Ch 01

Number skills

1
There are many factors
that affect the environment
of our planet. One of these
is world population.
The data supplied in the
table give the estimated or
projected world
Year World population
population for the middle
1950 2 555 078 074 of the year. In which
1960 3 039 332 401 ten-year period did (or
1970 3 707 610 112 will) the world population
increase the most?
1980 4 456 705 217
In which ten-year period
1990 5 283 755 345 is the percentage increase
2000 6 080 141 683 the largest?
2010 6 823 634 553
This chapter refreshes
your skills in working with
2020 7 518 010 600 numbers expressed as
2030 8 140 344 240 fractions, decimals,
2040 8 668 391 454 percentages or in index
form and applying those
2050 9 104 205 830
skills to real-life situations.

2 Maths Quest 9 for Victoria

Order of operations
Anton has calculated the answer to 5 + 6 × 4 as
44, while Marco insists that the answer is 29.
Who is correct?
In mathematics, it is important to ensure that
everybody obtains the same result from a calcu-
lation; so the order in which mathematical oper-
ations are worked is important.
The order of operations requires that:
1 all brackets are evaluated first, beginning
with the innermost brackets
2 then, all multiplication and division are
evaluated, working from left to right
3 and finally, any addition and subtraction,
working from left to right.
To obtain the correct answer to the calculation 5 + 6 × 4, we must complete the
operations of + and × in the correct order. That is, × first then +.
5+6×4
= 5 + 24
= 29

WORKED Example 1
Evaluate each of the following without using a calculator.
a 4 + 12 − 5 + 6 b 4 + 12 − (5 + 6) c 6 + 21 ÷ 7 d [4 × (5 + 8)] ÷ 2
THINK WRITE
a 1 Write the calculation. a 4 + 12 − 5 + 6
2 Perform the addition and subtraction = 16 − 5 + 6
from left to right. = 11 + 6
3 Write the answer. = 17
b 1 Write the calculation. b 4 + 12 − (5 + 6)
2 Evaluate the bracket first. = 4 + 12 − 11
3 Perform the addition and subtraction =5
from left to right and write the answer.
c 1 Write the calculation. c 6 + 21 ÷ 7
2 Perform the division. =6+3
3 Perform the addition. =9
d 1 Write the calculation. d [4 × (5 + 8)] ÷ 2
2 Remove the brackets by working the = [4 × 13] ÷ 2
innermost bracket first. = 52 ÷ 2
3 Divide and write the answer. = 26

In other examples you will need to read the question carefully to interpret the correct
order of operations and the correct way to write the calculation.

Chapter 1 Number skills 3
WORKED Example 2
Mum bought 2 packets of Easter eggs to hide
in the garden for her 4 children to ﬁnd. Each
packet contained 20 eggs. While she was
hiding them, the dog ate 4 eggs, Dad ate 3,
and 1 was squashed. If all the other eggs were
found, and each child found the same number
of eggs, how many eggs did each child have?

THINK WRITE
1 Write a mathematical sentence showing
what happened. Find the total number of
eggs and subtract the number that were
eaten or squashed. Then divide by the
number of children looking for eggs. [2 × 20 − (4 + 3 + 1)] ÷ 4
2 Use order of operations to solve the = (2 × 20 − 8) ÷ 4
problem. = [40 − 8] ÷ 4
= 32 ÷ 4
=8
3 Write the answer in a sentence. Each child found 8 eggs.

remember
remember
Evaluate in the following order.
1. Brackets ﬁrst, beginning with the innermost pair, then working through to the
outermost pair.
2. Multiplication and division in order from left to right.
3. Addition and subtraction in order from left to right.

1A Order of operations

Math
WORKED 1 Evaluate each of the following without using a calculator.
cad

Example
1 a 3 + 12 − 5 + 6 b 7 + 5 − 11 + 2 − 3 c 10 − 2 − 3 + 4 Order of
d 18 − 11 + 4 + 12 − 14 e 25 + 5 − 10 + 2 − 10 f 32 − 8 + 6 − 7 − 5 operations
g 10 × 6 × 4 × 2 h 18 × 4 × 3 × 0 i 80 ÷ 4 ÷ 5
j 25 ÷ 5 × 6 k 8×2÷4×3 l 72 ÷ 2 ÷ 6 × 3 Math
cad

m 16 + 2 × 5 n 80 ÷ 2 + 28 o 12 − 14 × 0
Ascending and
p (4 + 6) × 8 q (35 − 11) ÷ 6 r (7 + 2 − 3) × 8 descending
s 12 ÷ (9 − 3) t 75 ÷ (12 + 13) order


d 2 multiple choice
hca
a What is 12 × (4 + 2) ÷ 8 equal to?
Mat

Adding A 10 B 12 C 9 D 8 E 1
whole
numbers DIY b What is 36 ÷ 3 ÷ 4 + 2 equal to?
d
A 2 B 72 C 50 D 5 E 10
hca c What is 8 × 5 + 3 × (8 − 5) is equal to?
Mat

Subtracting A 49 B 192 C 59 D 129 E 339
whole
numbers DIY 3 Evaluate each of the following.
d
a 8 × 9 − 10 × 6 b 14 × (3 + 2) ÷ 7 c 72 ÷ (2 + 7 × 1)
hca
d 80 ÷ 5 − 60 ÷ 6 e 35 × (8 + 4 − 6 × 2) f (13 − 3) × 2 + 4 × 6
Mat

Multiplying g (17 − 12) ÷ 5 × 2 h (14 + 7 − 8) × 6 i [14 + (2 × 6 − 3)] × 4
whole j [(2 + 1) × 7 − 3 × 5] − 6 ÷ 3 k {[(3 + 9) ÷ 12] + 4 × 4} − 17
numbers DIY l {40 − [(8 + 2) × 3 − 5]} ÷ 5 m 16 ÷ 4 + 24 ÷ 6 + 5 × 5 − 19
hca
d n 108 ÷ 4 × (4 − 4) × 4 o {11 + (4 + 3) × 2 + 5 × 6 + (8 – 2) × 5} × 4
Mat

p [16 × 3 ÷ 2 + 40 ÷ 4 × 2 − 3 × 11 + 14] ÷ 5 + (6 × 2 + 4) × 2 − (7 × 5 + 2)
Dividing
whole WORKED 4 Takiko has brought 3 packs of nut biscuits to share with the 20 members of her class. If
numbers Example each pack contains 12 nut biscuits and 3 girls and 5 boys are allergic to nuts or don’t eat
DIY 2
biscuits, so don’t have any; how many nut buscuits will each of the other class members
receive?

5 The Wimbletons wanted to buy a
tennis racquet for each of their
3 children. The normal price of
a racquet is $100 but the shop
is offering a special deal. If two
racquets are bought at the same
time, the price is reduced by $25
for each one. If the Wimbletons
buy one at the normal price and two
on the special deal, how much do they
pay altogether? Write an equation to
show how you could have found the answer.

Integers
Integers include positive whole numbers, negative whole numbers and zero. They can
be represented on the number line.

–5 –4 –3 –2 –1 0 1 2 3 4 5

The rules for using integers are:
Rule 1 When adding integers with the same sign, keep the sign and add; −3 + −2 = −5.
Rule 2 When adding integers with different signs, ﬁnd the difference and use the sign
of the number further from zero; –3 + 4 = 1.
Rule 3 When subtracting integers, add the opposite; 5 − −7 = 12.
Rule 4 When multiplying integers, the following rules are obeyed.
(a) Positive × Positive = Positive 5 × 8 = 40
(b) Positive × Negative = Negative 5 × −8 = −40
(c) Negative × Positive = Negative −5 × 8 = −40
(d) Negative × Negative = Positive −5 × −8 = 40
Rule 5 When dividing integers, use the same rules as for multiplication.
(a) 16 ÷ 2 =8
(b) 16 ÷ −2 = −8
(c) −16 ÷ 2 = −8
(d) −16 ÷ −8 = 2

WORKED Example 3
Calculate each of the following without the use of a calculator and using the correct order
of operations.
a −15 × −5 ÷ 3 b 7 + −5 − −8 c 4 − 60 ÷ (−4 − 6)
THINK WRITE
a 1 Write the calculation. a −15 × −5 ÷ 3
2 Multiplication and division are the only = 75 ÷ 3
operations; so work from left to right. = 25
b 1 Write the calculation. b 7 + −5 − −8
2 Addition and subtraction are the only = 2 − −8
operations; so work from left to right. =2+8
= 10
c 1 Write the calculation. c 4 − 60 ÷ (−4 − 6)
2 Work the brackets. = 4 − 60 ÷ −10
3 Perform the division. = 4 − −6
4 Perform the subtraction. =4+6
= 10


WORKED Example 4
Insert operation signs to make this equation true.
5 K 3 K 4 K 1 = −2
(Trial and error is a suitable method.)
THINK WRITE
1 The answer (−2) is less than the first 5 − 3 − 4 − 1 = −3 ≠ −2
number in the question; so try subtraction.
2 The result of the first try (−3) is a little too 5 − 3 − 4 + 1 = −1 ≠ −2
small; so change the last sign to +.
3 The result of the second try (−1) is too big; 5−3−4×1
so try multiplying the last digit, which is 1, =5−3−4
remembering to use the order of operations. = −2

remember
remember
1. When adding integers with the same sign, keep the sign and add.
2. When adding integers with different signs, find the difference and use the sign
of the number further from zero.
3. When subtracting integers, add the opposite; for example 5 – –7 = 12.
4. When multiplying and dividing integers, like signs give positive answers,
unlike signs give negative answers.
5. When using order of operations, evaluate brackets before multiplication and
division, then evaluate addition and subtraction.

