University 4 Collecting, organising and displaying data
Compound bar charts Copy Copy Copy Copy
PercentageUniversofitCchildrenyoPprye-ssC-aRmebrviidegweA compound bar chart displays two or more sets of data on the same set of axes to make it easy
with growth problemsto compare the data. This chart compares the growth rates of children born to mothers with
different education levels.
Percentage of children under age 3 whose growth is impacted by mother’s education
60 57 No education Secondary or higher
50 47 47
40 39
36
Review 30 30 28
22 20 19 23
20 17 17
10 10 8
5
UniversitCyoPprye-ssC-aRmebrviidegwe
0 India* Madagascar Nigeria Cambodia Haiti Columbia Egypt Senegal
Country
* Children under age 5
(Adapted from Nutrition Update 2010: www.dhsprogram.com)
You can see that children born to mothers with secondary education are less likely to experience
growth problems because their bars are shorter than the bars for children whose mothers have
only primary education. The aim of this graph is to show that countries should pay attention to
the education of women if they want children to develop in healthy ways.
Exercise 4.8 Applying your skills
Review 1 Draw a bar chart to show each of these sets of data.
a Burgers Noodles Fried Hot chips Other
Favourite take- chicken 20 29
away food
84
No. of people 40 30
UniversitCyoPprye-ssC-aRmebrviidegwe
b African countries with the highest HIV/AIDS infection rates (2015 est)
Country % of adults (aged 15 to 49) infected
HIV is a massive global health Swaziland 28.8
issue. In 2017, the organisation
Avert reported that 36.7 million Botswana 22.2
people worldwide were living with
HIV. The vast majority of these Lesotho 22.7
people live in low- and middle-
income countries and almost 70% Zimbabwe 14.7
of them live in sub-Saharan Africa.
The countries of East and Southern South Africa 19.2
Africa are the most affected.
Review Since 2010, there has been a Namibia 13.3
29% decrease in the rate of new
infection in this region, largely Zambia 12.3
due to awareness and education
campaigns and the roll out of anti- Malawi 9.1
retroviral medication on a large
scale. (Source: www.Avert.org) Uganda 7.1
- Cambridge Mozambique 10.5
ess - Review
(Data taken from www.aidsinfo.unaids.org)
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REWIND 2 Here is a set of raw data showing the average summer temperature (in °C) for 20 cities in the
Middle East during one year.
Look at the earlier sections of this
chapter to remind yourself about 32 42 36 40 35 36 33 32 38 37
grouped frequency tables if you 34 40 41 39 42 38 37 42 40 41
need to.
a Copy and complete this grouped frequency table to organise the data.
In this example, the temperature
groups/class intervals will be Temperature (°C) 32–34 35–37 38–40 41–43
displayed as ‘categories’ with gaps Frequency
between each bar. As temperature
is continuous, a better way to deal b Draw a horizontal bar chart to represent this data.
with it is to use a histogram with
equal class intervals; you will see 3 The tourism organisation on a Caribbean island records how many tourists visit from the
these in chapter 20. region and how many tourists visit from international destinations. Here is their data for the
Review first six months of this year. Draw a compound bar chart to display this data.
Regional Jan Feb Mar Apr May Jun
visitors 12 000 10 000 19 000 16 000 21 000 2 000
International 40 000 39 000 15 000 12 000 19 000 25 000
visitors
Review Pie charts
A pie chart is a circular chart which uses slices or sectors of the circle to show the data. The
circle in a pie chart represents the ‘whole’ set of data. For example, if you surveyed the favourite
sports played by everyone in a school then the total number of students would be represented
by the circle. The sectors would represent groups of students who played each sport.
Like other charts, pie charts should have a heading and a key. Here are some fun examples
of pie charts:
How lions spend a typical day How elephants spend a typical day
5% 2% Sleeping 8% Eating
6% Socialising 10%
7%
Sleeping
Grooming Not forgetting
Attacking gazelles
Eating gazelles
80% 82%
How pandas spend a typical day
How jellyfish spend a typical day
Being adorable
9% 1%
Review Just floating there
Absorbing food
Ruining a perfectly
good day at the beach
90% 100%
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Worked example 7
The table shows how a student spent her day.
Activity School Sleeping Eating Online On the Complaining
3 phone about stuff
Number 7 8 1.5
of hours 2.5 2
Draw a pie chart to show this data.
7 + 8 + 1.5 + 3 + 2.5 + 2 = 24 First work out the total number of hours.
Review Then work out each category as a fraction of the whole and convert the fraction to
degrees:
(as a fraction of 24) (convert to degrees)
School =7 = 7 × 360 = 105°
24 24
Sleeping =8 = 8 × 360 = 120°
24 24
Eating = 1 5 = 15 = 15 × 360 = 22.5°
24 240 240
Online =3 = 3 × 360 = 45°
24 24
On the phone = 2 5 = 25 = 25 × 360 = 37.5°
24 240 240
Review Complaining =2 = 2 × 360 = 30°
24 24
It is possible that your angles, once Activity School Sleeping Eating Online On the Complaining
rounded, don’t quite add up to 7 phone about stuff
360°. If this happens, you can add Number 8 1.5 3
or subtract a degree to or from the of hours 105° 2.5 2
largest sector (the one with the 120° 22.5° 45°
highest frequency). Angle 37.5° 30°
A student’s day
School
Review Sleeping • Draw a circle to represent the
Eating whole day.
Online
On phone • Use a ruler and a protractor to
Complaining measure each sector.
• Label the chart and give it a title.
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Worked example 8
This pie chart shows how Henry spent one day of his school holidays.
Henry’s day
Sleeping
Other stuff
Computer games
Review a What fraction of his day did he spend playing computer games?
b How much time did Henry spend sleeping?
c What do you think ‘other stuff’ involved?
a 120 = 1 Measure the angle and convert it to a fraction. The yellow
360 3 sector has an angle of 120°. Convert to a fraction by
writing it over 360 and simplify.
b 210 × 24 = 14 hours Measure the angle, convert it to hours.
360
c Things he didn’t bother to list. Possibly eating, showering, getting dressed.
Review Exercise 4.9 1 The table shows the results of a survey carried out on a university campus to find out about
the use of online support services among students. Draw a pie chart to illustrate this data.
Category Number of students
Never used online support 180
Used online support in the past 120
Use online support presently 100
2 The table shows the home language of a number of people passing through an international
airport. Display this data as a pie chart.
Review Language Frequency
English 130
Spanish 144
Chinese 98
Italian 104
French 24
German 176
Japanese 22
94 Unit 1: Data handling
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3 The amount of land used to grow different vegetables on a farm is shown below.
Draw a pie chart to show the data.
Vegetable Squashes Pumpkins Cabbages Sweet potatoes
Area of land (km2) 1.4 1.25 1.15 1.2
4 The nationalities of students in an international school is shown on this pie chart.
Nationalities of students at a school
Review Brazilian
French
Indian
American
Chinese
a What fraction of the students are Chinese?
b What percentage of the students are Indian?
c Write the ratio of Brazilian students : total students as a decimal.
d If there are 900 students at the school, how many of them are:
i Chinese? ii Indian? iii American? iv French?
Review FAST FORWARD Line graphs
Graphs that can be used for Some data that you collect changes with time. Examples are the average temperature each month
converting currencies or systems of the year, the number of cars each hour in a supermarket car park or the amount of money in
of units will be covered in your bank account each week.
chapter 13. Graphs dealing with
time, distance and speed are The following line graph shows how the depth of water in a garden pond varies over a year.
covered in chapter 21. The graph shows that the water level is at its lowest between June and August.
Depth of water in a garden pond
60
55
50
45
Depth of
water (mm) 40
35
Review 30
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month of year
When time is one of your variables it is always plotted on the horizontal axis.
Unit 1: Data handling 95
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Tip Choosing the most appropriate chart
You may be asked to give You cannot always say that one type of chart is better than another – it depends very much on
reasons for choosing a the data and what you want to show. However, the following guidelines are useful to remember:
particular type of chart.
Be sure to have learned • Use pie charts or bar charts (single bars) if you want to compare different parts of a whole, if
the advantages and there is no time involved, and there are not too many pieces of data.
disadvantages in the table.
• Use bar charts for discrete data that does not change over time.
Tip • Use compound bar charts if you want to compare two or more sets of discrete data.
• Use line graphs for numerical data when you want to show how something changes over time.
Before you draw a chart
decide: The table summarises the features, advantages and disadvantages of each different types of chart/
• how big you want the graph. You can use this information to help you decide which type to use.
Review chart to be Chart/graph and their Advantages Disadvantages
• what scales you will use features
Review Attractive and appealing, can Symbols have to be broken
and how you will divide Pictogram be tailored to the subject. up to represent ‘in-between
these up Data is shown using symbols Easy to understand. values’ and may not be clear.
• what title you will give or pictures to represent Size of categories can be Can be misleading as it
the chart quantities. easily compared. does not give detailed
• whether you need a key The amount represented by information.
or not. each symbol is shown on a key. Clear to look at.
Easy to compare categories Chart categories can be
FAST FORWARD Bar chart and data sets. reordered to emphasise
You will work with line graphs Data is shown in columns Scales are given, so you can certain effects.
when you deal with frequency measured against a scale on work out values. Useful only with clear sets of
distributions in chapter 20. the axis. numerical data.
Review Double bars can be used for Looks nice and is easy to
two sets of data. understand. No exact numerical data.
Data can be in any order. Easy to compare categories. Hard to compare two
Bars should be labelled and No scale needed. data sets.
the measurement axis should Can shows percentage of ‘Other’ category can be a
have a scale and label. total for each category. problem.
Total is unknown unless
Pie charts Shows more detail of specified.
Data is displayed as a information than other Best for three to seven
fraction, percentage or graphs. categories.
decimal fraction of the Shows patterns and trends
whole. Each section should clearly. Useful only with numerical
be labelled. A key and totals Other ‘in-between’ data.
for the data should be given. information can be read Scales can be manipulated
from the graph. to make data look more
Line graph Has many different formats impressive.
Values are plotted against and can be used in many
‘number lines’ on the vertical different ways (for example
and horizontal axes, which conversion graphs, curved
should be clearly marked and lines).
labelled.
96 Unit 1: Data handling
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Exercise 4.10 1 Which type of graph would you use to show the following information? Give a reason for your choice.
a The number of people in your country looking for jobs each month this year.
b The favourite TV shows of you and nine of your friends.
c The number of people using a gym at different times during a day.
d The favourite subjects of students in a school.
e The reasons people give for not donating to a charity.
f The different languages spoken by people in your school.
g The distance you can travel on a tank of petrol in cars with different sized engines.
Review Applying your skills
2 Collect ten different charts from newspapers, magazines or other sources.
Stick the charts into your book.
For each graph:
a write the type of chart it is
b write a short paragraph explaining what each chart shows
c identify any trends or patterns you can see in the data.
d Is there any information missing that makes it difficult to interpret the chart? If so what is
missing?
e Why do you think the particular type and style of chart was used in each case?
f Would you have chosen the same type and style of chart in each case? Why?
Summary
ReviewDo you know the following? Are you able to … ?
• In statistics, data is a set of information collected to • collect data to answer a statistical question
answer a particular question. • classify different types of data
• use tallies to count and record data
• Categorical (qualitative) data is non-numerical. Colours, • draw up a frequency table to organise data
names, places and other descriptive terms are all categorical. • use class intervals to group data and draw up a grouped
• Numerical (quantitative) data is collected in the form of frequency table
numbers. Numerical data can be discrete or continuous.
