PDF- -Year 9 Mathematics Study Program - Assumption College - Cambridge Year 9 Textbook - Chapter 1

- mbridge math textbook Year 9

What you will learn

1A 1B 1C 1D 1E 1F 1G 1H 1I 1J 1K 1L 1M

Computation with integers REVISION Decimal places and significant figures Rational numbers REVISION Computation with fractions REVISION Ratios,

rates and best buys REVISION Computation with percentages and money REVISION Percentage increase and decrease REVISION Profits and discounts REVISION Income The PAYG income tax system Simple interest Compound interest and depreciation Using a formula for compound interest and depreciation

- 2014 ISBN: 9781107645264 Photocopying is restricted under law and this material must not be transferred to another party

NSW Syllabus

for the Australian Curriculum Strand: Number and Algebra

orders and calculates with integers,

applying a range of strategies to aid computation

(MA4–4NA) A student operates with fractions,

- decimals and percentages

(MA4–5NA) A student solves financial problems involving earning,

- spending and investing money

1–4NA)

Global financial crisis The global financial crisis of 2008 and 2009 was one of the most serious financial situations since the Great Depression in the 1930s

US interest rates were lowered to about 1%,

which created access to easy credit and ‘sub-prime’ lending

When the housing bubble burst,

house prices began to fall and lenders began foreclosing on mortgages if borrowers could not keep up with their repayments

As house prices collapsed,

financial institutions struggled to survive due to the increased number of bad debts

The crisis expanded to cause negative growth in the US general economy and in other countries

In Australia,

our sharemarket All Ordinaries Index collapsed by 55% from 6874 in November 2007 to 3112 in March 2009

2–4NA)

- 2014 ISBN: 9781107645264 Photocopying is restricted under law and this material must not be transferred to another party

- 1 Evaluate each of the following

a 5+6×2 c'12 ÷ (4 × 3) + 2 e 8 − 12 g −2 × 3 11 as a: 5 a mixed number

- b d'f h
- 12 ÷ 4 × 3 + 2 3 + (18 − 2 × (3 + 4) + 1) −4 + 3 −18 ÷ (−9)
- decimal
- (−4)2

2 Write

- 3 Evaluate each of the following

25 b a 32

- 3 7 or by: 4 9 a rewriting with the lowest common denominator b converting to decimals (to three decimal places where necessary)
- 4 Determine which is larger,
- 5 Arrange the numbers in each of the following sets in descending order

465 and 2

564 b 0

0456 and 0

- 654 6 Evaluate each of the following

89 − 3

- 7 Evaluate each of the following
- a 7 × 0
- 8 Evaluate each of the following
- 345 × 100 c'37

54 ÷ 1000

- 74 × 100 000 3
- 754 ÷ 100 000
- 9 Find the lowest common denominator for these pairs of fractions
- 1 1 and 3 5
- 1 1 and 6 4
- 1 1 and 10 5
- 10 Evaluate each of the following

2 3 + 7 7

- 11 Find: a 50% of 26
- 1 3 b 2 − 2 2
- b 10% of 600

2 3 × 3 4

1 ÷2 2

- c 9% of 90

- 2014 ISBN: 9781107645264 Photocopying is restricted under law and this material must not be transferred to another party

Throughout history,

mathematicians have developed number systems to investigate and explain the world in which they live

The Egyptians used hieroglyphics to record whole numbers as well as fractions,

the Babylonians use a place-value system based on the number 60 and the ancient Chinese and Indians developed systems using negative numbers

Our current base-10 decimal system (the Hindu-Arabic system) has expanded to include positive and negative numbers,

fractions (rational numbers) and also numbers that cannot be written as fractions (irrational numbers),

- for example,

All the numbers in our number system,

not including imaginary numbers,

- are called real numbers

? Try to use the correct vocabulary,

- for example,
- factors of 12
- … • 1,
- … • 1,
- … • 0,
- … • 2,

Markets used number systems in ancient times to enable trade through setting prices,

counting stock and measuring produce

… If a and b are positive integers – a + (−b) = a – b For example: 5 + (−2) = 5 − 2 = 3 – a − (−b) = a + b For example: 5 − (−2) = 5 + 2 = 7 – a × (−b) = −ab For example: 3 × (−2) = −6 – −a × (−b) = ab For example: −4 × (−3) = 12 a – a ÷ (−b) = − For example: 8 ÷ (−4) = −2 b a – −a ÷ (−b) = For example: −8 ÷ (−4) = 2 b Squares and cubes – a2 = a × a and a 2 = a (if a ≥ 0),

- for example,
- 62 = 36 and 36 = 6 – a3 = a × a × a and 3 a3 = a,
- for example,
- 43 = 64 and 3 64 = 4

HCF and primes – The lowest common multiple (LCM) of two numbers is the smallest multiple shared by both numbers,

- for example,
- the LCM of 6 and 9 is 18

– The highest common factor (HCF) of two numbers is the largest factor shared by both numbers,

- for example,
- the HCF of 24 and 30 is 6

– Prime numbers have only two factors,

- 1 and the number itself

– Composite numbers have more than two factors

– Do multiplication and division next,

- from left to right

– Do addition and subtraction last,

- from left to right

a −2 − (−3 × 13) + (−10)

a −2 − (−3 × 13) + (−10) = −2 − (−39) + (−10) = −2 + 39 + (−10) = 37 − 10 = 27

−a − (−b) = −a + b a + (−b) = a − b

b (−20 ÷ (−4) + (−3)) × 2 = (5 + (−3)) × 2 =2×2 =4

- −a ÷ (−b) =

a b Deal with the operations inside brackets before doing the multiplication

- 5 + (−3) = 5 − 3

- 2 Evaluate the following
- b 152 a 112 e 33 f 53
- 1 Write down these sets of numbers

a The factors of 16 b The factors of 56 c'The HCF (highest common factor) of 16 and 56 d'The first 7 multiples of 3 e The first 6 multiples of 5 f The LCM (lowest common multiple) of 3 and 5 g The first 10 prime numbers starting from 2 h All the prime numbers between 80 and 110 d'h

