correct to each of these numbers of significant figures:a 1 b 2 c 3 d 4

63.25 76.03+55.89 89.24–---------------------------------

15.013 5.78÷6.45 2.254×---------------------------------

3.567 12.67+( ) 9.67 4.007–( )67.23 56.873+

------------------------------------------------------------------------- 613.67 5.6002–------------------------------------

9.732 2.765+12.27 15.8×---------------------------------

10.976------------- 253

0.0076----------------+ 84.3


Example 14

Example 15


Significant digits



EstimationCalculators enable us to compute answers quickly, but errors can be made if we accidentally press the wrong key. Sometimes a calculator is not available, or we may require only a rough answer. Whether we are shopping at a supermarket, budgeting for a holiday, or sharing the cost of a meal, we can mentally calculate rough answers by estimating. We can then use the calculator if we need a more precise answer.

Note: The symbols ≈, � and � all mean ‘is approximately equal to’.

Working mathematicallyCommunicating: Significant figures1 The numerical facts in the following passage would be easier to read and remember if they

were rounded using significant figures. For example, ‘Australia has an area of 7.6 million km2’ (the number has been rounded to two significant figures). Round the other numbers in the passage and compare your answers to those of other students in the class.

Australia in 2001: Some facts and figures

Australia has an area of 7 682 300 km2. Its coastline measures 36 375 km. The population in 2001 was 19 386 740, of which 426 175 were Aboriginal or Torres Strait Islanders. The median age was 35.41. The most populous state was New South Wales, with 6 532 010 residents, of whom 4 085 378 lived in Sydney.

In 2001, there were 3 275 137 students enrolled in 9609 Australian schools, taught by 261 947 teachers. Of the 9 241 240 people in employment, 7 576 089 were wage and salary earners. Males worked on average 40.76 hours per week, earning $780.23. Females worked on average 38.14 hours per week, earning $520.57.

There were 7 012 265 households in Australia, of which 4 891 840 were owner-occupied. There were 4 037 274 households that had a computer, and 2 680 240 that had internet access. The average weekly household expenditure on food and alcoholic beverages was $127.34. There were 12 476 767 motor vehicles in Australia, and their average age was 10.47 years.

Australia is a multicultural country, with 4 507 800 of its citizens born overseas. A total of 120 888 people migrated to Australia in 2001. Tourism is one of Australia’s booming industries. In 2001, 5 061 270 overseas visitors contributed $25 205 781 450.13 to the Australian economy.

Example 16

1 Estimate the answer to 15.7 + 12.8 − 6.9

Solution15.7 + 12.8 − 6.9 ≈ 16 + 13 − 7

= 22(Actual answer is 21.6.)

2 Estimate the answer to 12.75 ÷ 0.032



Solution12.75 ÷ 0.032 = 12 750 ÷ 32 (changing 0.032 to 32)

≈ 12 000 ÷ 30

= 400

(Actual answer is 398.44 correct to to two decimal places.)

3 Estimate the answer to 57.63 × 22

Solution57.63 × 22 ≈ 60 × 20

= 1200

(Actual answer is 1267.86)

Points to remember when estimating• Round to the nearest 1, 10, 100, etc. if this is easier to work with.• First estimate your answer without using a calculator.• Then compare the magnitude (size) of your estimate (say 6000) with your calculator

result (say 6.21) to reveal whether you have made an error.

Working mathematicallyCommunicating and reasoning: Rounding numbersWork in groups of three or four to complete these questions.

1 A number has been rounded to 9.36a What is the smallest number that the original number could have been?b What is the largest number that can be rounded to 9.36?(Compare your answers with those of other students.)

2 Two students were asked to multiply 3.56 by 4.73 and express their answers to two decimal places. Their methods are shown below.Student A Student B 3.56 × 4.73 3.56 × 4.73

≈ 3.6 × 4.7 = 16.8388= 16.92 ≈ 16.84Discuss the methods used by both students and decide which is the more accurate answer. Give reasons.

