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Year 9 Mathematics Study Program - Assumption College

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Description

Computation and financial mathematics

Chapter

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

© David Greenwood et al

Cambridge University Press

NSW Syllabus

for the Australian Curriculum Strand: Number and Algebra

Substrands: COMPUTATION WITH INTEGERS FRACTIONS,

DECIMALS AND PERCENTAGES FINANCIAL MATHEMATICS

Outcomes A student compares,

orders and calculates with integers,

applying a range of strategies to aid computation

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

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

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

Prior to the crisis,

US interest rates were lowered to about 1%,

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

House prices in the US rose about 125% in the 10 years prior to the crisis

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

At the beginning of the crisis,

US household debt as a percentage of personal income was about 130%

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

A student solves financial problems involving compound interest

2–4NA)

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

Pre-test

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

2 Write

25 b a 32

465 and 2

564 b 0

0456 and 0

89 − 3

54 ÷ 1000

2 3 + 7 7

2 3 × 3 4

1 ÷2 2

© David Greenwood et al

Cambridge University Press

Number and Algebra

1A Computation with integers

R EVI S I ON Stage

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),

π and 2

All the numbers in our number system,

not including imaginary numbers,

Let’s start: Special sets of numbers Here are some special groups of numbers

Can you describe what special property each group has

? Try to use the correct vocabulary,

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

counting stock and measuring produce

Key ideas

The integers include …,

… 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),

© David Greenwood et al

Cambridge University Press

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

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

– Prime numbers have only two factors,

The number 1 is not a prime number

– Composite numbers have more than two factors

Order of operations – Deal with brackets first

– Do multiplication and division next,

– Do addition and subtraction last,

Example 1 Operating with integers Evaluate the following

a −2 − (−3 × 13) + (−10)

SOLUTION

EXPLANATION

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

Deal with the operations in brackets first

−a − (−b) = −a + b a + (−b) = a − b

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

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

REVISION

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

Exercise 1A

(−20 ÷ (−4) + (−3)) × 2

Key ideas

Chapter 1 Computation and financial mathematics

M AT I C A

Number and Algebra

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

8 b 100,

a 23 − 16 d'(−2)3 ÷ (−4) g

27 − 81

−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))

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

M AT I C A

R K I NG

M AT I C A

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

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

Cambridge University Press

M AT I C A

? −6 × 4 − (−7) + (−1)

© David Greenwood et al

R K I NG

Example 1

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

−6 − 2 −6 + (−10) −21 × 4 −36 ÷ 6

R K I NG

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

R K I NG

Margaret visits her daughter in Paris every 28 days

Mildred visits her son in Paris every 36 days

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

Chapter 1 Computation and financial mathematics

M AT I C A

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

What is the largest number of players in a group if the group size for both squads is the same

R K I NG

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

write down the possible values 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

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,

i Show that 6 is a perfect number

ii There is one perfect number between 20 and 30

Find the number

iii The next perfect number is 496

Show that 496 is a perfect number

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

The first two numbers in the sequence are 0 and 1

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

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

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

The time for a 100-metre sprint race,

94 seconds

Due to the experimental nature of science and engineering,

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

The number of cubic metres of gravel required for a road,

might be calculated as 3485 but is rounded to 3500

This number has been written using two significant figures

Let’s start: Plausible but incorrect

Key ideas

Johnny says that the number 2

What is wrong with Johnny’s theory

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

For example: – To two decimal places,

54 and 32

2 and 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

For example,

these numbers are all rounded to three significant figures: 2

5391 ≈ 2

© David Greenwood et al

Cambridge University Press

Number and Algebra

Example 2 Rounding to a number of decimal places Round each of these to two decimal places

1793 b 0

SOLUTIO N

EXPLANATI ON

The number after the second decimal place is 9,

so round up (increase the 7 by 1)

