Math Smart - 7

@ ABC is an isosceles triangle and ZBAC = 134^ Q PQR is an equilateral triangle and /-PTS = 46'

Find zlACB. Find APST.

















6.5.2 Finding unknown angles in

quadrilaterals



We have learnt some properties of the sides and angles of quadrilaterals earlier.
To solve geometrical problems, we need to make use of these properties of
quadrilaterals.

Quadrilateral Sides Diagonals and height
1. Rectangle Two pairs of Diagonals AC and BD
parallel sides. are equal and bisect
ABII DC, AD II BC each other.
Opposite sides are
equal.
AB = DC, AD = BC




2. Square Two pairs of Diagonals AC and
parallel sides. BD are equal and
ABIIDC,ADIIBC perpendicular. They
also bisect each other.
All sides are equal.
AB = BC=CD = AD





3. Parallelogram Two pairs of Diagonals AC and BD
parallel sides. bisect each other.
ABIIDQADIIBC
Height EF\s
Opposite sides are perpendicular to AD
equal. and BC.
AB = DC, AD = BC

4. Rhombus Two pairs of Diagonals /4Cand BD
parallel sides. are not equal but are
AB/IDQADIIBC perpendicular.
They also bisect each
All sides are equal.
other.
AB = BC=CD = AD

5. Kite No parallel sides. Diagonals AC and BD
are perpendicular.
Adjacent sides are
equal.

AB = AD, CB = CD








6. Trapeziums One pair of Diagonals AC and BD
A D parallel sides. are not equal and do
not bisect each other.
ADIIBC

























O ABCD is a square. Find the following angles.
a) ^ADC b) Ax

c) Z.CMD d) Ay



Solution
a) ABCD is a square. Each angle at the corner of
So, zl4DC=90°. a square is a right angle.

b) ABCD is a square. The diagonals of a square are
So, Ax = 90°. perpendicular to each other.

c) ABCD is a square. The diagonals of a square are
So, ACMD = 90°. perpendicular to each other.



UNIT 6 Angles and Their Properties

d) MCD is an isosceles triangle, 3^^ BD bisect each other. So, MD = MC
The diagonals of a square
^y._^ 180°-ZC/WD divide the angles at the
corners into 45° angles.
= 180°-90°
2
_ 90!
2
= 45°


Q PQRS is a rectangle. Find the following angles.

100° ^ ^ y
a) APSR b) zlx
c) AQOR d)

X

Solution
a) PQRS is a rectangle. Each angle at the corner of a rectangle
So, ZPS/? = 90°. is a right angle.

b) QS is a straight line.
Sum of angles on a straight line is 180°.
So, ZP05= 180°-100° When two straight lines
= 80°. cut each other, the angles
opposite each other are
PR is a straight line.
equal.
So, ^x=180°-80° Sum of angles on a straight line is 180°.
= 100°. AP0Q= A SOR
APOS=AQOR
c) PR is a straight line.
Sum of angles on a straight line is 180°.
So, ^Q0/?= 180°-100= Each pair of angles are
= 80°. vertically opposite angles.
d) ^OQP is an isosceles triangle.
180°-ZOO/? other.
zx=
So, OR = OS.
= 180°-80°
2
= 100°
2
= 50°.

Angles in parallelograms and trapeziums
Look at parallelogram ABCD below. Cut out Z.ABC along the dotted line EF, and
place it next to ^FCD.












C,B
We see they form a straight llne.The sum of the two angles is 180°. ^ABC ar\d
ABCD are a pair of interior angles of the parallelogram. They are between the
parallel lines, AB and DC. Another pair of interior angles which are between the
parallel lines, >16 and DC, are ^>1DCand aDAB.
Look at trapezium PQRS. AQRS is cut off along the dotted line TU, and placed next
to APQU.

Q




Q □


The sum of the two angles is also 180°.
The sum of a pair of
interior angles in a APQR and AQRS are a pair of interior angles of the trapezium. They are between
quadrilateral is 180°. the parallel lines, PQ and SR. The other pair of interior angles, AQPS and APSR,
also add up to 180°.
The sum of all interior
angles in a quadrilateral Look the square and the rectangle in Example 2. The sum of each pair of interior
= 180° X 2 = 360°. angles is 180°.