• construct single and back-to-back stem and leaf
Discrete data takes a certain value; continuous data can diagrams to organise and display sets of data
take any value in a given range. • draw up and use two-way tables to organise two or more
sets of data
• Primary data is data you collect yourself from a primary
source. Secondary data is data you collect from other • construct and interpret pictograms
• construct and interpret bar charts and compound
sources (previously collected by someone else).
bar charts
• Unsorted data is called raw data. Raw data can be organised
using tally tables, frequency tables, stem and leaf diagrams • construct and interpret pie charts.
Review and two-way tables to make it easier to work with.
• Data in tables can be displayed as graphs to show
patterns and trends at a glance.
• Pictograms are simple graphs that use symbols to
represent quantities.
• Bar charts have rows of horizontal bars or columns of vertical
bars of different lengths. The bar length (or height) represents
an amount. The actual amount can be read from a scale.
• Compound bar charts are used to display two or more
sets of data on the same set of axes.
• Pie charts are circular charts divided into sectors to
show categories of data.
• The type of graph you draw depends on the data and
what you wish to show.
Unit 1: Data handling 97
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Exam-style questions
1 Salma is a quality control inspector. She randomly selects 40 packets of biscuits at a large factory. She opens each
packet and counts the number of broken biscuits it contains. Her results are as follows:
Review 0021300231
1123012342
0000100123
3222101212
a Is this primary or secondary data to Salma? Why?
b Is the data discrete or continuous? Give a reason why.
c Copy and complete this frequency table to organise the data.
No. of broken biscuits Tally Frequency
0
1
2
3
4
d What type of graph should Salma draw to display this data? Why?
Review 2 The number of aircraft movements in and out of five main London airports during April 2017 is summarised
in the table.
Airport Gatwick Heathrow London City Luton Stansted
Total flights 23 696 39 660 6380 10 697 15 397
a Which airport handled most aircraft movement?
b How many aircraft moved in and out of Stansted Airport?
c Round each figure to the nearest thousand.
d Use the rounded figures to draw a pictogram to show this data.
3 This table shows the percentage of people who own a laptop and a mobile phone in four different districts in a large city.
Review District Own a laptop Own a mobile phone
A 45 83
B 32 72
C 61 85
D 22 68
a What kind of table is this?
b If there are 6000 people in District A, how many of them own a mobile phone?
c One district is home to a University of Technology and several computer so ware manufacturers. Which district
do you think this is? Why?
d Draw a compound bar chart to display this data.
98 Unit 1: Data handling
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Mode of transport Percentage
Metro 36
Bus 31
Motor vehicle 19
Cycle 14
Review Represent this data as a pie chart.
5 Study this pie chart and answer the questions that follow.
Sport played by students
Review Baseball
Review Cricket
Football
Netball
Hockey
The data was collected from a sample of 200 students.
a What data does this graph show?
b How many different categories of data are there?
c Which was the most popular sport?
d What fraction of the students play cricket?
e How many students play netball?
f How many students play baseball or hockey?
Past paper questions
1 The table shows the number of goals scored in each match by Mathsletico Rangers.
Number of goals scored Number of matches
04
1 11
26
33
42
51
62
Unit 1: Data handling 99
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University Copy Copy Copy Copy
UniFrvequeerncsyitCyoPprye-ssC-aRmebrviidegweDraw a bar chart to show this information. [3]
Complete the scale on the frequency axis.
Review
UniversitCyoPprye-ssC-aRmebrviidegwe
012345 6
Number of goals scored
[Cambridge IGCSE Mathematics 0580 Paper 33 Q1 d(i) October/November 2012]
2 Some children are asked what their favourite sport is.
The results are shown in the pie chart.
Review Swimming
Gymnastics
UniversitCyoPprye-ssC-aRmebrviidegwe 120° 80° Running
45° 60°
Tennis Hockey
Review i Complete the statements about the pie chart. [4]
The sector angle for running is ............................ degrees. [3]
The least popular sport is ............................
1 of the children chose ............................
6
Twice as many children chose ............................ as ............................
ii Five more children chose swimming than hockey.
Use this information to work out the number of children who chose gymnastics.
- Cambridge [Cambridge IGCSE Mathematics 0580 Paper 32 Q5a) October/November 2015]
ess - Review
100 Unit 1: Data handling
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Chapter 5: Fractions and standard form
Review Key words
UniversitCyoPprye-ssC-aRmebrviidegwe• Fraction
• Vulgar fraction
• Numerator
• Denominator
• Equivalent fraction
• Simplest form
• Lowest terms
• Mixed number
• Common denominator
• Reciprocal
• Percentage
• Percentage increase
• Percentage decrease
• Reverse percentage
• Standard form
• Estimate
Review In this chapter you
will learn how to:
UniversitCyoPprye-ssC-aRmebrviidegwe
EXTENDEDReview • find equivalent fractions The Rhind Mathematical Papyrus is one of the earliest examples of a mathematical document. It is thought
• simplify fractions to have been written sometime between 1600 and 1700 BC by an Egyptian scribe called Ahmes, though it
• add, subtract, multiply and may be a copy of an older document. The first section of it is devoted to work with fractions.
divide fractions and mixed Fractions are not only useful for improving your arithmetic skills. You use them, on an almost
numbers daily basis, often without realising it. How far can you travel on half a tank of petrol? If your
share of a pizza is two-thirds will you still be hungry? If three-fifths of your journey is complete
• find fractions of numbers how far do you still have to travel? A hairdresser needs to mix her dyes by the correct amount
• find one number as a and a nurse needs the correct dilution of a drug for a patient.
percentage of another
• find a percentage of a
number
• calculate percentage
increases and decreases
• increase and decrease by a
given percentage
• handle reverse percentages
(undoing increases and
decreases)
• work with standard form
• make estimations without a
calculator.
- Cambridge
ess - Review Unit 2: Number 101
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RECAP
You should already be familiar with the following fractions work:
Equivalent fractions
Find equivalent fractions by multiplying or dividing the numerator and denominator by the same number.
1×4 =4 1 and 4 are equivalent
24 8 28
40 ÷ 10 = 4 40 and 4 are equivalent
50 10 5 50 5
To simplify a fraction you divide the numerator and denominator by the same number.
18 = 18 ÷ 2 = 9
40 40 ÷ 2 20
Review
Mixed numbers
Convert between mixed numbers and improper fractions:
3 4 = (3 × 7) + 4 = 25 Number of data in that group, not
77 7 individual values.
Calculating with fractions
To add or subtract fractions make sure they have the same denominators.
7 + 1 = 21+ 8 = 29 = 1 5
8 3 24 24 24 Class intervals are equal and should
not overlap.
To multiply fractions, multiply numerators by numerators and denominators by denominators. Write the answer in
simplest form.
Multiply to find a fraction of an amount. The word ‘of’ means multiply.
3×3= 9 3 of 12 = 3 × 12
8 4 32 8 81
Review = 36
8
= 4 1
2
To divide by a fraction you multiply by its reciprocal.
12 ÷ 1 = 12 × 3 = 36 2÷ 1 = 2×2 = 4
31 52 51 5
Review Percentages or use a calculator
The symbol % means per cent or per hundred.
Percentages can be written as fractions and decimals.
45% = 45 = 9
100 20
45% = 45 ÷ 100 = 0.45
Calculating percentages
To find a percentage of an amount:
use fractions and cancel or use decimals
25% of 60 =
1 15 0.25 × 60 = 15 2 5 % × 6 0 = 15
25 × 60 =15
100 1
41
102 Unit 2: Number
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5.1 Equivalent fractions
REWIND A fraction is part of a whole number.
Before reading this next Common fractions (also called vulgar fractions) are written in the form a. The number on
section you should remind
yourself about Highest Common b
Factors (HCFs) in chapter 1. the top, a, can be any number and is called the numerator. The number on the bottom, b, can
be any number except 0 and is called the denominator. The numerator and the denominator are
REWIND separated by a horizontal line.
You have come across simplifying
in chapter 2 in the context of If you multiply or divide both the numerator and the denominator by the same number, the new
algebra. fraction still represents the same amount of the whole as the original fraction. The new fraction
is known as an equivalent fraction.
Review For example, 2 = 2 4= 8 and 25 = 25 ÷ 5 = 5.
33 4 12 35 35 ÷ 5 7
Notice in the second example that the original fraction 25 has been divided to smaller terms
35
and that as 5 and 7 have no common factor other than 1, the fraction cannot be divided any
further. The fraction is now expressed in its simplest form (sometimes called the lowest terms).
So, simplifying a fraction means expressing it using the lowest possible terms.
LINK Worked example 1
Percentages are particularly Express each of the following in the simplest form possible.
important when we deal with
money. How o en have you a3 b 16 c 21 d5
been in a shop where the 15 24 28 8
signs tell you that prices are
reduced by 10%? Have you a 3 = 3 3 = 1
considered a bank account 15 15 ÷3 5
and how money is added?
The study of financial ideas b 16 = 16 ÷ 8 = 2
forms the greater part of 24 24 ÷ 8 3
Review economics.
c 21 = 21÷ 7 = 3
28 28 ÷ 7 4
d 5
8
is already in its simplest form (5 and 8 have no common factors other than 1).
Notice that in each case you Worked example 2
divide the numerator and the
denominator by the HCF of both. Which two of 5, 20 and 15 are equivalent fractions?
6 25 18
You could have written:
5 Simplify each of the other fractions: 5 is already in its simplest form.
15= 5 6
18 6
Review 6 20 = 20 ÷ 5 = 4
This is called cancelling and is a 25 25 ÷ 5 5
shorter way of showing what you
have done. 15 = 15 ÷ 3 = 5
18 18 ÷ 3 6
So 5 and 15 are equivalent.
6 18
Unit 2: Number 103
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Exercise 5.1 1 By multiplying or dividing both the numerator and denominator by the same number, find
three equivalent fractions for each of the following.
a5 b3 c 12 d 18 e 110
9 7 18 36 128
2 Express each of the following fractions in its simplest form.
a7 b3 c9 d 15 e 500 f 24 g 108
21 9 12 25 2500 36 360
Review 5.2 Operations on fractions
Multiplying fractions
When multiplying two or more fractions together you can simply multiply the numerators and
then multiply the denominators. Sometimes you will then need to simplify your answer. It can
be faster to cancel the fractions before you multiply.
Worked example 3
Calculate: b 5×3 c 3 of 4 1
a 3×2 7 8 2
47
a 3×2=3 2= 6 = 3 Multiply the numerators to get the new
4 7 4 7 28 14 numerator value. Then do the same with
the denominators. Then express the
Notice that you can also cancel before fraction in its simplest form.
multiplying: Divide the denominator of the first
fraction, and the numerator of the
3 × 1 = 3× 1 = 3 second fraction, by two.
4 2 2× 7 14
Review 2 7
To multiply a fraction by an integer b 5 × 3 = 5 3 = 15 15 and 7 do not have a common
you only multiply the numerator 7 71 7 factor other than 1 and so cannot
by the integer. For example, be simplified.
5 × 3 = 5 3 = 15.
7 77
c 3 of 4 1
8 2
To change a mixed number to a Here, you have a mixed number (4 21). This needs to be changed to an improper
vulgar fraction, multiply the whole fraction (sometimes called a top heavy fraction), which is a fraction where the
number part (in this case 4) by
the denominator and add it to the numerator is larger than the denominator. This allows you to complete the
numerator. So:
multiplication.