1B Integers

1.1 WORKED 1 Calculate each of the following without the use of a calculator and using the correct
HEET Example
order of operations.
SkillS

3a
a −7 + 12 b −14 + 7 c −18 − 8 d 25 − 24 − 2
e −2 − 3 − 6 f −7 − 11 + 5 g 14 − 15 + 11 h 13 − 19 − 6 + 9
i 10 × 2 ÷ 5 j 6 × −3 × −2 k −4 × −3 × −5 l 64 ÷ −16 ÷ 4
m −12 × 4 ÷ 16 n −120 ÷ −10 × 2 o 36 ÷ −6 × −5 p −6 × −1 × −10 ÷ 4
HEET
1.2
WORKED 2 Calculate each of the following without the use of a calculator and using the correct
SkillS

Example
3b
a 8 + −7 + −3 b 15 − 18 + −8 c 6 + −7 + −10 d 6 − −7
d e −5 − −2 f 7 − −2 − 7 g 4 + −8 − −5 h −7 − −13
hca
i −9 + −9 − −9 j 4 + −6 + −2 − −1 k −3 − 6 − −10 + −5
Mat

Order of 3 multiple choice
operations
with integers a −7 − 8 + 2 − 3 is equal to:
reads
A −2 B −16 C −14 D −20 E 0
L Sp he b −12 × −8 ÷ −4 × 2 is equal to:
et
EXCE

Arithmetic A 12 B −12 C 48 D −48 E 316
timer c 9 + −5 − −4 + 2 − −1 is equal to:
A 9 B 1 C 23 D 11 E −1

WORKED 4 Calculate each of the following without the use of a calculator and using the correct
Example
3c
a −3 + 3 × 3 b −9 − 2 × 6 c 15 ÷ 5 − 5 d 7×0−5
e 6 × (0 − 6) f −14 × 2 − 2 × 10 g 2 − 6 × 3 h 8 + 2 × −5
i 3×8−5×7 j 12 × −3 − 4 k 0 × 3 × −6 + 6 l −90 ÷ −5 − 26
m 5 × (−3 + 5) + 7 n 128 ÷ −16 + 3 × −5 o (3 + 7) ÷ −2 + −4
p −60 ÷ −4 × 3 − 43 q 28 ÷ −2 × (2 − 5) r 56 ÷ 7 + 70 ÷ −10
s 94 ÷ 2 + 3 × −3 t 14 − 4 × (5 + −6)
5 multiple choice
a What does 5 × −4 − 10 × −6 equal?
A −40 B 40 C 80 D −80 E 180
b What does 5 × (−4 − 10) × −6 equal?
A 420 B −420 C 180 D −180 E −40
c (–2 – –4) × (8 × 5 − 4) is equal to:
A 244 B −216 C 16 D −48 E 72
d –64 ÷ 8 – –8 is equal to:
A 4 B −4 C 0 D −16 E 16
e The correct operation signs to make 2 K −5 K −2 K −5 = −3 a true statement are:
A ×, −, − B ×, +, + C ×, ÷, + D −, +, × E −, ×, +
WORKED 6 Insert operation signs to make these equations true:
Example
4
a 5 K 6 = 11 b 7 K −4 = −28 c −18 K −2 = 9
d −7 K −3 = −4 e 3K4K5=2 f 7 K 2 K 3 = 17
g −5 K −4 K 10 = 2 h 6 K 3K 3 = 0 i 8 K 5 K 2 = −2
j 2 K 3 K 5 K 4 = 26 k 16 K 8 K 8 = −6 l 12 K 18 K 2 = 21
m 12 K 18 K −2 = 21 n −8 K 4 K −2 = 0 o 10 K 3 K 4 K 2 = 0
p 5 K 2 K −3 K −3 = 2
GAME
7 Thanh stands on a cliff top 68 m above sea level and drops a stone into the water.

time
It stops on the bottom 27 m below sea level. How far has the stone fallen? Number skills
— 001
8 The temperature range in Melbourne on 29 April was 7°C. If the maximum temperature
was 15°C, what was the minimum temperature?


Golf scores
In golf, par is the number of strokes considered necessary to complete a hole in
expert play. A birdie is a score of one stroke under par and a bogey is one stroke
over par. An eagle is a score of 2 strokes under par while 3 strokes under par is
called an albatross. A double bogey is 2 strokes over par and a triple bogey is
3 strokes over par.
1 Use integers to represent:
a par b a birdie c a bogey d an eagle
e an albatross f a double bogey g a triple bogey.
2 Which score for a hole would be the most difficult to achieve?
3 Leon and Dion have finished a round of 18 holes with the following information
shown on their scorecards.

Leon Dion
pars 4 pars 6
birdies 3 birdies 2
bogeys 6 bogeys 4
eagles 1 eagles 0
double bogeys 2 double bogeys 2
albatrosses 0 albatrosses 1
triple bogeys 2 triple bogeys 3
Final score Final score

What integer represents each person’s final score as a number of strokes over,
under or at par?
4 Who wins this round of golf?
5 Two professional golfers achieve overall final scores for 18 holes of −8 and −6.
a What does this mean?
b Who achieved a better score for this round of golf?
c How many strokes did each player make for the 18 holes if the course is
considered to be a par 71 course?

Estimation and rounding
Rounding to a given number of decimal places
Ms Shopper’s bill at the supermarket comes to $94.68 and she pays $94.70. Mr
Shopper’s bill is $83.72 and he pays $83.70. The bills have been rounded to the nearest
5 cents because the 5-cent is the smallest coin used. Ms Shopper’s bill has been
rounded up because 68 cents is closer to 70 cents than to 65 cents. Mr Shopper’s bill
has been rounded down because 72 cents is closer to 70 cents than to 75 cents.
Measuring distances is another one of the many practical situations where it is
necessary to round an answer to a given number of decimal places. For example, the
distance between two towns is given to the nearest kilometre. It is not practical or
useful to the average motorist that the distance between Melbourne and Sydney by a
certain route is 1024.352 km. We give the distance simply as 1024 km.
The accuracy of measurement is limited by what is practical and by the accuracy of
the instrument being used to take the measurement. For example, with your ruler it would
not be possible to measure anything more accurately than to the nearest millimetre.
The measurement 5.6713 cm ≈ 5.7 cm because 5.6713 is closer to 5.7 than it is to
5.6. The rounded answer, 5.7, is the closest approximation to the exact answer.
To round an answer to a given number of decimal places, consider only the first
digit after the required number of decimal places.
If that digit is 0, 1, 2, 3 or 4, then leave it and all following digits out of the answer.
If that digit is 5, 6, 7, 8 or 9, then the last digit to be written is increased by
1 and all else is left out.
Many calculators are able to round off using the
FIX function.
On some scientific calculators, you need to press
MODE first.
On a TI graphics calculator, press MODE , arrow
down to the second row, then arrow across to
highlight the number that corresponds to the required
number of decimal places. Press ENTER to set this
rounding condition. To undo this operation, press MODE , arrow down to highlight
FLOAT and press ENTER .
Any rounded answer is not an exact answer but a close approximation.

WORKED Example 5
Round 15.439 657 to: a 1 decimal place b 3 decimal places.
THINK WRITE
a 1 Write the number. a 15.439 657
2 Look at the second decimal place to determine whether to ≈ 15.4
leave it or to round it up. The digit is 3; so rewrite the number
without all digits after the first decimal place.
b 1 Write the number. b 15.439 657
2 Look at the fourth decimal place to determine whether to ≈ 15.440
leave it or to round it up. The digit is 6; so increase the third
decimal place by 1. Note: Adding 1 to 9 gives 10, thus 439
becomes 440 and the zero must be included.

Note: The more decimal places, the closer the approximation is to the exact answer.


There were 70 000 people at the Melbourne Cricket Ground for Australia’s one-day match
against the West Indies.

Rounding to a given number of significant figures
Although the caption describes a crowd of 70 000, in reality there may have been
70 246 people. The number has been rounded to 1 significant figure because the rest
of the number, 246, has no impact on our image of the size of the crowd. When using
very large or very small numbers, rounding to a given number of significant figures is
often used.
To round to 1 significant figure means having only 1 non-zero digit beginning from
the left with the other digits being zeros. The number 367 rounded to 1 significant
figure is 400 because 367 is closer to 400 than to 300.
To write 452 correct to 2 significant figures, we need to consider whether 452 is
closer to 450 or 460. It is closer to 450, and 4 and 5 are the 2 significant figures.
The method of deciding whether to leave or round up is the same as rounding to a
number of decimal places.

WORKED Example 6
Round 347 629 to: a 1 significant figure b 3 significant figures.
THINK WRITE
a 1 Write the number. a 347 629
2 Look at the second significant figure to determine whether to ≈ 300 000
leave it or to round it up. The digit is 4, so rewrite the number,
replacing all digits after the first significant figure with zeros.
b 1 Write the number. b 347 629
2 Look at the fourth significant figure to determine whether to it ≈ 348 000
leave or to round it up. The digit is 6 so write the answer by
adding 1 to the third digit and replace all other digits with zeros.

Note: The more significant figures taken, the closer the approximation is to the exact
answer.
When the first non-zero significant figure appears after the decimal point, any zeros
before that figure are not significant.

WORKED Example 7
Round 0.004 502 6 to 3 significant figures.
THINK WRITE
1 Write the number. 0.004 502 6
2 The first significant figure is the 4. ≈ 0.004 50
Round to 3 significant figures
beginning with the 4. The last zero
must be included in the answer because
it is one of the significant figures.

Estimation
Rounding is also used when making an estimation or mental approximation of an
answer. Estimation is a method of checking the reasonableness of an answer or a calcu-
lator computation. We can estimate an answer by rounding the numbers in the question
to simple numbers that can be calculated mentally.

WORKED Example 8
Estimate answers to the following without calculating the exact answer.
a 31 × 58 b 46 679 + 2351 × 65
THINK WRITE
a 1 Write the calculation. a 31 × 58
2 Round each number to 1 significant figure. ≈ 30 × 60
3 Perform the mental calculation. = 1800
b 1 Write the calculation. b 46 679 + 2351 × 65
2 Round each number to 1 significant figure. ≈ 50 000 + 2000 × 70
3 Perform the mental calculation. = 50 000 + 140 000
= 190 000

remember
remember
1. When rounding to a given number of decimal places, count only those places
after the decimal point.
2. When rounding to a given number of significant figures, begin counting from
the first non-zero digit.
3. A quick mental estimation can be used to check the accuracy of calculations.
4. Rounding is often used to convey a concept of size rather than an exact number.