© David Greenwood et al

Cambridge University Press

R K I NG

(−20 ÷ (−4) + (−3)) × 2

Number and Algebra

- d h l'p

−9 + 18 −21 − (−30) −3 × (−14) −950 ÷ (−50)

- 5 Find the LCM of these pairs of numbers
- 6 Find the HCF of these pairs of numbers

8 b 100,

- 7 Evaluate the following

a 23 − 16 d'(−2)3 ÷ (−4) g

27 − 81

- b 52 − 3 8 e h

−3 × (−2) + (−4) 2 − 7 × (−2) 4 + 8 ÷ (−2) − 3 −7 − (−4 × 8) − 15 4 × (−3) ÷ (−2 × 3) 10 × (−2) ÷ (−7 − (−2)) (−3 + 7) − 2 × (−3) −18 ÷ ((−2 − (−4)) × (−3)) (7 − 14 ÷ (−2)) ÷ 2 20 ÷ (6 × (−4 × 2) ÷ (−12) − (−1))

- 4 Evaluate the following,
- showing your steps

a −4 − 3 × (−2) b c'−2 × (3 − 8) d'e 2 − 3 × 2 + (−5) f g (−24 ÷ (−8) + (−5)) × 2 h i −3 − 12 ÷ (−6) × (−4) j k (−6 − 9 × (−2)) ÷ (−4) l'm 6 × (−5) − 14 ÷ (−2) n o −2 + (−4) ÷ (−3 + 1) p q −2 × 6 ÷ (−4) × (−3) r s'2 − (1 − 2 × (−1)) t

- c (−1)2 × (−3)
- 9 − 3 125
- 13 + 23 − 33
- 27 − 9 − 3 1
- (−1)101 × (−1)1000 ×
- 8 Evaluate these expressions by substituting a = −2,
- b = 6 and c'= −3

a a2 − b b a − b2 c'2c + a d'b2 − c2 e a3 + c2 f 3b + ac g c'− 2ab h abc − (ac)2

c 2 − 5 × (−2) = 6 f (−2)2 + 4 ÷ (−2) = −22

- 9 Insert brackets into these statements to make them true

a −2 × 11 + (−2) = −18 b −6 + (−4) ÷ 2 = −5 d'−10 ÷ 3 + (−5) = 5 e 3 − (−2) + 4 × 3 = −3

Cambridge University Press

- 10 How many different answers are possible if any number of pairs of brackets is allowed to be inserted into this expression

? −6 × 4 − (−7) + (−1)

© David Greenwood et al

R K I NG

−3 + 2 11 − (−4) −11 × (−2) −100 ÷ (−10)

- c g k o

−6 − 2 −6 + (−10) −21 × 4 −36 ÷ 6

- 3 Evaluate the following

a 5 − 10 b e 2 + (−3) f i 2 × (−3) j m 18 ÷ (−2) n

- 11 Margaret and Mildred meet on a Eurostar train travelling from London to Paris

Margaret visits her daughter in Paris every 28 days

When will Margaret and Mildred have a chance to meet again on the train

- 12 a The sum of two numbers is 5 and their difference is 9

What are the two numbers

? b The sum of two numbers is −3 and their product is −10

What are the two numbers

? 13 Two opposing football teams have squad sizes of 24 and 32

For a training exercise,

each squad is to divide into smaller groups of equal size

- 15 If a and b are both positive numbers and a > b,

decide if the following are true or false

a a−b 0 c'−a ÷ (−b) > 0 d'(−a)2 − a2 = 0 e −b + a < 0 f 2a − 2b > 0

© David Greenwood et al

Cambridge University Press

- 14 a Evaluate: C R PS HE i 42 ii (−4)2 M AT I C A 2 b If a = 16,

write down the possible values of a

- c'If a3 = 27,
- write down the value of a

d'Explain why there are two values of a for which a2 = 16 but only one value of a for which a3 = 27

- e Find 3 −27

f Explain why −16 cannot exist (using real numbers)

g −22 is the same as −1 × 22

Now evaluate: i −22 ii −53 iii −(−3)2 iv −(−4)2 2 2 h Decide if (−2) and −2 are equal

i Decide if (−2)3 and −23 are equal

j Explain why the HCF of two distinct prime numbers is 1

k Explain why the LCM of two distinct prime numbers a and b is a × b

Number and Algebra

Enrichment: Special numbers 16 a Perfect numbers are positive integers that are equal to the sum of all their factors,

- excluding the number itself

i Show that 6 is a perfect number

ii There is one perfect number between 20 and 30

iii The next perfect number is 496

b Triangular numbers are the number of dots required to form triangles as shown in this table

- i Complete this table

Number of rows Diagram Number of dots (triangular number)

ii Find the 7th and 8th triangular numbers

c'Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding numbers

i Write down the first 10 Fibonacci numbers

ii If the Fibonacci numbers were to be extended in the negative direction,

what would the first four negative Fibonacci numbers be

Fibonacci numbers have many applications in nature,

such as in the structure of an uncurling fern frond

1B Decimal places and significant figures

Numbers with and without decimal places can be rounded depending on the level of accuracy required

When using numbers with decimal places it is common to round off the number to leave only a certain number of decimal places

- for example,
- might be 9

94 seconds

not all the digits in all numbers are considered important or ‘significant’