1 4 × 2+1 9 3 × 4 1 = 3 × 9 = 27 Notice that the word ‘of’ is replaced with the × sign.
2 2 2 8 2 8 2 16
4 = =
Review Exercise 5.2 Evaluate each of the following.
1 a 2×5 b 1×3 c 1×8 d 2 × 14
39 27 49 7 16
2 a 50 × 256 b 1 1 × 2 c 2 2 × 7 d 4 of 3 2
128 500 3 7 7 8 5 7
e 1 1 of 24 f 5 1 × 7 1 g 8 8 × 20 1 h 7 2 × 10 1
3 2 4 9 4 3 2
104 Unit 2: Number
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REWIND Adding and subtracting fractions
You will need to use the lowest You can only add or subtract fractions that are the same type. In other words, they must have the
common multiple (LCM) in this same denominator. This is called a common denominator. You must use what you know about
section. You met this in chapter 1. equivalent fractions to help you make sure fractions have a common denominator.
The following worked example shows how you can use the LCM of both denominators as the
common denominator.
Worked example 4
Write each of the following as a single fraction in its simplest form.
a 1+ 1 b 3+5 c 2 3 − 15
24 46 47
Review Notice that, once you have a a 1+ 1 Find the common denominator.
common denominator, you only 24
add the numerators. Never add the The LCM of 2 and 4 is 4. Use this as the common
denominators! = 2+ 1 denominator and find the equivalent fractions.
44
Then add the numerators.
=3
4
You will sometimes find that two b 3+5 Find the common denominator.
fractions added together can result 46
in an improper fraction (sometimes The LCM of 4 and 6 is 12. Use this as the common
called a top-heavy fraction). Usually = 9 + 10 denominator and find the equivalent fractions.
you will re-write this as a mixed 12 12
number. Add the numerators.
= 19
12 Change an improper fraction to a mixed number.
= 1172
Review The same rules apply for subtracting c 2 3 − 15
fractions as adding them. 47
= 11 − 12 Change mixed numbers to improper fractions to
47 make them easier to handle.
The LCM of 4 and 7 is 28, so this is the common
= 77 − 48 denominator. Find the equivalent fractions.
28 28 Subtract one numerator from the other.
= 77 − 48 Change an improper fraction to a mixed number.
28
= 29
28
= 1218
Tip Egyptian fractions
Egyptian fractions An Egyptian fraction is the sum of any number of different fractions (different denominators)
are a good example of
Review manipulating fractions but each with numerator one. For example 1 + 1 is the Egyptian fraction that represents 5. Ancient
they are not in the syllabus. 23 6
Egyptians used to represent single fractions in this way but in modern times we tend to prefer
the single fraction that results from finding a common denominator.
Exercise 5.3 Evaluate the following.
1 a 1+1 b 3+2 c 5−3 d 5+8
33 77 88 99
e 1+1 f 2−5 g 2 5 1 3 h 5 1 3 1
65 38 8 4 8 16
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2 a 4−2 b 6+ 5 c 11 + 7 1 d 11 − 7 1
3 11 4 4
e 3 1 4 1 f 5 1 + 3 1 + 4 3 g 5 1 − 3 1 + 4 3 h 1 1 + 2 2 − 1 1
2 3 4 16 8 8 16 4 3 5 4
Remember to use BODMAS here. i 3 + 2 × 14 j 3 1 − 2 1 × 4 k 3 1 − 1 1 + 7 3 l 2 1 − 3 1 + 4 1
73 8 2 4 3 6 2 4 4 3 5
Think which two fractions with a
numerator of 1 might have an LCM 3 Find Egyptian fractions for each of the following.
equal to the denominator given.
a3 b2 c5 d3
4 3 8 16
Review FAST FORWARD Dividing fractions
The multiplication, division, Before describing how to divide two fractions, the reciprocal needs to be introduced. The
addition and subtraction of reciprocal of any fraction can be obtained by swapping the numerator and the denominator.
fractions will be revisited in chapter
14 when algebraic fractions are So, the reciprocal of 3 is 4 and the reciprocal of 7 is 2 .
considered. 43 27
Also the reciprocal of 1 is 2 or just 2 and the reciprocal of 5 is 1 .
21 5
If any fraction is multiplied by its reciprocal then the result is always 1. For example:
1 × 3 = 1 , 3 × 8 = 1 and a × b = 1
31 83 ba
To divide one fraction by another fraction, you simply multiply the first fraction by the
reciprocal of the second.
Look at the example below:
a
b
Review a ÷ c = c
b d
d
Now multiply both the numerator and denominator by bd and cancel:
a ÷ c = a = a × bd
b d b b × bd
c
d c
d
= ad = a × d
bc b c
Worked example 5
Evaluate each of the following.
Review a 3÷1 b 13 ÷ 2 1 c 5÷2 d 6÷3
42 43 8 7
a1 Multiply by the reciprocal of 1. Use the rules you
3 1 3 2 3 1 2
4 ÷ 2 = 4 × 1 = 2 = 1 2
have learned about multiplying fractions.
2
106 Unit 2: Number
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b 13 ÷ 21 = 7 ÷ 7 Convert the mixed fractions to improper fractions.
4 343 Multiply by the reciprocal of 73.
=17 × 3
4 71 Write 2 as an improper fraction.
=3 Multiply by the reciprocal of 21.
4
c 5 ÷ 2 = 5 ÷ 2
8 8 1
= 5× 1
82
Review =5
16
To divide a fraction by an integer d 2
you can either just multiply the 6 ÷ 3= 6× 1
denominator by the integer, or 7 73
divide the numerator by the same =2 1
integer. 7
Exercise 5.4 Evaluate each of the following.
Worked example 6
1 1÷1 2 2÷3 3 4÷7 4 10 ÷ 5
73 57 9 11
5 41÷ 1 6 31 ÷52 7 77 ÷5 1 8 31 ÷31
57 53 8 12 42
9 Evaluate
Review a 2 1 − 1 2 ÷11 b 2 1 − 1 2 ÷ 1 1
3 5 3 3 5 3
Fractions with decimals
Sometimes you will find that either the numerator or the denominator, or even both, are
not whole numbers! To express these fractions in their simplest forms you need to
• make sure both the numerator and denominator are converted to an integer by finding an
equivalent fraction
• check that the equivalent fraction has been simplified.
Simplify each of the following fractions.
a 01 b 13 c 36
3 24 0 12
Review a 0.1 = 0.1× 10 = 1 Multiply 0.1 by 10 to convert 0.1 to an integer. To make sure the fraction is
3 3 10 30 equivalent, you need to do the same to the numerator and the denominator, so
multiply 3 by 10 as well.
b 13 = 1 3 × 10 = 13 Multiply both the numerator and denominator by 10 to get integers.
24 2 4 × 10 24 13 and 24 do not have a HCF other than 1 so cannot be simplified.
c 36 = 36 × 100 = 3600 = 300 Multiply 0.12 by 100 to produce an integer.
0.12 0.12 × 100 12 Remember to also multiply the numerator by 100, so the fraction is equivalent.
The final fraction can be simplified by cancelling.
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Exercise 5.5 Simplify each of the following fractions.
Remember that any fraction that 1 03 2 04 36 4 07
contains a decimal in either its 12 05 07 0 14
numerator or denominator will not
be considered to be simplified. 5 36 6 03× 5 7 04×15 8 2 8 × 1.44
15 12 16 07 06
What fraction can be used to Further calculations with fractions
represent 0.3?
You can use fractions to help you solve problems.
Review Remember for example, that 2 = 2 × 1 and that, although this may seem trivial, this simple fact
33
can help you to solve problems easily.
Worked example 7
Suppose that 2 of the students in a school are girls. If the school has 600 students, how
5
many girls are there?
Remember in worked example 3, 120
you saw that ‘of’ is replaced by ×. 2 of 600 = 2 × 600 = 2 × 600 = 2 ×120 = 240 girls
5 5 51
1
Worked example 8
Review Now imagine that 2 of the students in another school are boys, and that there are 360
5
boys. How many students are there in the whole school?
2 of the total is 360, so 1 of the total must be half of this, 180. This means that 5 of
55 5
the population, that is all of it, is 5 × 180 = 900 students in total.
Exercise 5.6 1 3 of the people at an auction bought an item. If there are 120 people at the auction, how
4
many bought something?
2 An essay contains 420 sentences. 80 of these sentences contain typing errors. What fraction
(given in its simplest form) of the sentences contain errors?
3 28 is 2 of which number?
7
4 If 3 of the people in a theatre buy a snack during the interval, and of those who buy a snack
5
5 buy ice cream, what fraction of the people in the theatre buy ice cream?
Review 7
5 Andrew, Bashir and Candy are trying to save money for a birthday party. If Andrew saves
1 of the total needed, Bashir saves 2 and Candy saves 1 , what fraction of the cost of the
4 5 10
party is left to pay?
6 Jroicseepwhitnhe2e21dsc6u12pscoufpws oatfecromokaekde rice for a recipe of Nasi Goreng. If 2 cups of uncooked E
41 cups of cooked rice, how many cups of uncooked
3
rice does Joseph need for his recipe ? How much water should he add ?
108 Unit 2: Number
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5.3 Percentages
A percentage is a fraction with a denominator of 100. The symbol used to represent percentage is %.
To find 40% of 25, you simply need to find 40 of 25. Using what you know about multiplying
100
fractions:
2 25
40 =514000 1
100 × 25 ×
= 2 × 25 5
1 5 ×
= 2 1
1
5 = 10
1
Review ∴ 40% of 25 = 10
Equivalent forms
A percentage can be converted into a decimal by dividing by 100 (notice that the digits
move two places to the right). So, 45% = 45 = 0.45 and 3.1% = 3.1 = 0.031.
100 100
A decimal can be converted to a percentage by multiplying by 100 (notice that the digits
move two places to the left). So, 0.65 = 65 = 65% and 0.7 × 100 = 70%.
100
Converting percentages to vulgar fractions (and vice versa) involves a few more stages.
Worked example 9
Convert each of the following percentages to fractions in their simplest form.
Review a 25% b 30% c 3.5%
a 25% = 25 = 1 Write as a fraction with a denominator of 100, then
100 4 simplify.
b 30% = 30 = 3 Write as a fraction with a denominator of 100, then
100 10 simplify.
Remember that a fraction that c 3.5% = 3 5 = 35 = 7 Write as a fraction with a denominator of 100, then
contains a decimal is not in its 100 1000 200 simplify.
simplest form.
Worked example 10
Convert each of the following fractions into percentages.
Review a1 b1
20 8
a 1 = 1 5 = 5 = 5% Find the equivalent fraction with a denominator of 100.
20 20 ×5 100 (Remember to do the same thing to both the numerator
and denominator).
5 = 0.05, 0.05 100 = 5%
100
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b 1 = 1 12 5 Find the equivalent fraction with a denominator of 100. (Remember
8 8 12 5 to do the same thing to both the numerator and denominator).
= 12.5 = 12.5% 12.5 = 0.125, 0.125 × 100 = 12.5%
100 100
Although it is not always easy to find an equivalent fraction with a denominator of 100, any
fraction can be converted into a percentage by multiplying by 100 and cancelling.
Worked example 11
Review Convert the following fractions into percentages:
a3 b8
40 15
a 3 × 100 = 30 = 15 = 7.5, b 8 × 100 = 160 = 53.3 (1d.p.),
40 1 4 2 15 1 3
so 3 = 7.5% so 8 = 53.3% (1d.p.)