1C Estimation and rounding

WORKED 1 Round the following to: i 1 decimal place ii 2 decimal places iii 3 decimal places.
Example
5
a 5.893 27 b 67.805 629 c 712.137 84 d 81.053 72 e 504.896 352
2 Round the following to 0 decimal places. (To 0 decimal places means to the nearest
whole number.)
a 25.68 b 317.19 c 1027.8 d 19.53

reads
3 Round the following to 1 decimal place.
L Sp he a 3047.2735 b 24.7392 c 8.2615 d 19.9804
et
EXCE

Rounding WORKED 4 Round the following to: i 1 ii 2 iii 3 iv 4 significant figures.
and Example
6
a 574 248 b 430 968 c 28 615 d 1 067 328 e 458 610
significant
figures DIY 5 Round the numbers in question 2 to 2 significant figures.
hca
d WORKED 6 Round the following correct to 3 significant figures.
Example
Mat

7
a 0.085 246 b 0.000 580 4 c 0.000 008 067 3
Rounding
d 0.006 765 73 e 0.000 026 973 f 0.000 352 1
WORKED 7 Estimate answers to the following without calculating the exact answer.
d Example
hca a 183 ÷ 58 b 78 × 11 c 632 + 169 d 1010 ÷ 98
8
Mat

Estimation e 17 × 19 f 476 ÷ 8 + 52 g (51 + 68) × 12 h 68 + 19 × 9
i 5 × (78 − 59) j 42 × 8 + 18 × 5 k 176 ÷ 18 + 689 ÷ 7
l 397 m 473 × 248 n 657 − 239 ÷ 49
o 12 345 + 549 × 146
8 multiple choice
a The number 49.954 correct to 1 decimal place is:
A 49.9 B 49.0 C 50 D 50.0 E 50.1
b The number 3 056 084 correct to 3 significant figures is:
A 3 050 000 B 3 056 000 C 3 057 000 D 306 E 3 060 000
c The number 0.008 065 3 correct to 3 significant figures is:
A 0.008 B 0.008 065 C 0.008 06 D 0.008 07 E 0.806
d A number rounded to 2 decimal places is 6.83. The original number could have been:
A 6.835 B 6.831 C 6.8372 D 6.85 E 6.8
SHE
ET 1.1
9 Each of the 178 students who attend the Year 9 Social has to pay $55. If the cost of
Work

hiring the band is $1000, estimate how much money would be available to pay for the
supper and the security people.

QUEST
S
MAT H

GE

1 In 1832, a young runner named Mensen Ehrnot reportedly ran nearly
EN

8950 km over a 59-day period. On each of those days he ran 16 hours
CH L and rested for 8 hours. Estimate how many kilometres he ran, on
AL average, per hour.
2 In the hundred consecutive whole numbers from 1 to 100, how many
times does each of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 occur?


1
1 Evaluate 9 − 13 − 14.
2 Evaluate 8 − 8 ÷ 4.
3 Evaluate (13 + 5 × 7) ÷ 12.
4 Evaluate −25 + −10 − −50.
5 Evaluate −84 ÷ 12 × 3.
6 Evaluate −18 + (−9 + 11) × 14.
7 Insert signs to make the following equation true. 5 K 21 K 7 K 5 = 20
8 Round 1.746 582 to 4 decimal places.
9 Round 0.006 059 9 to 4 significant figures.
10 Give an estimate for 78 + 43 + 55 − 86.

Decimal numbers
Decimal numbers are so much a part of everyday life that we need to be able to use
them, put them in order and convert them to simple fractions and percentages.
When using either your graphics calculator or a scientific calculator, enter the calcu-
lation as written and the calculator will perform the calculation using the correct order
of operations. There are, however, many things that we need to be able to do ourselves
with decimals without the aid of a calculator.

Ordering decimal numbers
Ascending order means from lowest to highest and descending order means from
highest to lowest. This is done by first writing each number with the same number of
decimal places, adding zeros where necessary. We then look at the left-most digit. The
greater this digit, the greater the decimal number. If the left-most digits are the same,
we move to the next digit, and so on.

WORKED Example 9
Write the following decimal numbers in ascending order:
0.66, 0.606, 0.6.
THINK WRITE
1 Write the numbers. 0.66, 0.606, 0.6
2 Write all numbers with the largest 0.660, 0.606, 0.600
number of decimal places, in this case
3, then compare.
3 Write the original numbers in 0.6, 0.606, 0.66
ascending order after looking at the
second and third decimal places.


Finite or terminating decimal numbers
Finite decimal numbers have a fixed or finite number of decimal places and can be
written as a fraction with a denominator that is a multiple of 10. If the decimal number
has 1 decimal place, the denominator of the fraction is 10; if there are 2 decimal places,
the denominator is 100; if there are 3 decimal places, the denominator is 1000 and so
on. In each case the numerator is the decimal number without the decimal point. These
fractions are simplified where possible.

WORKED Example 10
Convert each of the following to fractions in simplest form:
a 0.65 b 1.2 c 0.6275.
THINK WRITE
a 1 Write the decimal number. a 0.65
13
2 There are 2 decimal places, so write as a fraction 65
= ------------
-
with a denominator of 100 and simplify by 100
20
cancelling. (You may use a calculator to simplify.)
13
3 Write the answer. = -----
-
20
b 1 Write the decimal number. b 1.2
6
2 There is 1 decimal place, so write as a fraction with a 12
= -------
-
denominator of 10 and simplify by cancelling. (You 10
5
may use a calculator to simplify.)
1
3 Write the answer as a mixed number. = 1 --
-
5
c 1 Write the decimal number. c 0.6275
251
2 There are 4 decimal places, so write the fraction with 6275
= ---------------------
-
a denominator of 10 000 and simplify. (You may use 10 000
400
a calculator to simplify.)
251
3 Write the answer. = --------
-
400

Converting decimal numbers to percentages
To convert a decimal number to a percentage, we multiply the decimal number by 100
and include the % sign.

WORKED Example 11
Convert 0.357 to a percentage.
THINK WRITE
1 Write the decimal number. 0.357
2 Multiply the decimal number by 100 by moving the = (0.357 × 100)%
decimal point 2 places to the right. Remember to include = 35.7%
the percentage sign.

remember
remember
1. To order decimal numbers, write each with the same number of decimal places
and compare.
2. To write ﬁnite decimal numbers as fractions, make the denominator an
appropriate multiple of 10 and simplify where possible. The number of zeros in
the denominator should be the same as the number of digits after the decimal
point.
3. To convert a decimal number to a percentage, multiply by 100 and include the
percentage sign.

1D Decimal numbers
1 Calculate each of the following. Math 1.3
a 6.56 + 3.214 b 4.87 − 2.493 c 5.6 × 7.04 HEET

cad

SkillS
d 5.75 ÷ 0.25 e (4.5 + 2.1) × 3.5 f (8.6 − 4.4) ÷ 7 Operations
g 4.8 − 2.16 ÷ 0.18 h 3.2 × (6.4 + 0.78) i 7.2 ÷ 0.12 × 6 with
decimal
j 7.2 ÷ (0.12 × 6) k 5.8 × (3.1 ÷ 0.4) l 6.2 + 3.5 × 2 numbers
2 Calculate each of the following, rounding your answers to 2 decimal places. HEET
1.4

SkillS
a 6.46 × 2.356 b 8.12 × 5.4 ÷ 9.6 c 8 ÷ 0.35 + 2.1
d (6.509 + 4.804) ÷ 0.341 e 3.2 × 4.057 − 13.91 ÷ 2.43
WORKED 3 Write each of the following sets of decimal numbers in ascending order. 1.5
Example
a 0.66, 0.4, 0.71 b 2.3, 0.23, 23 c 0.7, 1.32, 1.04 HEET

SkillS
9
d 1.02, 1.1, 1.22 e 0.5, 0.56, 0.06 f 0.323, 0.4, 0.35
4 Write each of the following sets of decimal numbers in descending order.
a 0.24, 0.204, 0.2004 b 0.062, 0.081, 0.11 c 0.7, 0.77, 0.707
d 0.082, 0.09, 0.0802 e 1.2304, 1.23, 1.204 f 0.359, 0.39, 0.3592
5 multiple choice
a The expression 6.43 × 2.356 ÷ (2.1 − 0.365) correct to 2 decimal places is equal to:
A 0.36 B 6.85 C 87.31 D 8.73 E 6.84
b The false statement is:
A 0.67 < 0.7 B 0.506 < 0.51 C 0.735 > 0.73
D 0.203 < 1.3 E 0.085 > 0.85
c The expression −0.9 + 6.5 × 0.004 − 1.2 ÷ 0.6 is equal to:
A −1.074 B −2.874 C −2.64 D −20.874 E −0.84
d A good estimate for 5.2 × 0.2 + 1.18 ÷ 0.012 is:
A 101 B 11 C 110 D 99.373 E 1010
WORKED
Example
6 Convert each of the following to fractions in simplest form. XCE
L Spread
HEET
1.6
sheet
E

a 0.9 b 0.6 c 0.16 d 0.27 e 0.78
SkillS

10 Converting
f 0.15 g 0.08 h 1.5 i 2.84 j 0.125
decimals to
k 0.484 l 0.963 m 0.775 n 0.0625 o 0.8875 fractions
WORKED 7 Convert each of the following to percentages. L Spread 1.7
Example XCE HEET
a 0.72 b 0.31 c 0.89 d 0.57 e 0.9
sheet
E

SkillS

11
f 0.06 g 0.782 h 0.6175 i 0.0094 j 1.35 Converting
k 1.602 l 11 m 2.3 n 5.75 o 2.485 decimals to
percentages


8 multiple choice
a In simplest form and as a fraction 0.3125 is equal to:
3125 5
A ---------------
10 000
- B 31.25 C -----
16
- D 31 1 --
4
- E 13
-----
40
-

b As a percentage 0.0875 is equal to:
7
A 0.875% B 8.75% C 87.5% D ----- %
80
- E 875%
c The number 0.656 25 is equal to:
A 13-----
20
- B 53
-----
80
- C 11-----
16
- D 21 -----
32
- E 5
--
8
-

9 Francis is paid $11.50 an hour for babysitting. If he works for 7 hours over the
weekend, how much does he earn altogether?
10 Yvette babysits for
5 hours after school
each Friday. She is
paid $10 an hour.
a How much does
she earn each
week?
b If she banks $3.25
of the money each
week, how much
does she have left
to spend?