In such cases we are able to round numbers to within a certain For road construction purposes,

the volume of sand in these piles would number of significant figures (sometimes only need to be known to two or three significant figures

- abbreviated to sig

- for example,

might be calculated as 3485 but is rounded to 3500

This number has been written using two significant figures

Johnny says that the number 2

- 748 when rounded to one decimal place is 2
- 8 because: • the 8 rounds the 4 to a 5 • then the new 5 rounds the 7 to an 8

To round a number to a required number of decimal places: – Locate the digit in the required decimal place

– Round down (leave as is) if the next digit (critical digit) is 4 or less

– Round up (increase by 1) if the next digit is 5 or more

- 543 rounds to 1

54 and 32

- 9283 rounds to 32
- – To one decimal place,
- 248 rounds to 0

2 and 0

- 253 rounds to 0

To round a number to a required number of significant figures: – Locate the first non-zero digit counting from left to right

– From this first significant digit,

count out the number of significant digits including zeros

– Stop at the required number of significant digits and round this last digit

– Replace any non-significant digit to the left of a decimal point with a zero

these numbers are all rounded to three significant figures: 2

5391 ≈ 2

- 002713 ≈ 0
- 568 810 ≈ 569 000

© David Greenwood et al

Number and Algebra

1793 b 0

SOLUTIO N

EXPLANATI ON

- 1793 ≈ 256

The number after the second decimal place is 9,

so round up (increase the 7 by 1)

- 04459 ≈ 0

The number after the second decimal place is 4,

- so round down
- 4459 is closer to 4000 than 5000

8972 ≈ 4

- so round up

- 89 becomes 0

Example 3 Rounding to a number of significant figures Round each of these numbers to two significant figures

- a 2567 b 23 067

SOLUTIO N

- a 2567 ≈ 2600

The first two digits are the first two significant figures

The third digit is 6,

- so round up

Replace the last two non-significant digits with zeros

- b 23 067
- 453 ≈ 23 000

The third digit is 0,

- so round down
- 04059 ≈ 0

Locate the first non-zero digit,

So 4 and 0 are the first two significant figures

The next digit is 5,

- so round up

Example 4 Estimating using significant figures Estimate the answer to the following by rounding each number in the problem to one significant figure and use your calculator to check how reasonable your answer is

27 + 1329

- 0064 SOLUTIO N

27 + 1329

- 0064 ≈ 30 + 1000 × 0
- 006 = 30 + 6 = 36

Round each number to one significant figure and evaluate

Recall multiplication occurs before the addition

0064 = 35

Cambridge University Press

- 1 Choose the number to answer each question
- a Is 44 closer to 40 or 50

? b Is 266 closer to 260 or 270

- ? c'Is 7
- 89 closer to 7
- ? d'Is 0
- 043 closer to 0

04 or 0

- 2 Choose the correct answer if the first given number is rounded to 3 significant figures
- a 32 124 is rounded to 321,
- 3210 or 32 100 b 431
- 92 is rounded to 431,
- 432 or 430 c'5
- 8871 is rounded to 5

88 or 5

- 44322 is rounded to 0

443 or 0

44302 e 0

- 0019671 is rounded to 0

00197 or 0

- 00196 3 Using one significant figure rounding,
- 324 rounds to 300,
- 7 rounds to 2 and 9
- 6 rounds to 10
- a Calculate 300 × 2 ÷ 10

b Use a calculator to calculate 324 × 1

c'What is the difference between the answer in part a and the exact answer in part b

- 5 Round these numbers to the nearest whole number

814 b 73

148 c'129

- d 36 200
- 6 Use division to write these fractions as decimals rounded to three decimal places
- 1 2 13 400 a b c'd'3 7 11 29 Example 3

- 7 Round each of these numbers to two significant figures
- a 2436 b 35057
- 06049 e 107 892 f 0

00245 g 2

024 h 0

- 8 Round these numbers to one significant figure
- a 32 000 b 194

0006413

- 9 Estimate the answers to the following by rounding each number in the problem to one significant figure

- a 567 + 3126 b 795 − 35

8 × 42

2 d'965

- 98 + 5321 − 2763

23 − 1

92 × 1

827 f 17

43 − 2

047 × 8

165 g 0

0704 + 0

0482 h 0

023 × 0

027 ÷ 0

0032 k 0

078 × 0

98032 l'1

84942 + 0

972 × 7

032 j 41

© David Greenwood et al

Cambridge University Press

R K I NG

859 h 500

- 5749 l'2649
- 4 Round each of the following numbers to two decimal places

962 b 11

082 c'72

986 e 63

925 f 23

807 g 804

5272 i 821

- 2749 j 5810
- 2539 k 1004

Example 2

Number and Algebra

R K I NG

94 × 11

- 31 is calculated by first rounding each of the three numbers

- 2 c'924 12 150 m of fencing and 18 posts are used to create an area in the shape of an equilateral triangle

- 13 A tonne (1000 kg) of soil is to be equally divided between 7 garden beds

How much soil does each garden bed get

? Write your answer in tonnes rounded to the nearest kilogram

© David Greenwood et al

- 10 An electronic timer records the time for a running relay between two teams A and B

Team A’s time is 54

- 283 seconds and team B’s time is 53
- 791 seconds

Find the difference in the times for teams A and B if the times were written down using: a 1 decimal place b 4 significant figures c'2 significant figures d'1 significant figure

- 15 A scientific experiment uses very small amounts of magnesium (0
- 0025 g) and potassium (0

0062 g)

Why does it make sense to use two significant figures instead of two decimal places when recording numbers in a situation like this