1 Estimate answers to the following:a 19.7 + 32.1 b 183.7 + 97.034 c 47.62 − 18.54d 764.73 − 18.59 e 934 − 129 f 6.11 × 18.5g 53.7 × 8 h 76 × 12 i 93.7 ÷ 8.97j 753.6 ÷ 4.87 k 18.76 ÷ 8.9 l 874 ÷ 94.5

2 Estimate answers to the following:

a b 705.7 ÷ 0.0019 c 0.38 ÷ 0.215.9 11.7×2.45

------------------------


Worksheet 1-04

Estimation game



Recurring decimalsSome decimals repeat or recur. Other decimals terminate. Some decimals do neither.• Recurring decimals • Terminating decimals

0.777 777 7 … = 0.7, 0.17, 0.005, …0.1777 … = • Neither recurring nor terminating

0.171717 … = π = 3.141 592 653 …

Expressing fractions as decimalsTo change a fraction to a decimal, we divide the numerator by the denominator.

d 789 ÷ 37 × 3.9 e f

g (91.7 + 162.7) ÷ 48.1 h 13.1 + 4.8 × 9.2 i 87.3 × 9 + 108.3 × 7

j k (7.97)2 l (4.85)2 ÷ 23.756

3 Estimate which of the given answers is the correct one for each of the following:a 18 × 39

A 70.2 B 702 C 7020 D 70 200

b 5.32 × 198.2A 10.544 24 B 105.4424 C 1054.424 D 10 544.24

c 141.2 ÷ 0.21A 672.38 B 67.238 C 6.7238 D 0.672 38

d 6970 ÷ 68A 1025 B 102.5 C 10.25 D 1.025

e 215.72 ÷ 9 × 22A 0.5273 B 5.273 C 52.73 D 527.3

4 Which of the given answers is the best estimate for each of the following?a 145.7 + 93.7

A 240 B 24 C 2.4 D 2400

b 33.8 × 4.8A 1200 B 1500 C 150 D 120

c 9.475 ÷ 0.31A 300 B 3 C 3000 D 30

d (27.9 − 8.4) × 19.8A 800 B 400 C 80 D 1200

98.7 310.78 108.97–( )47.5

-------------------------------------------

34.7 56.9+24.3 11.1–---------------------------

0.7̇0.17̇0.1̇7̇

Example 17

Express as decimals:

a b

Solution0.375 0.636363 …

a = 8 3000 b = 11 7.00000

= 0.375 (a terminating decimal) = (a recurring decimal)

38--- 7

11------

38--- 7

11------

0.6̇3̇



Expressing decimals as fractions

Comparing fractions and decimalsWhen comparing fractions and decimals, change the fractions to decimals.

Example 18

Express these terminating decimals as fractions:a 0.2 b 0.34 c 0.235

Solutiona 0.2 = = b 0.34 = = c 0.235 = = 2

10------ 1

5--- 34

100--------- 17

50------ 235

1000------------ 47

200---------

Example 19

Which is larger, or 0.38?

Solution = 0.375 and 0.38 = 0.380

∴ 0.38 is larger.

38---

38---

Patterns in recurring decimalsStep 1: Set up the spreadsheet as shown below.

Step 2: Select cell D6 and drag down to copy the formula into cells D6 to D14.

Step 3: Write what you notice about the recurring decimals in column D.

Step 4: Print the spreadsheet and paste it in your book.

To save time typing, use this link to go to a partly completed spreadsheet.

A B C D E F G

1

2

3

4

5 Numerator Denominator Decimal

6 1 9 =B6/C6

7 2 9

8 3 9

9 4 9

10 5 9

11 6 9

12 7 9

13 8 9

14 9 9

15

Using technology

Skillsheet 1-05

Spreadsheets

Spreadsheet 1-03

Recurring decimals



Converting recurring decimals to fractionsTo convert a recurring decimal to a fraction, we can use an algebraic method and solve an equation.

1 Repeat the procedure described above for each of the following fraction families:a sixths b elevenths c thirteenthsd sevenths e twelfths f fourteenths.

2 Write a few sentences about your observations of patterns in recurring decimals.

1 Express these fractions as decimals and state whether they are recurring or terminating.

a b c d e

f 6 g h 4 i j

2 Express these decimals as fractions (or as mixed numerals): a 0.2 b 0.7 c 0.25 d 0.45e 0.48 f 0.005 g 0.025 h 0.305i 0.825 j 2.8 k 1.45 l 3.05m 2.065 n 0.000 21 o 4.625 p 10.01

3 Which number in each of the following pairs is larger?

a 0.52 and b and 0.615 c 0.69 and d and 0.8

4 a Arrange , 0.68, , 0.678, in ascending order.

b Arrange , , 0.34, 0.305, in descending order.

