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Number

Working with numbers

In previous years you have been introduced to new numbers and have found some interesting facts about familiar numbers. Now you will take a fresh look at some of that work and at the use of calculators.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■

apply a range of mental strategies to aid computation revise operations on whole numbers, integers, decimals and fractions divide two-digit and three-digit numbers by a two-digit number apply ‘order of operations’ to simplify expressions round numbers and estimate answers estimate and calculate squares, cubes, other powers, square roots and cube roots explore the properties of the square and square root of products: (ab) 2 and ab

Wordbank ■ ■ ■ ■ ■ ■ ■

mental calculation To operate with numbers ‘in your head’, without using pen and paper, or calculator. order of operations The rules for calculating an expression containing mixed operations, such as 14 − 2 × 4 + 1. decimal places The places after the decimal point in a number. square root The positive value which, if squared, will give the number required, for example 49 = 7 because 7 2 = 49. cube root The value which, if cubed, will give the number required, for example 3 8 = 2 because 2 3 = 8. improper fraction A fraction whose numerator is larger than its denominator, for example 7--4- . mixed numeral A numeral consisting of a whole number and a fraction, for example 1 3--4- .

Think! Can you think of a simple way of evaluating 18 2? What about

49 × 9 ?

W OR KI NG W I T H NUM B E R S

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CHAPTER 1

Start up Worksheet 1-01 Brainstarters 1

Skillsheet 1-01 Factors and divisibility

1 Find the answers to these without using a calculator: a 6×9 b 7×4 c 43 + 20 d 17 + 25 e 5×8 f 9×9 g 42 ÷ 7 h 36 ÷ 4 i 64 ÷ 8 j 16 − 9 k 6×6 l 45 ÷ 9 2 Round 2870 to the nearest hundred. 3 Rewrite these integers in ascending order: −6, 0, 9, 7, −1, −5, 3, −3 4 Find the highest common factor of: a 12 and 8 b 20 and 25 5 Find: a 52

c 6 and 18 b 82

c 152

d

e

f

43

h

3

8

j

3

125

100

g 33 i

3

–1

36

6 Find the lowest common multiple of: a 6 and 10 b 2 and 5

c 3 and 4

7 Convert each of these fractions to a decimal. a

2 --5

b

1 --4

c

3 --8

8 Rewrite these numbers in ascending order: 1.805, 1.085, 1.85, 1.05, 1.058, 1.508 9 Complete these pairs of equivalent fractions: 2 4 a --- = --3 ? 4 ? b --- = -----5 40 c

5 ? --- = -----8 32

10 Convert each of these decimals to a common fraction in its simplest form. a 0.003 b 0.8 c 0.05

Mental calculation shortcuts In the Skillbank sections of New Century Maths 7, you were provided with a variety of strategies for mental calculation to simplify numerical expressions. Some of them are shown in the table on the next page.

4

NEW CENTURY MATHS 8

Skill

Examples

Multiplying by a multiple of 10

5 × 80 = 5 × 8 × 10 = 40 × 10 = 400

Changing the order when adding or multiplying

15 + 37 + 18 + 45 + 22 = (15 + 45) + (18 + 22) + 37 = 60 + 40 + 37 = 137 7 × 4 × 5 = 7 × (4 × 5) = 7 × 20 = 140

Adding and subtracting 8 or 9

43 + 29 = 43 + 30 − 1 = 73 − 1 = 72 67 − 18 = 67 − 20 + 2 = 47 + 2 = 49

Doubling and halving numbers

47 × 2 = double 40 + double 7 = 80 + 14 = 94 1 --- × 144 = half of 140 + half of 4 = 70 + 2 = 172 2 1 --2

× 338 = half of 320 + half of 18 = 160 + 9 = 169

Multiplying and dividing by 4 or 8

17 × 8 Double 17 = 34, double 34 = 68, double 68 = 136. 17 × 8 = 136. 560 ÷ 4 Half 560 = 280, half 280 = 140. 560 ÷ 4 = 140

Estimating answers

43 + 125 + 66 + 32 ≈ 40 + 130 + 70 + 30 = (130 + 70) + (40 + 30) = 270 635 ÷ 18 ≈ 640 ÷ 20 = 64 ÷ 2 = 32

Multiplying and dividing by 5, 15, 20, 25, 50

18 × 5 = 9 × 2 × 5 = 9 × 10 = 90 300 ÷ 25 = 300 ÷ 100 × 4 = 3 × 4 = 12

Multiplying by 9, 11, 99, 101

17 × 11 = 17 × 10 + 17 × 1 = 170 + 17 = 187 25 × 9 = 25 × 10 − 25 × 1 = 250 − 25 = 225

Commonly used fractions and decimals

0.25 × 24 = 0.6˙ × 36 =

1 --4 2 --3

× 24 = 6 × 36 = ( 1--- × 36) × 2 = 12 × 2 = 24 3

Exercise 1-01 1 Use the mental calculation shortcuts shown in the table above to evaluate each of the following expressions. a 0.1 × 130 b 58 + 19 c 68 × 2 d 8 × 60 e Estimate 26 + 71 + 146 + 19 + 14 f 26 + 71 + 146 + 19 + 14 g 16 × 5 h 0.3˙ × 24 i 6 × 25 × 4 j 600 ÷ 25 k 168 ÷ 4 l 32 × 11 m 3 × 70 n 16 + 48 o 140 ÷ 5 p 0.5 × 38 q Estimate 88 + 43 + 27 + 7 + 102 r 88 + 43 + 27 + 7 + 102 2 Use mental calculation shortcuts to evaluate these: a 7 × 1000 b 14 × 15 c 400 ÷ 50 f k p u

392 ÷ 8 0.25 × 44 0.75 × 20 15 × 8

g l q v

16 × 25 80 ÷ 5 3×8×2 27 × 99

h m r w

4×7×5 22 × 8 5 × 900 28 + 35

d i n s x

1 --2

× 232

46 × 9 300 ÷ 20 12 × 50 63 × 2

e 74 − 28 j 27 × 4 o 16 × 101 t 15 + 39 y 1--2- × 826

W OR KI NG W I T H NUM B E R S

5

CHAPTER 1

The four operations The four basic operations of arithmetic are:

+

−

×

÷

addition

subtraction

multiplication

division

We will now review these operations.

Example 1 Complete this number grid: +

5

14

8 12

Solution + 8

5

+

5

14

8

13

22

12

17

26

14

5 + 8 14 + 8

13

22

5 + 12 14 + 12

12

17

26

Example 2 Simplify 504 ÷ 18.

