number - haesemathematics.com · find the hcf of 24 and 40: ... 5 find the lcm of: a 5, 8 b 4, 6 c...

Contents: A

B

C

D

E

F

G

Natural numbers

Divisibility tests

Integers and order of operations

Rational numbers (fractions)

Decimal numbers

Ratio

Prime numbers and index notation

2Number

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Y:\HAESE\SA_09-6ed\SA09-6_02\031SA09-6_02.CDR Wednesday, 18 October 2006 2:40:44 PM PETERDELL

OPENING PROBLEM “POCKET MONEY”

�

32 NUMBER (Chapter 2)

Numbers have been part of human thinking since man lived in caves. Today, we make ma-

chines to crunch numbers; we live in numbered streets, have telephone numbers, registration

numbers and tax-file numbers; we use numbers to measure the universe and plot courses

through time and space; we are “tagged” with a number when we are born and often after we

die!

An understanding of numbers and how to operate with them is an essential part of daily living.

Consider the following questions:

1 What is the ratio of the work Tim does to the work Becky does?

2 What fraction of the work does Tim do?

3 How much will they be paid in weeks 4, 5 and 6?

4 If they continue for 20 weeks, what would each person be paid for the final week?

5 How much would each person be paid for doing the dishes for a 20 week period?

The natural numbers are the counting numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, ......

The set of natural numbers is endless. As there is no largest natural number, we say that the

set of natural numbers is infinite.

The factors of a natural number are all the natural numbers which divide exactly into

it, leaving no remainder.

For example, the factors of 10 are: 1, 2, 5 and 10.

A number may have many factors. When a number is written as a product of factors it is

factorised.

For example, consider the number 20.

Tim and his sister Becky have agreed to dothe dishes for their parents from Monday toFriday. Tim does them on Monday,Wednesday and Friday, leaving Becky to

do them on Tuesday and Thursday. They negotiatewith their parents to be paid in the following way:cents for the first week, cents for the second week,cents for the third week and so on.

24 8

In this chapter we review integers, fractions, decimals, and ratio, and also investigateindex form.

NATURAL NUMBERSANATURAL NUMBERS

FACTORS


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Y:\HAESE\SA_09-6ed\SA09-6_02\032SA09-6_02.CDR Wednesday, 23 August 2006 10:18:05 AM PETERDELL

NUMBER (Chapter 2) 33

20 has factors 1, 2, 4, 5, 10, 20 and can be factorised into pairs, i.e., 1£20, 2£10, 4£5.

20 may also be factorised as the product of 3 factors, i.e., 20 = 2 £ 2 £ 5:

A multiple of any natural number is obtained by multiplying it by another natural number.

For example, the multiples of 3 are: 3, 6, 9, 12, 15, 18, ...... and these are obtained by

multiplying 3 by each of the natural numbers in turn,

i.e., 3 £ 1 = 3, 3 £ 2 = 6, 3 £ 3 = 9, 3 £ 4 = 12, etc.

1 List all the factors of:

a 9 b 12 c 19 d 60e 23 f 48 g 49 h 84

2 List the first five multiples of:

a 4 b 7 c 9 d 15

a Find the largest multiple of 9 less than 500.

b Find the smallest multiple of 11 greater than 1000.

a 9 50505 5 with 5 remainder

b 11 100109 0 with 10 remainder

So, the largest multiple is So, the smallest multiple is

9 £ 55 = 495: 11 £ 91 = 1001:

3 a Find the largest multiple of 7 which is less than 1000.

b Find the smallest multiple of 13 which is greater than 1000.

c Find the largest multiple of 17 which is less than 2000.

d Find the smallest multiple of 15 which is greater than 10 000.

The HCF (highest common factor) of two or more natural numbers is the largest

factor which is common to both of them.

MULTIPLES

EXERCISE 2A

Self TutorExample 1

Find the HCF of 24 and 40:

The factors of 24 are: f1, 2, 3, 4, 6, 8, 12, 24gThe factors of 40 are: f1, 2, 4, 5, 8, 10, 20, 40g) the HCF of 24 and 40 is 8.

Self TutorExample 2


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Y:\HAESE\SA_09-6ed\SA09-6_02\033SA09-6_02.CDR Friday, 15 September 2006 11:11:53 AM PETERDELL

DIVISIBILITY TESTSB

4 Find the HCF of:

a 6, 8 b 12, 18 c 24, 30 d 72, 120

e 6, 12, 15 f 20, 24, 36 g 12, 18, 27 h 24, 72, 120

The LCM (lowest common multiple) of two or more natural numbers is the smallest

multiple which is common to both of them.

Find the LCM of 6 and 8:

The multiples of 6 are: f6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ....gThe multiples of 8 are: f8, 16, 24, 32, 40, 48, 56, ....g) the common multiples of 6 and 8 are: 24, 48, .....

) the LCM of 6 and 8 is 24.

5 Find the LCM of:

a 5, 8 b 4, 6 c 8, 10 d 15, 18

e 2, 3, 4 f 3, 4, 5 g 5, 9, 12 h 12, 18, 27

6 Three bells chime at intervals of 4, 5 and 6 seconds respectively. If they all chime at

the same instant, how long before they all chime together again?

The following divisibility tests should be kept in mind when looking for prime factors:

A natural number is divisible by 2 if the last digit is even

3 if the sum of the digits is divisible by 3

4 if the last two digits are divisible by 4

5 if the last digit is 0 or 5

6 if the number is even and divisible by 3.

1 Which of the following are divisible by:

i 3 ii 4 iii 5 iv 6?

a 1002 b 12 345 c 2816 d 123 210

e 861 f 6039 g 91 839 h 123 456789

One number is by another if, when we divide, the answer isdivisible a natural number

(whole number).