? 16 Consider the two numbers 24 and 26

a Calculate: i 24 + 26 ii 24 × 26 b Find the sum (+) of the numbers after rounding each number using one significant figure

c'Find the product (×) of the numbers after rounding each number using one significant figure

d'What do you notice about the answers for parts b and c'as compared to part a

- ? Give an explanation

- 2 as a decimal correct to 8 decimal places
- 11 Using the decimal pattern from part a find the digit in the: i 20th decimal place ii 45th decimal place iii 1000th decimal place
- 1 Express as a decimal correct to 13 decimal places
- 7 Using the decimal pattern from part c'find the digit in the: i 20th decimal place ii 45th decimal place iii 1000th decimal place

- ? Use a calculator to help

Express

- 14 Should 2
- 14999 be rounded down or up if it is to be rounded to one decimal place
- ? Give reasons

M AT I C A

it was discovered that some numbers could not be expressed as a fraction

These special numbers,

- called irrational numbers,

when written as a decimal continue forever and do not show any pattern

you need to use special symbols such as 2 and π

- however,

the decimal places in a number terminate or if a pattern exists,

the number can be expressed as a fraction

These numbers are called rational numbers

This is 2 to 100 decimal places:

- 4142135623730950488016887 2420969807856967187537694 8073176679737990732478462 1070388503875343276415727

A non-terminating decimal is one in which the decimal places continue indefinitely

Real numbers Rational numbers (Fractions) (e

Irrational numbers (e

- pi (π ),

phi (ϕ) − the golden ratio)

345) (e

- √2 = 1

- 2 6 For example: = 3 9 A fraction is simplified by dividing the numerator and denominator by their highest common factor
- a If is a proper fraction,
- then a < b
- b 2 For example: 7
- a is an improper fraction,
- then a > b

b 10 For example: 3 A mixed number is written as a whole number plus a proper fraction

- 3 For example: 2 5 If

Cambridge University Press

the ancient and modern civilisations have used fractions to approximate π

1415926536

• Using single digit numbers,

what fraction best approximates π

? • Using single and/or double digit numbers,

find a fraction that is a good approximation of π

Compare with others students to see who has the best approximation

Key ideas

Fractions can be compared using a common denominator

This should be the lowest common multiple of both denominators

For example: 1 = 0

- 16 or 3 = 0

27 or 0

27 6 11

3 a 3 8

SOLUTIO N

EXPL ANATI ON

- 3 7 5 a 8 3

3 06 0 4 0

- 3 by dividing 8 into 3 using the 8 short division algoritm

3 3 = 3

Add a bar over the repeating pattern

- 3 8 4 6 1 5 3 b 13 5
- 5 01106 08 0 2 0 7 0 5 0

384615 13

- 384615 is an alternative

Example 6 Writing decimals as fractions Write these decimals as fractions

- 24 SOLUTIO N

- 24 100 6 = 25 2385 1000 477 = 200 77 =2 200
- 385 EXPL ANATI ON
- 385 1000 77 =2 200 2

The smallest place value is thousandths

Simplify to an improper fraction or a mixed number

Number and Algebra

Example 7 Comparing fractions Decide which fraction is larger

- 7 8 or 12 15 EXPLANATION

LCM of 12 and 15 is 60

Find the lowest common multiple of the two denominators (lowest common denominator)

- 35 > 32) to determine the larger fraction
- 7 35 8 32 = and = 12 60 15 60 7 8 ∴ > 12 15

REVISION

- d −72
- 2 Write these numbers as improper fractions
- 1 4 c'a 1 b 5 3 7
- 1 Write these numbers as mixed numbers
- 7 13 b c'a 5 3

Exercise 1C

M AT I C A

- 3 Simplify these numbers by cancelling

3 9 = 7

125 1000

- 4 Write down the missing number

5 20 = 6 9

- 5 = 11 77 11
- 6 Write these fractions as recurring decimals
- 3 7 a b c'11 9 5 10 g e f 3 6 9

2 5 37 16

- 15 8 7 h 32
- 5 12 29 h 11

© David Greenwood et al

- 5 Write these fractions as decimals
- 11 7 a b 4 20 5 4 f 3 e 2 8 5

Example 5a

3 = 5 15

375 l'2

- 8 Decide which is the larger fraction in the following pairs
- 3 5 13 3 7 8 a ,
- 4 6 20 5 10 15 e

26 11 ,

8 23 7 ,

15 40 12

- 7 Write these decimals as fractions

075 i 2

005 j 10

M AT I C A

- 9 Place these fractions in descending order

3 5 7 ,

8 12 18

1 5 3 ,

6 24 16

- 10 Express the following quantities as simplified fractions

a $45 out of $100 b 12 kg out of 80 kg c'64 baskets out of 90 shots in basketball d'115 mL out of 375 mL

1 5 4 ,

6 14 2 ,

2 3 5 ,

1 4 9 ,

- 12 The ‘Weather Forecast’ website says there is a 0
- 45 chance that it will rain tomorrow

The ‘Climate Control’ website says that 14 the chance of rain is

Which website 30 gives the least chance that it will rain

? 1 13 A jug has 400 mL of strength orange 2 juice

The following amounts of full-strength juice are added to the mix

Find a fraction to describe the strength of the orange drink after the full-strength juice is added

- a 100 mL b 50 mL c

The chance of rain can be expressed as a decimal,

- a fraction or a percentage
- 11 These sets of fractions form a pattern

Find the next two fractions in the pattern

- d 375 mL

M AT I C A

Enrichment: Converting recurring decimals to fractions 17 Here are two examples of conversion from recurring decimals to fractions