QUEST
S
M ATH

GE

1 Allison, Bhiba, Chris and Dinesh ordered one box of apples to share
EN

equally between them. However, no one was present when the box was
1
CH L delivered. Allison arrived and took -- of the apples. Later, Bhiba came
-
4
AL 1
and took 3-- of the apples left in the box. Then Chris came and did the
-

same. Finally Dinesh arrived and took his rightful share of the
remaining apples. If 9 apples remained in the box, how many apples
were in the box originally?
2 Mitchell has mown 0.6 of the lawn. He still has 50 m2 of lawn to mow.
What is the total area of the lawn?
3 A train 0.5 km long is travelling at a speed of 80 km/h. How long will it
take the train to go completely through a tunnel which is 1.5 km long?

17 Chapter 1 Number skills

What type of creature is a KATYDID
creature KATYDID
and where are its ears?
where are
Answer the decimal questions to
find the puzzle’s code.

= 3
– as a decimal = 8.6 = 2.8 + 3.6
4 – 4.9
= =

= 0.3 + 0.4 = 10% as a decimal = 0.5 × 8.4
= = =

5 23
= 93% as a decimal = –
2 = ––
50
as a decimal
= = =

= 5 × 0.3 = 12.7 = 8.34
– 9.87 – 6.54
=

7
= 1.2 – 0.8 = ––
20
as a decimal = 6.3 ÷ 0.63
= = =

= 22% as a decimal = 0.67 = 60% as a decimal
+ 0.53
= =

= 12 ÷ 0.5 = 1.1 × 0.8 = 0.2 × 20
= = =

= 4
– as a decimal = 1.64 ÷ 0.4 = 0.87
5 + 1.33
= =

= 51% as a decimal = 1.6
= 5.26 + 1.87
– 0.95
= =

= 4.5 = 6 × 0.8 = 17
–– as a decimal
4
× 1.2 = =

= 2.374 = 7.63 = 3
– as a decimal
+ 3.926 – 3.23 8
=

5.4 2.5 4.0 3.7 1.8 0.46 2.2 0.375 4.8 0.22 24 0.4 0.8 6.3 6.4 4.4

4.1 0.93 0.75 7.13 0.88 0.6 0.1 0.51 4.2 1.5 0.65 10 0.7 1.2 4.25 2.83 0.35


Fractions
There are many essential skills that you will need with fractions. You can review them
in the exercise below and by the matching SkillSHEET. You should be able to simplify
fractions and convert between improper fractions and mixed numbers. You should also
be able to use your calculator efﬁciently.

Graphics Calculator tip! Obtaining an a fraction
expressed as
answer

As with any calculation involving fractions, if you
wish to have an answer expressed as a fraction then
each calculation needs to end by pressing MATH ,
selecting 1: Frac and pressing ENTER .
M

For example, to simplify 28 on your graphics calcu-
-----
44
-
lator, enter 28 ÷ 44 then press MATH choose option
1: Frac, then press ENTER . This can be seen in the
M

screen at right.
Note: The graphics calculator gives all answers as improper fractions and will not give
answers as mixed numbers.
It is important that we know how to perform calculations using fractions both with
and without a calculator.
28
Without a calculator, we would simplify -----
44
- by dividing both the numerator and the
denominator by the highest common factor (HCF) of both. The HCF of 28 and 44 is 4.
7
28 28
----- = ---------
- 11
-
44 44
7
= -----
-
11

WORKED Example 12
Evaluate the following.
a 3 + 5
--
4
- --
6
- b 3
--
4
- × 5
--
6
- c 21 ÷
--
4
-
3
--
5
-

THINK WRITE
a 1 Write the fraction calculation. a 3
--
4
- + 5
--
6
-

2 Write both fractions with the same = 9
-----
12
- + 10
-----
12
-
denominator by using equivalent fractions.
3 Add the fractions and simplify the answer = 19
-----
12
-
by writing it as a mixed number.
= 1 -----
12
7
-

31
b 1 Write the fraction calculation and cancel b ----
- × 5
----
-
4 62
where applicable.
2 Multiply numerators and multiply = 5
--
8
-
denominators.

THINK WRITE
c 1 Write the fraction calculation. c 21 ÷
--
4
-
3
--
5
-

2 Change the mixed number to an improper fraction. = 9
--
4
- ÷ 3
--
5
-

93
3 Times and tip, (change the division sign to a multiplication = ----
- × 5
----
-
4 31
sign and tip the second fraction) and cancel.
4 Multiply numerators and multiply denominators; then = 15
-----
4
-
simplify the answer by writing the fraction as a mixed
number. = 33
--
4
-

Graphics Calculator tip! Fraction calculations
To perform the calculations in worked example 12 on
a graphics calculator, the following steps need to be
followed:
(a) Enter 3 ÷ 4 + 5 ÷ 6, press MATH , choose
1: Frac then press ENTER . The result is given
M

as 19 . The graphics calculator gives all answers as
-----
12
-
improper fractions.
(b) Enter 3 ÷ 4 × 5 ÷ 6, press MATH , choose 1: Frac then press ENTER .
M

(c) Enter (2 + 1 ÷ 4) ÷ (3 ÷ 5), press MATH , choose 1: Frac then press ENTER .
M

WORKED Example 13
3
Find --
7
- of 98.
THINK WRITE
3
1 Write the calculation. -- of 98
7
-
98 14
2 Change the ‘of’ to ×, write the whole number over 1 and = 3
----
- × ----------
-
71 1
cancel if applicable.
3 Multiply numerators and multiply denominators. = 42

Writing fractions with the same denominator allows us to compare the size of fractions.

WORKED Example 14
Write the fractions 2 , 8 ,
-- --
- -
3 9
5
--
6
- in ascending order.
THINK WRITE
2 8 5
1 Write the fractions. -- , -- , --
- - -
3 9 6
2 Write all fractions as equivalent fractions by ﬁnding the = 12 16 15
----- , ----- , -----
-
18 18 18
- -
lowest common denominator, in this case 18.
3 Rewrite the original fractions in the correct order. = 2, 5,
-- --
- -
3 6
8
--
9
-


Another way of writing fractions in order is to convert each fraction to a decimal
number before comparing them.
Converting fractions to decimal numbers
To convert a fraction to a decimal number, divide the numerator by the denominator.

WORKED Example 15
7
Convert --
8
- to a decimal number.
THINK WRITE
7
1 Write the fraction. --
8
-
0.875
2 Divide the numerator by the denominator.
8 ) 7.000
3 Write the fraction and the equivalent decimal number. 7
--
8
- = 0.875

Converting fractions to percentages
To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign.

WORKED Example 16
23
Convert -----
40
- to a percentage.
THINK WRITE
23
1 Write the fraction. -----
40
-
100 5
2 Multiply by 100, include the percentage sign and = ( -------- ×
23
2
----------
1
- )%
40
cancel if applicable.
3 Multiply the numerators and multiply the denominators. = 115
-------- %
2
-

4 Simplify by writing as a mixed number. = 57 1 %
--
2
-

remember
remember
1. To write fractions in simplest form, divide the numerator and the denominator by the
highest common factor (HCF) of both.
2. To change improper fractions to mixed numbers, divide the numerator by the denominator
and express the remainder as a fraction in simplest form.
3. To change a mixed number into an improper fraction, multiply the whole number by the
denominator, add the numerator and write the result over the denominator.
4. To add or subtract fractions, form equivalent fractions with the same denominator, then
add or subtract the numerators.
5. To multiply fractions, cancel if possible, then multiply the numerators, multiply the
denominators and simplify if appropriate.
6. To divide fractions, times and tip, then simplify if possible.
7. To add, subtract, multiply or divide mixed numbers, change the mixed numbers to improper
fractions ﬁrst. (When subtracting, an alternative method is to make the second fraction into
a whole number after writing the fractions with the same denominator.)
8. To write fractions in order, express them as equivalent fractions and compare.
9. To ﬁnd a fraction of an amount, multiply the fraction by the amount.
10. To convert a fraction to a decimal number, divide the numerator by the denominator.
11. To convert a fraction to a percentage, multiply the fraction by 100 and include the % sign.


1E Fractions
Math
HEET
1.8

cad

SkillS
1 Write the following fractions in simplest form. Simplifying
8 24 14 72 45
a -----
12
- b -----
30
- c -----
28
- d -----
81
- e -----
50
- fractions

f 35
-----
49
- g 24
-----
64
- h 14
-----
22
- i 21
-----
36
- j 36
--------
108
-
HEET
1.9

SkillS
108 75 16 25 33
k --------
144
- l --------
500
- m -----
20
- n --------
100
- o -----
99
-

2 Convert the following to mixed numbers in simplest form.
22 31 49 37 21 HEET
1.10
a -----
- b -----
- c -----
- d -----
- e -----
-

SkillS
5 7 4 6 9
68 55 80 98 94
f -----
16
- g -----
20
- h -----
15
- i -----
10
- j -----
12
-

3 Convert the following mixed numbers to improper fractions.
HEET
1.11
a 33
--
- b 44
--
- c 91
--
- d 55
--
-
9
e 3 -----
-

SkillS
4 5 3 6 10

f 56
--
7
-
11
9
g 2 -----
- h 15
--
8
- i 5 13
-----
20
- j 3 14
-----
17
-

WORKED 4 Evaluate the following.
Example GC pro
HEET
1.12
1
+ 3 2
− 5 2
× 3 8
÷ 5

gram
12 a --
- --
- b --
- -----
- c --
- --
- d -----
- --
-

SkillS
4 5 3 12 7 4 15 6
Fractions
e 5
--
7
- + 2
--
5
- f 9
-----
10
- − 5
-----
12
- g 7
-----
10
- × 11
-----
14
- h 5
--
8
- ÷ 2
--
5
-

i 7
--
8
- − 5
--
9
- + 10
1
-----
- j 21 + 31
--
4
- --
3
- k 5 -- − 4 -----
2
5
-
10
9
- l 37 × 11
--
8
- --
6
- HEET
1.13