EXERCISE 2B

Self TutorExample 3


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Y:\HAESE\SA_09-6ed\SA09-6_02\034SA09-6_02.CDR Wednesday, 23 August 2006 4:49:23 PM PETERDELL


2 Find ¤ if the following are divisible by 2:

a 43¤ b 592¤ c 3¤6 d ¤13


a 31¤ b 2¤3 c ¤42 d 32¤5


a 42¤ b 3¤4 c 514¤ d 68¤0


a 39¤ b 896¤ c 73¤5 d 64¤2


a 42¤ b 55¤ c 6¤8 d 41¤2

7 Find the digits X and Y if the number of form ‘X7Y6’ is divisible by 24.

8 Find the largest three digit number divisible by 3 and 4.

9 Find the smallest number with a remainder of 1 when it is divided by 3, 4 and 5.

10 How many three digit numbers are divisible by 7?

11 P Q R S is a four digit number. Find the number given the following clues:

Clue 1: All digits are different.

Clue 2: The product of digits R and S equals digit P.

Clue 3: Digit Q is one more than P.

Clue 4: The sum of all digits is 18:

Clue 5: The four digit number is divisible by 11.

Find ¤ if 53¤ is divisible by:

a 2 b 5 c 4 d 3

a To be divisible by 2, ¤ must be even.

) ¤ = 0, 2, 4, 6 or 8

b To be divisible by 5, ¤ must be 0 or 5.

) ¤ = 0 or 5

c To be divisible by 4, ‘3¤’ must be divisible by 4.

) ¤ = 2 or 6

d To be divisible by 3, 5 + 3 + ¤ must be divisible by 3.

) 8 + ¤ must be divisible by 3.

) ¤ = 1, 4 or 7 f

f

as

as and are divisible by32 36 4

the number must be 9, 12 or 15g

g

Check youranswers using

your calculator!

Self TutorExample 4


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INTEGERS AND ORDER OF OPERATIONSC

i.e., ::::::¡ 5, ¡4, ¡3, ¡2, ¡1, 0, 1, 2, 3, 4, 5 ::::::

We can show these numbers on a number line. Zero is neither positive nor negative.

+ (positive) = (positive)

¡ (positive) = (negative)

+ (negative) = (negative)

¡ (negative) = (positive)

1 Find the value of:

a 13 ¡ 8

e ¡13 ¡ 8

i 16 + 25

m ¡16 + 25

b 13 + ¡8

f ¡13 ¡¡8

j 16 ¡ 25

n ¡16 + ¡25

c 13 ¡¡8

g 8 ¡ 13

k 16 + ¡25

o ¡16 ¡¡25

d ¡13 + 8

h 13 + 8

l 16 ¡¡25

p 25 ¡ 16

(positive) £ (positive) = (positive)

(positive) £ (negative) = (negative)

(negative) £ (positive) = (negative)

(negative) £ (negative) = (positive)

Simplify:

a 4 + ¡9 b 4 ¡¡9 c ¡3 + ¡5 d ¡3 ¡¡5

a 4 + ¡9 b 4 ¡¡9 c ¡3 + ¡5 d ¡3 ¡¡5

= 4 ¡ 9 = 4 + 9 = ¡3 ¡ 5 = ¡3 + 5

= ¡5 = 13 = ¡8 = 2

We have previouslydeveloped the following

for handlingand of :rules addition

subtraction integers

We can classify integers as follows:

zero

integers

negative integersnatural numbers

(i.e., positive integers)

�5 ��

The negative natural numbers, zero and the positivenatural numbers form the set of all integers,

EXERCISE 2C

Self TutorExample 5

The following rules have also been developedfor multiplication with integers:


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Find the value of:

a 3 £ 4 b 3 £¡4 c ¡3 £ 4 d ¡3 £¡4

a 3 £ 4 = 12 b 3 £¡4 = ¡12 c ¡3 £ 4 = ¡12 d ¡3 £¡4 = 12


a 6 £ 7 b 6 £¡7 c ¡6 £ 7 d ¡6 £¡7

e 5 £ 8 f 5 £¡8 g ¡5 £ 8 h ¡5 £¡8

Simplify:

a ¡42 b (¡4)2 c ¡23 d (¡2)3

a ¡42 b (¡4)2 c ¡23 d (¡2)3

= ¡4 £ 4 = ¡4 £¡4 = ¡2 £ 2 £ 2 = ¡2 £¡2 £¡2

= ¡16 = 16 = ¡8 = ¡8


a ¡52 b (¡5)2 c (¡1)3 d ¡13

e 3 £¡2 £ 5 f ¡3 £ 2 £¡5 g ¡3 £¡2 £¡5 h 2 £ (¡3)2

i ¡2 £ (¡3)2 j ¡24 k (¡2)4 l (¡3)2 £ (¡2)2

(positive) ¥ (positive) = (positive)

(positive) ¥ (negative) = (negative)

(negative) ¥ (positive) = (negative)

(negative) ¥ (negative) = (positive)

Find the value of:

a 14 ¥ 2 b 14 ¥¡2 c ¡14 ¥ 2 d ¡14 ¥¡2

a 14 ¥ 2 b 14 ¥¡2 c ¡14 ¥ 2 d ¡14 ¥¡2

= 7 = ¡7 = ¡7 = 7


a 15¥ 3 b 15 ¥¡3 c ¡15 ¥ 3 d ¡15 ¥¡3

e 24¥ 8 f 24 ¥¡8 g ¡24 ¥ 8 h ¡24 ¥¡8

i4

8j

¡4

8k

4

¡8l

¡4

¡8

The followinghave also been developed

with integers:

rules

or divisionf

Self TutorExample 6

Self TutorExample 7

Self TutorExample 8


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² Perform the operations within brackets first.

² Calculate any part involving exponents.

² Starting from the left, perform all divisions

and multiplications as you come to them.

² Finally, restart from the left and perform all

additions and subtractions as you come

to them.