The number after the second decimal place is 4,

8972 ≈ 4

The number after the second decimal place is 7,

Increasing by 1 means 0

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

SOLUTIO N

EXPLANATI ON

The first two digits are the first two significant figures

The third digit is 6,

Replace the last two non-significant digits with zeros

The first two digits are the first two significant figures

The third digit is 0,

Locate the first non-zero digit,

So 4 and 0 are the first two significant figures

The next digit is 5,

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

EXPLANATI ON

27 + 1329

Round each number to one significant figure and evaluate

Recall multiplication occurs before the addition

The estimated answer is reasonable

By calculator (to one decimal place): 27 + 1329

0064 = 35

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

R K I NG

? b Is 266 closer to 260 or 270

04 or 0

Exercise 1B

M AT I C A

88 or 5

443 or 0

44302 e 0

00197 or 0

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

814 b 73

148 c'129

Example 4

00245 g 2

024 h 0

0006413

Check how reasonable your answer is with a calculator

8 × 42

2 d'965

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

962 b 11

082 c'72

986 e 63

925 f 23

807 g 804

5272 i 821

Example 2

M AT I C A

Number and Algebra

R K I NG

M AT I C A

94 × 11

Describe the type of rounding that has taken place if the answer is: a 900 b 893

Posts are used in the corners and are evenly spaced along the sides

Find the distance between each post

Write your answer in metres rounded to the nearest centimetre

How much soil does each garden bed get

? Write your answer in tonnes rounded to the nearest kilogram

© David Greenwood et al

Cambridge University Press

Team A’s time is 54

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

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

Minute amounts of reagents are commonly used in chemistry laboratories

Enrichment: nth decimal place 17 a b

Can you find any fraction whose decimal representation is non-terminating and has no pattern

Express

© David Greenwood et al

Cambridge University Press

R K I NG

Chapter 1 Computation and financial mathematics

M AT I C A

Number and Algebra

1C Rational numbers

R EVI S I ON Stage

Under the guidance of Pythagoras around 500 bc,

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

These special numbers,

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

So to write these numbers exactly,

you need to use special symbols such as 2 and π

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:

Let’s start: Approximating π

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

Real numbers Rational numbers (Fractions) (e

Irrational numbers (e

phi (ϕ) − the golden ratio)

Terminating decimals (e

Recurring decimals Non-terminating non-recurring decimals (e

345) (e

Equivalent fractions have the same value

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

© David Greenwood et al

Cambridge University Press

Key ideas

To simplify calculations,

the ancient and modern civilisations have used fractions to approximate π

To 10 decimal places,

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

Chapter 1 Computation and financial mathematics

Fractions can be compared using a common denominator

This should be the lowest common multiple of both denominators

A dot or bar is used to show a pattern in a recurring decimal number

For example: 1 = 0

27 or 0

27 6 11

Example 5 Writing fractions as decimals Write these fractions as decimals

3 a 3 8

SOLUTIO N

EXPL ANATI ON

3 06 0 4 0

Find a decimal for

3 3 = 3

Divide 13 into 5 and continue until the pattern repeats

Add a bar over the repeating pattern

384615 13

Writing 0

Example 6 Writing decimals as fractions Write these decimals as fractions

Write as a fraction using the smallest place value (hundredths) then simplify using the HCF of 4