272 727

6666…

- (1) Let x = 1

272 727…

- (1) 10x = 6
- 6666… (2) 100x = 127

2727…

(2) (2) − (1) 9x = 6 (2) − (1) 99x = 126 6 2 126 x= = x= 9 3 99

- 2 126 27 3 ∴ 0

6 = ∴ 1

- 27 = =1 =1 3 99 99 11 Convert these recurring decimals to fractions using the above method

© David Greenwood et al

R K I NG

M AT I C A

b 15 a is a mixed number with unknown digits a,

- b and c

- c'a 16 If is a fraction,

answer the given questions with reasons

b a Is it possible to find a fraction that can be simplified by cancelling if one of a or b is prime

? b Is it possible to find a fraction that can be simplified by cancelling if both a and b are prime

- ? Assume a ≠ b

a c'If is a fraction in simplest form,

- can a and b both be even

? b a d'If is a fraction in simplest form,

- can a and b both be odd
- 14 If x is an integer,

determine what values x can take in the following

x a The fraction is a number between (and not including) 10 and 11

- 3 x b The fraction is a number between (and not including) 5 and 8
- 7 34 c'The fraction is a number between 6 and 10

x 23 d'The fraction is a number between 7 and 12

x x e The fraction is a number between (and not including) 3 and 4

- 14 58 is a number between 9 and 15
- f The fraction x

M AT I C A

1D Computation with fractions

R EVI S I ON Stage

- subtraction,
- multiplication and division

Addition and subtraction of fractions is generally more complex than multiplication and division because there is the added step of finding common denominators

Let’s start: The common errors Here are incorrect solutions to four computations involving fractions

- 2 5 2 × 5 10 × = = 3 3 3 3
- 7 7 7 14 2 1 ÷ = ÷ = = 6 3 6 6 6 12

- 2 1 2 +1 3 = + = 3 2 3+ 2 5
- 1 2 3 4 1 1 − = 1 − = −1 2 3 6 6 6

In each case describe what is incorrect and give the correct solution

first convert each fraction to equivalent fractions that have the same denominator

– Choose the lowest common denominator

– Add or subtract the numerators and retain the denominator

- 1 2 3 4 7 1 For example: + = + = = 1 2 3 6 6 6 6 To multiply fractions,

multiply the numerators and multiply the denominators

– Cancel the highest common factor between any numerator and any denominator before multiplication

– Convert mixed numbers to improper fractions before multiplying

– The word ‘of’ usually means ‘multiplied by’

- 1 1 For example: of 24 = × 24
- 3 3 The reciprocal of a number multiplied by the number itself is equal to 1
- 1 1 – For example: the reciprocal of 2 is since 2 × = 1
- 2 2 3 5 3 5 the reciprocal of = since × = 1
- 5 3 5 3 To divide a number by a fraction,
- multiply by its reciprocal
- 2 5 2 6 4 ÷ = × = 3 6 31 5 5 – Whole numbers can be written using a denominator of 1

- 1 For example:

© David Greenwood et al

Number and Algebra

Example 8 Adding and subtracting fractions Evaluate the following

- 1 3 a + 2 5 SOLUTIO N a
- 1 3 5 6 + = + 2 5 10 10
- 11 1 or 1 10 10 2 5 5 29 b 1 +4 = + 3 6 3 6 10 29 = + 6 6 39 = 6 13 1 = or 6 2 2 2 5 4 5 Alternatively,
- 1 + 4 = 1 + 4 3 6 6 6 9 =5 6 3 =6 6 1 =6 2 2 3 17 11 − c'3 −2 = 5 4 5 4 68 55 = − 20 20 13 = 20 =
- 2 5 1 +4 3 6
- 2 3 3 −2 5 4

Change each mixed number to an improper fraction

Change to an equivalent fraction with 3 denominator 6,

then add the numerators and simplify

obtaining a common denominator for the ⎛ 2 4⎞ fractions ⎜ = ⎟

- ⎝ 3 6⎠ 9 3 =1 6 6

then rewrite as equivalent fractions with the same denominator

Chapter 1 Computation and financial mathematics

Example 9 Multiplying fractions Evaluate the following

- 2 5 a × 3 7

SOLUTIO N a

- 2 1 1 ×2 3 10 EXPL ANATI ON
- 2 5 2×5 × = 3 7 3×7 10 = 21

No cancelling is possible as there are no common factors between numerators and denominators

- 2 1 1 5 21 × b 1 ×2 = 3 10 1 3 10 2

Rewrite as improper fractions

Cancel common factors between numerators and denominators and then multiply numerators and denominators

- 7 1 or 3 2 2

- 4 12 a ÷ 15 25

SOLUTIO N a

- 17 1 ÷1 18 27

EXPL ANATI ON

- 4 12 4 25 ÷ = × 15 25 15 12

- 4 25 = × 12 3 3 15 =

- 17 1 35 28 ÷1 = ÷ 18 27 18 27 = =
- 12 25 we multiply by its reciprocal

- 35 27 × 28 4 2 18

- 15 7 or 1 8 8

- 1 2 2 b 1 − = − 3 5 3 5 =
- 6 Evaluate the following
- 4 2 a − b 5 5 8 5 e − f 9 6 Example 8c
- 7 Evaluate the following
- 3 1 a 2 −1 4 4 5 9 d'3 −2 8 10
- 5 4 + 7 7 2 3 + 5 10
- 3 4 b 2 + 5 5 5 5 e 2 +4 7 9 4 7 − 5 9 3 1 − 8 4
- 5 7 b 3 −2 8 8 2 5 e 2 −1 3 6
- 3 1 − 4 5 5 3 − 9 8
- 3 1 + 4 5 4 5 + 9 27 3 5 c'1 +3 7 7 5 3 f 10 + 7 8 16
- 2 3 − 5 10 5 5 h − 12 16 d
- 1 3 c'3 −2 4 5 7 3 f 3 −2 11 7