² If an expression contains one set of grouping symbols, i.e., brackets, work that part

first.

² If an expression contains two or more sets of grouping symbols one inside the other,

work the innermost first.

² The division line of fractions also behaves as a grouping symbol. This means that the

numerator and the denominator must be found separately before doing the division.

5 Simplify:

a 5 + 8 ¡ 3 b 5 ¡ 8 + 3 c 5 ¡ 8 ¡ 3

d 2 £ 10 ¥ 5 e 10 ¥ 5 £ 2 f 5 £ 10 ¥ 2

The word BEDMASmay help you

remember this order.

Simplify: a 3 + 7 ¡ 5 b 6 £ 3 ¥ 2:

a 3 + 7 ¡ 5 fWork left to right as only + and ¡ are involved.g= 10 ¡ 5

= 5

b 6 £ 3 ¥ 2 fWork left to right as only £ and ¥ are involved.g= 18 ¥ 2

= 9

Simplify: a 23 ¡ 10 ¥ 2 b 3 £ 8 ¡ 6 £ 5

a 23 ¡ 10 ¥ 2 b 3 £ 8 ¡ 6 £ 5

= 23 ¡ 5 f¥ before ¡g = 24 ¡ 30 f£ before ¡g= 18 = ¡6

Self TutorExample 9

Self TutorExample 10

RULES FOR BRACKETS:

ORDER OF OPERATIONS RULES:


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6 Simplify:

a 4 + 7 £ 3 b 7 £ 2 + 8 c 13 ¡ 3 £ 2

d 5 £ 4 ¡ 20 e 33 ¡ 3 £ 4 f 15 ¡ 6 £ 0

g 2 £ 5 ¡ 5 h 60 ¡ 3 £ 4 £ 2 i 15 ¥ 3 + 2

j 8 £ 7 ¡ 6 £ 3 k 5 + 2 + 3 £ 2 l 7 ¡ 5 £ 3 + 2

Simplify:

3 + (11 ¡ 7) £ 2

3 + (11 ¡ 7) £ 2

= 3 + 4 £ 2 fwork the brackets firstg= 3 + 8 f£ before +g= 11

7 Simplify:

a 12 + (5 ¡ 2) b (12 + 5) ¡ 2 c (8 ¥ 4) ¡ 2

d 8 ¥ (4 ¡ 2) e 84 ¡ (12 ¥ 6) f (84 ¡ 12) ¥ 6

g 32 + (8 ¥ 2) h 32 ¡ (8 + 14) ¡ 7 i (32 ¡ 8) + (14 ¡ 7)

j (16 ¥ 8) ¥ 2 k 16 ¥ (8 ¥ 2) l 18 ¡ (6 £ 3) ¡ 4

Simplify:

[12 + (9 ¥ 3)] ¡ 11

[12 + (9 ¥ 3)] ¡ 11

= [12 + 3] ¡ 11 fwork

outer brackets next

the inner brackets firstg= 15 ¡ 11 f g= 4

8 Simplify:

a 8 ¡ [(2 ¡ 3) + 4 £ 3] b [16 ¡ (9 + 3)]£ 2

c 13¡ [(8¡ 3) + 6] d [16 ¡ (12 ¥ 4)] £ 3

e 200 ¥ [4 £ (6 ¥ 3)] f [(12 £ 3) ¥ (12 ¥ 3)] £ 2

Simplify:12 + (5 ¡ 7)

18 ¥ (6 + 3)

12 + (5 ¡ 7)

18 ¥ (6 + 3)

=12 + (¡2)

18 ¥ 9fworking the brackets firstg

=10

2fsimplifying numerator and denominatorg

= 5





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9 Simplify:

a75

5 £ 5b

21

16 ¡ 9c

18 ¥ 3

14 ¡ 11d

7 + 9

6 ¡ 2

e53 ¡ 21

9 ¡ 5f

3 £ 8 + 6

6g

57

7 ¡ (2 £ 3)h

(3 + 8) ¡ 5

3 + (8 ¡ 5)

10 Using £, ¥, + or ¡ only, insert symbols between the following sets of numbers so that

correct equations result:

a 9 3 2 = 8 b 9 3 2 = 25 c 9 3 2 = 5

11 Insert grouping symbols, if necessary, to make the following true:

a 8 ¡ 6 £ 3 = 6 b 120 ¥ 4 £ 2 = 15 c 120 ¥ 4 £ 2 = 60

d 5 £ 7 ¡ 3 ¡ 1 = 15 e 5 £ 7 ¡ 3 ¡ 1 = 33 f 5 £ 7 ¡ 3 ¡ 1 = 19

g 3 + 2 £ 8 ¡ 4 = 36 h 3 + 2 £ 8 ¡ 4 = 11 i 3 + 2 £ 8 ¡ 4 = 15

In this course it is assumed that you have a scientific calculator. If you learn how to operate

your calculator successfully, you should experience little difficulty with future arithmetic

calculations.

There are many different brands (and types) of calculators. Different calculators do not have

exactly the same keys. It is therefore important that you have an instruction booklet for your

calculator, and use it.

Most modern calculators have the Order of Operations

rules built into them.

For example, consider 5 £ 3 + 2 £ 5:

If you key in 5 3 2 5 the calculator

gives an answer of 25, which is correct.

The calculator has in fact performed the two multiplications

before the addition.

However, if we consider12

4 + 2and key in 12 4 2 the calculator gives

an answer of 5, which is incorrect.

The calculator has divided 12 by 4 to give 3 and then 2 has been added to give 5.

Fortunately, many of the latest calculators also have grouping symbol keys, which allow this

second example to be done without needing to use the memory.

There are two grouping symbols keys: Left hand bracket

Right hand bracket

Try to rememberBEDMAS!