58--- 3

7--- 7

9--- 5

6--- 6

5---

45--- 5

12------ 1

3--- 5

11------ 7

20------

1120------ 5

8--- 2

3--- 7

9---

710------ 6

9--- 7

11------

825------ 1

3--- 13

40------


Example 18

Example 19

Example 20

Write each of the following as a fraction in simplest terms.

a b

Solutiona Let n =

n = 0.7777 . . . . (1)∴ 10n = 7.777 . . . . . (2) (one digit is recurring, so multiply by 10)

10n = 7.7777 [subtracting (1) from (2)]− n = 0.7777

9n = 7

∴ n = (dividing both sides by 9)

Since n = , =

0.7̇ 0.25̇3̇

0.7̇

79---

0.7̇ 0.7̇ 79---



b Let n = n = 0.2535353. . . (1) (two digits are recurring, so multiply by 100)

100n = 25.35353 . . . . (2)

100n = 25.35353 … [subtracting (1) from (2)]− n = 0.25353 …

99n = 25.1

n = (dividing both sides by 99)

= ×

=

Since n = , =

0.25̇3̇

25.199

----------

25.199

---------- 1010------

251990---------

0.25̇3̇ 0.25̇3̇ 251990---------

1 Express each of the following as a fraction in simplest form.

a b c

d e f

g h i

j k l

m n o

2 a Express as a fraction in simplest form.b Use your answer from part a to write as a fraction in simplest form.

0.2̇ 0.6̇ 0.15̇

0.38̇ 0.724̇ 0.6̇5̇

0.3̇1̇ 0.62̇4̇ 0.1̇53̇

0.8̇32̇ 0.03̇ 0.7̇5̇

1.7̇3̇ 2.02̇ 0.15̇7̇

0.3̇0.6̇


Working mathematicallyCommunicating and reasoning: Recurring decimals1 The fraction written as a recurring decimal is 0.333 … .

a Use this result to write as a recurring decimal.

b Use your calculator to convert to a recurring decimal. Write the display shown by your calculator.

c Is there a difference between the results you wrote for parts a and b? Explain.

2 What is the fraction value of 0.999 … = ? (Hint: = ).

3 a Express , , …, as decimals.

b What do you notice about the ‘ninths family’ when they are expressed as decimals.

4 Repeat the procedure used in Question 3 for the following fraction families:a sixths b elevenths c thirteenths.

13---

23---

23---

0.9̇ 13--- 0.3̇

19--- 2

9--- 9

9---



PercentagesA percentage is a type of rational number. A percentage is a fraction out of 100 and has the

symbol %. For example 45 per cent, written as 45%, means ‘45 out of 100’ or .

Converting percentages to fractions and decimals

Converting fractions and decimals to percentages

Ordering fractions, decimals and percentages

45100---------

To convert a percentage to a fraction, divide by 100 and then simplify (if possible).

To convert a percentage to a decimal, divide by 100.

1 Express these as fractions (in simplest form):a 35% b 72% c 95% d 8% e 40%f 80% g 18% h 120% i 325% j 105%k 140% l 48% m 155% n 675% o 150%

2 Express these as fractions (in simplest form):

a 12 % b 2 % c 8 % d 4 % e 10 %

f 13 % g 33 % h 5 % i 16 % j 37 %

k 66 % l % m 6 % n 1 % o 41 %

3 Express these as decimals:a 55% b 133% c 46% d 5% e 2.4%f 200% g 9.5% h 5.25% i 12% j 0.3%

k 12 % l 25 % m 8 % n 10.9% o 350%

12--- 1

2--- 1

3--- 1

4--- 1

2---

34--- 1

3--- 1

2--- 2

3--- 1

2---

23--- 1

2--- 2

3--- 1

2--- 2

3---

12--- 1

4--- 1

3---

Exercise 1-13

Spreadsheet 1-04

Equivalent fractions,

percentages and decimals

Worksheet1-07

Fractions squaresaw

To convert either a fraction or a decimal to a percentage, multiply by 100%.