The smallest place value is thousandths

Simplify to an improper fraction or a mixed number

© David Greenwood et al

Cambridge University Press

Number and Algebra

Example 7 Comparing fractions Decide which fraction is larger

LCM of 12 and 15 is 60

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

Write each fraction as an equivalent fraction using the common denominator

Then compare numerators (i

REVISION

R K I NG

Exercise 1C

SOLUTION

M AT I C A

3 9 = 7

125 1000

5 20 = 6 9

2 5 37 16

© David Greenwood et al

Example 5b

Example 5a

Cambridge University Press

R K I NG

3 = 5 15

M AT I C A

375 l'2

26 11 ,

8 23 7 ,

15 40 12

R K I NG

Example 7

075 i 2

005 j 10

M AT I C A

3 5 7 ,

8 12 18

1 5 3 ,

6 24 16

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 ,

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

The chance of rain can be expressed as a decimal,

Find the next two fractions in the pattern

© David Greenwood et al

Cambridge University Press

R K I NG

Example 6

Chapter 1 Computation and financial mathematics

M AT I C A

Number and Algebra

R K I NG

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

272 727

Let x = 0

6666…

272 727…

2727…

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

6 = ∴ 1

© David Greenwood et al

Cambridge University Press

R K I NG

M AT I C A

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

Write it as an improper 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

a c'If is a fraction in simplest form,

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

determine what values x can take in the following

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

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

M AT I C A

Chapter 1 Computation and financial mathematics

1D Computation with fractions

R EVI S I ON Stage

Computation with integers can be extended to include rational and irrational numbers

The operations include addition,

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

Fractions are all around you and part of everyday life

Key ideas

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

To add or subtract fractions,

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

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’

For example: 6 can be written as 6

© David Greenwood et al

Cambridge University Press

Number and Algebra

Example 8 Adding and subtracting fractions Evaluate the following

EXPL ANATI ON The lowest common denominator of 2 and 5 is 10

Rewrite as equivalent fractions using a denominator of 10

Add the numerators

Change each mixed number to an improper fraction

Remember the lowest common denominator of 3 and 5 6 is 6

Change to an equivalent fraction with 3 denominator 6,

then add the numerators and simplify

Alternative method: add whole numbers and fractions separately,

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

Convert to improper fractions,

then rewrite as equivalent fractions with the same denominator

Subtract the numerators

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

Example 9 Multiplying fractions Evaluate the following

SOLUTIO N a

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

Multiply the numerators and denominators

Rewrite as improper fractions

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

Example 10 Dividing fractions Evaluate the following

SOLUTIO N a

EXPL ANATI ON

To divide by

Cancel common factors between numerators and denominators then multiply fractions

Rewrite mixed numbers as improper fractions

Multiply by the reciprocal of the second fraction

© David Greenwood et al

Cambridge University Press

Number and Algebra

© David Greenwood et al

Example 8b

M AT I C A

Example 8a

R K I NG

Cambridge University Press

R K I NG

29 3 22

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

REVISION

Exercise 1D

M AT I C A

M AT I C A

R K I NG

R K I NG

M AT I C A

If 2 2 8 tonnes are removed,

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

How long is the reading time

Cambridge University Press

Example 9

Chapter 1 Computation and financial mathematics

Number and Algebra

R K I NG

M AT I C A

Write down its reciprocal

b b b A mixed number is given by a

Write an expression for its reciprocal

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

Enrichment: Fraction operation challenge 20 Evaluate the following

⎛ 1 3⎞ ⎛ 1 3⎞ ⎜⎝ 1 5 − 4 ⎟⎠ × ⎜⎝ 1 5 − 4 ⎟⎠

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Cambridge University Press

R K I NG

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

Three-fifths of her time was spent cleaning up and the rest was spent serving customers

How much time did she spend serving customers

M AT I C A

Chapter 1 Computation and financial mathematics

1E Ratios,

R EVI S I ON Stage

Fractions,

ratios and rates are used to compare quantities

A lawnmower,

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)