+

+

×× =

=÷

(

)

USING YOUR CALCULATOR WITH INTEGERS


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Let us once again consider the example12

4 + 2:

If we key in 12 4 2 we obtain the answer 2, which is correct.

In this case the calculator has divided 12 by the sum of 4 and 2.

The (or ) key is used to enter negative numbers into the calculator.

To enter ¡5 into the calculator, key in 5 or 5 and ¡5 will appear on the

display.

Calculate: a 41 £¡7 b ¡18 £ 23

a Key in 41 7 Answer: ¡287

b Key in 18 23 Answer: ¡414

12 Evaluate each of the following using your calculator:

a 17 + 23 £ 15 b (17 + 23) £ 15 c 128 ¥ 8 + 8

d 128 ¥ (8 + 8) e 34 £¡8 f ¡64 ¥¡16

g89 + ¡5

¡7 £ 3h ¡25 + 32 ¥¡4 i

¡15 ¡ 5

6 ¡ (8 ¥ 4)

Rational numbers appear in many forms.

For example, 4, ¡2, 0, 10%, ¡47 , 1:3, 0:6 are all rational numbers,

i.e., they can be written in the forma

bas: 4

1 , ¡21 , 0

1 , 110 , ¡4

7 , 1310 , 2

3 :

+�

+�

( )�

( )�

Calculate: a 12 + 32 ¥ (8 ¡ 6) b75

7 + 8

a 12 32 8 6 Answer: 28

b 75 7 8 Answer: 5

+

+

� =

=÷

÷ (

( )

)

RATIONAL NUMBERS (FRACTIONS)D

THE SIGN CHANGE KEY

+ =÷ ( )



× =( )�

( )� × =

can be written as a ratio of two integers in the forma

b:Rational numbers


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Note that 4, ¡2 and 0 are integers, and 10%, ¡47 , 1:3 and 0:6 are fractions.

We can extend our classification of numbers

as shown alongside:

45 is a proper fraction fas the numerator is less than the denominatorg76 is an improper fraction fas the numerator is greater than the denominatorg

234 is a mixed number fas it is really 2 + 3

4g12 , 3

6 are equivalent fractions fas both fractions represent equivalent portionsg

To add (or subtract) two fractions we convert each one to an equivalent fraction with

a common denominator and add (or subtract) the new numerators.

1 Find:

a 314 + 6

14 b 716 + 2

16 c 58 + 3

4 d 25 + 1

6

e 23 + 4 f 11

2 + 58 g 11

2 + 216 h 21

3 + 334

45

numerator

denominatorbar (which also means )divide

rational numbers

integers fractions

negative integers zero positive integers

A consists oftwo whole numbers, aand a , separated by abar symbol. For example,

common fraction

numerator

denominator

TYPES OF FRACTIONS

ADDITION AND SUBTRACTION

Find: 34 + 5

6

34 + 5

6 fLCD = 12g= 3£3

4£3 + 5£26£2 fto achieve a common denominator of 12g

= 912 + 10

12

= 1912

= 1 712


EXERCISE 2D


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2 Find:

a 79 ¡ 2

9 b 34 ¡ 5

6 c 47 ¡ 1

2 d 1 ¡ 58

e 5 ¡ 318 f 24

5 ¡ 113 g 21

2 ¡ 416 h 21

3 ¡ 334

To multiply two fractions, we multiply the numerators together and multiply the

denominators together and then simplify if possible, i.e.,

3 Calculate:

a 34 £ 5

6 b 47 £ 1

2 c 2 £ 34 d 2

3 £ 4

e 112 £ 5

8 f 138 £ 4

11 g (213 )2 h (11

2 )3

To divide by a number, we multiply by its reciprocal,

i.e.,

Find: 123 ¡ 14

5

123 ¡ 14

5

= 53 ¡ 9

5 fwrite as improper fractionsg= 5£5

3£5 ¡ 9£35£3 fto achieve a common denominator of 15g

= 2515 ¡ 27

15

= ¡215 or ¡ 2

15

Find: a 14 £ 2

3 b (312)2

a 14 £ 2

3 b (312 )2

= 14 £ 2

1

2 3 = 312 £ 31

2

= 16 = 7

2 £ 72

= 494 or 121

4

Remember to cancelany common factorsbefore completingthe multiplication.


MULTIPLICATION


DIVISION

a

b

¥c

d

=

a

b

£d

c

.

a

b

£c

d

=

ac

bd

.


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4 Evaluate:

a 34 ¥ 5

6 b 34 ¥ 2

3 c 712 ¥ 3

4 d 23 ¥ 4

e 2 ¥ 34 f 11

2 ¥ 58 g 3

4 ¥ 212 h 21

3 ¥ 334

5 Calculate:

a 438 + 22

5 b (23)4 c 6 ¡ 3 £ 34 d 3

4 £ 112 ¥ 2

e6 £ 3 £ 1

234

f 1 ¥ (12 + 35 ) g 1 ¥ 1

2 + 35 h

4 ¡ 12

3 £ 23

6 Solve the following problems:

a Bob eats 14 of a pie and later eats 2

5 of the pie. What fraction remains?

b The cost of a Holden is 213 of the cost of a Rolls Royce. If the current price of a

Rolls Royce is $211 250, what is the cost of a Holden?

c The price of a beach coat is 57 of the price of a matching swimsuit. What does the

beach coat cost if the swimsuit sells for $52:50?

d A family spends 13 of its weekly budget on rent, 1

4 on food, 18 on clothes, 1

12 on

entertainment and the remainder is banked.

How much is banked if the weekly income is $864:72?

Remember that‘of ’ means ‘ ’.£

c

d

d

c

The reciprocal

of is !Find: a 3 ¥ 2

3 b 213 ¥ 2

3

a 3 ¥ 23 b 21

3 ¥ 23

= 31 ¥ 2

3 = 73 ¥ 2

3

= 31 £ 3

2 = 73 £ 3

2

= 92 = 7

2

= 412 = 3 1

2

1

1


Anna scores 35 of her team’s goals in a netball match.