Example 21

Arrange , 78%, 0.75 and in ascending order.45--- 9

11------



SolutionExpress each number as a decimal:

= 0.8 78% = 0.78 0.75 = = 0.80

Arranging the decimals in order, we have:0.75, 0.78, 0.80,

∴ The ascending order is:

0.75, 78%, ,

45--- 9

11------ 0.8̇1̇

0.8̇1̇

45--- 9

11------

1 Express these as percentages:

a b c d e

f g h i j

k l 4 m n o

2 Express these as percentages:a 0.5 b 1.86 c 0.35 d 1.7e 0.008 f 2.39 g 0.2 h 0.25

3 Express these as percentages:

a b c d e

f g 0.72 h 0.61 i j 1.3

k l m n 3.52 o

4 Which amount is the larger in each of these pairs?

a 55% and 0.57 b and 57% c 18% and

5 Which amount is the smaller in each of these pairs?

a 49% and b and 40% c 130% and

6 a Arrange , 55%, , 0.59 in ascending order.

b Arrange 72%, , 0.79, in descending order.

c Arrange 0.764, 77%, , , 7.95% in order from largest to smallest.

38--- 9

20------ 3

4--- 21

2--- 17

20------

711------ 8

25------ 1

5--- 12

20------ 11

4---

235--- 21

3--- 2

3--- 5

8---

31100--------- 4

5--- 13

20------ 7

8--- 14

25------

740------ 5

6---

235--- 52

3--- 47

8--- 29

40------

35--- 1

7---

1225------ 9

20------ 11

3---

47--- 1

2---

34--- 17

20------

1925------ 11

40------

Exercise 1-14

Skillsheet 1-06

Fractions, decimals,

percentages

SkillBuilder 4-04

Changing fractions to percentages

Example 21



Percentage of a quantity

Blood: It takes all typesThe blood groups in Australia are as follows:

1 If Australia’s population is 20 million, how many people are in each blood group?

2 Do you know what your blood group is? How would/did you find out?

O A

+ − + −

40% 9% 31% 7%

B AB

+ − + −

8% 2% 2% 1%

Just for the record

Blood donor at the Red Cross Blood Bank.

Example 22

Find 15% of $8

Solution

The key on your calculator may also be used. To find 15% of $8, press the keys

8 15 .

15% × 8 = × $8

= $1

= $1.20

or 15% × 8 = 15 ÷ 100 × 8

= $1.20

or 15% × 8 = 0.15 × 8

= $1.20

15100---------

15---

%

+ % =

Skillsheet 1-07Mental

percentages

Skillsheet 1-08Ratios

1 Find:a 20% of 70 b 143% of $800 c 25% of $15 d 110% of 18 kge 15% of 16 m f 30% of 80 L g 26% of 600 g h 20% of 950 km

2 Find:

a 2 % of $90 b 3.5% of 24 m c 6.4% of $456

d 8 % of 48 L e 112 % of 72 kg f 14.2% of $550

g 0.5% of $8000 h 4.5% of 20 km i 133 % of 12 t

12---

13--- 1

2---

13---


SkillBuilder 4-07–4-08Finding the

percentage of a quantity



Percentage increase and decreaseThe amount by which goods are measured or decreased in price is often given as a percentage.

3 A total of 92 000 people attended the state-of-origin match between NSW and Queensland at Stadium Australia. If 72% of the crowd supported NSW:a what percentage of the crowd supported Queensland?b how many people in the crowd supported NSW?

4 Amanda scored 60% in a geography test that had been marked out of 40. What was Amanda’s mark out of 40?

5 Savuth started working three years ago on a salary of $22 000 a year. She now earns 160% of her starting salary. What is Savuth’s salary now?

6 A food company advertises 50% more chips for the same price per packet of sliced potato chips. What size packet (in grams) will you now get for the price of a 70 g packet?

7 A total of 75 000 copies of a novel were sold in the first six months after its release. After a further three months, sales increased by 15%.a How many more books were sold in the later three months?b What were the total sales for the nine months?

8 House prices have risen by 4.5% over the last 12 months. What is the increase in value of a house that was bought 12 months ago for $385 000?

9 A 55 g serving of a breakfast cereal contains 8.4% dietary fibre. What is the amount of dietary fibre in the 55 g serving?