Example 3


0 AeCD is a parallelogram ./D/46= 124°.

Find 124°
a) zlx, b) ABCD.

A BAD
Solution
another
a) Zx+124° = 180° Interior angles between the parallel lines/4D and 6C.
^x= 180°- 124°
= 56°
b) 56° + ABCD = 180° Interior angles between the parallel lines 46and DC.
^6CD= 180°-56°
= 124°




Angles and Their Properties

0 /C/.MN Is a rhombus. A/CNM = 68°. Find:

a) Ly, b) ^LMN.

Solution Interior angles between the
a) ^NKL + 68° = 180° parallel lines KL and NM.
^N/C/. = 180°-68° N
= 112°

/_y +112° = 180° Interior angles between the parallel lines KN and LM.
Ay =180°-112°
What do you notice
= 68°
about the opposite
b) ALMN + 68° = ^80^ Interior angles between the parallel lines/(N and/.M. angles in the rhombus?
./tMA/= 180°-68°
= 112° rr
105
0 PQ/J5 is a trapezium. aPQR- 105°. Find Az.

Solution ll
Interior angles between
105° + Zz=180° I, ,
the parallel lines PQand S/<.
Zz=180°-105° ^
= 75°



Check My
Understanding


0 ABCD is a square. Find the following angles,

a) AABC b) AO ON.
c) ADMC d) Ad


/m\


D

0 EFCH Is a rectangle. Find the following angles.

a) AFGH b) AEOH
c) Ax d) Ay





0 WXYZ is a parallelogram. AXYZ = 117°. Find » »■

a) AW, b) AmY.

Q KLMN is a trapezium.

^LMN = 57°. Find Az.








Q In the diagram, PQ/?5 is a rectangle.

a) Calculate the size of angle a and b respectively.
+ 20)°
b) Determine the value of x.
c) If PQ = 10 mm, what is the length of SR7
d) If QR is half the length of PQ, what is the length of PS7
>-<30°
e) What is the perimeter of rectangle PQRS7




In
0 the diagram,/4F, DFand Sf are straight lines.
a) Show that ACDf is an isosceles triangle.
b) If Cf is 5 cm, what is the length of DC?






0 the diagram ABCD is a kite.
In
Calculate the unknown angles.


















0*Challenge!
This floor tile has the shape of a rhombus.
(23X-5)
a) Calculate the size of angle TUR.
b) Calculate the value of X.
c) Hence, show that the sum of the angles in a
quadrilateral is 360®.









Angles and Their Properties
I

Revision





0 Use your protractor to measure the size of each angle.

c








































Q Use your ruler and protractor to draw the following angles,

a) 130° b) 347° c) 209° d) 92°
e) 57° f) 101° g) 17° h) 358=
Q Calculate the size of the unknown angles.

a) X b)





0
0



d)



(4x-9)

e) f)

















Q Calculate the size of the unknown angles.


a) b)










d)











e) f)







n
g) h)

IL




I) What type of quadrilateral is ABCD7 j) Determine the size of angle C
Hence, what are the lengths of sides
DC and AD7
B











Angles and Their Properties

0 ABCD is a square. @ EFGH is a rectangle.
Find the following angles, Find the following angles,
a) ^BCD b) AX a) AfHG b) Ax
c) ABMC d) A/ c) AFOC d) Ay

A B E





110°
D


Q M/XVZ is a parallelogram. AXWZ:: 123'^ @ XtMN is a trapezium. A/CZ-M = 128'
Find Find Am.
a) AXKZ, b) Az.