- 5 Evaluate the following
- 1 3 a 3 +1 4 4 1 2 d'2 +4 3 5
- 5 2 5 ÷ = × 3 7 3
- 3 1 + 9 9 3 4 + 8 5

Example 8b

- 4 Evaluate the following
- 2 1 a + b 5 5 1 4 e + f 3 7

M AT I C A

Example 8a

Cambridge University Press

R K I NG

- 3 4 + = + 2 3 6 6
- 10 15 13 2 ,

29 3 22

- 1 Find the lowest common denominator for these pairs of fractions
- 1 1 3 5 2 1 a ,

d 2 3 7 9 5 13 1 1 9 4 5 7 e ,

h 2 4 11 33 12 30 2 Convert these mixed numbers to improper fractions

- 1 4 1 b 7 c'10 d'a 2 3 5 4 3 Copy and complete the given working

REVISION

- c g k o
- 6 5 ×1 21 9 1 h 1 ×4 4 d

- 25 7 ×1 44 15 1 1 p 1 ×1 5 9
- 9 Write down the reciprocal of these numbers
- 5 1 a 3 b c'7 8 Example 10

- 2 5 × 15 8 5 ×9 6 10 2 ×1 21 5 2 1 2 ×2 3 4
- 1 1 m 1 ×1 2 2
- 3 5 × 5 6 2 f 8× 3 5 j 1 × 16 8 1 1 n 1 ×2 2 3 b
- 2 3 × 5 7 3 e 6× 4 1 i 2 ×6 2 a
- 8 Evaluate the following
- 10 Evaluate the following
- 4 3 ÷ 7 5 3 9 e ÷ 4 16 5 i 15 ÷ 6 4 m ÷8 5 1 q 6 ÷1 2 a
- 3 2 ÷ 4 3 4 8 f ÷ 5 15 2 j 6÷ 3 3 n ÷9 4 1 r 1 ÷8 3
- c g k o s
- 5 7 ÷ 8 9 8 4 ÷ 9 27 3 12 ÷ 4 8 ÷6 9 1 1 2 ÷1 4 2
- 3 4 ÷ 7 9 15 20 h ÷ 42 49 3 l'24 ÷ 8 1 p 14 ÷ 4 5 2 1 t 4 ÷5 3 3
- 4 6 1 × ÷ 9 25 150 4 3 3 2 × ÷3 13 8 4
- 11 Evaluate these mixed-operation computations
- 2 1 7 4 3 9 a × ÷ b × ÷ 3 3 9 5 5 10 1 3 3 1 13 1 e 5 × ÷1 d'2 × ÷1 5 7 14 3 24 6

- 12 To remove impurities a mining company 1 5 filters 3 tonnes of raw material

what quantity of material remains

? 13 When a certain raw material is processed 1 3 it produces 3 tonnes of mineral and 2 7 8 tonnes of waste

How many tonnes of raw material were processed

? 1 1 The concentration (proportion) of the desired mineral within 14 In a 2 hour maths exam,

of that time is an ore body is vital information in the minerals industry

- 2 6 allocated as reading time

How long is the reading time

- ? © David Greenwood et al

Number and Algebra

- 1 1 3 2 2 −1 = 2 −1 2 3 6 6 1 =1 6 Try this technique on the following problem and explain the difficulty that you encounter
- 1 1 2 −1 3 2 a 18 a A fraction is given by

b b b A mixed number is given by a

- c'19 If a,
- b and c'are integers,
- simplify the following

b a a b a × b ÷ a b b a a c'a abc bc d'× ÷ e ÷ b a b a a

- a a ÷ b b a b b ÷ × b c'a

Enrichment: Fraction operation challenge 20 Evaluate the following

- 1 2 1 a 2 −1 × 2 3 5 7
- 1 1 1 1 × 1 − 2 ÷ 10 4 5 2
- 4 1 2 1 c'1 × 4 + ×1 5 6 3 5
- 3⎞ 5 ⎛ 2 ⎜⎝ 1 3 + 1 4 ⎟⎠ ÷ 3 12

- 2⎞ ⎛ 1 2⎞ ⎛ 1 g ⎜2 −1 ⎟ × ⎜2 +1 ⎟ ⎝ 4 3⎠ ⎝ 4 3⎠
- 3⎞ ⎛ 1 3⎞ ⎛ 1 ⎜⎝ 3 2 + 1 5 ⎟⎠ × ⎜⎝ 3 2 − 1 5 ⎟⎠
- 3⎞ ⎛ 2 3⎞ ⎛ 2 ⎜⎝ 2 3 − 1 4 ⎟⎠ × ⎜⎝ 2 3 + 1 4 ⎟⎠
- 2⎞ ⎛ 1 1⎞ ⎛ 1 ⎜⎝ 4 2 − 3 3 ⎟⎠ ÷ ⎜⎝ 1 3 + 2 ⎟⎠
- 1 ⎛ 1 1⎞ ÷ 1 +1 ⎟ 6 ⎜⎝ 3 4⎠

- 17 Here is an example involving the subtraction of fractions where improper fractions are not used

R K I NG

- 1 15 A road is to be constructed with 15 m3 of 2 1 crushed rock

If a small truck can carry 2 m3 of 3 crushed rock,

how many truckloads will be needed

? 1 16 Regan worked for 7 hours in a sandwich 2 shop

M AT I C A

- rates and best buys

R EVI S I ON Stage

ratios and rates are used to compare quantities

- for example,

might require 1 of a litre of oil to make a petrol mix of 2 parts oil 6 to 25 parts petrol,

which is an oil to petrol ratio of 2 to 25 or 2 : 25

The mower’s blades might then spin at a rate of 1000 revolutions per minute (1000 rev/min)