Solution Method 1: Long division

Method 2: Preferred multiples

28 18 ) 5 0 4 −3 6

18 ) 5 0 4 −1 8 0

10 times

324 −1 8 0

10 times

144 −1 4 4

18 into 50 goes 2 18 into 144 goes 8

0

144 −90

5 times

54 −54

3 times

0 ∴ 504 ÷ 18 = 28

6

NEW CENTURY MATHS 8

28 times

Exercise 1-02 1 Copy and complete the following number grids: a

+

17

23

48

95

Example 1

b top row minus left-hand column

35

−

46

38

77

43

81

57

59

68

91 112

34 c

×

12

15

20

37

d top row divided by left-hand column

8

÷

10

3

18

5

33

15

120 180 135

2 Find the answers to the following: a 285 + 633 b 581 + 1023 d 688 − 35 f 899 − 389 h 158 × 7 i 601 × 36

c 3417 + 45 g 1436 − 802 i 246 × 25

3 Find the answers to the following: a 780 ÷ 12 b 512 ÷ 16 d 672 ÷ 42 e 756 ÷ 21

c 525 ÷ 35 f 364 ÷ 52

Example 2

W OR KI NG W I T H NUM B E R S

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CHAPTER 1

Working mathematically Reasoning and communicating: Doubling numbers Calculators always carry out calculations in the same way. People, however, can use calculator answers to discover patterns and relationships between numbers. 1 a Use a calculator to double each of these numbers. (Write the answers.) 2358 4229 7490 63 236 180 b Choose your own numbers to double and write the answers. 2 Use your answers from Question 1 to explain what happens to numbers when you double them. a What happens to the number of digits? b What happens to the number of zeros at the end? 3 Double these numbers and write the answers. 9 99 999 9999 99 999 etc. a Can you see a pattern in the answers? b How long before your calculator breaks the pattern? What does your calculator do? 4 What do you notice if you triple some numbers?

Integers Worksheet 1-02 Integer review

Integers are the positive and negative whole numbers and zero. You have previously learned the rules for operating with integers using the number line. Negative numbers can be entered into a calculator using the sign change key +/– or (–) .

Example 3 Skillsheet 1-02 Integers

1 Find the answer to −1 + 5.

Solution -2

Skillsheet 1-03 Integers using diagrams

-1

0

On a calculator:

1

2

3

4

+/–

1

+

5

5

=

The answer is 4.

2 Find the answer to −3 − 2.

Solution -6

-5

-4

On a calculator:

-3

-2

-1

+/–

3

–

The answer is −5.

8

NEW CENTURY MATHS 8

0

2

1

=

• Adding a negative number is the same as subtracting its opposite. • Subtracting a negative number is the same as adding its opposite. • positive × positive = positive × + − positive × negative = negative negative × positive = negative + + − negative × negative = positive (The above is also true for dividing with integers.) − − + • When multiplying or dividing two numbers which have the same sign, the answer is positive. • When multiplying or dividing two numbers which have different signs, the answer is negative.

Example 4 Find the answer to 4 − (−2).

Solution 4 − (−2) = 4 + 2 =6 On a calculator: 4 The answer is 6.

(subtracting a negative number is the same as adding its opposite) –

+/–

2

=

Example 5 Find the answer to: a −3 × 5

b −6 ÷ (−2)

Solution

a −3 × 5 = −15 On a calculator: +/–

3

b −6 ÷ (−2) = 3 On a calculator: 5

=

The answer is −15.

+/–

6

+/–

2

=

The answer is 3.

Exercise 1-03 1 Find the answers to the following: a 3 − 10 b 6 − 13 2 Find the answers to the following: b 6 − (−4) a −5 + (-8) e 6 − 15 f −7 + 8 i −18 + 10 − 3 j −7 + 3 + 8

Example 3

c 12 − 3 − 11

d −2 − 7 Example 4

c −12 − (−5) g −13 + 13 k 18 − 15 + 3 − 6

d −15 + 3 h 6−5−4 l −2 − 12 − 3 + 18

W OR KI NG W I T H NUM B E R S

SkillBuilder 3-03

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CHAPTER 1

Example 5

SkillBuilder 3-15

3 Work out answers to each of the following: a −5 × 4 b 3 × (−6) d 26 ÷ (−13) e −15 ÷ (−3) g 5 × (−9) h −10 × 7 j 64 ÷ (−4) k −25 ÷ (−5) n (−2) × (−2) × 7 m 18 ÷ (−2) ÷ (−3)

c f i l o

4 Find the answers to the following: a 11 − 7 − 4 b 8+3−5 d 12 ÷ (−3) + 4 e −8 ÷ 4 ÷ (−2) g 25 + 10 − 15 h −8 × (−3) × 5

c −3 × 2 + 5 f 6−3−8 i 6 × (−2) × (−1)

(−4) × (−8) −14 ÷ 2 −12 × (−4) −75 ÷ (−5) (−5)2

5 We have a number of ways of saying ‘add’, such as ‘plus’ and ‘increase’. Find other words which mean to ‘subtract’ and to ‘multiply’.

Rounding and estimation Worksheet 1-03 Estimation game

There are many situations in which it is impractical or impossible to give an exact answer. If the length of a wall is measured or calculated to be 4.831 metres, we may approximate it to 4.83 m or 4.8 m. In Year 7, we looked at rounding a number to a certain number of decimal places. To round a decimal: • cut the number at the required decimal place • look at the digit immediately to the right of the speciﬁed place • if this digit is 0, 1, 2, 3 or 4, leave the number in the speciﬁed place unchanged • if the digit is 5, 6, 7, 8 or 9, add 1 to the number in the speciﬁed place

Example 6 Round 5.261 correct to one decimal place.

Solution 5.2 61 cut The next digit is 6, so add 1 to the 2 in the tenths place, to give 3. So 5.261 is 5.3 (correct to one decimal place).

Example 7 Round 4.8239 correct to two decimal places.

Solution 4.82 39 cut The next digit is 3, so the number 2 does not change. So 4.8239 is 4.82 (correct to two decimal places).

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NEW CENTURY MATHS 8

Example 8 a Estimate the answer to 6.03 × 12.16 − 53.99 b Use your calculator to ﬁnd the exact answer, then round it to two decimal places.

Solution

a 6.03 × 12.16 − 53.99 ≈ 6 × 12 − 54 = 72 − 54 = 18. Estimated answer = 18. – = b On a calculator: 6.03 12.16 53.99 gives 19.3348. Rounded answer = 19.33 (correct to two decimal places).

Note: Most scientiﬁc calculators have a FIX mode that rounds the number on its display to a given number of decimal places. You may like to investigate the FIX mode.

Exercise 1-04 1 Round each of these, correct to one decimal place. a 3.851 b 4.0736 c 0.3333 d 7.34

e 15.0801

2 Round each of these, correct to three decimal places. a 9.7043 b 13.45671 d 53.09423 e 68.91093

c 0.08281 f 100.003011

Example 6

f

3.991 Example 7

3 Round each of these, correct to the number of decimal places shown in the brackets. a 38.055 [2] b 99.005 [1] c 86.539 [1] d 3.0983 [3] e 4.70771 [4] f 3.198 [2] g 32.999 [1] h 19.769312 [4] 4 For each of these questions, make an estimate of the answer and then use your calculator to evaluate the answer to the number of decimal places shown in brackets. a 1.9 × 5.3 + 8.66 [1] b (19.75 − 14.3) ÷ 5.1 [2] c 301.603 × 98.5 [2] 2 2 2 d 7.09 × 10.38 [1] e 9.9 ÷ 4.71 [1] f 3.61 × 2.08 × 11.431 [2]

Spreadsheet 1-01 Rounding decimals Example 8

Order of operations You should remember when carrying out calculations that there is a certain order in which the operations are done.

The order of operations First: Grouping symbols (innermost brackets ﬁrst) Second: × or ÷ (working left to right) Third: + or − (working left to right)

Worksheet 1-04 Order of operations puzzle Skillsheet 1-04 Order of operations

Scientiﬁc calculators are also programmed to perform calculations using the ‘order of operations’ rules.