»■ »•









^*Chaiienge! (E) *ChallGnge!
Calculate the size of the unknown angles. In the diagram, ABCE is a rectangle.
State your reasons clearly in the working. Calculate the size of angles p, q, rand s.
State your reasons clearly in the working.
A _B
Vp y/s*
25\


\ r /
n K/ r
E D C


(D *Challenge! ^^*Challenge!
The figure below is made up of different In the diagram, AD, BE and FC are straight lines.
overlapping triangles. Calculate the size of Calculate the size of the unknown angles.
the unknown angles. .8

^*Challenge!
In the diagram, lines AB, CD, DE and FG
are straight lines.
Calculate the size of the unknown angles.
State your reasons clearly in the working.









Challengel Draw arrow heads on the diagram
to show that AB is parallel to CD.
There are four pairs of equal angles.
Show the four pairs of equal angles on the diagram.




*Challengel
In the diagram, /.D- AF=AS^. Df and BE
are straight lines. What is the relationship
between line segments DE and BR









Mathematics Connect % y O Spotlight



Find out more about
Angles in Action!
using a digital theodolite.
We measure lines and
angles on paper using mm
rulers and protractors. But
how do we measure angles
a road makes or how steep
a slope is?

A theodolite is an
instrument used for
measuring angles
accurately. Theodolites are
used to survey landscapes
to draw accurate maps.
They have also been
adapted for use in
meteorology and even
rocket launches.

Help

Types of angles Sheet
Supplementary angles
are two angles that add
Adjacent angles have a . ^ ft QO '
n
BO 90 JOo
up to 180°.
common vertex and one
common side.



Vertically opposite angles
Complementary angles A protractor is an instrument have a common vertex and
are two angles that add used to measure angles. their sides are formed by
up to 90°. The angles are measured in the same lines.
degrees {°) .













An acute angle A right angle is An obtuse angle A straight line is A reflex angle
A revolution
measures between equal to 90°. measures between equal to 180°. measures between
is equal to 360'
O'and 90°. 90° and 180°. 180° and 350°.


Angle properties





360'
The sum of angles on a The sum of the interior
straight line is 180°. angles of a triangle is 180'
^o+^b=180° ^0+ Z.b + ^c= 180°
(zis on a str. line) {L sum of A)




The sum of angles at a
point is 360°.
^0+ zlb+ zlc = 360°
(zs at a point)

The sum of the interior
Vertically opposite angles of a quadrilateral
angles are equal. is 360°.
^0 = i-b ^a + ^b + ^c+ ^cl= 360°
(vert. opp. L%) (sum of As in a quad.)

Lines





n 1 Parallel lines
Intersecting lines Perpendicular lines


Transversal lines
Angles are formed when a transversal
crosses two or more parallel lines.
The angles in matching colours are
equal in size.



Properties in triangles and quadrilaterals


Name Shape Sides Angles Name Shape Sides Angles



All sides All Two pairs Opposite
Scalene
different angles Parallelogram of equal angles
triangle
lengths different sides equal




Two Two Four Opposite
Isosceles
equal equal Rhombus equal angles
triangle
sides angles sides equal




Three All All sides All
Equilateral
equal angles Trapezium different angles
triangle
sides 60° lengths different




Four All Two Two pairs
Isosceles
Square equal angles equal of equal
trapezium
sides 90° sides angles


n II
Two
All Two pairs One pair
pairs of
Rectangle angles Kite of equal of equal
equal
90° sides angles
IL sides




Angles and Their Properties

UNIT 7













You will learn about:
Decision making in data handling

Collecting and representing data
Construct and use frequency tables to gather discrete data
Construct and use frequency tables to gather grouped data




There are lots of reasons for collecting data. Plastic in ocean Frequency
For example, scientists collect data on how
much plastic enters our oceans. They organise
Bottles
and represent this data to study the impact of
plastic pollution on the environment, which in Drinking straws
turn impacts the human race.
Toothbrushes

CHAPTER 7.1
In this chapter

Pupils should be able to:
• decide which data
would be relevant to an
enquiry and collect and
organise the data
• design and use a data
collection sheet or
questionnaire for a Data is often collected by observing, questioning or measuring. Data is often
simple survey organised in tables and represented using graphs or charts. In other words, data is
information that helps to inform people to make decisions.

We can use a data collection sheet or a questionnaire to help us collect
Information about a product or an event.