Key ideas

- $100 000 is to be divided up for three lucky people into a ratio of 2 to 3 to 5 (2 : 3 : 5)

• Clearly write down your method and answer

There may be many different ways to solve this problem

• Write down and discuss the alternative methods suggested by other students in the class

Ratios are used to compare quantities with the same units

– The ratio of a to b is written a : b

– Ratios in simplest form use whole numbers that have no common factor

The unitary method involves finding the value of one part of a total

– Once the value of one part is found,

then the value of several parts can easily be determined

– The rate is usually written with one quantity compared to a single unit of the other quantity

- a 38 : 24

1 1 :1 2 3

SOLUTION

EXPLANATION

- a 38 : 24 = 19 : 12

- 1 1 5 4 :1 = : 2 3 2 3 15 8 = : 6 6 = 15 : 8

- 14 = 20 : 14 = 10 : 7

Example 12 Dividing into a given ratio $300 is to be divided into the ratio 2 : 3

Find the value of the larger portion using the unitary method

Total number of parts is 2 + 3 = 5 5 parts = $300 1 1 part = of $300 5 = $60

Calculate the value of 3 parts

a 120 km every 3 hours SOLUTIO N 120 a 120 km per 3 hours = km/h 3 = 40 km/h 1 b 5000 revolutions per 2 minutes 2 = 10 000 revolutions per 5 minutes 10 000 = rev/min 5 = 2000 rev/min

- 1 5000 revolutions in 2 minutes 2 EXPL ANATI ON Divide by 3 to write the rate compared to 1 hour

© David Greenwood et al

Chapter 1 Computation and financial mathematics

Example 14 Finding best buys a Which is better value

- ? 5 kg of potatoes for $3
- 80 or 3 kg for $2
- 20 b Find the cost of 100 g of each product then decide which is the best buy
- 400 g of shampoo A at $3
- 20 or 320 g of shampoo B at $2
- 85 SOLUTION

- a Method A

5 kg bag

- 1 kg costs $3
- 80 ÷ 5 = $0
- 76 3 kg bag
- 1 kg costs $2
- 20 ÷ 3 = $0
- 73 ∴ The 3 kg bag is better value

5 kg bag

- $1 buys 5 ÷ 3
- 32 kg 3 kg bag
- $1 buys 3 ÷ 2
- 36 kg ∴ The 3 kg bag is better value

- b Shampoo A
- 100 g costs $3
- 20 ÷ 4 = $0
- 80 Shampoo B
- 100 g costs $2

85 ÷ 3

Alternatively,

divide by 400 to find the cost of 1 g then multiply by 100

divide by 320 to find the cost of 1 g then multiply by 100

- ∴ Shampoo A is the best buy

- : 12 = 1 : 4
- : 28 = 16 : 36
- c 5 : 8 = 15 : g 8:
- = 640 : 880
- d 7 : 12 = 42 : h
- : 4 = 7
- 2 Consider the ratio of boys to girls of 4 : 5

a What is the total number of parts

? b What fraction of the total are boys

? c'What fraction of the total are girls

? d'If there are 18 students in total,

- how many of them are boys

? e If there are 18 students in total,

- how many of them are girls

© David Greenwood et al

- 1 Write down the missing number
- : 10 b 3:7= a 2:5=

Cambridge University Press

R K I NG

Odometers in cars record the distance travelled

- c 4 kg costs $10
- 6 Write each of the following as a ratio in simplest form

- a 80c : $8 b 90c : $4
- 50 c'80 cm : 1

2 m d'0

- 7 kg : 800 g e 2
- 5 kg : 400 g f 30 min : 2 hours g 45 min : 3 hours h 4 hours : 50 min i 40 cm : 2 m : 50 cm j 80 cm : 600 mm : 2 m k 2
- 5 hours : 1

5 days l'0

- 09 km : 300 m : 1
- 2 km Example 12
- 7 Divide $500 into these given ratios using the unitary method
- a 2:3 b 3:7 c'1:1
- d 7 : 13
- 8 420 g of flour is to be divided into a ratio of 7 : 3 for two different recipes

- 9 Divide $70 into these ratios
- a 1:2:4 b 2:7:1 Example 13
- 10 Write these rates in simplest form

a 150 km in 10 hours 1 b 3000 revolutions in 1 minutes 2 1 c'15 swimming strokes in of a minute 3 d'56 metres in 4 seconds e 180 mL in 22

- 5 hours 1 f 207 heartbeats in 2 minutes 4

The correct ratio of ingredients in a recipe has to be maintained when the amount of product to be made is changed

© David Greenwood et al

- d 52 : 39 5 1 h 1 :3 6 4 l'0
- 42 : 28 3 3 :1 8 4 1

- b 5 kg costs $15
- 5 Simplify these ratios

a 6 : 30 b 8 : 20 1 2 1 1 e 1 :3 f 2 :1 4 5 2 3 i 0

M AT I C A

- 4 Find the cost of 1 kg if: a 2 kg costs $8
- 3 A car is travelling at a rate (speed) of 80 km/h

a How far would it travel in: 1 i 3 hours

- ? ii hour
- ? 2 1 iii 6 hours

? 2 b How long would it take to travel: i 400 km

- ? ii 360 km
- ? iii 20 km

- 12 Determine the best buy in each of the following

a 2 kg of washing powder for $11

- 70 or 3 kg for $16
- 5 kg of red delicious apples for $4

80 or 2

- 2 kg of royal gala apples for $7
- 4 litres of orange juice for $4
- 20 or 3 litres of orange juice for $5
- 7 GB of internet usage for $14 or 1
- 5 GB for $30
- 90 with different service providers