W OR KI NG W I T H NUM B E R S

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CHAPTER 1

Example 9 Find answers for each of the following: a 6+5×2 b 18 ÷ (2 + 1)

c 5×2+3×9

Solution

a 6 + 5 × 2 = 6 + 10 = 16 On a calculator: +

6

5

d 2 × [25 − (24 ÷ 8)]

b 18 ÷ (2 + 1) = 18 ÷ 3 =6 On a calculator: 2

=

18

(

2

+

1

c 5 × 2 + 3 × 9 = 10 + 27 = 37

On a calculator: 5

2

+

3

=

9

d 2 × [25 − (24 ÷ 8)] = 2 × [25 − 3] = 2 × 22 = 44 On a calculator:

2

(

–

25

(

24

)

8

)

=

)

=

Example 10 Evaluate: 8 a -----------------39 – 23

8 + 16 b --------------12 – 8

Solution 8 a ------------------ Divide 8 by all of 39 − 23. 39 – 23 ( On a calculator: 8 39 The answer is 0.5 or

–

23

)

12

–

=

1 --- . 2

8 + 16 b --------------- Divide all of 8 + 16 by all of 12 − 8. 12 – 8 On a calculator: (

8

+

16

)

(

8

The answer is 6.

Exercise 1-05 Example 9

1 Calculate: a 8+5×2 c 6×5−2 e 3×6+2×5 g 34 − 18 ÷ 3 i 15 − (20 ÷ 2)

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NEW CENTURY MATHS 8

b d f h j

7−2×3 12 − 6 ÷ 3 7 + 15 ÷ 3 (34 − 29) × 6 5 × 10 + 16 ÷ 2

)

=

k m o q s

−3 × 6 − 2 × 5 (38 − 14) ÷ (7 + 5) 72 ÷ (−4 + 16) − 7 [(38 − 14) ÷ 6] ÷ 4 −6 × [22 − (4 ÷ 2)] + 1

l n p r t

26 ÷ (14 + 12) (7 − 10) × 20 ÷ 5 −14 ÷ [3 + 2 × 2] 48 − (29 + 3) + (26 − 5 × 4) [36 − (2 × 4)] ÷ [3 × (5 + 2) + 7]

2 Simplify each of the following. Give your answers to one decimal place where necessary. 15 + 5 19 + 5 45 × 2 a --------------b --------------c --------------------5×8 18 – 6 100 + 10 -66 d --------------14 + 4

e

41 – 13 ------------------15 + 8

[ 28 – ( 5 × 3 ) ] g ---------------------------------( 56 – 30 ) ÷ 2

7 × ( 11 – 2 ) h ------------------------------------------30 – [ ( 7 × 2 ) – 1 ]

f

4+5×2 --------------------------16 + 10 × 4

i

96 ÷ 3 – 2 -----------------------18 ÷ 3 + 2

Example 10

CAS 1-01 BODMAS

Just for the record The abacus The abacus is often called the ‘first computer’. It was invented by the Chinese in the 14th century and it is still used today to add, subtract, multiply, divide and to solve mathematical problems involving fractions and square roots. The word ‘abacus’ comes from the Greek word abax meaning ‘calculating board’. The abacus An abacus uses place value is composed of three sections: the upper beads, to represent numbers. the lower beads and the horizontal centre bar called the ‘beam’. Only the beads which have been moved to touch the two sides of the beam represent numbers. Each vertical row of beads represents a power of 10 (that is 10 000, 1000, 100, 10, 1). The beads below the beam represent one unit of that row. The beads above the beam represent five units of that row. Study the examples shown:

Abacus showing 15 (One 10 unit bead and one 5 unit bead)

1 0 0 0 0

1 0 0 0

1 0 0

1 0

1

Abacus showing 517 (One 500 unit bead, one 10 unit bead, one 5 unit bead and two 1 unit beads)

Represent 23, 56 and 466 on an abacus. W OR KI NG W I T H NUM B E R S

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CHAPTER 1

Decimals Addition and subtraction Worksheet 1-05 Decimal review Skillsheet 1-05 Decimals

Make sure you keep place-value columns correct by placing the decimal points underneath each other.

Example 11 Evaluate: a 2.1 + 44.3 + 13.25

b 13.85 − 5.6

Solution

a 2.1 + 44.3 + 13.25 2.1 44.3 + 13.25

b 13.85 − 5.6 13.85 − 5.6 8.25

59.65

Multiplication and division Example 12 1 Evaluate: a 12 × 0.1

b 5.31 × 1.3

c 6.25 ÷ 5

Solution Multiply without decimal points ﬁrst. Then make sure you have the same number of decimal places in the answer as there were at the start of the question. a 12 × 0.1 b 5.31 × 1.3 (question has one decimal place) (question has three decimal places) Multiplying without decimal points: Multiplying without decimal points: 12 531 × 1 × 13 12 1593 5310 Answer: 1.2 (answer has one decimal place) 6903 c 6.25 ÷ 5 = 1.25 1.25 5) 6.25

Answer: 6.903 (answer has three decimal places)

2 Simplify 12.4 ÷ 0.04.

Solution When dividing by a decimal fraction, make the decimal fraction a whole number by moving the decimal point the appropriate number of places to the right. In this case: 0.04 → 4 Move the decimal point in the other number the same number of places: 12.4 → 1240. Divide the new ﬁrst number by the new second number: 12.4 ÷ 0.04 = 1240 ÷ 4 = 310.

14

NEW CENTURY MATHS 8

Exercise 1-06 1 Write each of these as a fraction in its simplest form: a 0.3 b 0.07 c 0.03 e 0.4 f 0.82 g 0.35 2 Work out these calculations: a 1.3 + 0.8 d 3.92 − 0.49 g 20.03 − 1.06 j 65.001 − 13.06

d 0.009 h 0.026 Example 11

b e h k

42.51 + 3.6 3.6 − 0.46 12.56 − 9.88 9 − 0.004

3 Find the answers to the following: a 4.2 × 3 b 12.61 × 2 d 18.5 ÷ 0.5 e 1.3 × 0.6 g 6.24 ÷ 1.2 h 0.12 ÷ 1.2 j 0.87 × 12 k 0.252 ÷ 2.1

c f i l

18.4 − 6.9 12 + 0.56 + 3.4 4.123 + 71.05 + 6.3 3.671 − 1.289

c f i l

24.8 ÷ 4 0.06 × 0.4 238 ÷ 1.4 1.7 ÷ 1.5

Example 12

Number grids Exercise 1-07 1 Complete each of these number grids by ﬁnding the missing numbers. (Round decimals to two places, when required.) a