Data collection sheet


Tally marks are often used to record numerical information on the data collection
sheet. Tally marks are used to keep count by drawing marks. Every fifth mark is
drawn across the previous four marks so that you can easily count groups of five.




#


The traffic department wants to reduce traffic congestion
at the intersection of 1" Avenue and Main Street. Tiana Date: 1 July 2019 Day: Monday
collected data on the number and types of vehicles that use Start time: 09:15 End time: 11:30
this intersection at different times of the day. Total survey time: 1 hr 15 mins

How many bicycles were observed? Location: Intersection of 1 "Avenue and Main Street
Weather: Clear skies Surveyor: Tiana
How many vehicles were observed in total?
Which type of vehicle was observed most?
Vehicle type Tally Total
Which type of vehicle was least common?
Car im ifH ifH III 23
The day on which Tiana collected the data was not a
Motorbike mm iiii 14
rainy day. Do you think that weather conditions could
affect the data collected? Give an example to support Bus ill 3
your answer. Truck MM 13
III
Tiana collected the data on a Monday. Do you think Bicycle mm mmmmi 31
that the data would be different if it was collected on
Other 0
a Sunday? Give an example to support your answer.
Total 84









160 UNIT? Data handling
I

Questionnaire


A questionnaire is a set of questions that allows a researcher to ask different
people the same set of questions. This way we can find out a range of opinions
and response to a single product. This allows us to compare their answers on the
same basis. We can find out about the majority's consensus or extreme responses.

A questionnaire can include
closed questions and
• open-ended questions.

Closed questions provide a list of answers to choose from. The person answering
the question is limited to the given choices. Their answers may not be listed.
However, it is easy to organise data from closed questions.

Open-ended questions have no suggested answers. The person answering the
question can give a qualitative response. It can be difficult to organise data from
open-ended questions as you may get very diverse opinions.







Aim; Conduct a survey using a questionnaire.
Use these steps to guide you.


Decide what you want to find out from the masses. Ask a question.
1

Decide what data you need to collect. There should always be a
reason for collecting this data.


^ Design your questionnaire.


< ^ ' Conduct a survey using your questionnaire to collect and record
your data.
Show your questionnaire
to your teacher before
Organise your data in a table, graph or chart. you continue.
J

Design data collection sheets that can be used to find out the following.

a) Which make of cellphone is the most popular in a class in a school?
b) How many red, blue, white and silver cars pass an intersection in one hour?
c) What is the least popular mode of transport to school?
d) How many learners in a class enjoy reading more than watching television?
e) How many learners come from homes that separate glass, plastic, paper and metal for recycling?

Priya wanted to know which method of communication she used most. She designed a data collection
sheet and used it to record all communications over five days.



nn
g#- - Monday Tuesday Wednesday Thursday Friday Total

Text message -Hfl- II 1 m -m II 20
Voice call II II II II II 10

Video call 1 1
Email II -Htf 1 nil 12
Total 7 5 8 9 13 43


a) Priya made a mistake when she added up the tallies. Find and correct the mistake.
b) How many different methods of communication did Priya use?
c) How many voice calls did she make over the five days?
d) Which method of communication did she use most over the five days?

e) Which method of communication did she use least over the five days?
f) How many more voice calls than video calls did she make over the five days?
g) On which day did she use all methods of communication?
h) On which day did she text the most?

























162 UNIT 7 Data handling

CHAPTER 7.2



equencyHBiies In this chapter

Pupils should be able to:
for Grouped and • construct and use

frequency tables
Ungrouped Data to gather discrete
data, grouped where
appropriate in equal
class intervals
Frequency is the number of times an event occurs. A frequency table is used to
organise data by showing the number of times each value occurs.







Number of books read in a year Height of learners in a class
Books Tally Frequency Height (cm) Tally Frequency
4 II 2 100-109 II 2
5 II 2 110-119 nil 4

6 nil 4 120-129 m Mil 9
7 m 5 130-139 -mt m III 13
8 nil 4 140-149 m 5

9 n 2 150-159 1 1
Total 34
10 1
Total 20 ^Heights are rounded to the nearest whole number.
What information does the first table provide?