40 15
Exercise 5.7 1 Convert each of the following percentages into fractions in their simplest form.
Later in the chapter you will see a 70% b 75% c 20% d 36% e 15% f 2.5%
that a percentage can be greater
than 100. g 215% h 132% i 117.5% j 108.4% k 0.25% l 0.002%
2 Express the following fractions as percentages.
Review a3 b7 c 17 d3 e8 f5
5 25 20 10 200 12
Finding one number as a percentage of another
To write one number as a percentage of another number, you start by writing the first number
as a fraction of the second number then multiply by 100.
Worked example 12
a Express 16 as a percentage of 48.
16 = 16 × 100 = 33.3% (1d.p.) First, write 16 as a fraction of 48, then multiply
48 48 by 100.
16 = 1 × 100 = 33.3% (1d.p.) This may be easier if you write the fraction in its
48 3 simplest form first.
b Express 15 as a percentage of 75.
Review 15 × 100 Write 15 as a fraction of 75, then simplify and multiply
75 by 100. You know that 100 divided by 5 is 20, so you
don’t need a calculator.
= 1 × 100 = 20%
5
110 Unit 2: Number
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c Express 18 as a percentage of 23.
You need to calculate 18 × 100, but this is not easy using basic fractions because
23
you cannot simplify it further, and 23 does not divide neatly into 100. Fortunately,
you can use your calculator. Simply type:
1 8 ÷ 2 3 × 1 0 0 = 78.26% (2 d.p.)
Exercise 5.8 Where appropriate, give your answer to 3 significant figures.
Review 1 Express 14 as a percentage of 35.
2 Express 3.5 as a percentage of 14.
3 Express 17 as a percentage of 63.
4 36 people live in a block of flats. 28 of these people jog around the park each morning. What
percentage of the people living in the block of flats go jogging around the park?
5 Jack scores 19 in a test. What percentage of the marks did Jack get?
24
6 Express 1.3 as a percentage of 5.2.
7 Express 0.13 as a percentage of 520.
Review Percentage increases and decreases
Suppose the cost of a book increases from $12 to $15. The actual increase is $3. As a fraction
of the original value, the increase is 3 = 1 . This is the fractional change and you can write
12 4
this fraction as 25%. In this example, the value of the book has increased by 25% of the original
value. This is called the percentage increase. If the value had reduced (for example if something
was on sale in a shop) then it would have been a percentage decrease.
Note carefully: whenever increases or decreases are stated as percentages, they are stated as
percentages of the original value.
Worked example 13
The value of a house increases from $120 000 to $124 800 between August and
December. What percentage increase is this?
$124 800 − $120 000 = $4800 % First calculate the increase.
% increase = increase × 100% Write the increase as a fraction of the
original and multiply by 100.
original Then do the calculation (either in your head
= 4800 × 100% or using a calculator).
120 000
Review Exercise 5.9 Applying your skills
Where appropriate, give your answer to the nearest whole percent.
1 Over a five-year period, the population of the state of Louisiana in the United States
of America decreased from 4 468 976 to 4 287 768. Find the percentage decrease in the
population of Louisiana in this period.
2 Sunil bought 38 CDs one year and 46 the next year. Find the percentage increase.
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3 A theatre has enough seats for 450 audience members. After renovation it is expected that
this number will increase to 480. Find the percentage increase.
4 Sally works in an electrical component factory. On Monday she makes a total of 363
components but on Tuesday she makes 432. Calculate the percentage increase.
5 Inter Polation Airlines carried a total of 383 402 passengers one year and 287 431 the
following year. Calculate the percentage decrease in passengers carried by the airline.
6 A liquid evaporates steadily. In one hour the mass of liquid in a laboratory container
decreases from 0.32 kg to 0.18 kg. Calculate the percentage decrease.
Review Increasing and decreasing by a given percentage
If you know what percentage you want to increase or decrease an amount by, you can find the
actual increase or decrease by finding a percentage of the original. If you want to know the new
value you either add the increase to or subtract the decrease from the original value.
Worked example 14
Increase 56 by: a 10% b 15% c 4%
Remember that you are always a 10% of 56 = 10 × 56 First of all, you need to calculate 10% of 56 to work out
considering a percentage of the 100 the size of the increase.
original value.
= 1 × 56 = 5 6 To increase the original by 10% you need to add this
10 to 56.
56 + 5.6 = 61.6
Review If you don’t need to know the actual increase but just the final value, you can use
this method:
If you consider the original to be 100% then adding 10% to this will give 110% of
the original. So multiply 56 by 110 , which gives 61.6.
100
b 115 × 56 = 64.4 A 15% increase will lead to 115% of the original.
100
c 104 × 56 = 58.24 A 4% increase will lead to 104% of the original.
100
Worked example 15
In a sale all items are reduced by 15%. If the normal selling price for a bicycle is $120
calculate the sale price.
100 − 15 = 85 Note that reducing a number by 15% leaves you with
85% of the original. So you simply find 85% of the
Review 85 × $120 = $102 original value.
100
Exercise 5.10 1 Increase 40 by:
a 10%
b 15% c 25% d 5% e 4%
2 Increase 53 by: b 84% c 13.6% d 112%
a 50% e 1 %
2
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3 Decrease 124 by:
a 10% b 15% c 30% d 4% e 7%
c 0.3% d 100%
4 Decrease 36.2 by:
a 90% b 35.4% e 1 %
2
Applying your skills
5 Shajeen usually works 30 hours per week but decides that he needs to increase this by
10% to be sure that he can save enough for a holiday. How many hours per week will Shajeen
need to work?
Review 6 12% sales tax is applied to all items of clothing sold in a certain shop. If a T-shirt is advertised
for $12 (before tax) what will be the cost of the T-shirt once tax is added?
7 The Oyler Theatre steps up its advertising campaign and manages to increase its audiences by
23% during the year. If 21 300 people watched plays at the Oyler Theatre during the previous
year, how many people watched plays in the year of the campaign?
8 The population of Trigville was 153 000 at the end of a year. Following a flood, 17%
of the residents of Trigville moved away. What was the population of Trigville after
the flood?
9 Anthea decides that she is watching too much television. If Anthea watched 12 hours
of television in one week and then decreased this by 12% the next week, how much
time did Anthea spend watching television in the second week? Give your answer in
hours and minutes to the nearest minute.
Reverse percentages E
Review Sometimes you are given the value or amount of an item after a percentage increase or
decrease has been applied to it and you need to know what the original value was. To solve
this type of reverse percentage question it is important to remember that you are always
dealing with percentages of the original values. The method used in worked example 14 (b)
and (c) is used to help us solve these type of problems.
Worked example 16
A store is holding a sale in which every item is reduced by 10%. A jacket in this sale is sold for $108.
How can you find the original price of the Jacket?
90 × x = 108 If an item is reduced by 10%, the new cost is 90% of the original (100–10). If x is the
100 original value of the jacket then you can write a formula using the new price.
x = 100 × 108 Notice that when the × 90 was moved to the other side of the = sign it became its
90 100
original price = $120. reciprocal, 100 .
90
Review Important: Undoing a 10% decrease is not the same as increasing the reduced value by 10%. If you increase the sale
price of $108 by 10% you will get 110 × $108 = $118.80 which is a different (and incorrect) answer.
100
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Exercise 5.11 1 If 20% of an amount is 35, what is 100%? E
2 If 35% of an amount is 127, what is 100%?
3 245 is 12.5% of an amount. What is the total amount?
4 The table gives the sale price and the % by which the price was reduced for a
number of items. Copy the table, then complete it by calculating the original prices.
Sale price ($) % reduction Original price ($)
52.00 10
Review 185.00 10
5
4700.00 5
2.90 12
24.50 8
10.00 7
12.50 15
9.75 20
25
199.50
99.00
5 A shop keeper marks up goods by 22% before selling them. The selling price of ten
items are given below. For each one, work out the cost price (the price before the mark up).
a $25.00 b $200.00 c $14.50 d $23.99 e $15.80
f $45.80 g $29.75 h $129.20 i $0.99 j $0.80
6 Seven students were absent from a class on Monday. This is 17.5% of the class.
Review a How many students are there in the class in total?
b How many students were present on Monday?
7 A hat shop is holding a 10% sale. If Jack buys a hat for $18 in the sale, how much
did the hat cost before the sale?
8 Nick is training for a swimming race and reduces his weight by 5% over a 3-month
period. If Nick now weighs 76 kg how much did he weigh before he started training?
9 The water in a pond evaporates at a rate of 12% per week. If the pond now contains
185 litres of water, approximately how much water was in the pond a week ago?
5.4 Standard form
When numbers are very small, like 0.0000362, or very large, like 358 000 000, calculations can be
time consuming and it is easy to miss out some of the zeros. Standard form is used to express
very small and very large numbers in a compact and efficient way. In standard form, numbers
are written as a number multiplied by 10 raised to a given power.
Remember that digits are in place Standard form for large numbers
order:
The key to standard form for large numbers is to understand what happens when you multiply by
Review1000s 100s 10s units 10ths 100ths 1000ths powers of 10. Each time you multiply a number by 10 each digit within the number moves one place
order to the left (notice that this looks like the decimal point has moved one place to the right).
3 0 00• 0 0 0
3.2
3.2 × 10 = 32.0 The digits have moved one place order to the left.
3.2 × 102 = 3.2 × 100 = 320.0 The digits have moved two places.
3.2 × 103 = 3.2 × 1000 = 3200.0 The digits have moved three places.
... and so on. You should see a pattern forming.
114 Unit 2: Number
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Any large number can be expressed in standard form by writing it as a number between 1 and 10
multiplied by a suitable power of 10. To do this write the appropriate number between 1 and 10
first (using the non-zero digits of the original number) and then count the number of places you
need to move the first digit to the left. The number of places tells you by what power, 10 should
be multiplied.
Worked example 17
Write 320 000 in standard form. Start by finding the number between 1 and 10 that has the same digits in
3.2 the same order as the original number. Here, the extra 4 zero digits can be
excluded because they do not change the size of your new number.
Review 54 3 2 1 Now compare the position of the first digit in both numbers: ‘3’ has to move 5
place orders to the left to get from the new number to the original number.
3 2 0 0 0 0.0
3.2
320 000 = 3.2 × 105 The first digit, ‘3’, has moved five places. So, you multiply by 105.
REWIND Calculating using standard form
The laws of indices can be found in
chapter 2. Once you have converted large numbers into standard form, you can use the index laws to carry
out calculations involving multiplication and division.
Although it is the place order that is
changing; it looks like the decimal Worked example 18
point moves to the right.
Solve and give your answer in standard form.
When you solve problems in
standard form you need to check a (3 × 105 ) × (2 106 ) b (2 × 103 ) × (8 107 )
your results carefully. Always be c (2.8 × 106 ) ÷ (1.4 104 ) d (9 × 106 ) + (3 108 )
sure to check that your final answer
Review is in standard form. Check that all a (3 × 105 ) × (2 × 106 ) = (3 × 2) × (105 106 ) Simplify by putting like terms
conditions are satisfied. Make sure = 6 × 105 6 together. Use the laws of indices
that the number part is between = 6 × 1011 where appropriate.
1 and 10.
Write the number in standard form.