Example 23

Solution

a Increase $80 by 15% b Decrease $126 by 12%

a Increase= 15% of $80 = 0.15 × 80 = $12

∴ The new amount= $80 + $12 = $92

or The new amount= 115% of $80(100% + 15% = 115%)

= 1.15 × 80 = $92

b Decrease= 12% of $126 = 0.12 × 126 = $15.12

∴ The new amount= $126 − $15.12 = $110.88

or The new amount= 88% of $126(100% − 12% = 88%)

= 0.88 × 126 = $110.88

1 Increase: a 60 kg by 20% b 40 km by 12% c 72 by 15%d $2400 by 2.5% e 80 kg by 7% f 56 L by 14%

2 Decrease:a 90 L by 8% b $650 by 12.5% c $165 by 33 %

d 5.2 km by 50% e $142 by 25% f $950 by 8.75%

13---




Expressing quantities as percentagesQuantities can be compared by expressing one quantity as a percentage of another quantity.

3 Meredith’s weekly wage of $765 is increased by 9.25%. Find her new weekly wage.

4 Matthew buys a BMX bike for $365 and sells it, making a 20% profit.a How much profit did Matthew make?b For what price did he sell the bike?

5 A clothing store offers discounts of 40% off marked prices on its winter stock. Find the discount price of a jumper marked at $170.

6 A car dealer is offering new-season sales. What will you pay for a car marked at $18 700 if a discount of 15% is given?

7 A CD player usually sells for $295, but a discount warehouse offers it at a 12% discount. Calculate the warehouse’s price for the item.

8 Due to heavy rain over a period of time, the price of fresh fruit and vegetables increases by 15%. How much will cauliflowers cost, if the price before the increase was $2.99 each? (Round your answer to the nearest cent.)

9 Maria’s annual salary of $55 680 is increased by 6%. What is her new annual salary?

10 A store owner buys a TV for $600 from a supplier. The store owner applies a mark-up of 70% to calculate the retail price, but then offers a discount of 30%.a Find the price of the TV after the 70% mark-up.b Find the price of the TV after the 30% discount is given.c What profit did the store owner make on the TV?

11 A hardware store offers a trade discount of 15% to plumbers, builders and tilers.a If goods to the value of $480 are bought by a builder, calculate the discount price.b If goods are paid for in cash, a second discount of 4% is given on the discount price. What

is the final amount paid by the builder if she pays cash?

12 Jumpers priced at $165 are reduced by 15% at a July sale. At an end-of-season sale, this price is reduced by a further 40%. What is the final price of the jumpers?

SkillBuilder 4-18

Percentage increase or decrease

Spreadsheet 1-05

Percentage increase and

decrease

SkillBuilder 4-19

An application of percentage increase/decrease

Example 24

1 Jason buys a CD player for $250 and sells it for $425. Find:a the profit b the percentage profit (on cost price).

Solutiona Profit = $425 − $250 = $175

b Percentage profit= × 100%

= × 100% = 70%

2 A pair of jeans marked at $140 is sold at a discount for $84. What is the percentage discount?

profitcost price-----------------------

175250---------



SolutionDiscount= $140 − $84

= $56

∴ Percentage discount= × 100%

= × 100%

= 40%

discountmarked price-------------------------------

56140---------

1 Express each of the following as a percentage:a 24 out of 30 b 76 out of 80 c 42 out of 60d $4 out of $60 e $24 out of $400 f $30 out of $90

2 Express each of these as a percentage. (Remember to make the units of both quantities the same.)a 20 min out of 1 h b 600 ml out of 1 L c 115 m out of 800 md 75c out of $2 e 25 s out of 2 min f 72 cm out of 5 m

3 Which is the better mark: 17 out of 25 for a science test, or 28 out of 40 for a history test?

4 A TV costing $500 is sold for $850. Find:a the profitb the percentage profit (on cost price).

5 A car costing $7000 is sold for $6200. Calculate:a the lossb the loss as a percentage of cost price (correct to one decimal place).

6 Alana’s watch gains 2 minutes every 3 hours. What percentage gain in time is this (correct to two decimal places)?

7 Edmond’s wage increases from $460 a week to $512 a week. Find:a the increase in Edmond’s wageb the percentage increase in Edmond’s wage (correct to one decimal place).

8 At a sale, Garrett and Katie paid $200 for a cutlery set marked at $350. Calculate:a the amount of the discountb the percentage discount (correct to one decimal place).

9 Dwayne scored 31 goals last season. This season he has scored 43 goals. What is his percentage increase? (Give your answer to the nearest whole number.)

10 In 1996, the most popular second language in Australia was Italian, with 367 300 Italian speakers. If Australia’s population at the time was 15 969 000, what percentage of the population could speak Italian? (Give your answer correct to one decimal place.)