- 13 Find the cost of 100 g of each product below then decide which is the best buy
- a 300 g of coffee A at $10
- 80 or 220 g of coffee B at $8
- 58 b 600 g of pasta A for $7
- 50 or 250 g of pasta B for $2
- 2 kg of cereal A for $4
- 44 or 825 g of cereal B for $3
- 17 The dilution ratio for a particular chemical with water is 2 : 3 (chemical to water)

how much water is needed to dilute the chemical

- 19 Julie is looking through the supermarket catalogue for her favourite cookies-and-cream ice-cream

She can buy 2 L'of triple-chocolate ice-cream for $6

- 30 while the cookies-and-cream ice-cream is usually $5

40 for 1

- 2 L'container of cookies-andcream ice-cream for it to be of equal value to the 2 L'triple-chocolate container

- 16 When a crate of twenty 375 mL soft-drink cans is purchased,
- the cost works out to be $1
- 68 per litre

A crate of 30 of the same cans is advertised as being a saving of 10 cents per can compared with the 20-can crate

Calculate how much the 30-can crate costs

- 15 If a prize of $6000 was divided among Georgia,

- how much did each girl get

18 Amy,

Candice and Diane invested money in the ratio of 2 : 3 : 1 : 4 in a publishing company

If the profit was shared according to their investment,

- and Amy’s profit was $2400,

find the profit each investor made

- 14 Kirsty manages a restaurant

How many watermelons did she buy if,

- on one day,

the total number of watermelons and mangoes was 200

R K I NG

Example 14a

- 11 Hamish rides his bike at an average speed of 22 km/h

How far does he ride in: 1 3 a 2 hours

- ? b hours
- ? c'15 minutes

- 23 If a : b is in simplest form,

state whether the following are true or false

- a a and b must both be odd
- b a and b must both be prime

c'At least one of a or b is odd

- d'The HCF of a and b is 1
- 24 A ratio is a : b with a < b and a and b are positive integers

Write an expression for the: a total number of parts b fraction of the smaller quantity out of the total c'fraction of the larger quantity out of the total

Enrichment: Mixing drinks 25 Four jugs of cordial have a cordial-to-water ratio as shown and a given total volume

Total volume

1 2 3 4

- 1:5 2:7 3:5 2:9
- 600 mL 900 mL 400 mL 330 mL

a How much cordial is in: i jug 1

- ? ii jug 2

? b How much water is in: i jug 3

- ? ii jug 4

? c'If jugs 1 and 2 were mixed together to give 1500 mL of drink: i how much cordial is in the drink

? ii what is the ratio of cordial to water in the drink

? d'Find the ratio of cordial to water if the following jugs are mixed

i Jugs 1 and 3 ii Jugs 2 and 3 iii Jugs 2 and 4 iv Jugs 3 and 4 e Which combination of two jugs gives the strongest cordialto-water ratio

© David Greenwood et al

- 5 kg of cereal A costs $4

80 and 1

- 5 kg of cereal B costs $2

Write down at least two different methods to find which cereal is a better buy

Cambridge University Press

R K I NG

- 21 A quadrilateral (with angle sum 360°) has interior angles in the ratio 1 : 2 : 3 : 4

Find the size of each angle

- 20 The ratio of the side lengths of one square to another is 1 : 2

Find the ratio of the areas of the two squares

M AT I C A

R EVI S I ON Stage

the interest given on term deposits and discounts on goods

We know from our previous studies that a percentage is a fraction that has a denominator of 100

‘Per cent’ comes from the Latin word per centum and means ‘out of 100’

- 5% 1 • Tom 4 • Adam 0
- 26 • Mai The left over a Which person gets the most cake and why

? b How much cake does Mai get

? What is her portion written as a percentage,

- decimal and fraction

- multiply by 100%

- divide by 100%

x% x x% = = 100% 100 A percentage of a number can be found using multiplication

- 25 × $26 = $6
- 50 To find an original amount,

use the unitary method or use division

For example: if 3% of an amount is $36: – Using the unitary method: 1% of the amount is $36 ÷ 3 = $12 ∴ 100% of the amount is $12 × 100 = $1200 – Using division: 3% of the amount is $36 0

- 03 × amount = $36 amount = $36 ÷ 0

03 = $1200

decimals and fractions a Write 0

- 45 as a percentage
- b Write 25% as a decimal
- 1 c'Write 3 % as a fraction
- 4 SOLUTIO N

EXPL ANATI ON

- 45 × 100% = 45%

Multiply by 100%

- b 25% = 25% ÷ 100% = 0

This moves the decimal point 2 places to the left

- 1 1 c'3 % = 3 % ÷ 100% 4 4 13 1 = × 4 100 13 = 400

Divide by 100%

- 50 as a percentage

SOLUTIO N

- 50 × 100% 250 5 = 20%
- 50c out of $2

- 50 = 250c) and write as a fraction

Multiply by 100%,

- cancelling first

SOLUTIO N

EXPL ANATI ON 3

- 15 × $35 20 100 105 = 20 = $5
- 15% of $35 =

Chapter 1 Computation and financial mathematics

Example 18 Finding the original amount Determine the original amount if 5% of the amount is $45

Method B

- 05 × amount = $45 amount = $45 ÷ 0

05 = $900

To use the unitary method,

find the value of 1 part or 1% then multiply by 100 to find 100%

- 9 1 Divide these percentages by 100% to write them as fractions

- 100 Simplify where possible
- a 3% b 11% c'35% d'8%

R K I NG

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