What information does the second table provide?
What does the first column in each table tell you?
Where do the numbers in the last column come from?
How many learners read six books?
How many learners are 140 cm to 149 cm tall?
How many learners had their heights measured?

How many learners were surveyed in total?
What is the most common height among the learners in class?
How many learners are shorter than 130 cm?
What is the difference between the two frequency tables? How are the
data recorded differently?

We can construct frequency tables for grouped data and ungrouped data.

7.2.1 Frequency tables for ungrouped

data


Ungrouped data is data that is not organised into groups. In other words, the
data are discrete. In an ungrouped data frequency table, data can be presented in
a list.




These are the ages of 20 learners in a camp.
15 16 13 15 16 15 18 13 13 14
17 14 13 16 17 18 13 14 16 15

Organise the data in a frequency table using tally marks.
Label the third column:
Label the second Frequency
column:
Add the tally marks in each
Label the first column: Tally
row to get the frequency.
Age
Go through the list, one
Add the frequencies to
Arrange from the data value at a time. Use
lowest value to the tally marks to fill in the get the total number of
highest value. tally column. learners in the survey.



Age Tally Frequency
13 m 5
14 III 3
15 nil 4
16 nil 4
Check that this total 17 n 2
is equal to the total 18 n 2
number of data values.
Total 20

7.2.2 Frequency tables for grouped


data


Grouped data is data that has been bundled together into classes. The difference
between the lowest value and the highest value in each class is called the class
interval.

For example, 0-9 is a class with a class interval of 10. This means that there are 10
values that fall within this class.
We use classes when the data values are spread out. By organising the data into
classes, we reduce the number of rows in the frequency table.






164 UNIT7f Data handling

The ages of 40 people were recorded.

43 24 33 26 35 15 27 34 19 20
42 49 34 56 37 19 21 50 39 29
54 57 30 28 26 18 20 34 3 33
9 10 27 12 47 11 7 25 37 34

Organise the data in a frequency table.
Solution
This data should be organised into a grouped frequency table because there is a
wide range of data values. If the data was organised in an ungrouped frequency
table, you would need 31 rows.
Grouped frequency tables usually have about five to ten rows.


The class intervals must not
overlap.
Each data value can only be
in one class interval.




Age range Tally Frequency
0-9 III 3
In this example, there
II
10-19 m 7
are six class intervals.
1
20-29 mm 11
The size of each class
1
30-39 mm 11 interval is 10.
40-49 nil 4 All the class inten/als
are equal.
50-59 nil 4
Total 40


Check My

Und
m

Simone wants to find out which colour is the most popular colour for cars. She recorded the colours
of 40 cars in the parking area of a shopping centre.
red red blue green white red
blue red red blue white green
red white white blue red white
blue blue green black white blue
red silver silver blue red red
silver white white red blue green
red blue silver white





165

a) Organise the data in a frequency table.

Colour of car Tally Frequency

Red
Blue

Green
White
Silver
Black

b) How many cars were blue?
c) What is the difference between the number of silver cars and green cars?
d) Which car colour was least popular?
e) Which car colour was most popular?
f) Give a reason why it was easier to use the frequency table to decide which car colour was
most popular.

Jim rolled one die. He recorded the number after each roll.








a) Construct a frequency table to organise the data from rolling one die.
b) Was your frequency table a grouped frequency table, or an ungrouped frequency table?
Give a reason for your choice.
Jim rolled two dice at the same time. He added the numbers shown on each die.
He recorded the sum after each roll.

2 10 11 5
10 7 7 4

9 7 6 12
c) Construct a frequency table to organise the data from rolling two dice.
d) Was your frequency table a grouped frequency table, or an ungrouped frequency table?
Give a reason for your choice.
e) Jim rolled five dice at the same time and recorded the sum of all five numbers after each roll.
Would you construct a grouped frequency table or an ungrouped frequency table to organise
that data? Give a reason for your answer.