11 Jenny earns $680 a week. She spends $42 a week on fares for buses and trains. What percentage of her wage does Jenny spend on fares? (Give your answer correct to one decimal place.)

12 At the supermarket, packets of bread rolls marked at $2.79 are discounted to $1. What is the percentage discount?

Exercise 1-17

Example 24

SkillBuilder 4-13

Two quantities as a percentage



13 At a sale, a jumper priced at $180 is marked down to $120. A further discount of 15% is then given. Calculate:a the selling price of the jumperb the total discountc the discount as a percentage of the marked price.

14 A tennis racquet is bought for $75. If a profit of $90 is made when it is sold, calculate:a the selling price of the tennis racquetb the profit as a percentage of the cost price.

15 A bedroom suite is sold for $8400, making a profit of $3600. Find:a the cost price of the bedroom suiteb the percentage profit.

PowerPlus breakfast cerealA popular breakfast cereal, PowerPlus, lists nutritional information about its contents on the packet, including the masses of nutrients in a 50 g serve (about 1 bowl). Set up a spreadsheet as shown, containing these details.

To save time typing, use this link to go to a partly completed spreadsheet.

1 The total mass of nutrients in cell C16 is 50 g. What is the mass of the other ingredients not listed? Put this value in cell C15.

2 Calculate the amount of each of the nutrients as a percentage of a serve. What formula will you use in cell D8? (If you use =C8/C16*100, then you are correct!)

3 Check to see if your calculations are correct by adding the values in column D. What should the values sum to in cell D16?

A B C D E F

1

2

3

4 The masses of various nutrients in a 50 g serve of PowerPlus cereal are shown in this table.

5

6

7 Nutrients Mass (grams) Percentage (%)

8 Protein 10.8

9 Fat 0.3

10 Complex carbohydrates 2.6

11 Sugars 7.5

12 Dietary fibre 1.4

13 Sodium 0.25

14 Potassium 0.1

15 Other ingredients

16 Total

17

Using technologySpreadsheet

1-06PowerPlus

breakfast cereal



The unitary methodThe unitary method involves finding the value of one part or item when given the value of several parts or items.

Example 25

1 If of a number is 81, what is the number?

Solution of the number = 81

∴ of the number = 81 ÷ 3 = 27

∴ of the number = 27 × 4 = 108

The number is 108.

2 A watch has been reduced in price by so that it now costs $180. What was the original price of the watch?

SolutionIf the price has been reduced by , then of the price = $180.

∴ of the price = 180 ÷ 2 = $90

∴ of the price = $90 × 3 = $270

∴ The original price of the watch is $270.

6% of Vimila’s weekly wage is $30. Calculate her weekly wage.

Solution6% is $30

∴ 1% is $30 ÷ 6 = $5

∴ 100% is $5 × 100= $500

Vimila’s total wage is $500.

34---

34---

14---

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3---

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Example 26

1 Find the number if:

a of the number is 72 b of the number is 88 c of the number is 48

d of the number is 52 e of the number is 160 f of the number is 36

2 The price of a car has been reduced by , so that it now costs $6000. What was the original price of the car?

3 The price of a ring is reduced by and it now costs $1800. What was the ring’s original price?

4 Sean pays of his weekly wage in rent. If his rent is $240 per week, calculate Sean’s weekly wage.

5 In a basketball game, Anita scored 15 points, which was of the team’s score. What was the total number of points scored by the team?

23--- 4

5--- 3

4---

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8--- 3

10------

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15---

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Worksheet 1-08

The unitary method



6 A new car loses of its value in the first year. If its value is now $25 600, what was the

original price of the car?

7 If 35% of Ramy’s mass is 21 kg:a what is Ramy’s mass? b what is 95% of his mass?

8 An 8% discount represents a saving of $50 on the marked price.a What is the marked price?b How much saving does a 15% discount represent?

9 Find (to the nearest whole number) the complete amount in each of the following if:a 11% is 45 b 25% is 165 c 2.5% is 80 mm d 7% is 56

e 55% is $20 f 12.5% is 48 g 82% is 425 kg h 33 % is 24

i 18.5% is 24.5 j 120% is 60 m k 225% is 44 l 135% is 78 km

10 In a game of cricket, Chris scored 42 runs, which was 21% of the team’s score. How many runs did the team score?

11 Last year Ha paid 32% of her salary in tax. If her tax was $13 341, what was her salary?

12 If 15% of Shaheid’s weekly wage is $72.75, what is his weekly wage?

13 A jeweller charges $55 for valuing a diamond ring. Find the value of the ring if this charge represents 1.5% of its worth.