How many goals did she shoot if the team shot 70 goals?

Anna shoots 35 of 70 = 3

5 £ 70

=3 £ 70

5

= 42 goals.1

14



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e Renee used 13 of a length of pipe and later used 2

3 of what remained.

What fraction of the pipe is left?

f 910 of the weight of a loaf of bread comes from the flour used in making the bread.

If 29 of the weight of the flour is protein, what fraction of the weight of a loaf of

bread is protein?

g A farmer has 364 ewes and each ewe has either one

or two new-born lambs. If there are 468 lambs in

total, what fraction of the ewes have twin lambs?

7 A tree is losing its leaves. Two thirds fall off in the

first week, two thirds of those remaining fall off in the

second week and two thirds of those remaining fall off

in the third week. Now there are 37 leaves. How many

leaves did the tree have originally?

The fraction key is used to enter common fractions into the calculator.

8 Find, using your calculator:

a 13 + 1

4 b 13 + 5

7 c 38 ¡ 2

7 d 14 ¡ 1

3

e 29 £ 3

2 f 57 £ 2

3 g 67 £ 1

5 h 35 ¥ 2

9

i 114 + 21

2 £ 34 j 13

7 £ 218 + 11

4 k 317 ¥ (23

4 £ 47)

FRACTIONS ON A CALCULATOR

a

Find, using your calculator: a 14 ¡ 2

3 b 1 112 + 21

4 c 14 ¥ 2

3

a 14 ¡ 2

3

Key in 1 4 2 3 Display Answer: ¡ 512

b 1 112 + 21

4

Key in 1 1 12 2 1 4

Display Answer: 313

c 14 ¥ 2

3

Key in 1 4 2 3 Display Answer:38

+

� =

=

=÷

a

a

a a

aa a

a

3 8

3 1 3

-5 12

You may perform operations on fractions using your calculator, but you rely onyour calculator and forget how to manually perform operations with fractions.

must not



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The number 4:63 is a quick way of writing 4 + 610 + 3

100 ,

which can also be written as an improper 463100 or as a mixed number 4 63

100 .

Likewise, 14:062 is the quick way of writing 14 + 6100 + 2

1000 .

Numbers such as 4:63 are commonly called decimal numbers.

a Write 5:704 in expanded fractional form.

b Write 3 + 210 + 4

100 + 110 000 in decimal form.

c State the value of the digit 6 in 0:036 24

a 5:704

= 5 + 710 + 4

1000

b 3 + 210 + 4

100 + 110 000

= 3:2401

c In 0:036 24, the 6 stands for 61000 .

1 Write the following in expanded fractional form:

a 2:5 b 2:05 c 2:0501 d 4:0052 e 0:0106

2 Write the following in decimal form:

a 3 + 210 b 7

10 + 8100 c 6

10 + 31000

d 7100 + 9

1000 e 4 + 110 000 f 5 + 3

100 + 210 000

3 State the value of the digit 6 in the following:

a 7608 b 762 c 0:619 d 0:0762 e 0:000 164

4 Evaluate:

a 10:76 + 8:3 b 16:21 + 13:84 c 16:21 ¡ 13:84

d 2:5 ¡ 0:6 e 12 ¡ 7:254 f 0:26 + 3:09 + 0:985

g 0:0039 + 0:471 h 7:9 ¡ 8:6 i 0:25 + 0:087 ¡ 0:231

j 0:421 ¡ 1 k ¡0:258 + 3 l ¡2:7 ¡ 3:61

DECIMAL NUMBERSE

DECIMAL NUMBERS


EXERCISE 2E

fraction


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5 Evaluate:

a 13:7 £ 100 b 0:5 ¥ 100 c 15 ¥ 0:5 d 0:5 £ 8

e 0:5 £ 0:08 f 3000 £ 0:6 g 3:6 £ 0:6 h 1 ¥ 0:02

i (0:3)3 j 500 £ (0:2)2 k 0:64 ¥ 160 l 0:0775 ¥ 2:5

6 Write the following in simplest fraction form:

a 0:5 b 0:6 c 3:25d 0:625 e 0:0375 f 0:0084

7 Write the following in decimal form:

a 12 b 3

4 c 35 d 17

50 e 23

f 29 g 9

40 h 56 i 7

8 j 1780

k 37125 l 5

12 m 320 n 4

11 o 1125

Write 0:075 in simplest fraction form. 0:075 = 751000

= 75¥251000¥25

= 340

Evaluate:

a 24 £ 0:8 b 3:6 ¥ 0:02

a 24 £ 8 = 192 fdelete decimal point, by £10g) 24 £ 0:8 = 19:2 f¥10 by shifting decimal 1 place leftg

b 3:6 ¥ 0:02

= 3:60 ¥ 0:02 fshift both decimal points the same

= 360 ¥ 2 number of places to the rightg= 180



Write the following in decimal form: a 740 b 3

7

a 740

= 7 ¥ 40

= 0:175

b 37

= 3 ¥ 7

= 0:428571

fCalculator: 7 40 g=÷


When fractions areconverted to decimalsthey either terminate

or recur.


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a

b Sliced ham sells at $13:60 per kg. How much would 500 g of ham cost?

c Legs of lamb cost $4:85 per kg. How much would a leg weighing 4 kg cost?

d The cost of electricity is 43:4 cents per kilowatt for each hour. What would

i 500 kilowatts of electricity cost for one hour

ii it cost to burn a 60 watt globe for 10 hours?

e At a milk processing factory, cartons are filled

from large refrigerated tanks that each hold 8400litres. How many cartons can be filled from one

tank given that each carton holds 600 mL?

f

g How many dozen bottles of wine can be filled from

a 9000 litre fermentation tank if each bottle holds

750 mL?

h A manufacturer wishes to make brass medallions,

each of mass 17:5 g, from 140 kg of brass. How

many medallions can be made?