SkillS
m 63 ÷ 31
--
4
- --
2
- n (4 1 + 3 3 ) ×
--
4
- --
5
-
4
--
5
- o −1 −
--
6
-
2
--
3
- + 5
--
6
- p 1
--
2
- × −1 ÷ −1 ÷
--
3
- --
4
-
1
--
5
-

WORKED 5 Find the following.
Example
5 3 5 2 1
13 a --
8
- of 72 b --
4
- of 28 c --
6
- of 36 d --
3
- of 81 e --
5
- of 65
4 7 1 11 3
f --
9
- of 117 g -----
10
- of 150 h --
7
- of 98 i -----
12
- of 192 j -----
16
- of 480

WORKED 6 Write each of the following sets of fractions in ascending order.
Example
1 1 3 3 1 7 1 3 1
14 a -- , -- , --
- - -
4 2 8
b ----- , -- , -----
- -
10 3 20
- c -- , ----- , --
-
6 20 5
- -

7 2 13 1 2 7 11 19
d ----- , -- , ----- , --
- -
10 3 20 2
- - e -- , ----- , ----- , -----
- -
5 20 25 50
- - f 1 1 , 1 5 , 1 12 ,
--
4
- --
6
-
7
-----
-
11
-----
16
-

g 7
30 15 3 5
3 1 2
----- , ----- , -- , --
- - - - h 5
-
18 19 17
5
-
5
----- , ----- , -----
- i −1 , −1 ,
--
8
- --
5
-
1
--
4
-

j − 19 , − 31 , − --------
-----
25
- -----
40
-
79
100
- k 10
1
----- ,
- − 1 , -- , − 1
-- -
-
9 8
1
--
7
- l −2 ,
--
3
-
7 2
----- , -- ,
10 3
- - −3 ,
--
4
-
4
--
5
-

WORKED 7 Convert each of the following fractions to a decimal number. GC pro L Spread
Example XCE
gram

sheet
E

3 4 9 14
15 a --
4
- b --
5
- c -----
20
- d -----
25
-
Converting Converting
e 31
-----
- f 7
--------
- g 5
-----
- h 3
-----
-
fractions to fractions to
40 100 16 80 decimals decimals
3 141 5
i -----
10
- j --------
200
- k -----
32
- l 11
--
2
-


reads WORKED 8 Convert each of the following fractions to a percentage.
L Sp he Example
et
EXCE

1 1 3 7
16 a --
4
- b --
5
- c --
8
- d -----
16
-
Converting
fractions to 79 18 59 11
e --------
100
- f -----
25
- g -----
80
- h -----
20
-
percentages
5 7 5 4
d i --
- j --
- k --
- l -----
-
hca 6 9 7 11
Mat

Converting 9 multiple choice
fractions to a The fraction 112
--------
- is equal to:
decimals or 192
percentages 9 25 13 5 7
A -----
16
- B -----
48
- C -----
24
- D --
8
- E -----
12
-

b The expression (− ----- −
11
5
-
12
5
----- )
- × ( 5 − 5 ) is equal to:
--
8
- --
9
-

A − -----------
575
9504
- B − -----------
25
9504
- C −1 -----------
317
3168
- D −1 -----------
857
3168
- E 0
c The false statement below is:
A 1 > -----
--
5
-
3
20
- B 2 <3
--
3
- --
4
- C 5
--
8
- < 4
--
7
- D 4
--
5
- > 3
--
4
- E 5
--
6
- > 5
--
7
-

5
d The fraction --
8
- as a decimal number is:
A 0.58 B 1.6 C 0.625 D 62 1
--
2
- E 5.8
2
e The fraction --
3
- as a percentage is:
5
A 0.6% B 66.67% C 66 ----- %
11
- D 66 2 %
--
3
- E 67%

10 Easisell High had 100 boxes of lollies to sell to raise money for the Children’s
Hospital. If Year 9 sold 1 of the boxes and Year 10 sold 1 of what was left:
--
2
- --
5
-
a how many did each year level sell?
b how many boxes were left for the other year levels to sell?
c what fraction of the total was left for the other year levels to sell?
11 Alexa made a cake for
the family to share. As
soon as it was iced,
Mum and Alexa each
ate 1 of it, Dad ate 1 ,
6
-
--
4
-
--
and Freddi and Elliot
1
ate ----- each. What
-
12
fraction of the cake
was left for the next
day?

Percentages
In this section, we will be converting percentages to decimal numbers and to simple
fractions. We will be finding the percentage of an amount, expressing one amount as a
percentage of another, finding the full amount given some other amount as a percentage
of it, and increasing and decreasing amounts by a given percentage.

Converting percentages to decimal numbers
To convert a percentage to a decimal number, divide the percentage by 100.

% means ‘out of 100’.

WORKED Example 17
Convert 56.25% to a decimal number.
THINK WRITE
1 Write the percentage. 56.25%
2 Divide the percentage by 100. The = 0.5625
number will be smaller so change the
position of the decimal point 2 places to
the left.

Converting percentages to fractions in simplest form
To convert a percentage to a fraction in simplest form, place the percentage over
100 and simplify where appropriate. If the percentage includes a fraction, change to
an improper fraction then divide the percentage by 100 and simplify where
appropriate.

WORKED Example 18
2
Convert 22 -- % to a fraction in simplest form.
9
-

THINK WRITE
1 Write the percentage. 22 2 %
--
9
-

2 Change the percentage to an improper = 200
--------
9
- ÷ 100
fraction and divide by 100.
= 200
--------
9
- × 1
--------
100
-

3 Simplify. = 2
--
9
-

Finding a percentage of an amount
To find a percentage of an amount, change the percentage to a fraction, the ‘of’ to ×
and perform the operation.


WORKED Example 19
Find 34% of 950.
THINK WRITE
1 Write the calculation. 34% of 950
950 19
2 Change the percentage to a fraction, the ‘of’ to = 34
----------
- × -------------
-
100 2 1
× and perform the operation.
= 17 × 19
= 323

Expressing one amount as a percentage of another
To express one amount as a percentage of another is the same as to convert a fraction to
a percentage. Write a fraction with the first amount as the numerator and the second
amount as the denominator, then multiply the fraction by 100.

WORKED Example 20
Write 2.4 as a percentage of 12.8.
THINK WRITE
2.4
1 Write a fraction with the first amount as the ---------
12.8
-
numerator and the second amount as the
denominator.
2 Change the fraction to a percentage by = ( ---------- ×
2.4
12.8
100
--------
1
- )%
multiplying by 100 and including the %.
3 Simplify. = 18.75%
Note: The answer could also be left as a fraction.

Finding the full amount, given a percentage of it
If Fred knows that his cheque for $50 000 was 25% of his uncle’s estate, can he work
out the value of the estate? To do this, first calculate 1% of the total amount, then
multiply by 100 to find 100% of the amount. This is known as the unitary method of
percentages.

WORKED Example 21
Find the number, if 62% of the number is 186.
THINK WRITE
1 Write the given information. 62% is 186.
2 Find 1% by dividing both the percentage and 1% is 3.
the number by the percentage. (That is, ÷ 62.)
3 Find 100%, or the whole amount, by multiplying 100% is 300.
the percentage and the number by 100.
4 Write the answer in a sentence to show what 62% of 300 is 186.
is meant.

Increasing and decreasing by a given percentage
To increase an amount by a given percentage, add the increase to 100% to find a new
percentage and then find the new percentage of the original amount. Similarly, to
decrease an amount by a percentage, subtract the decrease from 100% and find the new
percentage of the original amount.

WORKED Example 22
Increase 300 by 17%.
THINK WRITE
1 Add the increase to 100% to find the (17 + 100)%
new percentage of 300. = 117%
2 Write the calculation using the new 117% of 300
percentage which is greater than 100%
because it is an increase.
117 300 3
3 Write the percentage as a fraction out = ---------- × ----------
- -
100 1 1
of 100, multiply by the amount and
cancel if appropriate.
4 Simplify. = 351
5 Write a sentence. If 300 is increased by 17%, it becomes 351.

Alternatively, the percentage increase could be found and added to the original amount.
To decrease an amount by a given percentage, subtract the decrease from 100% to
find a new percentage and then find the new percentage of the original amount. For
example, to decrease 300 by 17% is to find 83% of 300. The answer must be less than
the original amount because it has been decreased.

remember
remember
1. To convert a percentage to a decimal number, divide by 100.
2. To convert a percentage to a fraction in simplest form, divide by 100 or write
the percentage as a fraction out of 100, then simplify.
3. To find a percentage of an amount, divide the percent by 100 and multiply by
the amount.
4. To express one amount as a percentage of another, divide the first amount by
the second and multiply by 100.
5. If we need to find an amount, given the percentage of that amount, find 1% and
multiply the result by 100.
6. To increase an amount by a given percentage, add the percentage to 100% and
find the resulting percentage of the amount.
7. To decrease an amount by a given percentage, subtract the percentage from
100% and find the resulting percentage of the amount.