166 UNIT 7 |Data handling

Tony measured the size of 22 angles.

15° 16° 9° 21° 32° 37° 25° 36° 40° 8° 32°
13° 21° 32° 29° 32° 29° 32° 7° 4° 18° 17°

Construct a frequency table to organise this data. Use four equal class intervals.

Range of angle ° Tally Frequency









Total
® The points scored in a shooting competition are recorded.

18 24 19 3 24 11 25 10 25 14 25 14 25 9
16 26 21 27 13 23 5 26 22 12 27 20 7 28
21 20 22 16 12 25 7 25 19 17 15 8

a) Construct a frequency table using the class intervals 1-5, 6-10, 11-15, 16-20, 21-25 and 26-30.
b) Use the same data to construct a frequency table with ten class intervals.
c) Which frequency table is better? Give a reason for your answer.







Spotlight


^^OUNTIF(C2:C10,"*ed"
Computer software
such as Excel can
generate frequency
tables by counting the Product n Status D \
cells that contains a
Product 1 Destination Scan 3l
particular text.
1 Product 2 Delivered
Product 3 Arrival Scan
Product 4 Destination Scan
1 Product 4 Delivered
1 Product 5 In Transit
1 Product 6 Received
Product 7 In Transit
Figure 7.1 Example of a frequency table in Excel
Spreadsheet using a 'countif function. The instruction
given to the software is to count the number of cells
with words ending with 'ed'.

Revision



o Meg wants to find out the type of music her classmates listen to most often. Design a data
collection sheet to help her find out.
e A company that makes and delivers ready-made food investigated the reasons for
deliveries that were not delivered on time. This data collection sheet was used to record
and organise the data.


Deliveries not on time
1 Delivery person: Sue Gin Weekending: 4/16

1 Dsy
Reason Total
Monday Tuesday Wednesday Thursday Friday
Delays in the
II 2
kitchen
Traffic 1 II 1 4
Car problems i 1 1 1 3
Delayed at
III II 5
previous delivery
Total 2 1 3 3 5 14


a) How many different reasons were there for late delivery?
b) How many times were deliveries late during the five days?
c) What was the most common cause for late delivery?
d) How many times was the delivery late due to car problems over the five days?
e) On how many days did car problems result in late delivery?

f) On which day did delays in the kitchen result in late delivery?
g) On Wednesday, how many late deliveries were caused by traffic?
© State whether each question is an example of an open-ended or a closed question.

a) How much did you spend on cellphone data this week? Answer:
b) Do you travel to school by bus? Yes No

O Write an improved question for each of the following questions.
a) Do you agree that it is better to have a four-door car, than a two-door car?
b) How often do you eat fish?
sometimes occasionally often

0 There is a lot of litter in the school playground. School management would like to find
out what can be done to reduce the litter problem. Propose one question that could be
used in a questionnaire to collect this information.

© Mike wants to find out what his classmates are planning to do once they leave school.
Design a questionnaire that Mike could use to find this information. The questionnaire
should have five questions.


168 UNIT 7

O During a football competition, the number of goals scored by each team was recorded.






Construct a frequency table to organise this data.

An airport records the masses of 30 passengers' bags in kilograms.

7.3 12.4 8.4 9.0 9.4 8.7 11.4 7.1 8.8 10.5
6.9 11.1 12.5 9.8 7.8 8.8 10.9 8.8 8.2 8.0
12.1 10.2 8.4 7.4 10.3 9.3 7.8 8.7 8.2 9.2
a) Round the masses to the nearest whole number. Then complete the grouped frequency table.

Mass (kg) Tally Frequency

6-7
8-9
10-11

12-13
Total 1

b) What was the most common mass recorded?

c) What was the mass of the heaviest bag?
d) What was the mass of the lightest bag?
e) How many bags were heavier than 7 kg?
f) How many bags were lighter than 12 kg?
g)* Challenge! Passengers whose bags are heavier than 11 kg must pay an additional fee of $100.
i) How many passengers had to pay the additional fee?
ii) How much money did the airport make in total from the additional fee charged?