Sometimes we wish to round numbers to a certain number of decimal places.

If, for example, an answer correct to 3 decimal places is required, we are interested

in the number in the 4th decimal place.

² If the number in the 4th decimal place is 0, 1, 2, 3 or 4, leave the first 3 decimal

digits unchanged.

² If the number in the 4th decimal place is 5, 6, 7, 8, or 9, increase the third decimal

digit by one.

ROUNDING DECIMAL NUMBERS

RULES FOR ROUNDING OFF DECIMALS

A amotorist used litres of petrol over period of one week. What did it cost herfor petrol selling at cents per litre?

40131:7

A alarge mixer at bakery holds tonnes ofdough. How many bread rolls, each of mass

g, can be made from the dough in the mixer?

0:75

150

Jon bought four drinks for $1:30 each. How much

change did he get from a $10 note?

cost = $1:30 £ 4

= $5:20

) change = $10 ¡ $5:20

= $4:80



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Y:\HAESE\SA_09-6ed\SA09-6_02\048SA09-6_02.CDR Friday, 15 September 2006 12:50:21 PM PETERDELL

RATIOF


9 Find, giving your answers correct to 2 decimal places where necessary:

a (16:8 + 12:4) £ 17:1 b 16:8 + 12:4 £ 17:1 c 127 ¥ 9 ¡ 5

d 127 ¥ (9 ¡ 5) e 37:4 ¡ 16:1 ¥ (4:2 ¡ 2:7) f16:84

7:9 + 11:2

g27:4

3:2¡ 18:6

16:1h

27:9 ¡ 17:3

8:6+ 4:7 i

0:0768 + 7:1

18:69 ¡ 3:824

A ratio is an ordered comparison of quantities of the same kind.

Calculate, to 2 decimal places:

a (2:8 + 3:7)(0:82 ¡ 0:57) b 18:6 ¡ 12:2 ¡ 4:3

5:2

a 2:8 3:7 0:82 0:57

Screen: 1:625 Answer: 1:63

b 18:6 12:2 4:3 5:2

Screen: 17:080 769 23 Answer: 17:08

+ �

� �

× =

=÷

(

(

() )

)


1 Write as a ratio, without simplifying your answer:

a $10 is to $7 b 2 L is to 5 L c 80 kg is to 50 kg

d $2 is to 50 cents e 500 mL is to 2 L f 800 m is to 1:5 km

Write as a ratio, without simplifying your answer:

a Jack has $5 and Jill has 50 cents.

b Mix 200 mL of cordial with 1 L of water.

a Jack : Jill = $5 : 50 cents fwrite in the correct orderg= 500 cents : 50 cents fwrite in the same unitsg= 500 : 50 fexpress without unitsg

b cordial : water = 200 mL : 1 L fwrite in the correct orderg= 200 mL : 1000 mL fwrite in the same unitsg= 200 : 1000 fexpress without unitsg


EXERCISE 2F


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If we have a ratio and multiply or divide both parts by the same non-zero number,

we obtain an equal ratio.

Express the following ratios in simplest form:

a 45 : 15 b 212 : 1

2 c 0:4 : 1:4

a 45 : 15

= 45 ¥ 15 : 15 ¥ 15

= 3 : 1

b 212 : 1

2

= 52 : 1

2

= 52 £ 2 : 1

2 £ 2

= 5 : 1

c 0:4 : 1:4

= 0:4 £ 10 : 1:4 £ 10

= 4 : 14

= 4 ¥ 2 : 14 ¥ 2

= 2 : 7

2 Express the following ratios in simplest form:

a 3 : 6 b 34 : 1

4 c 0:5 : 0:2 d 18 : 24

e 212 : 11

2 f 1:5 : 0:3 g 20 c to $1:20 h 2 L to 500 mL

3 Use your calculator to find 123 ¥ 11

3 . Simplify 123 : 11

3 . What do you notice?

4 Use your calculator to simplify the ratios:

a 13 : 1

2 b 212 : 11

3 c 314 : 2

3 d 123 : 1 3

10

Ratios are equal if they can be expressed in the same simplest form.

A proportion is a statement that two ratios are equal.

5 Find ¤ if:

a 4 : 5 = 12 : ¤ b 3 : 9 = ¤ : 18 c 2 : 3 = 10 : ¤

d 5 : 10 = ¤ : 18 e 16 : 4 = 12 : ¤ f 21 : 28 = 12 : ¤

SIMPLIFYING RATIOS


EQUAL RATIOS

Find ¤ if: a 3 : 5 = 6 : ¤ b 15 : 20 = ¤ : 16

a 3 : 5 = 6 : ¤

) ¤ = 5 £ 2

i.e., ¤ = 10

b 15 : 20 = 15 ¥ 5 : 20 ¥ 5

= 3 : 4

) 3 : 4 = ¤ : 16

) ¤ = 3 £ 4 = 12

£

£

£

£2

4

2

4



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a A hospital employs nurses and doctors in the ratio 7 : 2. If 84 nurses are employed,

how many doctors are employed?

b A farmer has pigs and chickens in the ratio 3 : 8. If she has 360 pigs, how many

chickens does she have?

c The price of a TV is reduced from $500 to $400. A DVD player costing $1250 is

reduced in the same ratio as the TV. What does the DVD player sell for?

Quantities can be divided in a particular ratio by considering the number of parts the

whole is to be divided into.


a Divide: i $50 in the ratio 1 : 4 ii $35 in the ratio 3 : 4

b A fortune of $400 000 is to be divided in the ratio 5 : 3. What is the larger share?

c The ratio of girls to boys in a school is 5 : 4. If there are 918 students at the

school, how many are girls?