1F Percentages

1.14 WORKED 1 Convert each of the following percentages to a decimal number.
HEET Example
a 62% b 41% c 38% d 93%
SkillS

17
e 10% f 2% g 36.7% h 21.25%
i 250% j 315.7% k 800% l 0.6%
d
hca WORKED 2 Convert each of the following percentages to a fraction in simplest form.
Mat

Example
18
a 97% b 42% c 40% d 70%
Percent-
ages e 55% f 50% g 25% h 30%
i 62 1 %
--
2
- j 33 1 %
--
3
- k 47 1 %
--
2
- l 81 %
--
3
-
reads
L Sp he 9
m 81 ----- % n 28 4 % o 44 4 % 2
p 16 -- %
et
EXCE

- --
- --
- -
11 7 9 3
Converting
percent- WORKED 3 Find the following.
ages to Example a 71% of 8 b 65% of 320 c 52% of 1700
fractions 19
and decimals d 13% of 54 e 83% of 27 f 24% of 175
reads g 12.5% of 104.48 h 42.5% of 55 i 58 1 % of 15.6
--
-
L Sp he 3
et
EXCE

8
Finding j 88 -- % of 3.69
9
- k 23 1 % of 150
--
2
- l 33 1 % of 300
--
3
-
the
percentage WORKED 4 Write each of the following as a percentage, giving your answer as an exact decimal
Example
of an
20
number where appropriate.
amount a 45 out of 60 b 27 out of 100 c 6 out of 20
reads
L Sp he d 32 out of 50 e 37.5 out of 60 f 0.3 out of 12
et
EXCE

g 21 out of 48 h 9.6 out of 15 i 18 out of 25
One amount j 0.63 out of 1.25 k 15.5 out of 60 l 62.8 out of 80
as a
percentage 5 multiple choice
of another
a The percentage 123.5% as a decimal number is:
A 1.235 B 12.35 C 0.1235 D 12 350 E 123.5
1
b As a fraction in simplest form 67 -- % is:
4
-
27 11 53 269 2
A -----
40
- B -----
16
- C -----
80
- D --------
400
- E --
3
-

c Which of the following is largest?
29 49
A 57% B 0.6 C -----
50
- D 61% E -----
80
-

d What is 23 1 % of 45?
--
5
-

A 10.4625 B 10.575 C 10.44 D 10.5 E 10.4
e What percentage of 60 is 35?
A 58 1 %
--
3
- B 171 3 %
--
7
- C 58.3% D 21% E 25%

WORKED 6 Find the number in each of the following examples.
Example
21
a 12% of the number is 156 b 23% of the number is 368
c 15% of the number is 690 d 82% of the number is 328
e 16% of the number is 1.44 f 120% of the number is 5.4
g 13% of the number is 32.5 h 68% of the number is 138.72
i 2.5% of the number is 22.5 j 31 1 % of the number is 37.5
--
4
-

WORKED 7 Increase each of the following numbers by the given percentage. L Spread
Example XCE
a 45 by 15% b 5800 by 42% c 65 by 20%

sheet
E
22
d 72 by 70% e 106 by 53% f 670 by 3% Increasing or
g 880 by 62 1 %
--
2
- h 2.5 by 27% i 84 by 41 2 %
--
3
- decreasing by a
percentage
8 Decrease each of the following numbers by the given percentage.
a 45 by 15% b 76 by 35% c 120 by 40%
d 2722 by 53% e 6530 by 30% f 104 by 7%
g 1.2 by 11% h 640 by 42 1 %
--
2
- i 96 by 16 2 %
--
3
-

9 multiple choice
GAME
a If 20% of a number is 80, what is the number?

time
A 400 B 16 C 25 D 96 E 64 Number skills
b If 480 is decreased by 27 1 %, the result is:
--
2
- — 002
A 612 B 132 C 348 D 72.5 E 1745
c If 60 is increased by 15%, by what percentage does the result have to be decreased
to obtain 54?
A 15% B 21 17 %
-----
23
- C 25%% 6
D 78 ----- %
23
- E 20%
10 If Fred knows that his cheque for $50 000 was 25% of his uncle’s estate, what was the
value of his uncle’s estate? SHE
ET 1.2

Work
11 The Sunﬂower Clothing Store was having a 15% off sale. If Sam wanted to buy a new
pair of jeans, how much would they cost if the original price was $75?


Career profile
GREG McINTYRE — Production Manager

Name: Greg McIntyre photographers and so on) and adding
Profession: Production Manager GST.
Qualifications: Certificate of Business Studies 2. Schedules are produced by working
(Advertising) backwards from the delivery deadline. I
Employer: Idea Communications determine the sequence of tasks, how long
each task will take and the studio members
I decided to undertake the Business Studies required. All staff time sheets must
(Advertising) course because I was interested correlate with the task and the time
in advertising and commercial art. It was also allocated. Work is often ‘juggled’,
recommended to me by my high school art depending on priority, to ensure that
teacher. At Idea Communications I oversee deadlines are met.
all work produced in the art studio and by 3. Invoice approvals are checked against
outside suppliers. I check that production purchase orders and take into account
specifications are complied with, that budgets unquoted items such as freight, tax and
are adhered to and deadlines are met. All new corrections.
jobs require ‘opening’ which entails filling in It seems to me that mathematical skills are
production brief forms, estimating costs and vital to a productive enterprise of any kind.
preparing schedules. Quotations are obtained Computers and calculators are a useful aid,
for all outside supplier costs. A choice is but without an understanding of the
made, but not always because of cost, and principles, not much can be achieved.
orders are issued.
A lot of the work I do involves calculations Questions
with fractions, decimals and percentages. For 1. What does ‘opening’ a new job entail at
example: Idea Communications?
1. Estimates involve calculating the number 2. Why is it more beneficial for Greg to ‘work
of hours each task or job will involve, backwards from the delivery deadline?’
multiplying by the hourly rate for staff 3. What other courses offer an advertising
required, adding outside costs (printer, certificate or degree?

Interesting times
Candice has the choice of investing $1000 at 6 2 % p.a. or at 6.65% p.a.
--
3
-
1 Write 6 2 % as a decimal number.
--
3
-
2 Write 6.65% as a decimal number.
3 Which of the two interest rates is greater?
4 Using the greater interest rate, how much interest would Candice earn in 1 year?
5 If she kept the money (including the interest) in the bank for 3 years, how much
would she have at the end of the 3 years? (Hint: It is not $1200.00.)

2
1 Evaluate −3 − 4 ÷ −2 × (6 − 9).
2 Evaluate 4 + 1 − ----- .
--
7
- --
4
-
11
14
-

3 Evaluate 11 × 15 ÷ 5 .
-----
20
- -----
22
- --
6
-
4 Round 0.003 950 01 to 4 signiﬁcant ﬁgures.
5 Convert 67% to a decimal.
6 Express 17 as a percentage.
-----
20
-
7 Write 0.62 as a simple fraction.
8 Evaluate 49.312 − 183.8 + 701.6511.
9 Change 5 6 to an improper fraction.
--
7
-
10 Decrease $349 by 83%.

Index notation, square roots and
higher order roots
Index notation
An index is a power to which a number is raised. It is
the number of times that the base number is multi-
plied together. In the case of 24, 2 is the base and 4 is
the index.
On a calculator this is calculated by using the ^
function or the xy function. The keys to press on a
graphics calculator to calculate 24 are 2^4 ENTER.

WORKED Example 23
3
Express  4 as a fraction in simplest form.
--
-
 9
THINK WRITE
3
1 Write the calculation.  4
--
-
 9
43
2 Remove the brackets (optional). = -----
93

3 Simplify both numerator and denominator. = 64
--------
729
-


Graphics Calculator tip! Simplifyingpower
raised to a
fractions

Fractions need to be entered using the ÷ key. However,
to ensure that both the numerator and the denominator
are raised to the given power, enter the fraction with
brackets. If you wish the answer to be expressed as a
fraction, remember to press MATH and select
1: Frac before pressing ENTER . The screen at right
M

shows the calculation for worked example 23.

Square roots
The square root of a number is a value which, when multiplied by itself, gives the
original number. For example 64 = 8 because 8 × 8 = 64.
Taking the square root of a number is the opposite
operation to squaring a number. That is, 82 = 64 and
64 = 8.
Generally, on a scientific calculator you would
enter 64 then the key. On a graphics calculator,
you would press 2nd [ ] then 64 and ENTER .

WORKED Example 24
Evaluate 458 and write the answer, correct
to 2 decimal places.
THINK WRITE
1 Write the given square root. 458
2 Use a calculator to find the square root.
3 Round the answer as required. ≈ 21.40

Higher order roots
The cube root of a given number is a value, which when written 3 times and multiplied,
is equal to the given number. For example, 3 27 = 3 because 3 × 3 × 3 = 27.
The fourth root of a given number is a value which, when written 4 times and multi-
plied, is equal to the given number. For example, 4 625 = 5 because 5 × 5 × 5 × 5 = 625.
1
x --
-
On a scientific calculator this is done by using the or x y key.

Graphics Calculator tip! Finding cube roots
and higher order roots
To find the cube root of a number, press MATH ,
select 4: 3 (, enter the number under the root sign
and press ENTER .
The screen to the right shows the calculations for
the following cube roots: 3 27 , 3 2197 , 3 12.56 .

To find a higher order root, press the number of the
root required (4 for fourth root, 5 for fifth root and so
on) then MATH , select option 5: x , enter the number
under the root sign and press ENTER . (Note: This option
can also be used with square roots and cube roots.)
The screen at right shows the calculations for the
following higher order roots: 4 625 , 5 781 , 6 42 .

WORKED Example 25
Calculate 5 649 correct to 3 decimal places.
THINK WRITE
Write the given root term. 5 649
1
2 Use a calculator to find the answer.
3 Write the answer correct to 3 decimal = 3.651
places.

The square root of a fraction can be evaluated by finding the square root of both the
numerator and the denominator. This also applies to higher order roots.

WORKED Example 26
36
Express -----
49
- as a fraction.
THINK WRITE
36
1 Write the given square root. -----
49
-

2 Rewrite as the square root of both the numerator = 36
---------
-
49
and the denominator.

3 Evaluate, keeping the answer in fraction form. = 6
--
7
-

36
On a graphics calculator, -----
49
- would be entered as

2nd [ ] 36 ÷ 49). Press MATH and select 1: Frac
M

(to express your answer as a fraction) then press
ENTER .

remember
remember
1. Use a calculator to evaluate numbers with indices and to find square roots and
higher order roots.
2. If the number is a fraction, calculate the numerator and denominator
separately.