You may be asked to convert your answer to an ordinary number. To convert 6 × 1011
into an ordinary number, the ‘6’ needs to move 11 places to the left:
6.0 × 1011
11 10 9 8 7 6 5 4 3 2 1
= 6 0 0 0 0 0 0 0 0 0 0 0 .0
b (2 × 103 ) × (8 × 107 ) = (2 × 8) × (103 107 ) The answer 16 × 1010 is numerically
= 16 × 1010 correct but it is not in standard
form because 16 is not between
16 × 1010 = 1 6 × 10 × 1010 1 and 10. You can change it to
= 1 6 × 1011 standard form by thinking of 16 as
1.6 × 10.
Review
c (2.8 × 106 ) ÷ (1.4 × 104 ) = 2.8 × 106 = 2.8 × 106 Simplify by putting like terms
1.4 × 104 1.4 104 together. Use the laws of indices.
= 2 × 106 4
= 2 × 102
Unit 2: Number 115
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To make it easier to add up the d (9 × 106 ) + (3 108 )
ordinary numbers make sure they
are lined up so that the place When adding or subtracting numbers in standard form it is often easiest to re-write
values match: them both as ordinary numbers first, then convert the answer to standard form.
9 × 106 = 9 000 000
300 000 000 3 × 108 = 300 000 000
So (9 × 106 ) + (3 × 108 ) = 300 000 000 + 9 000 000
+ 9 000 000
= 309 000 000
= 3.09 × 108
Review Exercise 5.12 1 Write each of the following numbers in standard form.
When converting standard form back a 380 b 4 200 000 c 45 600 000 000 d 65 400 000 000 000
to an ordinary number, the power e 20 f 10 g 10.3 h5
of 10 tells you how many places
the first digit moves to the left (or 2 Write each of the following as an ordinary number.
decimal point moves to the right),
not how many zeros there are. a 2.4 ×106 b 3 1 ×108 c 1 05 107 d 9 9 ×103 e 7.1 × 101
Remember that you can write 3 Simplify each of the following, leaving your answer in standard form.
these as ordinary numbers before
adding or subtracting. a (2 × 1013 ) × (4 1017 ) b (1.4 × 108 ) × (3 104 ) c (1.5 × 1013 )2
d (12 × 105 ) × (11 102 ) e (0.2 × 1017 ) × (0.7 1016 ) f (9 × 1017 ) ÷ (3 1016 )
g (8 × 1017 ) ÷ (4 1016 ) h (1.5 × 108 ) ÷ (5 104 ) i (2.4 × 1064 ) ÷ (8 1021)
j (1.44 × 107 ) ÷ (1.2 1 4 ) l (4.9 ×105) × (3.6 109 )
k (1.7 × 108 )
(3.4 × 105 )
4 Simplify each of the following, leaving your answer in standard form.
Review a (3 × 104 ) + (4 103 ) b (4 × 106 ) − (3 105 ) c (2.7 × 103 ) + (5.6 105 )
d (7.1 × 109 ) − (4.3 107 ) e (5.8 × 109 ) − (2.7 103 )
Review LINK Standard form for small numbers
Astronomy deals with very You have seen that digits move place order to the left when multiplying by powers of 10. If
large and very small numbers you divide by powers of 10 move the digits in place order to the right and make the number
and it would be clumsy and smaller.
potentially inaccurate to write
these out in full every time Consider the following pattern:
you needed them. Standard
form makes calculations and 2300
recording much easier.
2300 ÷ 10 = 230
2300 ÷ 102 = 2300 ÷ 100 = 23
2300 ÷ 103 = 2300 ÷ 1000 = 2 3
. . . and so on.
The digits move place order to the right (notice that this looks like the decimal point is
moving to the left). You saw in chapter 1 that if a direction is taken to be positive, the opposite
direction is taken to be negative. Since moving place order to the left raises 10 to the power of a
positive index, it follows that moving place order to the right raises 10 to the power of
a negative index.
Also remember from chapter 2 that you can write negative powers to indicate that you divide,
and you saw above that with small numbers, you divide by 10 to express the number in
standard form.
116 Unit 2: Number
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Worked example 19
Write each of the following in standard form.
a 0.004 b 0.000 000 34 c (2 × 10−3 ) × (3 10−7 )
a Start with a number between 1 and 10, in this case 4.
1 23 Compare the position of the first digit: ‘4’ needs to move 3 place orders to the right to get from the
new number to the original number. In worked example 17 you saw that moving 5 places to the
0.0 0 4 left meant multiplying by 105, so it follows that moving 3 places to the right means multiply by 10−3.
4.0
= 4 × 10–3 Notice also that the first non-zero digit in 0.004 is in the 3rd place after the decimal point and
that the power of 10 is −3.
Review
Alternatively: you know that you need to divide by 10 three times, so you can change it to a
fractional index and then a negative index.
0.004 = 4 ÷ 103
1
= 4 × 103
= 4 × 10 3.
b 0.000 00034 = 3.4 ÷ 107 1 234567
= 3 4 × 10−7
0. 0 0 0 0 0 0 3 4 = 3.4 × 10–7
Notice that the first non-zero digit in 0.000 000 34 is in the 7th place after the decimal point
and that the power of 10 is −7.
c (2 × 10−3 ) × (3 10−7 ) Simplify by gathering like terms together.
= (2 × 3) × (10−3 × 10−7 ) Use the laws of indices.
= 6 × 10−3 + −7
= 6 × 10−10
Review Exercise 5.13 1 Write each of the following numbers in standard form.
When using standard form with a 0.004 b 0.00005 c 0.000032 d 0.0000000564
negative indices, the power to
which 10 is raised tells you the 2 Write each of the following as an ordinary number.
position of the first non-zero digit
after (to the right of) the decimal a 3 6 × 10−4 b 1 6 × 10−8 c 2 03 10−7 d 8 8 × 10−3 e 7 1 × 10−1
point.
3 Simplify each of the following, leaving your answer in standard form.
For some calculations, you might
need to change a term into a (2 × 10−4 ) × (4 10−16 ) b (1.6 × 10−8 ) × (4 10−4 ) 10−3 )
standard form before you multiply c (1.5 × 10−6 ) × (2.1 10−3 ) d (11 × 10 5 ) × (3 1 2 )
or divide. e (9 × 1017 ) ÷ (4.5 10−16 ) f (7 × 10 21) ÷ (1 1016 )
g (4.5 × 108 ) ÷ (0.9 10 4 ) h (11 × 10−5 ) × (3 × 102 ) ÷ (2
Remember that you can write
these as ordinary numbers before 4 Simplify each of the following, leaving your answer in standard form.
adding or subtracting.
Review a (3.1 × 10−4 ) + (2.7 10−2 ) b (3.2 × 10−1) − (3.2 10−2 )
c (7.01 × 103 ) + (5.6 1 1) d (1.44 × 10−5 ) − (2.33 10−6 )
Applying your skills
5 Find the number of seconds in a day, giving your answer in standard form.
6 The speed of light is approximately 3 108 metres per second. How far will light travel in:
a 10 seconds b 20 seconds c 102 seconds
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7 Data storage (in computers) is measured in gigabytes. One gigabyte is 230 bytes.
a Write 230 in standard form correct to 1 significant figure.
b There are 1024 gigabytes in a terabyte. How many bytes is this? Give your answer in
standard form correct to one significant figure.
5.5 Your calculator and standard form
Standard form is also called On modern scientific calculators you can enter calculations in standard form. Your calculator
scientific notation or exponential will also display numbers with too many digits for screen display in standard form.
notation.
Keying in standard form calculations
Review
You will need to use the × 10x button or the Exp or EE button on your calculator. These are known
as the exponent keys. All exponent keys work in the same way, so you can follow the example below
on your own calculator using whatever key you have and you will get the same result.
When you use the exponent function key of your calculator, you do NOT enter the ‘× 10’ part of
the calculation. The calculator does that part automatically as part of the function.
Worked example 20
Using your calculator, calculate:
a 2.134 × 104 b 3.124 × 10–6
a 2.134 × 104 Press: 2 . 1 3 4 × 10x 4 =
= 21 340 This is the answer you will get.
b 3.124 × 10–6 Press: 3 . 1 2 3 Exp − 6 =
= 0.000003123 This is the answer you will get.
ReviewDifferent calculators work in Making sense of the calculator display
different ways and you need Depending on your calculator, answers in scientific notation will be displayed on a line with an
to understand how your own exponent like this:
calculator works. Make sure you
know what buttons to use to enter This is 5.98 × 10–06
standard form calculations and or on two lines with the calculation and the answer, like this:
how to interpret the display and
convert your calculator answer into This is 2.56 × 1024
decimal form.
If you are asked to give your answer in standard form, all you need to do is interpret the display
and write the answer correctly. If you are asked to give your answer as an ordinary number
(decimal), then you need to apply the rules you already know to write the answer correctly.
Review Exercise 5.14 1 Enter each of these numbers into your calculator using the correct function key and write
down what appears on your calculator display.
a 4.2 × 1012 b 1.8 × 10–5 c 2.7 × 106
d 1.34 × 10–2 e 1.87 × 10–9 f 4.23 × 107
g 3.102 × 10–4 h 3.098 × 109 i 2.076 × 10–23
118 Unit 2: Number
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2 Here are ten calculator displays giving answers in standard form. iv
i ii iii
v vi vii viii
ix x
a Write out each answer in standard form.
b Arrange the ten numbers in order from smallest to largest.
Review 3 Use your calculator. Give the answers in standard form correct to 5 significant figures.
a 42345 b 0.0008 ÷ 92003 c (1.009)5
d 123 000 000 ÷ 0.00076 e (97 × 876)4 f (0.0098)4 × (0.0032)3
g 8543 × 9210 9754
0.000034 h (0.0005)4
4 Use your calculator to find the answers correct to 4 significant figures.
a 9.27 × (2.8 × 105) b (4.23 × 10–2)3 c (3.2 × 107) ÷ (7.2 × 109)
d (3.2 × 10–4)2 e 231 × (1.5 × 10–6) f (4.3 × 105) + (2.3 × 107)
g 3 24 107 h 3 4.2 ×10−8 i 3 4.126 ×10−9
Review 5.6 Estimation
REWIND It is important that you know whether or not an answer that you have obtained is at least roughly
For this section you will need as you expected. This section demonstrates how you can produce an approximate answer to a
to remember how to round an calculation easily.
answer to a specified number of
significant figures. You covered this To estimate, the numbers you are using need to be rounded before you do the calculation.
in chapter 1. Although you can use any accuracy, usually the numbers in the calculation are rounded to one
significant figure:
3.9 × 2.1 ≈ 4 × 2 = 8
Notice that 3.9 × 2.1 = 8.19, so the estimated value of 8 is not too far from the real value!
Worked example 21
Estimate the value of:
4.6 + 3.9 b 42.2 5.1
a 398
Tip a 4.6 + 3.9 ≈ 5 4 Round the numbers to 1 significant figure.
398 400
Review Note that the ‘≈’ symbol
is only used at the point = 9 = 45 = 0 45
where an approximation is 20 10
made. At other times you
should use ‘=’ when two Check the estimate: Now if you use a calculator you will find the
numbers are exactly equal. exact value and see that the estimate was good.
4.6 + 3.9
398 = 0.426 (3sf)
Unit 2: Number 119
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b 42.2 − 5.1 ≈ 40 5 In this question you begin by rounding each
= 35 value to one significant figure but it is worth
≈ 36 noting that you can only easily take the square
=6 root of a square number! Round 35 up to 36 to
get a square number.
A good starting point for the questions in the following exercise will be to round the numbers to
1 significant figure. Remember that you can sometimes make your calculation even simpler by
modifying your numbers again.