14 When pottery is fired, it loses 35% of the weight it had as clay. Find the weight of clay that was used to make a bowl weighing 780 g.

15---

13---

Example 26

Spreadsheet 1-07

Percentages and the unitary

method

Worksheet1-09

Percentage problems

1 The fraction can be written as the sum of reciprocals of whole numbers as follows:

= + or = + + +

Write each of the following fractions as the sum of reciprocals of whole numbers.

a b c d e f

2 Write down four possible ways of expressing as the sum of reciprocals of whole numbers.

3 Consider Pythagoras’ theorem for this right-angled triangle:

The sum of the reciprocals of the consecutive odd numbers 3 and 5 is

+ =

The 8 and 15 lead to the Pythagorean result82 + 152 = 172

a Show that the Pythagorean result is also true for the consecutive odd numbers 7 and 9.

(Hint: Find + and then state the Pythagorean result that follows.)

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5--- 3

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817

15

82 + 152 = 172

13--- 1

5--- 8

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Power plus



b Give two other examples.

4 Consider the rectangle below, which is made up of 12 small squares.

We can shade of the rectangle in one colour, of the rectangle in a different colour, and

of the rectangle in another colour, writing the result as

+ + = 1 whole

Note that , , are the reciprocals of the integers 2, 3 and 6.

In how many ways can you write the one whole as the sum of different reciprocals of integers for a rectangle made up of:a 12 squares? b 18 squares?

5 An irrational number is a number that is not rational, so it cannot be expressed in the form

, where a and b are integers and b ≠ 0.

Points to remember about an irrational number when expressed as a decimal:• It is not terminating: its digits run indefinitely.• It does not have a repeating group of figures (it is non-recurring).• It can never be written exactly as a fraction.State whether the following are rational or irrational:

a b c

d e f 0.3434526…

6 a Explain why you cannot find the square root of a negative number.b Explain why you can find the cube root of a negative number.

7 Explain why must lie between 2 and 3.

8 a To obtain a better approximation for , find the average of the 2 and 3.b Average this approximation with either the 2 or the 3 (whichever is closer in value). This

gives a second approximation for .c Use the same method to find a third approximation for .

9 If a unit fraction is defined as a fraction with a numerator of 1 (e.g. ), find seven different unit fractions whose sum is 1.

10 The product 5 × 4 × 3 × 2 × 1 can be written as 5! This is read as ‘five factorial’.6 factorial= 6!

= 6 × 5 × 4 × 3 × 2 × 1= 720

a Evaluate:i 4! ii 5! iii 10! iv 7 × 6! v 7!

b i Does 7 × 6! = 7! ? Why?ii Given that 8! = 40 320, find 9!

12--- 1

3--- 1

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3--- 1

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ab---

7 4 3.4̇1̇

11 0.9

7

7

77

112------



c Evaluate:i 4! + 3! ii 6! − 4! iii 4! × 3! iv 8! ÷ 7! v

11 Using a scale of 1 unit to 1 cm, draw an accurate diagram based on this sketch.a Use Pythagoras’ theorem to calculate

the lengths of AE, AF, …, AI. Which answer is a rational number?

b By measuring the lengths AE, AF,…, AI, find the approximations to the irrational numbers obtained in part a.

c Check the accuracy of your approximations by using your calculator.

12 Accurately construct a square with an area of 10 square units. (Hint: You will need to accurately construct an interval of .)

10!6! 4!×----------------

1 unit

1 unit

1

11

1

1

1

I

H

G

F

ED

C

BA

32

10

Language of mathsaccuracy approximate ascending cent

compare convert decimal decimal place

decrease denominator descending discount

equivalent estimate fraction improper fraction

increase integer loss mixed numeral

numerator operation profit ratio

rational recurring reciprocal round

significant figures simplify terminating unitary method

1 Select five words from the list above and write their meanings using your own words.

2 Find a non-mathematical meaning for:a reciprocal b recurring c terminating

3 What is a mixed numeral? Give an example.

4 Which word means ‘to determine roughly’ or ‘to calculate approximately’ the value or size of something?

5 a In the number 12.075 count the number of:i decimal places ii significant figures.

b What is the difference between decimal places and significant figures?