Index notation, square roots
1G and higher order roots
1.15 1 Calculate the following.
HEET a 27 b 35 c 106 d 44 e 53
SkillS

2
f 1.7 g 2.54 h 3.13 i 3.052 j 0.83
WORKED 2 Express the following as fractions in simplest form.
Example
2 3 6 3 4 7 2
23 a  4
--
- b  2
--
- c  1
--
- d  -----
- e  -----
-
 5  3  2  10  12
5 7 3 4 5
f  8
--
- g  1
--
- h  3
--
- i  6
--
- j  3
--
-
 9  2  4  7  5
reads
L Sp he 3 Evaluate the following.
et
EXCE

Square a 441 b 0.09 c 81 d 1.44 e 2116
roots DIY
f 0.0529 g 676 h 132.25 i 0.0576 j 7.29

hca
d WORKED 4 Evaluate the following, correct to 2 decimal places.
Example
Mat

Index
24 a 465 b 65.87 c 2354 d 0.986 e 19.9
notation, square f 8624 g 1.75 h 56.78 i 21.45 j 5.6
roots and higher
order WORKED 5 Calculate the following.
roots DIY Example
25 a 3 13.824 b 5 5.378 24 c 4 70.7281 d 6 729
6 Calculate the following, correct to 3 decimal places.
a 8 46 b 3 869 c 5 149.0642 d 7 8975
WORKED 7 Express the following as fractions in simplest form.
Example
1 9 121 4
26 a -----
81
- b -----
16
- c --------
169
- d --------
121
-

289 0.36 0.25 0.49
e --------
729
- f ---------
9.61
- g ---------
1.44
- h ---------
2.25
-

8 Calculate the following:
a 2 4 + 53 b 6.12 – 2.13 c 0.84 × 1.23 d 64 ÷ 43
9 multiple choice
a To 3 signiﬁcant ﬁgures, 3 4583 is:
A 67.7 B 16.617 C 67.698 D 17.0 E 16.6
b Rounded to 3 decimal places, (1.2)5 is equal to:
A 2.488 B 2.49 C 1.037 D 1.04 E 6
reads
L Sp he 10 A large number can be
et
EXCE

expressed in the form
Scientific 3.56 × 104. What is
notation
DIY
the number?
11 The planet Jupiter is
approximately
ET 1.3 7.78 × 108 km
SHE
from the sun. Write the
Work

number of kilometres
as a whole number.

Calculator computations
A calculator can be used to compute more difficult examples. Generally we enter the
calculation as it is seen; however, we have to remember to bracket everything which is
under a square root sign as well as bracketing the numerator and denominator in a frac-
tional calculation.

WORKED Example 27
Calculate the following, expressing the answers correct to 3 decimal places.
3.5 4 – 9.8 3
a 6.5 2 + 1.7 3 b ----------------------------
-
10.7 2 – 53

THINK WRITE

a 1 Write the calculation. a 6.5 2 + 1.7 3
2 Perform the calculation using either
a scientific or graphics calculator.
(On a graphics calculator, press
2nd [ ] 6.5 ^ 2 + 1.7 ^ 3), then
ENTER .)

3 Write your answer correct to = 6.868
3 decimal places.
3.5 4 – 9.8 3
b 1 Write the calculation. b ----------------------------
-
10.7 2 – 53
2 Perform the calculation using a
calculator.
(On a graphics calculator, press
(3.5 ^ 4 − 9.8 ^ 3) ÷ 2nd [ ]
10.7 ^ 2 − 53) then ENTER .)

3 Write your answer correct to = −100.889
3 decimal places.

remember
remember
1. Use a calculator to evaluate more difficult examples.
2. Rounding to a given number of significant figures begins at the first non-zero
digit.
3. Rounding to a given number of decimal places begins with the first digit after
the decimal point.


1H Calculator computations

hca
d WORKED 1 Calculate the following, expressing the answers correct to 3 decimal places.
Example
Mat

Calculator
27 a 516 – 204 b 516 – 204 c 516 – 204
computations
d 3
65 – 97 e 9.6 × 4.1 + 6.8 f 3
7.8 2
DIY
46.7 5.9 – 2.4 1 1
g ---------------------
- h --------------------
- i ------ + ------
- -
21 – 18.6 3.7 9.7 3.4
–2.7 – 3.9 1
j −6.53 + 2.66 k --------------------------
- l ---------------------
-
– 4.6 × – 3.2 59 + 75
65 + 35 75 + 9.2 416 – 324
m -----------------
- n -----------------------
- o ---------------------------
-
72 – 98 61 – 3.7 5.8 + 7.2
2 Calculate the following, correct to 4 significant figures.
1 56 + 99
a -----------------
- b ------------------ c 5.32 − 4.43
47 – 29 28 + 11
12 2 + 8 2 – 7 2 4.6 4 – 2.1 5
d 1.28 2 + 3.15 e ------------------------------
- f -------------------------
2 × 12 × 6 3.4 3 – 1.9 5
25.8 2 – 4.1 2 150 – 29.3 963 96.5 + 67.5
g --------------------------------
- h ------------------------ × ------------
- - i ------------------------------
-
5.9 3 – 6.4 4 4.1 2 2.7 ( 5.1 – 2.8 ) 2
3 multiple choice
a What is 3 49.6 – 5.6 × – 3.1, correct to 4 significant figures, equal to?
A 24.40 B −5.026 C 21.03 D −13.69 E 22.65
b What is ( 5.1 – 2.78 ) ( 1.6 3 + 1.05 ) , correct to 2 decimal places, equal to?

A 17.66 B −0.68 C 7.84 D 5.15 E 20.08

World population
World At the start of the chapter we looked at estimates of the world population over
Year population a 100-year period.
1950 2 555 078 074 1 Calculate the increase in population for each 10-year period.
1960 3 039 332 401 2 Which 10-year period has the largest increase in population?
1970 3 707 610 112 3 What is the predicted increase in population from 1950 to 2050?
4 Round each of the populations given in the table to 2 significant figures.
1980 4 456 705 217
5 Use your answers to part 4 to calculate the percentage increase in
1990 5 283 755 345 population for each 10-year period.
2000 6 080 141 683 6 Which 10-year period has the highest percentage increase in population?
2010 6 823 634 553 7 What is the predicted percentage increase in population from 1950 to 1960?
8 Write a few sentences describing your results. How does the increase in
2020 7 518 010 600
world population affect the environment?
2030 8 140 344 240 9 Parts 5, 6 and 7 used rounded results to calculate the percentage increase.
2040 8 668 391 454 Design a spreadsheet to perform these calculations with the original
2050 9 104 205 830 population figures.

Which piece of Australian currency
curr
features scenes from aviation history?
eatures from aviation history?
Use a calculator to answer the
questions to find the puzzle’s code.

= 2.37 = 6.2 + 3.05 + 1.79 = the mean of 3.672
5.06 + 8.43 + 0.95 21.8, 15.24, =
x 4.85
+ 8.98 + 3.781 + 2.62 7.62 and 31.8
= =

= 2.68
= 15.73 – 8.218 = 38% x 35.4
x 5.45 = 84.7 less
= = 57.359
=
= the average of 21.3
2.7, 3.6, 9.4 and = = 16.48 ÷ 4
2.5
3.8 = 18.9
= = 3.2
=
=
= the product of = 21% of 126.4 = 103.8 x 0.345
2.75 and 3.84
= = = = the average of
16.74, 13.29
and 4.89
= the mean of =
= 12.4 ÷ 1.6
6.78, 13.24,
34.293 less =
17.84
= 8.91 and 3.74
= =
= the sum of 8.73,
= 72% of 35.8 = the sum of 8.34, 9.21, 2.643, 8.4,
= 18.237 – 4.87 9.27, 5.416, 2.976 and 1.425
= = 18.203 and 1.94 =
=
= the sum of 3.74,
= 5.28 x 3.65
8.94, 0.87, 2.64, = 10.34 multiplied = 18.9 ÷ 2.4
3.17 and 1.43 by 1.265 =
= = =

= the difference = 2.3846 = 24% of 158.6
5.078
= 81% of 25.8
between 7.054
and 21.9 + 4.9424 = =
=

7.512 14.846 25.776 16.41 13.367 14.606 11.64 27.341 26.821 16.453 5.90625 4.12

20.898 8.52 20.79 17.8092 33.384 10.56 12.405 43.169 19.272

4.875 13.452 8.1675 26.544 38.064 35.811 13.0801 19.115 7.875 7.75


Applications
The skills learnt in this chapter can now be applied to problems relating to real-life
situations. In this part the problems will be simpliﬁed to basic mathematical
expressions so we can use mathematical skills to determine the answers.

WORKED Example 28
The temperature at Mt Buller drops
steadily at night by 1.8°C per hour.
The temperature at 6 pm is 4.5°C.
What is the temperature at 2 am?

THINK WRITE
1 Find the number of hours between 6 pm Hours between 6 pm and 2 am: 6 + 2 = 8
and 2 am.
2 Write a mathematical expression for the Temperature drop: 8 × 1.8
total drop in temperature.
3 Write a mathematical expression for the Temperature at 2 am: 4.5 − 8 × 1.8
temperature drop from 4.5°C.
4 Use the order of operations to solve the = 4.5 − 14.4
problem. = −9.9
5 Write the answer in a sentence. The temperature at 2 am is −9.9°C.

WORKED Example 29
In a school of 460 students, half of them buy
their lunch from the canteen, while 2 bring
--
5
-
lunch from home. The rest do not eat lunch.
How many students do not eat lunch?

THINK WRITE
1 Find the number of students who buy lunch. Students who buy lunch: 1
--
2
- × 460
--------
1
- = 230
2 Find the number of students who bring Students who bring lunch: 2
--
5
- × 460
--------
1
- = 184
their lunch.
3 Find the number who eat lunch by adding Students who eat lunch: 230 + 184 = 414
these amounts.
4 Find the number who do not eat lunch by Students who don’t eat lunch: 460 – 414 = 46
subtracting these amounts from the total
number of students.
5 Write the answer in a sentence. The number of students who do not eat lunch
is 46.

WORKED Example 30
Tessa’s wage is increased by 3.2%. If her old wage is $675 per week, what is her new wage?
THINK WRITE
1 Find the new percentage by adding the 100% + 3.2% = 103.2%
percentage increase to 100%, which is
the original wage.
2 Find the new wage. New wage: 103.2% of 675
= 696.60
3 Answer the question in a sentence. The new wage is $696.60.

remember
remember
1. Read the question carefully.
2. Highlight or underline important information.
3. Write a mathematical expression or calculation for the given situation.
4. Use mathematical skills to evaluate the expression.
5. Write a sentence to answer the question.