Review Exercise 5.15 1 Estimate the value of each of the following. Show the rounded values that you use.
a 23.6 b 43 c 7.21 0.46
63 0.087 × 3.89 9 09
d 4.82 6.01 e 48 f (0.45 + 1.89)(6.5 – 1.9)
2.54 1.09 2.54 4.09
g 23.8 20.2 h 109.6 45 1 i (2.52)2 48.99
4.7 + 5.7 19.4 13.9
j 223.8 45.1 k 9.26 99.87 l (4.1)3 × (1.9)4
2 Work out the actual answer for each part of question 1, using a calculator.
ReviewSummary Are you able to. . . ?
Do you know the following? • find a fraction of a number E
• find a percentage of a number
• An equivalent fraction can be found by multiplying or • find one number as a percentage of another number
dividing the numerator and denominator by the same • calculate a percentage increase or decrease
number. • find a value before a percentage change
• do calculations with numbers written in
• Fractions can be added or subtracted, but you must
make sure that you have a common denominator first. standard form
• To multiply two fractions you multiply their numerators • find an estimate to a calculation.
and multiply their denominators.
Review
• To divide by a fraction you find its reciprocal and then
multiply.
• Percentages are fractions with a denominator of 100.
• Percentage increases and decreases are always
percentages of the original value.
• You can use reverse percentages to find the original E
value.
• Standard form can be used to write very large or very
small numbers quickly.
• Estimations can be made by rounding the numbers in a
calculation to one significant figure.
120 Unit 2: Number
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Exam-style questions
1 Calculate 5 1 + 1 giving your answer as a fraction in its lowest terms.
6 4 8
Review 2 93 800 students took an examination.
19% received grade A.
24% received grade B.
31% received grade C.
10% received grade D.
11% received grade E.
The rest received grade U.
a What percentage of the students received grade U?
b What fraction of the students received grade B? Give your answer in its lowest terms.
c How many students received grade A?
3 During one summer there were 27 500 cases of Salmonella poisoning in Britain. The next summer there was an
increase of 9% in the number of cases. Calculate how many cases there were in the second year.
4 Abdul’s height was 160 cm on his 15th birthday. It was 172 cm on his 16th birthday. What was the percentage increase
in his height?
Past paper questions
1 Write 0.0000574 in standard form. [1]
[Cambridge IGCSE Mathematics 0580 Paper 22 Q1 May/June 2016]
Review 2 Do not use a calculator in this question and show all the steps of your working.
Give each answer as a fraction in its lowest terms.
Work out
a 3− 1 [2]
4 12
b 2 1 × 4 [2]
2 25
[Cambridge IGCSE Mathematics 0580 Paper 11 Q21 October/November 2013]
3 Calculate 17.5% of 44 kg. [2]
[Cambridge IGCSE Mathematics 0580 Paper 11 Q10 October/November 2013]
Review 4 Without using your calculator, work out [3]
5 3 −2 1 .
85
Give your answer as a fraction in its lowest terms.
You must show all your working.
[Cambridge IGCSE Mathematics 0580 Paper 13 Q17 October/November 2012]
5 Samantha invests $600 at a rate of 2% per year simple interest. [2]
Calculate the interest Samantha earns in 8 years.
[Cambridge IGCSE Mathematics 0580 Paper 13 Q5 October/November 2012]
Unit 2: Number 121
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10 5
Write down all the steps in your working. [2]
[Cambridge IGCSE Mathematics 0580 Paper 13 Q6 October/November 2012]
7 Maria pays $84 rent. [2]
The rent is increased by 5%. Calculate Maria’s new rent.
[Cambridge IGCSE Mathematics 0580 Paper 13 Q10 October/November 2012]
Review 8 Huy borrowed $4500 from a bank at a rate of 5% per year compound interest. [3]
He paid back the money and interest at the end of 2 years.
How much interest did he pay?
[Cambridge IGCSE Mathematics 0580 Paper 13 Q13 May/June 2013]
9 Jasijeet and her brother collect stamps.
When Jasjeet gives her brother 1% of her stamps, she has 2475 stamps left.
Calculate how many stamps Jasjeet had originally [3]
[Cambridge IGCSE Mathematics 0580 Paper 22 Q14 October/November 2014]
10 Without using a calculator, work out 2 5 × 3 . [3]
87
Show all your working and give your answer as a mixed number in its lowest terms.
[Cambridge IGCSE Mathematics 0580 Paper 22 Q14 May/June 2016]
Review
Review
122 Unit 2: Number
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formulae
Review Key words
• Expansion
• Linear equation
• Solution
• Common factor
• Factorisation
• Variable
• Subject
Review In this chapter you Leonhard Euler (1707–1783) was a great Swiss mathematician. He formalised much of the algebraic
will learn how to: terminology and notation that is used today.
• expand brackets that Equations are a shorthand way of recording and easily manipulating many problems. Straight
have been multiplied by a lines or curves take time to draw and change but their equations can quickly be written. How
negative number to calculate areas of shapes and volumes of solids can be reduced to a few, easily remembered
symbols. A formula can help you work out how long it takes to cook your dinner, how well your
• solve a linear equation car is performing or how efficient the insulation is in your house.
• factorise an algebraic
expression where all terms
have common factors
• rearrange a formula to
change the subject.
RECAP
You should already be familiar with the following algebra work:
Expanding brackets (Chapter 2)
y(y – 3) = y × y – y × 3
Solving equations (Year 9 Mathematics)
Review Expand brackets and get the terms with the variable on one side by performing inverse operations.
2(2x + 2) = 2x – 10
4x + 4 = 2x – 10 Remove the brackets first
4x – 2x = –10 – 4 Subtract 2x from both sides. Subtract 4 from both sides.
2x = –14 Add or subtract like terms on each side
x = –7 Divide both sides by 2 to get x on its own.
Unit 2: Algebra 123
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Factorising (Year 9 Mathematics)
You can think of factorising as ‘putting the brackets back into an expression’.
To remove a common factor:
• find the highest common factor (HCF) of each term. This can be a variable, it can also be a negative integer
• write the HCF in front of the brackets and write the terms divided by the HCF inside the brackets.
2xy + 3xz = x(2y + 3z)
–2xy – 3xz = –x (2y + 3z)
Changing the subject of a formula (Year 9 Mathematics)
You can rearrange formulae to get one letter on the left hand side of the equals sign. Use the same methods you use to
solve an equation.
b= 1 l=A
Review A = lb A b
6.1 Further expansions of brackets
REWIND You have already seen that you can re-write algebraic expressions that contain brackets by
You dealt with expanding brackets expanding them. The process is called expansion. This work will now be extended to consider
in chapter 2. what happens when negative numbers appear before brackets.
The key is to remember that a ‘+’ or a ‘−’ is attached to the number immediately following it and
should be included when you multiply out brackets.
Worked example 1
Tip Expand and simplify the following expressions.
Review Watch out for negative a −3(x + 4) b 4(y − 7) − 5(3y + 5) c 8(p + 4) − 10(9p − 6)
numbers in front of
brackets because they a −3(x + 4) Remember that you must multiply
always require extra care. the number on the outside of the
Remember: −3(x + 4) = −3x − 12 bracket by everything inside and that
+×+=+ the negative sign is attached to the 3.
+×−=− b 4(y − 7) − 5(3y + 5)
−×−=+ 4(y − 7) = 4y − 28 Because −3 × x = −3x and
−5(3y + 5) = −15y − 25 −3 × 4 = −12.
LINK
4( y − 7) − 5(3y + 5) = 4y − 28 − 15y − 25 Expand each bracket first and
Physicists o en rearrange = −11y − 53 remember that the ‘−5’ must
formulae. If you have a keep the negative sign when it
formula that enables you is multiplied through the second
work out how far something bracket.
has travelled in a particular
time, you can rearrange the Collect like terms and simplify.
formula to tell you how long it
Review will take to travel a particular c 8(p + 4) − 10(9p − 6) It is important to note that when you
distance, for example. 8(p + 4) = 8p + 32 expand the second bracket ‘−10’ will
−10(9p − 6) = −90p + 60 need to be multiplied by ‘−6’, giving
a positive result for that term.
8( p + 4) − 10(9p − 6) = 8 p + 32 − 90 p 60 Collect like terms and simplify.
= − 82p + 92
124 Unit 2: Algebra
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Exercise 6.1 1 Expand each of the following and simplify your answers as far as possible.
Try not to carry out too many steps a −10(3p + 6) b −3(5x + 7)
at once. Show every term of your c −5(4y + 0.2) d −3(q − 12)
expansion and then simplify. e −12(2t − 7) f −1.5(8z − 4)
2 Expand each of the following and simplify your answers as far as possible.
a −3(2x + 5y) b −6(4p + 5q)
c −9(3h − 6k) d −2(5h + 5k − 8j)
e −4(2a −3b − 6c + 4d) f −6(x2 + 6y2 − 2y3)
3 Expand each of the following and simplify your answers as far as possible.
Review a 2 − 5(x + 2) b 2 − 5(x − 2)
c 14(x − 3) − 4(x − 1) d −7(f + 3) − 3(2f − 7)
e 3g − 7(7g − 7) + 2(5g − 6) f 6(3y − 5) − 2(3y − 5)
4 Expand each of the following and simplify your answers as far as possible.
a 4x(x − 4) − 10x(3x + 6) b 14x(x + 7) − 3x(5x + 7)
c x2 − 5x(2x − 6) d 5q2 − 2q(q −12) − 3q2
e 18pq − 12p(5q − 7) f 12m(2n − 4) − 24n(m − 2)
5 Expand each expression and simplify your answers as far as possible.
a 8x – 2(3 – 2x) b 11x – (6 – 2x)
c 4x + 5 – 3(2x – 4) d 7 – 2(x – 3) + 3x
e 15 – 4(x – 2) – 3x f 4x – 2(1 – 3x) – 6
g 3(x + 5) – 4(5 – x) h x(x – 3) – 2(x – 4)
i 3x(x – 2) – (x – 2) j 2x(3 + x) – 3(x – 2)
k 3(x – 5) – (3 + x) l 2x(3x + 1) – 2(3 – 2x)
Review You will now look at solving linear equations and return to these expansions a little later in
the chapter.
6.2 Solving linear equations
REWIND I think of a number. My number is x. If I multiply my number by three and then add one, the
It is important to remind yourself answer is 13. What is my number?
about BODMAS before working
through this section. (Return to To solve this problem you first need to understand the stages of what is happening to x and then
chapter 1 if you need to.) undo them in reverse order:
LINK This diagram (sometimes called a function machine) shows what is happening to x, with
the reverse process written underneath. Notice how the answer to the problem appears
Accounting uses a great deal quite easily:
of mathematics. Accountants
use computer spreadsheets to x ×3 + 1 13
calculate and analyse financial
Review data. Although the programs 4 ÷ 3 12 – 1
do the calculations, the user
has to know which equations
and formula to insert to tell the
program what to do.
So x = 4
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A more compact and efficient solution can be obtained using algebra. Follow the instructions in
the question:
1 e number is x: x
Even if you can see what the 2 Multiply this number by three: 3x
solution is going to be easily you
must show working. 3 en add one: 3x + 1
4 e answer is 13: 3x + 1 = 13
Review is is called a linear equation. ‘Linear’ refers to the fact that there are no powers of x other
than one.
e next point you must learn is that you can change this equation without changing the
solution (the value of x for which the equation is true) provided you do the same to both sides at
the same time.