+

4.1

2.07

9.36

b top row minus left-hand column −

26

c

×

8.6

2.1

0.6

17.6

18

5.8 5.4 11.93

d

×

2.04

12.11

e top row divided by left-hand column

70.07

÷

0.65

0.5

20.14

0.81

0.07

2 Complete each of these number grids, rounding answers to two decimal places. a

+

1.6

b top row minus left-hand column

1.11

− 4.7

12.8

c

× 36.12

94.6

28.7

18.2

5.9

24.78

15.9 3.8 W OR KI NG W I T H NUM B E R S

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CHAPTER 1

d

×

0.8

0.7

e

÷

0.1512

2.5

3.528

0.8

40 4

3 Select an operation (+, −, ×, or ÷) to use with each of these number grids. Find a set of numbers that will correctly ﬁll the grid each time. a

b 45

15

18

c 9

21

60

24

48

48

Skillbank 1A Time before and time after SkillTest 1-01 Time before and after

1 Examine these examples. a What is the time 4 hours and 25 minutes after 6:30pm? 6:30pm + 4 hours = 10:30pm Count: ‘6:30, 7:30, 8:30, 9:30, 10:30’ 10:30pm + 25 minutes = 10:55pm. b What is the time 7 hours and 40 minutes after 11:45am? 11:45am + 7 hours = 6:45 pm Count: ‘11:45, 12:45, 1:45, 2:45, 3:45, 4:45, 5:45, 6:45’ 6:45pm + 40 minutes = 6:45pm + 15 minutes + 25 minutes = 7:00pm + 25 minutes = 7:25pm. or 15 minutes 7 hours 25 minutes = 7 hours 40 minutes

11:45am

12:00noon

7:00pm

7:25pm

c What is the time 10 hours and 15 minutes after 1850 hours? 1850 hours + 10 hours = 0450 hours (next day). Count: ‘1850, 1950, 2050, 2150, 2250, 2350, 0050, 0150, 0250, 0350, 0450’ 0450 hours + 15 minutes = 0450 hours + 10 minutes + 5 minutes = 0500 hours + 5 minutes = 0505 hours. or 10 minutes 10 hours 5 minutes = 10 hours 15 minutes

1850 hours 1900 hours

2 Now ﬁnd the time of day. a 3 hours 20 minutes after 9:05am c 4 hours 35 minutes after 6:15pm e 2 hours 45 minutes after 0325 hours g 8 hours 30 minutes after 12:40am i 6 hours 25 minutes after 0435 hours k 9 hours 50 minutes after 2:30pm

16

NE W C E N T U R Y M A T H S 8

0500 hours

b d f h j l

0505 hours

5 hours 40 minutes after 7:30pm 11 hours 10 minutes after 11:45am 7 hours 5 minutes after 1705 hours 4 hours 55 minutes after 10:20pm 2 hours 15 minutes after 2050 hours 3 hours 10 minutes after 8:25am

3 Examine these examples. a What is the time 3 hours and 15 minutes before 11:20am? 11:20am − 3 hours = 8:20am Count back: ‘11:20, 10:20, 9:20, 8:20’ 8:20am − 15 minutes = 8:05am. b What is the time 2 hours and 40 minutes before 7:20pm? 7:20pm − 2 hours = 5:20pm Count back: ‘7:20, 6:20, 5:20’ 5:20pm − 40 minutes = 5:20pm − 20 minutes − 20 minutes = 5:00pm − 20 minutes = 4:40 pm. or 20 minutes 2 hours 20 minutes = 2 hours 40 minutes

4:40pm 5:00pm

7:00pm

7:20pm

c What is the time 8 hours and 45 minutes before 1115 hours? 1115 hours − 8 hours = 0315 hours Count back: ‘1115, 1015, 0915, 0815, 0715, 0615, 0515, 0415, 0315’ (or 11 − 8 = 3). 0315 hours − 45 minutes = 0315 hours − 15 minutes − 30 minutes = 0300 hours − 30 minutes = 0230 hours or 30 minutes 8 hours 15 minutes = 8 hours 45 minutes

0230 hours

0300 hours

4 Now ﬁnd the time of day: a 1 hour 15 minutes before 7:20pm c 3 hours 20 minutes before 3:30pm e 2 hours 10 minutes before 1455 hours g 5 hours 25 minutes before 4:15am i 4 hours 20 minutes before 2005 hours k 3 hours 55 minutes before 5:30pm

1100 hours

b d f h j l

1115 hours

4 hours 40 minutes before 11:20am 5 hours 35 minutes before 8:25am 3 hours 45 minutes before 0740 hours 9 hours 30 minutes before 9:45pm 2 hours 15 minutes before 0615 hours 4 hours 40 minutes before 12:00 noon

Powers Remember that powers are used as a shorthand way of writing repeated multiplication. We write 2 × 2 × 2 × 2 as 24.

Skillsheet 1-06 Indices

Example 13 Evaluate 53.

Solution 53 = 5 × 5 × 5 = 125

Squares can be found on the calculator using the x 2 key. Other powers can be found on the calculator using the power key

xy

or

^

W OR KI NG W I T H NUM B E R S

.

17

CHAPTER 1

Example 14 Use your calculator to ﬁnd: a 142

b 64

c 25

Solution a On a calculator: 14 142 = 196.

x2

=

gives the answer 196.

b On a calculator: 6 64 = 1296.

xy

4

=

gives the answer 1296.

c On a calculator: 2 25 = 32.

xy

5

=

gives the answer 32.

Exercise 1-08 Example 13

1 Evaluate each of the following: a 52 b 23 c 62 2 4 g 8 h 4 i 103

d 34 j 92

2 Find the missing power each time. a 2 =8 b 3 = 27 d 4 = 4096 e 5 = 125 Example 14

3 Calculate: a 3 × 22 e 43 ÷ 2 i 62 × 82

b 2 × 32 f 43 ÷ 22 j 24 + 2

4 a Find (2 × 3)2. b Find: i 22 2 c Does (2 × 3) = 22 × 32?

e 71 k 35

f l

15 63

c 10 = 100 f 3 = 243 c 22 × 32 g 45 × 53 k 33 − 32

d 52 × 6 h 32 × 52 l 53 + 25

ii 32 Explain your answer.

ii 42 iii 52 5 a Find: i (4 × 5)2 2 2 2 b Does (4 × 5) = 4 × 5 ? Explain your answer. 6 Use what you found in Questions 4 and 5 to complete this pattern: (3 × 8)2 = × . 7 Write three examples of your own to show that (ab)2 = a2b2. 8 Copy and complete the following: a 182 = (6 × 3)2

18

b 222 = (2 × 11)2

= 62 ×

=

=

=

NEW CENTURY MATHS 8

×

c 302 = ( = =

× 10)2 ×

d 162 = (2 × =

e 282 = (

)2 ×

=

=

× 7)2

152 = (

f

×

×

=

=

)2

×

=

Working mathematically Applying strategies and reasoning: Crossnumber puzzle Choose the correct clue from each pair and complete the puzzle. Across 1 6 1. 37 × 6 or 37 × 2 2. 22 × 3 or 23 × 3 2 3. 6543 or 5432 4. 282 or 292 3 8 5. 5 × 16 − 1 or 5 × 16 + 1 9. 457 × 9 or 579 × 4 9 10. 27 or 28 11. 33 × 25 or 33 × 22 4 7 12. 82 − 52 or 72 − 52 14. 29 + 3 or 29 − 3 5 Down 1. 63 or 53 4. 87 or 78 6. 122 or 152 7. 72 or 92 8. 72 × 32 or 62 × 32

10. 12. 13. 15. 16.

12

15

13

16

10

14

11

123 or 231 840 ÷ 24 or 840 ÷ 35 11 × 12 or 13 × 14 16 × 3 or 16 × 6 25 × 13 or 52 × 13.