The ratio of walkers to guides on a demanding bushwalk is to be 9 : 2.

How many guides are required for 27 walkers?

walkers : guides = 27 : ¤

) 9 : 2 = 27 : ¤

) ¤ = 2 ££

£

33

3

i.e., ¤ = 6

i.e., 6 guides are needed.

or 9 parts is 27

) 1 part is 27 ¥ 9 = 3

) 2 parts is 3 £ 2 = 6

i.e., 6 guides are needed.


USING RATIOS TO DIVIDE QUANTITIES

An inheritance of $60 000 is to be divided between Donny and Marie in the

ratio 2 : 3. How much does each receive?

There are 2 + 3 = 5 parts.

) Donny gets 25 of $60 000

= 25 £ 60 000

= $24 000

35 of $60 000

= 35 £ 60 000

= $36 000

and Marie gets



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INVESTIGATION 1 “POCKET-MONEY” SPREADSHEET


d A glass contains alcohol and water in the ratio 1 : 4. A second glass has the same

quantity of liquid but this time the ratio of alcohol to water is 2 : 3: Each glass

is emptied into a third glass. What is the ratio of alcohol to water for the final

mixture?

e One full glass contains vinegar and water in the ratio of

1 : 3. Another glass of twice the capacity of the first

has vinegar and water in the ratio 1 : 4. If the contents

of both glasses were mixed together what is the ratio of

vinegar to water?

1 Open a new spreadsheet

and enter the following

headings and data:

2

3 Fill the formulae in:

a B4 across to column D b E2 and E3 across to column F

c F1, F2, F3 and F4 across to column X.

4

5 Investigate the split of pocket money:

a after one year (Hint: Fill across further.)

b if Tim works 4 days to Becky’s 1 day (Hint: enter 4 in B2 and 1 in B3.)

c if week 1 pocket money is set at 5 cents. (Hint: enter 0:05 in E4.)

Re-read the on page . A spreadsheetcan be used to analyse the problem and help calculate theanswers. We can also use a spreadsheet to answer ‘What ifthe situation changes?’ questions.

Opening Problem 32 SPREADSHEET

What to do:

week 1

2 cents for week 1

Complete your spreadsheet by entering the following formulae:

Use your spreadsheet to answer the questions posed in the onpage .

Opening Problem

32


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Rather than write 2 £ 2 £ 2, we write such a product as 23.

23 reads “two cubed ” or

“the third power of two” or

“two to the power of three”.

If n is a positive integer, then an is the product of n factors of a,

i.e., an= a£ a£ a£ a£ a£ a£ ::::::£ a| {z }

n factors

1 Find the integer equal to:

a 23 b 33 c 25 d 53

e 22 £ 3 f 22 £ 33 £ 5 g 23 £ 3 £ 72 h 24 £ 52 £ 11

A prime number is a natural number which has exactly two distinct factors, itself and 1.

A composite number is a natural number which has more than two factors.

Notice that these definitions indicate that one (1) is neither prime nor composite.

17 is a prime number since it has only 2 factors, 1 and 17.

26 is a composite number since it has more than two factors. 1, 2, 13 and 26, are its factors.

Every composite number can be written as a product of prime factors in one and only

one way (apart from order).

For example, 72 = 2 £ 2 £ 2 £ 3 £ 3 or 72 = 23 £ 32 (in exponent form).

PRIME NUMBERS AND INDEX NOTATIONG

23 index or power

base number

Find the integer equal to: a 34 b 24 £ 32 £ 7

a 34 b 24 £ 32 £ 7

= 3 £ 3 £ 3 £ 3 = 2 £ 2 £ 2 £ 2 £ 3 £ 3 £ 7

= 81 = 1008


EXERCISE 2G

PRIMES AND COMPOSITES

One method for factorising a composite number into prime factors is to continue to dividethe number by primes, firstly by and when is exhausted, by and when is exhausted,by and so on.

2 2 3 35


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a Write 32 as powers of a prime.

b Write 264 as a product of prime factors in exponent form.

a 32 = 2 £ 2 £ 2 £ 2 £ 2

= 25b 2 264

2 1322 663 33

11 111

264 = 2 £ 2 £ 2 £ 3 £ 11

) in exponent form

264 = 23 £ 31 £ 111

2 List the set of all primes less than 30.

3 Write the following as powers of a prime:

a 8 b 27 c 64 d 625

e 343 f 243 g 1331 h 529

4 Express the following as the product of prime factors in exponent form:

a 56 b 240 c 504 d 735

e 297 f 221 g 360 h 952

5 The most abundant number in a set of numbers is the number which has the highest

power of 2 as a factor.

For example, the most abundant number of f1, 2, 3, 4, 5, 6, 7, 8, 9, 10g is 8 as

8 = 23:

The most abundant number of f41, 42, 43, 44, 45, 46, 47, 48, 49, 50g is 48 as

48 = 24 £ 3:

Find the most abundant number of the set

f151, 152, 153, 154, 155, 156, 157, 158, 159, 160g.

The Power Key ( or or ) is used to find the power of a number.

6 Find, using your calculator:

a 54 b 83 c 125 d (¡4)9

e (¡7)4 f (¡3)3 g 2:86 h 190


USING YOUR CALCULATOR

^ xy

ax

yx

Find 75 using your calculator.

Press: 7 5 Answer: 16 807


^ =


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INVESTIGATION 2 THE 3-DIGIT PROBLEM

DISCUSSION


7 Find, correct to 3 decimal places:

a (2:6 + 3:7)4 b (8:6)3 ¡ (4:2)3 c 12:4 £ (10:7)4

d

µ3:2 + 1:92

1:47

¶3

e

µ0:52

0:09 £ 6:14

¶4

f648

(3:62)4

8 What is the last digit of 3100?

(Hint: Consider 31, 32, 33, 34, 35, 36 ...... and look for a pattern.)