Follow the reverse process shown in the function machine above but carry out the instruction on
both sides of the equation:
3x + 1 = 13 (Subtract one from both sides.)
3 + 1 −1 = 13 −1
3 12
3x = 12 (Divide both sides by three.)
33
x=4
Review Always line up your ‘=’ signs because this makes your working much clearer.
Sometimes you will also find that linear equations contain brackets, and they can also contain
unknown values (like x, though you can use any letter or symbol) on both sides.
e following worked example demonstrates a number of possible types of equation.
Worked example 2
An equation with x on both sides and all x terms with the same sign:
a Solve the equation 5x − 2 = 3x + 6
5x − 2 = 3x 6 3x Look for the smallest number of x’s and subtract
5x − 2 − 3x = 3x + 6 this from both sides. So, subtract 3x from both
2 −2=6 sides.
2 −2+2=6+2 Add two to both sides.
28 Divide both sides by two.
2x = 8
Review 22
x=4
An equation with x on both sides and x terms with different sign:
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b Solve the equation 5x + 12 = 20 − 11x
5x + 12 = 20 11x 11x This time add the negative x term to both sides.
5x + 12 + 11x = 20 − 11x Add 11x to both sides.
By adding 11x to both sides you
will see that you are left with a 16 x + 12 = 20 Subtract 12 from both sides.
positive x term. This helps you to
avoid errors with ‘−’ signs! 16 x + 12 − 12 = 20 − 12 Divide both sides by 16.
16 x = 8
Unless the question asks you to 16 x = 8
give your answer to a specific 16 16
degree of accuracy, it is perfectly
acceptable to leave it as a fraction. x= 1
2
Review An equation with brackets on at least one side:
c Solve the equation 2(y − 4) + 4(y + 2) = 30
2( y − 4) + 4( y + 2) = 30 Expand the brackets and collect like terms together.
2y − 8 + 4y + 8 = 30 Expand.
6y = 30 Collect like terms.
6y = 30 Divide both sides by 6.
66
y=5
An equation that contains fractions:
d Solve the equation 6 p = 10
7
Review 6 p × 7 = 10 7 Multiply both sides by 7.
7
Divide both sides by 6.
6p = 70 Write the fraction in its simplest form.
p = 70 = 35
63
Exercise 6.2 1 Solve the following equations.
Review a 4x + 3 = 31 b 8x + 42 = 2
c 6x −1 = 53 d 7x − 4 = − 66
e 9y + 7 = 52 f 11n − 19 = 102
g 12q − 7 = 14 h 206t + 3 = 106
i 2x 1 = 8 j 2x +1 = 8
3 3
k 3 x + 11 = 21 l x+3 = x
5 2
m 2x 1 = 3x n 3x + 5 = 2x
3 2
2 Solve the following equations.
a 12x + 1 = 7x + 11 b 6x + 1 = 7x + 11 c 6y + 1 = 3y − 8
f 1x−7= 1x+8
d 11x + 1 = 12 − 4x e 8 − 8p = 9 − 9p
24
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Tip 3 Solve the following equations. b 2(2p + 1) = 14 E
a 4(x + 1) = 12 d 5(m − 2) = 15
Review Some of the numbers in c 8(3t + 2) = 40 f 2(p − 1) + 7(3p + 2) = 7(p − 4)
each equation are powers e −5(n − 6) = −20 h 3(2x + 5) – (3x + 2) = 10
of the same base number. g 2(p − 1) − 7(3p − 2) = 7(p − 4)
Re-write these as powers b 4(x – 2) + 2(x + 5) = 14
and use the laws of indices 4 Solve for x. d −2(x + 2) = 4x + 9
from chapter 2 a 7(x + 2) = 4(x + 5) f 4 + 2(2 − x) = 3 – 2(5 – x)
c 7x – (3x + 11) = 6 – (5 – 3x)
e 3(x + 1) = 2(x + 1) + 2x b 23x+4 = 32
d 52(3x+1) = 625
5 Solve the following equations for x f 93x+4 = 274x+3
a 33x = 27
c 8.14x+3 = 1
e 43x = 2x+1
6.3 Factorising algebraic expressions
REWIND You have looked in detail at expanding brackets and how this can be used when solving some
If you need to remind yourself how equations. It can sometimes be helpful to carry out the opposite process and put brackets back
to find HCFs, return to chapter 1. into an algebraic expression.
Review Consider the algebraic expression 12x − 4. This expression is already simplified but notice that
12 and 4 have a common factor. In fact the HCF of 12 and 4 is 4.
Now, 12 = 4 × 3 and 4 = 4 × 1.
So, 12x − 4 = 4 × 3x − 4 1
= 4(3x −1)
Notice that the HCF has been ‘taken out’ of the bracket and written at the front. The terms inside
are found by considering what you need to multiply by 4 to get 12x and −4.
The process of writing an algebraic expression using brackets in this way is known as
factorisation. The expression, 12x − 4, has been factorised to give 4(3x−1).
Some factorisations are not quite so simple. The following worked example should help to make
things clearer.
Worked example 3
Factorise each of the following expressions as fully as possible.
a 15x + 12y b 18mn − 30m c 36p2q − 24pq2 d 15(x − 2) − 20(x − 2)3
a 15x + 12y The HCF of 12 and 15 is 3, but x and y have no
15x + 12y = 3(5x + 4y) common factors.
Because 3 × 5x = 15x
and 3 × 4y = 12y.
Review b 18mn − 30m The HCF of 18 and 30 = 6 and HCF of mn and m
18mn − 30m = 6m(3n − 5) is m.
Because 6m × 3n = 18mn and 6m × 5 = 30m.
128 Unit 2: Algebra
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Make sure that you have taken c 36p2q − 24pq2 The HCF of 36 and 24 = 12 and p2q and pq2 have
out all the common factors. If common factor pq.
you don’t, then your algebraic
expression is not fully factorised. 36p2q − 24pq2 = 12pq(3p − 2q) Because 12pq × 3p = 36p2q and
12pq × −2q = − 24pq2.
Take care to put in all the bracket
symbols. Sometimes, the terms can have an expression in brackets that is common to
both terms.
d 15(x − 2) − 20(x − 2)3 The HCF of 15 and 20 is 5 and the HCF of
(x − 2) and (x − 2)3 is (x − 2).
15(x − 2) − 20(x − 2)3 =
5(x − 2)[3 − 4(x − 2)2] Because 5(x − 2) × 3 = 15(x − 2) and
5(x − 2) × 4(x − 2)2 = 20(x − 2)3.
Review Exercise 6.3 1 Factorise.
a 3x + 6 b 15y − 12 c 8 − 16z d 35 + 25t
e 2x − 4 f 3x + 7 g 18k − 64 h 33p + 22
i 2x + 4y j 3p − 15q k 13r − 26s l 2p + 4q + 6r
2 Factorise as fully as possible.
Once you have taken a common a 21u − 49v + 35w b 3xy + 3x c 3x2 + 3x d 15pq + 21p
factor out, you may be left with e 9m2 − 33m f 90m3 − 80m2 g 36x3 + 24x5 h 32p2q − 4pq2
an expression that needs to be
simplified further. 3 Factorise as fully as possible.
a 14m2n2 + 4m3n3 b 17abc + 30ab2c c m3n2 + 6m2n2 (8m + n)
f 3(x − 4) + 5(x − 4)
d 1a 3b e 3 x4 + 7 x i 7x3y – 14x2y2 + 21xy2
22 48
Review g 5(x + 1)2 − 4(x + 1)3 h 6x3 + 2x4 + 4x5
j x(3 + y) + 2(y + 3)
6.4 Rearrangement of a formula
FAST FORWARD Very o en you will find that a formula is expressed with one variable written alone on one side
You will look again at rearranging of the ‘=’ symbol (usually on the le but not always). The variable that is written alone is known
formulae in chapter 22. as the subject of the formula.
Another word sometimes used Consider each of the following formulae:
for changing the subject is
‘transposing’. s = ut + 1 at2 (s is the subject)
2
F = ma (F is the subject)
x = −b ± b2 − 4ac (x is the subject)
2a
Review Now that you can recognise the subject of a formula, you must look at how you change the
subject of a formula. If you take the formula v = u + at and note that v is currently the subject,
you can change the subject by rearranging the formula.
To make a the subject of this formula:
v = u + at Write down the starting formula.
v − u = at Subtract u from both sides (to isolate the term containing a).
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Remember that what you do to v u = a Divide both sides by t (notice that everything on the le is divided by t).
one side of the formula must t
be done to the other side. This
ensures that the formula you You now have a on its own and it is the new subject of the formula.
produce still represents the same
relationship between the variables. This is usually re-written so that the subject is on the le :
a=v u
t
Notice how similar this process is to solving equations.
Worked example 4
Make the variable shown in brackets the subject of the formula in each case.
Review a x + y = c (y) b x + y = z (x) c a c b = d (b)
a x+y=c
⇒ is a symbol that can be used to ⇒y=c−x Subtract x from both sides.
mean ‘implies that’.
b x+y=z Subtract y from both sides.
Square both sides.
⇒ x =z−y
⇒ x = ( z − )2
Review c acb=d Multiply both sides by c to clear the fraction.
⇒ a − b = cd Make the number of b’s positive by adding b to both sides.
⇒ a = cd + b Subtract cd from both sides.
⇒ a − cd = b Re-write so the subject is on the left.
So b = a − cd
Exercise 6.4 Make the variable shown in brackets the subject of the formula in each case.
1 a a+b=c (a) b p−q=r (r) c =g (h)
d ab + c = d (b) (a) (n)
e a = c (a) f an − m = t
b (y) (x)
(b) (b)
2 a an − m = t (m) b a(n − m) = t c xy = t
(x) (a) z
x a=c (r) (b)
d b (a) e x(c − y) = d (b) f a−b=c
(b)
3a p−r =t b x a=c c a(n − m) = t (m)
q b (z)
(b)
d a=c e xba=c f xy = t (y)
bd z
Review 4a b c b ab = c c a b=c
d b + c = c (b) e x−b =c f x =c
y
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Applying your skills
5 A rocket scientist is trying to calculate how long a Lunar Explorer Vehicle will take to descend
towards the surface of the moon. He knows that if u = initial speed and v = speed at time
t seconds, then:
v = u + at
where a is the acceleration and t is the time that has passed.
If the scientist wants to calculate the time taken for any given values of u, v, and a, he must
rearrange the formula to make a the subject. Do this for the scientist.
Review 6 Geoff is the Headmaster of a local school, who has to report to the board of Governors on
how well the school is performing. He does this by comparing the test scores of pupils across
an entire school. He has worked out the mean but also wants know the spread about the
mean so that the Governors can see that it is representative of the whole school. He uses a
well-known formula from statistics for the upper bound b of a class mean:
b = a + 3s
n
where s = sample spread about the mean, n = the sample size, a = the school mean and
b = the mean maximum value.
If Geoff wants to calculate the standard deviation (diversion about the mean) from values of
b, n and a he will need to rearrange this formula to make s the subject. Rearrange the formula
to make s the subject to help Geoff.
7 If the length of a pendulum is l metres, the acceleration due to gravity is g m s−2 and T is the
period of the oscillation in seconds then:
T = 2π l
g
Review Rearrange the formula to make l the subject.