6 From the list above, write as many pairs of words that have opposite meanings as you can.

7 ‘Per cent’ means ‘per hundred’. Can you think of any other ‘cent’ words that mean a hundred of something?

8 What is the difference between rounding up and rounding down?

Worksheet 1-10

Numbers crossword



Topic overview• Write in your own words what you have learned in this chapter.• Write which parts of this chapter were new to you.• Copy and complete:

The things I understand about exploring numbers that I did not understand before are …The things I am still not confident in doing in this chapter are …

Give an example of each difficulty you still have.• Copy and complete whichever applies to you:

The steps I will take to overcome my problems with this chapter are …The sections of work that I found difficult in this chapter were …The sections of work that I found easy in this chapter were …The sections of work that I enjoyed doing were …

• Copy the overview below into your book. Use bright colours and, if necessary, add further information to complete your summary of this section of work.

Worksheet 1-11

No calculators

EXPLORINGNUMBERS

3–7

5–8

3–––––100

0.56 8

2.7

−6

• rounding• terminating• recurring decimals

• ordering, −5 � −3• four operations

−7, 81, −5

0.374

Order of operations• ( )2 power• ( ) brackets• ×, ÷• +, −

Estimation64.3 ÷ 7.6 ≈ ? Ordering

5%, 0.953–5

1–3

proper• Types

improper

• ordering, �

• four operations

1–2

3–4

7–2

3–5

1–4

+

3–5

1–2

−1–2

× 3

2–3

1–2

÷Decimals

Fractions

Integers

37%Percentages

Unitarymethod

Unitarymethod

• find 15% of $60• increase by a %• decrease by a %• one quantity as a

% of another• profit• loss

• 80% = 80 ÷ 100= 0.8

• 0.52 = 0.52 × 100= 52%

• = × 100

= 80%

• 75% =

=

4–5

4–5

3–4

75–––100



1 a Copy and complete: = =

b What number could be placed in the box so that is a fraction between 7 and 8?

c The expression , where is a whole number, has a value between 7 and 9.

What is a possible value for ?

d The fraction has a value between 3 and 5. What is a possible value for ?

2 Simplify:

a − b 2 + 3 c 1 − 3

d × 2 e 6 ÷ f + × (−1 )

g h ÷ i +

3 Evaluate the following, giving your answers to the number of decimal places indicated.

a 15.5 ÷ 3 − 9.78 × 0.023 (2) b (1)

c 2.13 − (−1.8)3 (3) d 0.0043 (4)

4 Evaluate the following, giving your answers to 4 significant figures.

a 8.752 × 34.5 b c −

5 Estimate answers to the following:

a 450 ÷ 63 + 23 × 4.6 b

6 Express each of the following as a fraction in simplest form:

a b 0. 1 c

7 3 % =

A 3.5 B 0.35 C 0.305 D 0.035

8 Arrange , 43%, 0.48, , in descending order.

9 In a school of 970 students, 8.7% of students receive financial assistance.a How many students receive assistance?b How many students do not receive assistance?

10 A clothing company offers a discount of 10.5% to its employees. How much will a jacket marked at $264.50 cost an employee?

11 a There are 14 g of nitrogen in 63 g of nitric acid. What percentage of nitrogen is this?b Increase $1.50 by 12% and then decrease this amount by 12%. Calculate the gain or

loss as a percentage of the original amount. Explain your answer.

12 In a cricket match, John made 57 runs, which was 22% of the team total. How many runs did the team make?

Chapter 1 Review

Ex 1-02510------ 15------

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Ex 1-06

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1 34---

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+----------------------------- 4

5---

2 12---

4 23--------- 4

5---

Ex 1-08

23 17.2×56 2.32–

---------------------------

Ex 1-09

5.76 7.96 3.2×+

2.73------------------------------------------- 1

4.8------- 3

2.7-------

Ex 1-10

112.45 37.8×48.5

-----------------------------------

Ex 1-12

0.15̇ 8̇ 7̇ 0.25̇3̇

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Ex 1-1437--- 1

2--- 19

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Ex 1-16

Ex 1-15

Ex 1-17

Ex 1-18

Topic testChapter 1

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Documents

Yr 9 New Century Maths Chapter 01- Exploring Numbers