Square roots and cube roots The square root ( give that number. The cube root ( 3

) of a given number is the positive value which, if squared, will ) of a given number is the value which, if cubed, will give that number.

Skillsheet 1-07 Square roots and cube roots

Example 15 Find the square root of 36.

Solution 36 = 6 because 62 = 6 × 6 = 36

On a calculator:

36

=

Example 16 Find the cube root of 125.

Solution 3

125 = 5 because 53 = 5 × 5 × 5 = 125

On a calculator:

3

125

=

W OR KI NG W I T H NUM B E R S

19

CHAPTER 1

Example 17 Estimate the value of

40 .

Solution There is no exact answer for the square root of 40, because there isn’t a number which, if squared, equals 40 exactly. Instead, we estimate and ﬁnd a number whose square is close to 40. Looking at the square numbers, 52 = 25, 62 = 36, 72 = 49, we can tell that 40 must lie somewhere between 6 and 7. Because 40 is closer to 36 than to 49, the square root must be closer to 6. As an estimate, 40 = 6.3 . On the calculator, the answer is 6.324555..., a more accurate answer than our estimate above.

Exercise 1-09 1 Copy and complete the following table: 1

Number

2

3

4

5

6

7

8

10

11

16

Number squared

512

Number cubed Example 17

9

2 Between which two numbers does 80 lie? (Choose one from the answers given.) A 40 and 41 B 9 and 10 C 79 and 81 D 8 and 9 3 Between which two numbers does A 22 and 23 B 5 and 6

45 lie? (Choose from the answers given.) C 6 and 7 D 8 and 9

4 Between which two whole numbers does

31 lie?

5 Give estimates for each of the following. a Example 15

Example 16

b

56

c

105

210

d

3

100

e

3

f

576

6 Find the square root of: a 4 b 121 e 784 f 256

c 81 g 289

d 900 h 1089

7 Find the cube root of: a 8 b 343 e 512 f 1728

c 2197 g 8000

d 216 h 2197

3

800

8 Give the answer to each of these to one decimal place: a e

3

37

b

495

f

9 a Find

100

c

502

d

3

6.5

2000

g

1.1

h

3

1103

36 .

b Find: c We know that

20

3

i

ii

4 36 =

NEW CENTURY MATHS 8

4 × 9.

9 Does

36 =

4 × 9?

Explain your answer.

12

10 a Find:

i

b We know that

225 225 =

ii 25 × 9 .

25 Does

iii 225 =

9

25 × 9 ?

Explain your answer.

11 Use what you found in Questions 9 and 10 to complete each of the following: a

64 = = =

c

900 = = = 2025 = = 9 =

e

b

484 = = =

121 × ×

× 100 ×

d

324 = = =

81 × ×

× × 5

f

16 × 4 ×

1764 = = 6 =

12 Write three examples of your own to show that

ab =

× × 7

a × b.

13 Evaluate each of the following. (Give answers to one decimal place where necessary.) 3× 3

a

b

144 d ------------9

e

3

2+ 3

c

11 × 2

f

7÷ 2 3

4×3 4×3 4

Fractions 2 ← numerator --7 ← denominator Fractions can be entered into a calculator using the fraction key:

a b/c .

Skillsheet 1-08 Fractions

Some types of fractions • proper: the numerator is smaller than the denominator. For example 1--2- ,

5 78 ------ , -----------12 1200

• improper: the numerator is larger than the denominator. For example 5--3- ,

Skillsheet 1-09 Fractions and decimals

11 123 ------ , --------5 74

• mixed numeral: a whole number and a common fraction. For example 1 3--5- , 4 7--8-

Example 18 Change these improper fractions into mixed numerals: a

7 --2

b

27 -----4

b

27 -----4

Solution a

7 --2

=7÷2 =

3 1--2-

On a calculator: 7

a b/c

2

=

= 27 ÷ 4 = 6 3--4-

On a calculator: 27

a b/c

4

=

W OR KI NG W I T H NUM B E R S

21

CHAPTER 1

Example 19 Change these mixed numerals into improper fractions: a 2 1--3-

b 4 2--5-

Solution 2×3 +1 2 1--3- = 6----------3

a

=

4×5 +2 --------------4 2--5- = 20 5

b

=

7 --3

On a calculator: 2

1

a b/c

On a calculator: =

3

a b/c

Pressing d/c ( improper fraction.

22 -----5

or

a b/c

SHIFT

4

d/c

a b/c

2

a b/c

=

5

d/c

a b/c ) converts a mixed numeral into an

2nd F

Example 20 Simplify these fractions: 10 -----25

a

36 -----60

b

Solution To simplify fractions, we divide the numerator and the denominator by a common factor. 10 -----25

a

=

10 ÷ 5 --------------25 ÷ 5

=

2 --5

36 -----60

b

On a calculator: 10

25

a b/c

=

=

36 ÷ 6 --------------60 ÷ 6

=

6÷2 --------------10 ÷ 2

=

3 --5

36 -----60

or

On a calculator: 36

=

36 ÷ 12 -----------------60 ÷ 12

=

3 --5

=

60

a b/c

Exercise 1-10 Example 18

Example 19

1 Write each of these improper fractions as a mixed numeral: a

3 --2

b

11 -----3

c

9 --4

d

11 -----5

e

20 -----3

f

47 -----11

g

100 --------21

h

73 -----15

2 Write each of these mixed numerals as an improper fraction: a 3 1--2-

b 4 1--3-

e 6 3--4-

f

7 1--5-

c 5 1--4-

d 5 2--3-

g 10 1--7-

h 15 3--4-

3 Arrange these fractions in order, starting with the smallest. 1 --- , 4 Example 20

3 --- , 4

1 --- , 8

3 --- , 8

5 --- , 8

7 --8

4 Simplify the following: a

5 -----10

b

4 -----12

c

12 -----26

d

18 -----24

e

15 -----25

f

17 -----34

g

32 -----48

h

60 --------100

i

44 -----77

j

150 --------310

k

21 -----35

l

18 -----16

22

NEW CENTURY MATHS 8

5 Copy and complete each of the following: =

a

1 --2

d

15 -----60

g

24 ---------

-----6

=

1 ------

=

6 -----10

=

b

2 --3

e

-----8

h

54 ---------

--------12

=

3 --4

=

9 -----15

=

c

4 --5

f

21 -----28

i

--------30

16 ---------

=

-----4

=

15 -----10

Operations with fractions Addition and subtraction To add or subtract fractions, the fractions must have common denominators. Worksheet 1-06 Fraction review

Example 21 Evaluate: a

1 --3

+

5 --6

b

5 --7

−

c 1 2--3- + 4 1--5-

2 --3

d 3 3--4- − 1 1--2-

Worksheet 1-07 Fractagons

Solution a

1 --3

+

5 --6

=

2×1 -----------2×3

=

2 --6

=

7 --6

+

b +

5 --6

5 --6

5 --7

−

=

3×5 -----------3×7

=

15 -----21

=

1 -----21

2 --3

−

−

7×2 -----------7×3

14 -----21

= 1 1--6On a calculator: 1 c

a

b/

+

3

c

On a calculator: 5

=

5

–

c

6

a b/c

2

a b/c

3

+

4

a b/c

1

a b/c

5

=

a b/c

3

a b/c

4

–

1

a b/c

1

a b/c

2

=

a

b/

a b/c

7

2

a b/c

3

=

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

10 -----15

2 --3

+

+

3 -----15

1 --5

-----= 5 13 15

On a calculator: 1 d 3 3--4- − 1 1--2=3−1+

3 --4

=2+

2 --4

3 --4

−

−

1 --2

= 2 1--4On a calculator: 3

W OR KI NG W I T H NUM B E R S

23

CHAPTER 1

Multiplication and division To multiply fractions, multiply the numerators together and multiply the denominators together. Convert any mixed numerals to improper fractions ﬁrst. To divide by a fraction, multiply by its reciprocal. Convert any mixed numerals to improper fractions ﬁrst.