Two-stroke lawnmowers run on petrol and oil mixed in a certain ratio

Let’s start: The lottery win

Key ideas

Work out how the money is to be divided

• 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

A rate compares related quantities with different units

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

For example: 50 km per 1 hour or 50 km/h

Ratios and rates can be used to determine best buys when purchasing products

Example 11 Simplifying ratios Simplify these ratios

1 1 :1 2 3

SOLUTION

EXPLANATION

The HCF of 38 and 24 is 2 so divide both sides by 2

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Cambridge University Press

Number and Algebra

Write as improper fractions using the same denominator

Then multiply both sides by 6 to write as whole numbers

Multiply by 100 to convert both numbers into whole numbers

Divide by the HCF

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

SOLUTIO N

EXPL ANATI ON

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

Use the ratio 2 : 3 to get the total number of parts

Larger portion = 3 × $60 = $180

Calculate the value of 3 parts

Calculate the value of each part

Example 13 Simplifying rates Write these rates in simplest form

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

First multiply by 2 to remove the fraction

Then divide by 5 to write the rate using 1 minute

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

Example 14 Finding best buys a Which is better value

EXPLANATION

Price per kg

5 kg bag

Divide each price by the number of kilograms to find the price per kilogram

Then compare

Method B

Amount per $1

5 kg bag

Divide each amount in kilograms by the cost to find the weight per $1 spent

Then compare

85 ÷ 3

Alternatively,

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

Alternatively,

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

REVISION

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,

? e If there are 18 students in total,

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R K I NG

Exercise 1E

M AT I C A

Number and Algebra

Odometers in cars record the distance travelled

Hint: convert to the same units first

2 m d'0

5 days l'0

Find the smaller amount

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

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

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R K I NG

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

Example 11

R K I NG

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

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

M AT I C A

a 2 kg of washing powder for $11

80 or 2

Example 14b

If you have 72 litres of chemical,

how much water is needed to dilute the chemical

Concentrations of substances in water or solvents are ratios

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

40 for 1

What saving does there need to be on the price of the 1

Cambridge University Press

M AT I C A

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

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R K I NG

Leanne and Maya in the ratio of 5 : 2 : 3,

18 Amy,

Belinda,

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,

find the profit each investor made

M AT I C A

Each day she buys watermelons and mangoes in the ratio of 3 : 2

How many watermelons did she buy if,

the total number of watermelons and mangoes was 200

R K I NG

Example 14a

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

Chapter 1 Computation and financial mathematics

Number and Algebra

M AT I C A

state whether the following are true or false

c'At least one of a or b is odd

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

Cordial-to-water ratio

Total volume

1 2 3 4

a How much cordial is in: i jug 1

? b How much water is in: i jug 3

? 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

80 and 1

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

Cambridge University Press

R K I NG

Find the size of each angle

R K I NG

Find the ratio of the areas of the two squares

M AT I C A

Chapter 1 Computation and financial mathematics

1F Computation with percentages and money

R EVI S I ON Stage

We use percentages for many different things in our daily lives

Some examples are loan rates,

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’

Let’s start: Which is the largest piece

Key ideas

Four people receive the following portions of a cake: • Milly 25

? b How much cake does Mai get

? What is her portion written as a percentage,

To change a decimal or a fraction into a percentage,

To change a percentage into a fraction or decimal,

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

For example: 25% of $26 = 0

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 = $1200

© David Greenwood et al

Cambridge University Press

Number and Algebra

Example 15 Converting between percentages,

decimals and fractions a Write 0

EXPL ANATI ON

Multiply by 100%

This moves the decimal point 2 places to the right

Divide by 100%

This moves the decimal point 2 places to the left

Divide by 100%

Write the mixed number as an improper fraction and 1 multiply by the reciprocal of 100 (i

Example 16 Writing a quantity as a percentage Write 50c out of $2

SOLUTIO N

EXPL ANATI ON 1

Convert to the same units ($2

Multiply by 100%,

Example 17 Finding a percentage of a quantity Find 15% of $35

SOLUTIO N

EXPL ANATI ON 3

Write the percentage as a fraction out of 100 and multiply by $35

Note: ‘of’ means to ‘multiply’

© David Greenwood et al

Cambridge University Press

Chapter 1 Computation and financial mathematics

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

EXPLANATION

Method B

Division 5% of amount = $45 0

05 = $900

Exercise 1F

To use the unitary method,

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

Write 5% as a decimal then divide both sides by this number to find the original amount

REVISION

For example,

R K I NG

Method A

Unitary 5% of the amount = $45 1% of the amount = $9 100% of the a