1I Applications

WORKED 1 a The temperature in Young drops steadily by 1.7°C per hour. The temperature at
Example
28
5 pm is 8°C. What is the temperature at 2 am?
b The temperature in Doblin at 4 am is −5°C. The temperature rises steadily by 2.6°C
per hour. What is the temperature at 10 am?
c The winter temperature at Dubbo drops from 13°C at 3 pm to −6°C at 4 am. How
many degrees does the temperature drop?
d If the minimum temperature in Canberra on Monday is −7°C and it rises by 16°C to
the maximum temperature, what is the maximum temperature that day?
e The temperature range on Tuesday in Milday was 24°C. If the maximum tempera-
ture was 10°C, what was the minimum temperature?

2 a Roger wins $150 on a poker machine but then loses $340 the rest
of the night. How much worse off is he than when he started?
b A bank statement shows a balance of −$53.76. Fran deposits
$156.80. What is the new balance?

3 a What is the cost of 36 litres of petrol at 71.9c per litre?
b Potatoes cost $1.80 per kilogram. How much will 8 kilograms of potatoes cost?
c Donna buys 5 exercise books at $1.35 each and 4 pens at 45c each. What is her
change from $10?
d Tony is paid $15.60 per hour. How long must Tony work in order to earn $140.40?
e Steve earns a salary of $41 000 per annum. What is his weekly salary if 1 year is
52.143 weeks?
f Janice pays for the following amounts of petrol on a trip from Sydney to
Melbourne: 23 L at 81.9c per L, 36 L at 85.8c per L, 31 L at 90.5c per L. What was
Janice’s total petrol bill for the journey?
g A certain body is 68% ﬂuid. The body’s volume is 5.8 L. How much of this body is
ﬂuid? Give the answer in millilitres.
3 1
WORKED 4 A school has 570 students. For sport ----- of the students choose soccer,
10
- --
3
- choose foot-
Example
29
ball and the rest play tennis. How many play tennis?
5 a Yan has a choice of 3 of $40
--
5
-
or 2 of $39. Which choice
--
3
-
would give him more money?
b Of the 1881 people who live
in Galaxy, 4 are women. How
--
9
-
many of the inhabitants are
men?
c At a factory, 1 out of 150
light bulbs are faulty. If the
factory makes 1200 light
bulbs, how many are faulty?
d Teri is paid $480 per week.
Of this, ----- goes to rent, 2 to
9
- --
-
1 20 5
food, ----- to other essentials
10
-
and the rest is saved. How
much per week does Teri
save?
e In a biscuit mixture, 3 is --
8
-
sugar. If there are 120 grams
of sugar in the mixture; how
many grams of mixture is
there?
WORKED
Example 6 Jay earns $12.50 per hour and works a 35-hour week. He obtains a 4.5% pay increase.
30 What is his new weekly wage?

7 a In a township, 68% of the people go to church. There are 5500 people in the town.
How many go to church?
b In an election, 43% vote Liberal, 41% Labor, 8% Democrat and the rest vote for
minor parties.
i What percentage of voters vote for minor parties?
ii If there are 15 500 registered voters, of which 95% vote at the election, how
many voters do not vote for either of the 2 major parties (Liberal or Labor)?
c Roald achieves 39 out of 60 for a test while Serena achieves 25 out of 40. Who per-
formed better and by what percent?
d A car is discounted by 15%. If the
customer pays $16 150, what was
the price of the car before it was
discounted?
8 The distance from Earth to our closest
neighbouring galaxy, the Large
Magellanic Cloud, is approximately
1.608 × 1018 km. Write this distance as
a basic numeral.
9 multiple choice
a The temperature at Fresia is 4.8°C
at midday. The temperature falls
steadily by 1.3°C per hour. What is
the temperature in Fresia at 1 am
the following morning?
A −12.2° B –12.1° C −12° D −9.5° E 3.11°
b At a school, 3 of Year 9 students study history, 1 study geography and the rest
--
5
- --
4
-
study commerce. If there are 180 students in the whole of Year 9, how many
students study commerce?
A 153 B 80 C 27 D 21 E 45
c The Johnson family pay council rates at 1.008 12 cents in the dollar. If their land is
valued at $60 000, how much do they pay in rates?
A $60 487.20 B $604.87 C $6048.70 D $604.90 E $60.50
d Jeans are discounted by 20%. The discounted price is $30 less than the usual price.
How much are the discounted jeans?
A $24 B $45 C $120 D $150 E $50


summary
Copy the sentences below. Fill the gaps by choosing the correct word or
expression from the word list that follows.
1 The order of operations is first, followed by multiplication or
division left to right, then finally addition or subtraction, left to right.
When using a scientific calculator, work the whole question from left to
right.
2 Rounding to a given number of decimal places begins with the first digit
after the decimal .
3 Rounding to a given number of figures begins with the first
non-zero digit of the complete number.
4 To convert a decimal number to a fraction, place the digits
over 10, 100, 1000 . . . (depending on the number of decimal places), and
simplify if appropriate.
5 To convert a fraction to a decimal number, divide the by the
denominator.
6 To convert a decimal number to a percentage, multiply the
number by 100.
7 To convert a percentage to a decimal number, the
percentage by 100.
8 To convert a fraction to a percentage, multiply the by 100.
9 To convert a percentage to a fraction, put the over 100 and
simplify if appropriate (or divide by 100 if the percentage has a fraction).
10 To find a percentage of an amount, divide the percentage by 100 and
multiply by the .
11 To express one amount as a percentage of another, divide the first
amount by the second amount and multiply by .
12 To find the full amount, given some other quantity is a percentage of the
amount, determine 1% by dividing, then by 100 to find the
full amount.
13 To increase an amount by a given percentage, the percentage
to 100% and find that percentage of the amount.
14 To an amount by a given percentage, subtract the percentage
from 100% and find that percentage of the amount.

WORD LIST
add decimal 100 amount
decrease significant multiply fraction
divide numerator percentage
point brackets finite


CHAPTER
review
1 Evaluate each of the following.
a (16 + 12) ÷ 7 b 7 × 6 − 12 ÷ 3 c [3 × (7 − 5)] ÷ 2 d 12 + 8 × 5 ÷ 4 1A
a 5 × 9 + (60 − 6) ÷ 9 b −6 − 4 × −2 1B
c [(8 × −4 + 7) + 7] ÷ −3 d 36 ÷ −6 − 4 × −5 + 6
3 Insert operation signs to make the equation true: 6 K −5 K 3 K 2 = −12 1B
4 Round 5689.7143 to:
a 2 decimal places b 2 signiﬁcant ﬁgures. 1C
5 Round 2156.586 to:
a the nearest ten b the nearest tenth. 1C
6 multiple choice
A rounded number is 13.50. The original number could have been: 1C
A 13.5 B 13.52 C 13.507 D 13.495 E 13.4938
7 Evaluate these expressions.
a 0.375 + 4 × 9.06 b 14.4 ÷ 1.2 − 0.65 × 23 1D
8 Convert each of the following decimal numbers into a fraction in simplest form.
a 0.875 b 0.24 c 0.55 d 0.365 e 0.248 1D
f 0.13 g 0.75 h 0.575 i 0.372 j 0.4
9 Evaluate these expressions.
a 1 +3 ×5
--
4
- --
5
- --
8
- b 2 × ( ----- − 1 ) ÷
--
3
-
7
12
- --
4
-
5
-----
18
- 1E
10 Convert each of the following to a percentage.
a 0.71 b 2.4 c 15
-----
16
- d 1
--
6
- 1E
11 multiple choice
Which of the following statements is false? 1E
A 3
--
4
- < 19
-----
25
- B 16
5
-----
- > 3
-----
10
- C 13
-----
40
- < 7
-----
20
- D 8
-----
11
- < 18
-----
25
- E 6
--
7
- > 17
-----
20
-

12 Evaluate the following.
a 3 of 56
--
7
- b 65% of 590 c 0.89 of 420 1F
13 Convert each of the following to a decimal number.
a 12% b 34.6% c 7--
8
- d 33
-----
4
- 1F
14 Convert each of these percentages to a fraction in simplest form.
a 52% b 130% c 28 4 %--
7
- 1F
15 Decrease 240 by 24%.
1F
16 If 45 is 3% of a number, what is the number?
17 Place the following in ascending order. 1F
29 6
a 57%, 0.6, b 0.245, 23 1 %
-----
50
- ----- ,
25
- --
2
-
1F


18 Place the following in descending order.
1F a 4
-- ,
- 82%, 0.83 b 5
----- ,
- 41%, 0.416
5 12

1G
a 1296 b 4.2025 c 3 1.331
1G a 2.52 × 3.53 b 2.5 × 102 × 3.5 × 103
21 Calculate each of the following correct to:
1H i 3 decimal places ii 3 significant figures.
87 + 54 3 789 – 614
a -----------------
- b ---------------------------
-
87 – 54 654 – 437
22 multiple choice
1I The temperature in Warialda at 7 pm is 3.5°C and at 3 am is −3.7°C. What is the average
drop in temperature per hour between the two readings?
A 0.72° B 0.02° C 0.025° D 0.9° E 0.8°
23 multiple choice
1I In the first semester test Dominic scored 37 out of 50, while in the second semester test he
scored 59 out of 80. Which of the following statements correctly compares his second
semester result with his first semester result?
A increased by 1 %
--
4
- B decreased by 1 %
--
4
- C same result
D increased by 28.7% E decreased by 28.7%
24 Rhonda pays $27.93 for petrol. If the price of petrol is 66.5 cents per litre, how many litres
1I did Rhonda put into her car?
25 A dam is 58% full. It presently holds 302 470 L. How many more litres of water will fill the
1I dam to 100% of capacity?
26 Simon visits his parents in a country town 1575 km
1I away from his home. He drives 16 of the way
-----
25
-
the first day and intends to reach the town
the next day.
a How far does he need to drive
on the second day?
b Simon drives for 12 hours
on the first day. What is
his average speed on
that day?
c Simon’s car averages
15 km per litre of
petrol. What is his
petrol cost for
the trip if the
price of petrol
is 69.9 cents
test
yourself per litre?
CHAPTER

1

Ch 01

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Ch 01