Example 22 Evaluate: 3 --5

a

×

b 1 1--2- × 3 2--5-

2 --7

4 --5

c

÷

d 2 1--2- ÷ 1 1--3-

2 --3

Solution 3 --5

×

=

6 -----35

a

2 --7

1 1--2- × 3 2--5-

b

=

3 --2

=

51 -----10

×

17 -----5

1 = 5 ----10

On a calculator: 3

a b/c

c

4 --5

÷

=

2

=

5

On a calculator: 2

=

7

a b/c

1

2 --3

a b/c

2

3

a b/c

5

=

2

1

a b/c

1

a b/c

2 1--2- ÷ 1 1--3-

d

=

5 --2

÷

4 --3

6 --5

=

5 --2

×

3 --4

= 1 1--5-

=

15 -----8

4 ----5

×

1

a b/c

3 ----21

= 1 7--8On a calculator: 4

5

a b/c

On a calculator: 2

=

3

a b/c

2

a b/c

3

=

1

a b/c

Exercise 1-11 Example 21

SkillBuilder 2-05–2-17 Adding and subtracting fractions

1 Evaluate: a

1 --5

+

3 --5

b

3 --8

+

1 --8

c

2 --5

+

3 -----10

d

2 --3

+

1 --5

e

3 --7

+

2 --3

f

3 --5

−

1 --4

g

1 --2

−

1 --4

h 1 1--2- −

i

2 1--3- −

j

2 5--6- − 1 1--2-

k 2 1--3- + 1 1--6-

l

3 1--3- − 1 2--3-

n 2 3--4- −

o 3 1--5- + 1 3--4-

m 4 2--5- + 1 3--4Example 22

2 a

24

1 --2

×

1 --3

b

NEW CENTURY MATHS 8

2 --5

×

3 --7

3 --4

5 --8

c

3 --4

÷

1 --2

2 --5

d

2 --5

÷

3 --4

e 1 1--2- × 2 1 -----11

i

× 8 1--4-

f

3 1--4- ×

4 --5

g 1 2--3- ÷

j

2 3--8- ÷

11 -----16

k

2 --3

×

1 --4

h 3 1--2- ÷ 1 1--4-

1 --3

+

1 --2

2 3--4- −

l

3 a

1 --2

×8

b

1 --5

× 15

c

3 --4

× 24

d

3 --5

× 60

e

2 --3

× 33

f

4 --7

× 21

1 --2

×

SkillBuilder 2-24 Multiplying mixed fractions

3 --5

Skillbank 1B Time differences 1 Examine these examples. a What is the time difference between 11:40am and 6:15pm? From 11:40am to 5:40pm = 6 hours Count: ‘11:40, 12:40, 1:40, 2:40, 3:40, 4:40, 5:40’ From 5:40am to 6:00pm = 20 minutes From 6:00pm to 6:15pm = 15 minutes 5 hours + 20 minutes + 15 minutes = 6 hours 35 minutes 20 minutes 6 hours 15 minutes or

11:40am 12:00noon

6:00pm

SkillTest 1-02 Time differences

= 6 hours 35 minutes

6:15pm

b What is the time difference between 8:30pm and 1:20am? From 8:30pm to 12:30am = 4 hours Count: ‘8:30, 9:30, 10:30, 11:30, 12:30’ From 12:30am to 1:00am = 30 minutes From 1:00am to 1:20am = 20 minutes 4 hours + 30 minutes + 20 minutes = 4 hours 50 minutes 30 minutes 4 hours 20 minutes = 4 hours 50 minutes or

8:30pm

9:00pm

1:00am

1:20am W OR KI NG W I T H NUM B E R S

25

CHAPTER 1

c What is the time difference between 1645 hours and 2320 hours? From 1645 hours to 2245 hours = 6 hours (22 − 16 = 6) From 2245 hours to 2300 hours = 15 minutes From 2300 hours to 2320 hours = 20 minutes 6 hours + 15 minutes + 20 minutes = 6 hours 35 minutes 15 minutes 6 hours 20 minutes = 6 hours 35 minutes or

1645 hours

1700 hours

2 Now ﬁnd the time difference between: a 11:10am and 7:40pm c 4:45pm and 8:10pm e 1:05pm and 12:30am g 0425 hours and 0935 hours i 7:55am and 3:50pm

2300 hours

b d f h j

2320 hours

6:20pm and 12:00 midnight 2:30am and 10:55am 9:35am and 11:15am 1440 hours and 2025 hours 2:45pm and 10:10pm

Applying number Exercise 1-12 1 Michael went shopping and bought the following items: an exercise book at $2.70, two pens at $1.60 each, a drink at $1.50 and a packet of chips for $2.65. a How much did Michael spend in total? b If Michael paid with a $20 note, how much change did he receive? 2 Jessica’s car holds 45 litres of petrol. If the price of petrol is 92.6 cents per litre, how much will Jessica need to pay to ﬁll the tank? 3 Traci needs to build a wooden rectangle similar to the one shown. How much timber would be left from a 3.4 m length of timber?

0.5 m 0.8 m

4 Lendal spent left?

3 --4

of his pocket money. If his pocket money is $14, how much does he have

5 A mobile phone plan charges $20 per month plus $0.18 for each phone call. How much will Thao need to pay if she made 92 calls in one month? 6 Katy, Josh and Kylie shared a $500 000 lotto win. How much did they each receive? 7 In 1912, Donald Lippincott from the USA ran 100 m in 10.6 seconds while, in 2002, Tim Montgomery, also from the USA, ran 100 m in 9.78 seconds. a If he could maintain the same speed, how far (to the nearest metre) could Donald have run in one minute? b How far could Tim have run in one minute? c After one minute, how far ahead of Donald would Tim be?

26

NEW CENTURY MATHS 8

8 Copy these shopping dockets and ﬁll in the missing sections: a

Fruity Fruit Shop

b

2 kg of potatoes at $2.15 per kg 1 1-2- kg of carrots at $2.99 per kg 5 kg of oranges at $3.55 per kg Total Amount tendered:

3 sponge cakes at $3.88 each 6 1.25 L bottles of lemonade at $1.65 each 1 loaf of sliced bread at $2.55 1 2 L carton of milk at $2.85 2 videos at $29.99 each

$50 Total Amount tendered:

Change

$100

Change

9 From a jar containing 160 lollies, Lindy takes 3--8- of the lollies and shares them equally among her four children. How many lollies does each child receive? 10 Elly made a dress for herself and the expenses were: • 3 metres of material at $15.60 per metre. • 2 1--2- metres of lace at $1.85 per metre. • 2 1--4- metres of ribbon at $1.05 per metre. • 6 buttons at 35 cents each. Elly saw a similar dress for sale at $126.50. How much did Elly save by making the dress herself? 11 Calculate the area of each of these triangles. a b 20.4 m 33.7 m 11.1 cm

23.6 cm

12 Danielle uses half a sheet of contact to cover her books, and Christina uses two-ﬁfths of the same sheet of contact. What fraction of the original sheet remains? Worksheet 1-08 Magic squares Did you know that your calculator can talk? Not out loud, but it can give you written messages.