9 What is the last digit of the number 7200? (Hint: Consider 71, 72, 73, 74, 75, etc.)

10 Find two different odd numbers such that they add to 36 and their difference when

factorised into primes has a sum of primes of 10.

11 3a £ a3 = 518¤ i.e., ‘five thousand one hundred and eighty ......’ Find ¤.

Is 333

= (33)3

= 273

= 19683

or is 333

= 3(3)3

= 327

= 7625 597 484 987?

1

2 Choose another three digits and repeat the process outlined above.

3

Digits chosen Sum of 6 orderings Sum in prime factored form

e.g., 2, 7, 8 374 2 £ 11 £ 17...

......

4 Find the prime factorisation of all sums and record your results in your table.

5 Find the HCF of all the sums.

6 Prove algebraically that you will always obtain this HCF for all possible choices of 3different digits. (Hint: Any two digit number with digits a and b has form 10a+b,

for example, 37 = 10 £ 3 + 7:)

What to do:

Choose a further three digits and repeat.

Choose any three different digits from to . Write down the sixpossible two digit numbers using these digits. For example, if wechoose the digits , and , we would write down: , , , , ,

which add to . Find the sum of the six numbers and record yourresults in a table like that of question .

1 9

2 7 8 27 28 72 78 8287 374

3

333


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Y:\HAESE\SA_09-6ed\SA09-6_02\055SA09-6_02.CDR Friday, 8 September 2006 2:17:30 PM PETERDELL

INVESTIGATION 4 TOWER OF HANOI PUZZLE

INVESTIGATION 3 PRIME OR COMPOSITE?


For example,p

251 + 15:84. Now divide 251 by the primes 2, 3, 5, 7, 11, 13 (all less

than 15:84). We discover that none of these primes divide exactly into 251, thus 251 is a

prime number!

1 Use this method to determine which of the following numbers are prime:

a 129 b 371 c 787 d 4321

If you do not have a commercial version of the game, you can make your own using coins

or cardboard discs with increasing diameters.

Start with a small number of discs arranged in a pyramid on one of the three needles (or

piles). The object of the puzzle is to move the discs to another needle (or pile) by moving

only one disc at a time and not placing a larger disc on a smaller disc.

Hint: Build up a table of

results like this:Number of discs 1 2 3 4 5 6 7Minimum number of moves 1 3

You should be able to complete the puzzle with 5 discs in 31 moves.

1 How many moves would be needed to shift 6 discs?

2 What about 7 discs?

3 Can you find a formula to express the number of moves required if n discs are used?

4 Read again the Tower of Hanoi legend which says the world will end when all 64discs have been moved. If the priests can move 5 discs per minute, how many years

would it take to move all 64 discs?

Is prime or composite?

A systematic approach is to divide by all the primes up to the squareroot of starting with the smallest prime first.

251

251251

What to do:

This problem allegedly gets its namefrom the Tower of Bramah in Hanoi,where the Lord Bramah instructedhis priests to place golden discs

of varying diameters on one of three diamondneedles. The priests were to transfer them as fastas possible to another needle, one disc at a time,never putting a large disc on a small disc andnever putting the discs aside not on a needle.When the discs were completely transferred, theworld would come to an end.

64

What to do:


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REVIEW SET 2B

REVIEW SET 2A


1 a Evaluate (¡5) £ (¡3):

b Evaluate 18 ¥ (6 ¡ 9).

c Find 2 if 692 is divisible by 2.

d Evaluate 35 £ 11

3 :

e Write 400 mL is to 2 L as a ratio in simplest form.

f Write 132 as a product of primes in exponent form.

g Write 38 as a decimal.

h What natural number has factorisation 23 £ 32 £ 5?

2 Find the:

a HCF and LCM of 24 and 36

b largest multiple of 17 which is less than 1000.

3 Find 512 of $48:60.

4 Use your calculator to find correct to 2 decimal places:

a 2:76 £ 5:7 ¡ 2:9 ¥ 4 b 2:3 ¥ 9:2 c 14 ¥ 2

3

5 a If a butcher sells 23 of his hamburger patties on Monday and 1

2 of the remainder

on Tuesday, what fraction of the patties remains unsold?

b A cordial drink is mixed using one part cordial to five parts water.

If 500 mL of cordial is used, how much water should be added?

c At a Falcons vs Crocs basketball match, the ratio

of supporters is 5 : 3. If there are 4000 in the

crowd, how many are Falcons supporters?

1 a Evaluate 1:2 £ 0:4 b Evaluate 234 £ 11

3 .

c Find the value of¡8 £¡5

¡2: d Evaluate 1

2 + 13 :

e Write 725 in decimal form. f Evaluate 5 £ 3 ¡ 4 £ 5:


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g Write 2 m is to 40 cm as a ratio in simplest form.

h Write 1188 as a product of primes in index form.

i Find 2 if 2722 is divisible by 3:

j Evaluate 22 £ 33 £ 52:

2 Find:

a the HCF and LCM of 15 and 27

b the smallest multiple of 13 which is greater than 2000.

3 Find 38 of 2 hours and 16 minutes (in minutes).

4 Use your calculator to find:

a 345 ¡ 11

9 b 4:65 ¡ 7:87 ¥ 3:25 (correct to 3 significant figures)

5 a How many 2:4 m lengths of wire can be cut from a roll 156:4 m long?

How much wire is wasted?

b A profit of $85 000 is to be split amongst two business owners in the ratio 3 : 2.

What is the smaller share?

c A painter paints 13 of a house on one day and

14 of it on the next day.

What fraction of the house is yet to be painted?


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number - haesemathematics.com · find the hcf of 24 and 40: ... 5 find the lcm of: a 5, 8 b 4, 6 c...

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