Summary
Do you know the following? Are you able to . . . ? E
• Expanding brackets means to multiply all the terms • expand brackets, taking care when there are
inside the bracket by the term outside. negative signs
• A variable is a letter or symbol used in an equation or • solve a linear equation
formula that can represent many values. • factorise an algebraic expressions by taking
• A linear equation has no variable with a power greater out any common factors
than one.
• rearrange a formulae to change the subject by
• Solving an equation with one variable means to find the treating the formula as if it is an equation.
value of the variable.
Review E
• When solving equations you must make sure that you
always do the same to both sides.
• Factorising is the reverse of expanding brackets.
• A formula can be rearranged to make a different variable
the subject.
• A recurring fraction can be written as an exact fraction.
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Exam-style questions
1 Given that T = 3p − 5, calculate T when p = 12.
2 In mountaineering, in general, the higher you go, the colder it gets. This formula shows how the height and
temperature are related.
Temperature drop (°C) = height increase (m)
200
a If the temperature at a height of 500 m is 23 °C, what will it be when you climb to 1300 m?
b How far would you need to climb to experience a temperature drop of 5 °C?
3 The formula e = 3n can be used to relate the number of sides (n) in the base of a prism to the number of edges (e)
that the prism has.
a Make n the subject of the formula.
b Find the value of n for a prism with 21 edges.
Review Past paper questions [2]
1 Factorise 2x − 4xy. [Cambridge IGCSE Mathematics 0580 Paper 22 Q2 Feb/March 2016]
2 Make r the subject of this formula. [2]
v= 3 p+r
[Cambridge IGCSE Mathematics 0580 Paper 22 Q5 October/November 2014]
3 Expand the brackets. y(3 − y3)
[2]
4 Factorise completely. 4xy + 12yz
[Cambridge IGCSE Mathematics 0580 Paper 13 Q9 October/November 2012]
5 Solve the equation. 5(2y − 17) = 60
[2]
6 Solve the equation (3x − 5) = 16.
[Cambridge IGCSE Mathematics 0580 Paper 13 Q13 October/November 2012]
7 Factorise completely. 6xy2 + 8y
[3] E
[Cambridge IGCSE Mathematics 0580 Paper 22 Q12 May/June 2013]
[2]
[Cambridge IGCSE Mathematics 0580 Paper 13 Q5 May/June 2013]
[2]
[Cambridge IGCSE Mathematics 0580 Paper 13 Q9 May/June 2013]
Review
132 Unit 2: Algebra
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Review Key words
• Perimeter
• Area
• Irrational number
• Sector
• Arc
• Semi-circle
• Solid
• Net
• Vertices
• Face
• Surface area
• Volume
• Apex
• Slant height
Review In this chapter you
will learn how to:
Review The glass pyramid at the entrance to the Louvre Art Gallery in Paris. Reaching to a height of 20.6 m, it is a
• calculate areas and beautiful example of a three-dimensional object. A smaller pyramid – suspended upside down – acts as a
perimeters of two- skylight in an underground mall in front of the museum.
dimensional shapes
When runners begin a race around a track they do not start in the same place because their
• calculate areas and routes are not the same length. Being able to calculate the perimeters of the various lanes allows
perimeters of shapes that the officials to stagger the start so that each runner covers the same distance.
can be separated into two
or more simpler polygons A can of paint will state how much area it should cover, so being able to calculate the areas of
walls and doors is very useful to make sure you buy the correct size can.
• calculate areas and
circumferences of circles How much water do you use when you take a bath instead of a shower? As more households are
metered for their water, being able to work out the volume used will help to control the budget.
• calculate perimeters and
areas of circular sectors
• understand nets for three-
dimensional solids
• calculate volumes and
surface areas of solids
• calculate volumes and
surface area of pyramids,
cones and spheres.
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RECAP
You should already be familiar with the following perimeter, area and volume work:
Perimeter
Perimeter is the measured or calculated length of the boundary of a shape.
The perimeter of a circle is its circumference.
You can add the lengths of sides or use a formula to calculate perimeter.
ls
d
b
Review P = 2(l + b) P=4s C = πd
Area
The area of a region is the amount of space it occupies. Area is measured in square units.
The surface area of a solid is the sum of the areas of its faces.
The area of basic shapes is calculated using a formula.
l r
s bh
A = s2 A = lb b A = πr2
A = 1 bh
2
Review Volume
The volume of a solid is the amount of space it occupies.
Volume is measured in cubic units.
The volume of cuboids and prisms can be calculated using a formula.
l πr2 h 1 bh h
2
h
b V = Area of cross section × height
V=lbh
Review
134 Unit 2: Shape, space and measures
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7.1 Perimeter and area in two dimensions
LINK Polygons
When geographers study A polygon is a flat (two-dimensional) shape with three or more straight sides. The perimeter of
coastlines it is sometimes very a polygon is the sum of the lengths of its sides. The perimeter measures the total distance around
handy to know the length of the outside of the polygon.
the coastline. If studying an
island, then the length of the The area of a polygon measures how much space is contained inside it.
coastline is the same as the
perimeter of the island.
Review Two-dimensional shapes Formula for area
Quadrilaterals with parallel sides
h Area = bh
b h h
rhombus
b b
rectangle parallelogram
Triangles
h Area = 1 bh or bh
h h 22
Review b b b Area = 1 (a b)h or (a b)h
a 22
Trapezium
It is possible to find areas of
a other polygons such as those
on the left by dividing the
hh shape into other shapes such as
triangles and quadrilaterals.
b b
Here are some examples of other two-dimensional shapes.
Review kite regular hexagon irregular pentagon
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Tip Units of area
You should always give If the dimensions of your shape are given in cm, then the units of area are square centimetres
units for a final answer if it and this is written cm2. For metres, m2 is used and for kilometres, km2 is used and so on. Area is
is appropriate to do so. It always given in square units.
can, however, be confusing
if you include units Worked example 1
throughout your working.
a Calculate the area of the shape shown in the diagram.
ReviewThe formula for the area of a 5 cm 6 cm
triangle can be written in different
ways:
1 × b × h = bh
22
OR = 1 b × h 7 cm
2
This shape can be divided into two simple polygons: a rectangle and a triangle.
OR = b × 1 h Work out the area of each shape and then add them together.
2
Choose the way that works best
for you, but make sure you write it
down as part of your method.
5 cm 5 cm 6 cm
Review You do not usually have to redraw 7 cm triangle
the separate shapes, but you might rectangle
find it helpful.
Area of rectangle = bh = 7 × 5 = 35cm2 (substitute values in place of b and h)
REWIND
At this point you may need to Area of triangle = 1 bh = 1 × 5 × 6 = 1 × 30 = 15cm2
remind yourself of the work you 22 2
did on rearrangment of formulae in
chapter 6. Total area = 35 + 15 = 50 cm2
b The area of a triangle is 40 cm2. If the base of the triangle is 5 cm, find the height.
A= 1×b×h Use the formula for the area of a
2 triangle.
40 = 1 × 5 × h Substitute all values that you know.
2
Rearrange the formula to make h the
⇒ 40 × 2 = 5 × h subject.
⇒ h = 40 × 2 = 80 = 16 cm
55
Review
136 Unit 2: Shape, space and measures
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Exercise 7.1 1 By measuring the lengths of each side and adding them together, find the perimeter of each
of the following shapes.
LINK a b
c d
Agricultural science involves
work with perimeter, area and
rates. For example, fertiliser
application rates are o en
given in kilograms per hectare
(an area of 10 000 m2). Applying
too little or too much fertiliser
can have serious implications
for crops and food production.
Review
2 Calculate the perimeter of each of the following shapes.
a 2.5 cm b
5 cm
5.5 cm 3 cm
Review c 7 cm 4 cm
4 cm 10 cm d 2 cm
e 2.5 m 4 cm
10 cm 2 cm
f 9 cm
2.8 m 8 km 9 km
1.9 m
Review
8.4 m
3 km 3 km
7.2 m
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3 Calculate the area of each of the following shapes.
ab
11 cm
5m
Review 5 cm d
c 3.2 cm
4m
5m 1.4 cm
e f
2m 2.8 cm
Review 8m h 6m
g 6 cm
6m
5 cm 8m
10 cm j 6 cm
i
4 cm
6 cm
4 cm
12 cm
Draw the simpler shapes separately 4 The following shapes can all be divided into simpler shapes. In each case find the total area.
and then calculate the individual
Reviewareas, as in worked example 1. ab
4 m 5.1 m
7.2 m
1.2 m
8m 8m 2.1 m 4.5 m
5m
138 Unit 2: Shape, space and measures
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c 4.9 cm d 2.1 cm
5.3 cm
8.2 cm 5.4 cm
7.2 cm 3.4 cm 7.2 cm
7.8 cm
Reviewe f g 1.82 cm 3.71 cm
12 cm 19.1 cm
8.53 cm
18 cm 38.2 cm
12 cm 3.8 cm
7.84 cm
2.4 cm
Write down the formula for the area 5 For each of the following shapes you are given the area and one other measurement.
in each case. Substitute into the Find the unknown length in each case.
formula the values that you already
know and then rearrange it to find a bb
the unknown quantity.
24 cm2 h
8 cm
Review 289 cm2 17 cm
c a d 15 cm e 6 cm h
b
132 cm2 14 cm 75 cm2 18 cm 200 cm2
16 cm 6 cm
6 How many 20 cm by 30 cm rectangular tiles would you need to tile the outdoor area
shown below?
Review 2.6 m 0.9 m
1.7 m
4.8 m
7 Sanjay has a square mirror measuring 10 cm by 10 cm. Silvie has a square mirror which
covers twice the area of Sanjay’s mirror. Determine the dimensions of Silvie’s mirror correct
to 2 decimal places.
Unit 2: Shape, space and measures 139
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8 For each of the following, draw rough sketches and give the dimensions: E
a two rectangles with the same perimeter but different areas
b two rectangles with the same area but different perimeters
c two parallelograms with the same perimeter but different areas
d two parallelograms with the same area but different perimeters.
9 4(y – 2)
3x + 2 2(x + 1) + 3 NOT TO
SCALE
ReviewYou will need to use some of the 3y + 4
algebra from chapter 6. Find the area and perimeter of the rectangle shown in the diagram above.
Circles
‘Inscribing’ here means to draw a
circle inside a polygon so that it just
touches every edge. ‘Circumscribing’
means to draw a circle outside a
polygon that touches every vertex.
Review REWIND
You learned the names of the parts
of a circle in chapter 3. The diagram Archimedes worked out the formula for the area of a circle by
below is a reminder of some of the inscribing and circumscribing polygons with increasing numbers of
parts. The diameter is the line that sides.
passes through a circle and splits it
into two equal halves. The circle seems to appear everywhere in our everyday lives. Whether driving a car, running
on a race track or playing basketball, this is one of a number of shapes that are absolutely
circumference essential to us.
radius Finding the circumference of a circle
O is the O Circumference is the word used to identify the perimeter of a circle. Note that the diameter =
centre 2 × radius (2r). The Ancient Greeks knew that they could find the circumference of a circle by
diameter multiplying the diameter by a particular number. This number is now known as ‘π’ (which is the
Greek letter ‘p’), pronounced ‘pi’ (like apple pie). π is equal to 3.141592654. . .
Review
The circumference of a circle can be found using a number of formulae that all mean the
same thing:
FAST FORWARD Circumference = π × diameter
π is an example of an irrational = πd (where d = diameter)
number. The properties of irrational
numbers will be discussed later in = 2πr (where r = radius)
chapter 9.
140 Unit 2: Shape, space and measures
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