Calculator talk

Try this calculation on it: 623 × 411 − 213 × 303 + 1296 ÷ 4 × 579 − 288 − 16 Turn it upside down to read the word. (Hint: You should not eat your food like this!) W OR KI NG W I T H NUM B E R S

Worksheet 1-09 Cross number puzzles

27

CHAPTER 1

Exercise 1-13 Skillsheet 1-10 Spreadsheets

1 Turn your calculator upside down and make a list of the numbers that match these letters: O I Z E h S g L B G D 2 What number would make your calculator display these words? a hEEL b SLIDE c OhIO

d gLOSS

3 Find the answers by turning your calculator upside down after each of the following calculations: a 121 × 217 − 8550 is liked by all children. b The number that multiplies by itself to give 196 says G’day. c The 5 × 77 × 8 is a very difﬁcult instrument to play. d 8.0808 ÷ 20 tells you what Father Christmas said to the child who pulled his beard. e Some people like to eat a pickled 52 043 ÷ 71. f (12 500 ÷ 0.625 × 5 − 6000 + 152) × 4 is the name of an exciting word game. 4 Find the word answers to these questions: a What is made in the factory where Mavis is the manager? (343 409 − 534) × 2 ÷ 13 + 295 b The waves and tides have damaged many of these: (145 420.4 × 12 ÷ 0.24 + 910 500) ÷ 16 c This is how Drew told his Mum he would avoid detention for not doing his homework: ‘… (864.5 ÷ 3.5 × 20 + 0.9) × 19 …’. d High on the cliff overlooking the beach was the: (17 967 − 15 680) × 16 + 1146 Vue hotel. 5 Do each question on your calculator. Turn it upside down to read the answer to the given clue. a b c d e f g h i j k l m n o p q r s

28

Question

Clue

9 × 22 × (45 654 − 45 463) 202 × 7 × 73 × 137 13 456 704 ÷ 123 456 × 31 3 × 17 × 73 × 101 × 137 8237 × 41 (43 505 + 43 210) ÷ 123 000 13 003 × 823 ÷ 200 14 × (659 × 2 + 1) × 29 1667 × 7 × 3 (123 456 + 10 421) × 4 8922 + 20 132 + 6285 (300 + 67) × 67 × 2 9 × (123 456 + 173 807) ÷ 50 4 × 131 × (11 000 − 733) 5 × 49 × 358 005 ÷ 12 345 0.12 × 0.37 × (2 × 53 × 151 + 1) 0.73 × 1.01 × 1.37 × 0.4 0.01 × (692 + 62) × 7 − 0.7 (1 + 62 × 11) × 9

Good book Greetings Delight T’aint Cricket legend One only Defeated feminist Snake talk Not tight Top brass Silly birds Mutiny captain Beat him The mind Dirty For torture Santa Claus Find out Alternative

NEW CENTURY MATHS 8

t u v w x y

2 205 459 ÷ 32 ÷ 372 × 33 79 × (822 + 9) (777 + 10) × 7 − 0.082 2 × (1702 − 41) (72.62 + 3.31) × 0.5 × 0.7 × 3 (1.12 × 2.32 + 0.0178) × 241 × 50

Tree bits TV awards Top man Accounts Bad business By the sea shore

Power plus 1 Calculate the answers to three decimal places: a c

3

5.6 2 + 1.8 3

1 2 b ------- ÷ ------5 3

6.2 – 5.4 -----------------------------11.01 + 6.04

d

5.9 2 + 8.1 2 --------------------------13.6 ÷ 2.04

2 Solve this crossnumber puzzle using these four pieces of information as a guide: • p + q = 680 • f + k = 342 • k = 161 • k + m = 193 Across 1 2 3 1. 1 less than 11 down 5 3. k 5. p + q + k + f 6 7 6. k + m − 80 8. p + q − k + 102 8 9 10. k + 1020 12. 2k + 2m + 15 10 13. m Down 1. Equals 1 across 2. q + k + f + p 3. 4m − 7 4. m + k 7. p + q + f + 2k 8. 2f + 2k 9. f − 70 11. A dozen

12

4

11

13

W OR KI NG W I T H NUM B E R S

29

CHAPTER 1

3 Make your own calculator talking puzzles. Step 1: Enter a number into the calculator so that it spells a word when the calculator is turned upside down. 5508 spells BOSS Step 2: Create a string of operations starting with your number: [(5508 − 500) ÷ 8 + 374] ÷ 32 = 31.25 Step 3: Write the reverse operation string which will be your talking clue: (31.25 × 32 − 374) × 8 + 500 Step 4: Make up a question, riddle or rhyme: What do you call a gorilla armed with a machine gun? Create a calculation and a word problem that makes your calculator give these talking answers: a ShELLOIL b hOLES c hELLO d EggShELLS e ShE’BOIL f two of your own invention Worksheet 1-10 Scientific notation

4 Scientiﬁc notation (standard form) Scientiﬁc notation is a special way of representing very large or very small numbers. This is how your calculator handles this problem: Screen 2.56

Screen 1.678

04

means that is

2.56 × 104 2.56 × 10 × 10 × 10 × 10 = 25 600

-03

means that is

1.678 × 10−3 1.678 ÷ 10 ÷ 10 ÷ 10 = 0.001 678

a Can you see a quick way of writing the answer each time? b Write each of the following calculator displays as an ordinary number: i 2.4

04

iii 9.33 v 9.6 vii 1.001 ix 2.4

ii 4.55

05

-02

iv 6.667

02

-06

vi 8.9

-02 08

5 Write each of these in scientiﬁc notation: a 12 000 c 0.007 e 0.0005 g 1 000 000 i 0.000 000 011 1

viii 5.698 x 5.7011

b d f h j

-06 07 -03

345 000 000 4000 0.000 41 0.000 335 (1.2 × 104) × (6 × 10−3)

6 a What is the largest number that can be displayed on your calculator? b What is the smallest?

30

NEW CENTURY MATHS 8

Worksheet 1-11 Numbers crossword

Language of maths calculator decimal place evaluate integer operation round

cube denominator factor tree long division order of operations simplify

cube root division fraction mixed numeral power square

decimal estimate improper fraction numerator proper fraction square root

1 What are the four arithmetic operations? 2 If you round a decimal to the nearest hundredth, how many decimal places is this? 3 What are the ‘order of operations’ rules? 4 What type of numeral can an improper fraction be converted to? 5 How do you write the cube root of −64? 6 What is the cube root of −64?

Topic overview

FRACT IO N

• What parts of this chapter did you remember from last year? • Are there parts of this chapter that you still don’t understand? Discuss any problems with your teacher or a friend. • Copy and complete this topic overview which has been started for you. Check your work with other students and your teacher.

S Numerators ominator s Den

OPERAT+IO– NS

× ÷

6

ALS

. ....

+/–

CIM

DE

7

R

8

CAL CU LA TO 3

5

NUMBERS 9

-3 2 -1 0 1 2 3 IN TE GER S

Powers

4

a b/c

..

0

3

..

2

1

W OR KI NG W I T H NUM B E R S

31

CHAPTER 1

Chapter 1 Ex 1-01

Review

1 Evaluate each of these expressions without using a calculator. a 7 × 30

b 0.25 × 16

c 44 − 29

d 53 × 9

e 2 × 154

f

g 0.1 × 400

h 25 × 11

920 ÷ 20 k 0.6˙ × 60

j

612 ÷ 4

l

8 × 18

m 120 ÷ 15

n 0.5 × 100

o 23 × 50

p 37 × 8

i

Ex 1-01

Ex 1-02

Topic test Chapter 1

18 × 15

2 a Estimate the answer to 45 + 73 + 11 + 160 + 25. b Find the exact answer to 45 + 73 + 11 + 160 + 25 without using a calculator. 3 Copy and complete the following number grids: a

+

16

21

39

88

b top row minus left-hand column

27

−

59

2

81

48

103

51

56

68

99 101

55 c

Ex 1-02

×

7

13

29

61

d top row divided by left-hand column

6

÷

13

4

21

9

35

18

36 108 180

4 Evaluate each of these expressions without using a calculator. a 48 + 126 + 56 b 109 + 53 + 1002 c 783 − 52 d 652 − 388 e 27 × 12 f 44 × 17 g 231 × 28 h 1347 × 6 i 812 ÷ 7 j 840 ÷ 12 k 396 ÷ 18 l 2139 ÷ 23 m 103 + 2099 + 56 n 236 × 15 o 4803 − 178 p 759 ÷ 11

32

NEW CENTURY MATHS 8

5 Evaluate: a −4 + 6 c 5 − (−2) e −3 − 8 g 24 ÷ (−2) i −36 ÷ 9 k −10 × 5 + 20 m −28 ÷ 7 ÷ (−2) o 13 − 15 − 6

Ex 1-03

b d f h j l n p

13 − 18 11 − 20 −2 × 7 −15 ÷ (−5) 24 ÷ (−6) × (−2) (−2) × (−2) × (−2) −12 ÷ 3 × (−5) −18 × (−3) ÷ (−6)

6 Round each of these numbers correct to the number of decimal places shown in the brackets. a 0.473 [1] b 13.1051 [2] c 98.0873 [3] d 69.97 [1] e 0.952 [2] f 6.0738 [3] g 100.099 [1] h 12.309 16 [4]

Ex 1-04

7 Evaluate: a 22 − 5 × 2 c 12 + 16 ÷ 4 − 8 e −16 ÷ 8 + 23 g 80 − [(4 + 5) × 8] i 40 − 18 ÷ 3 × 2 k (84 − 10) × 6 ÷ 4

Ex 1-05

b d f h j l

4÷2×5 (13 + 8) × 11 16 × 8 − 5 × 15 −56 ÷ (3 + 5 × 5) (36 −2) × (21 − 9) [38 + (6 × 5)] ÷ [4 × (5 − 4)]

8 Evaluate these expressions, giving your answers rounded to two decimal places. 8–3 33 – 2 × 5 a --------------b -----------------------15 ÷ 4 17 – 6 c

438 – 15 × 14 --------------------------------( 69 + 13 ) × 7

22 × ( 8 – 6 ) d ----------------------------24 ÷ 3 + 8

e

72 ÷ ( -4 + 16 ) ---------------------------------( 38 – 16 ) ÷ 6

f

9 Evaluate: a 2.51 + 6.8 c 37.4 − 6.9 e 2.6 × 4 g 4.2 ÷ 0.2 i 0.26 ÷ 0.8 k (2.5)2 m 12.5 × 3.01 o 16.4 ÷ 0.3 × 3 10 Calculate: a 73 d 42 × 5

Ex 1-05

-14 ÷ ( 5 + 2 × 2 ) ----------------------------------------( 11 – 6 ) × 6 Ex 1-06

b d f h j l n

13.3 + 0.82 + 5.6 8 − 0.03 3.5 × 0.5 0.071 × 1.3 9.6 ÷ 0.12 32.13 ÷ 5.1 (−1.1)2 Ex 1-08

64

b e 25 ÷ 4

115

c f 43 × 34

W OR KI NG W I T H NUM B E R S

33

CHAPTER 1

Ex 1-08

11 Copy and complete: 202 = (____ × 5)2 = ____2 × 52 = ____ × ____ = ____

Ex 1-09

12 Calculate: a

81

b

e

2.25

f

400

c

-1

g

3

Ex 1-09

13 Without using a calculator, write an estimate for

Ex 1-09

14 Copy and complete the following:

3

27

d

10 000

h

3

-125 3× 3

31 .

3136 = ___ × 49 = ____ × ____ = ____ Ex 1-10

Ex 1-10

Ex 1-10

Ex 1-11

Ex 1-12

15 Convert each of these improper fractions to a mixed numeral. a

15 -----4

b

22 -----5

c

7 --3

d

78 -----11

e

33 -----10

f

26 -----5

g

41 -----7

h

66 -----13

16 Convert each of these mixed numerals to an improper fraction. a 4 1--2-

b 3 2--3-

e 7 2--3-

f

8 1--2-

c 6 3--4-

d 11 2--5-

g 15 4--5-

h 3 5--8-

17 Reduce these fractions to their simplest form. a

6 --8

b

12 -----14

c

18 -----36

d

28 -----48

e

30 -----70

f

13 -----13

g

90 --------130

h

52 -----26

2 -----10

c

1 --3

d

4 --7

3 --4

g 2 1--5- + 1 1--2-

18 Evaluate: a

2 --7

+

3 --7

b

9 -----10

−

e

7 --8

−

2 --3

f

1 1--2- +

i

4 --5

×

2 --3

j

6 --7

×

5 --8

k

1 --6

+

÷

1 --5

1 --3

+

1 --2

h 1 5--8- − l

7 -----11

6 --7

÷ 1 1--2-

19 a Tamara earns $579.50 for working 38 hours a week. How much does she earn each hour? b A light aircraft can climb 320 metres every minute. If it climbed for 4.5 minutes after take-off, what height did it reach? c One Friday, the manager of a store added together all the sales ﬁgures of the staff. They were: Mario $1230, Sue $957.60, Theo $883.50, Frank $1448.40, Samantha $1101. What was the total of the sales ﬁgures?

34

NEW CENTURY MATHS 8

d A gardener took 300 watermelons to market and sold three-quarters of them for $2.30 each. The rest were sold for $1.90 each. i How many watermelons were sold for $2.30 each? ii Calculate the total amount received by the gardener.

e Mark is paid $6.75 per hour. How much does he earn if he works for 16 hours? A petrol tanker holds 20 000 L of fuel. If 1--4- of the tank is emptied, how much fuel is left in the tank? g Copy this shopping docket and ﬁll in the missing sections. f

2 shirts at $49.95 each 3 belts at $35.90 each 4 pairs of socks at $6.99 each Total Amount tendered:

$250

Change

W OR KI NG W I T H NUM B E R S

35

CHAPTER 1

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