0 Lea wants to find out the most common length for a movie.
She records the lengths of the following movies in minutes.

110 99 126 110 100 152 130 137 135 95
89 160 90 111 126 103 121 89 101 83
122 98 140 119 95 96 91 109 164 124
117 87 94 89 91 109 88 124 130 106

a) Help Lea organise the data in a grouped frequency table.
b) How many movies did Lea record?
c) What was the most common movie length in minutes?
d) How long was the longest movie?
e) How long was the shortest movie?

f) How many movies were longer than 120 mln?
g) How many movies were shorter than 90 min?


169

<E> Study the class intervals in the frequency table. Emma has not chosen the class Intervals
correctly. Suggest how Emma could correct the class intervals.

Class intervals Tally Frequency
0-10 -mt+m III 13
10-20 41

20^0 mm mm m m 30
40-50 mmm 15
50-60 1 1
Total 100









Mathematics Connect






7 Day Forecast O
Wed Thu Fri Sat Sun Men Tue

Stormy Windy Cloudy Partly Light rain Sunny Sunny
and humid cloudy

A




28°C 27°C 28°C 29°C 29°C 30°C 30°C




Weather is linked to our everyday activities. People often turn to weather
forecasts when preparing for a variety of activities. Nowadays, we can easily
access weather information and predictions from different sources for free.
By gauging and taking note of the weather, we see patterns taking place. These
patterns come and go, thus providing the foundation for predicting weather.
Meteorological agencies around the globe collect as much data as possible about
the present condition of the atmosphere. Real-time observations of air pressure,
temperature, wind speed and direction, cloud type, precipitation and so on are
collected. Meteorologists then feed the data input from atmospheric observations
into computer systems that make use of modelling techniques to interpret the
data and predict future weather.









170 UNIT 7 Data handling
1

Collecting and recording data

We can use a data collection sheet or a questionnaire to help us collect
information about a product or an event.

Data collection sheet Questionnaire
Tally marks are often used to record numerical A questionnaire is a set of questions that allows a
information in a data collection sheet. Tally marks researcher to ask different people the same set of
are used to keep count by drawing marks. Every fifth questions. It can include closed and open-ended
mark is drawn across the previous four marks. questions.
Closed questions provide a list of answers to choose
from so that it is easy to organise data.
Open-ended questions have no suggested answers
Date: 1 |uly 2019 Day: Monday
so it can be difficult to organise data as there may be
Start time: 09:15 End time: 11:30
Total survey time: 1 hr 15 mins diverse opinions.
Location: Intersection of 1" Avenue and Main Street
Weather: Clear skies Surveyor: liana
Decide what you want to find out from the
masses. Ask a question.
Vehicle type Tally Total
Car ilHJfH JWifH III 23
V Decide what data you need to collect. There
Motorbike iittijtt nil 14
should always be a reason for collecting this data.
Bus 111 3
Truck mm 13 Design your questionnaire.
III
Bicycle mm iwt iwtiwt i 31
Conduct a survey using your questionnaire to
4
Other 0
collect and record your data.
Total 84
Organise your data in a table, graph or chart.

Frequency tables

Ungrouped data Grouped data
When data values are spread out, we bundle the
Ungrouped data is discrete data that is not
organised into groups. In an ungrouped data data into classes in a grouped data frequency table.
frequency table, data can be presented in a list. The difference between the lowest value and the
highest values in each class is called the class interval.
The class intervals must not overlap. Each data value
Age Tally Frequency
can only be in one class interval.
13 Nil 5
14 III 3 Age range Tally Frequency
15 nil 4 0-9 III 3
16 nil 4
10-19 ■m II 7
17 II 2 20-29 4Hh4ttt 1 11
18 II 2 30-39 m-m 1 11
Total 20
40-49 nil 4
50-59 nil 4
Total 40
■ iirsfc

171

A father left 17 camels to his three
Fract ons sons In his will.


When he passed away, the sons
opened up the will.

You will learn about:
The will stated that the eldest son
In this would get 2 of the cameis,the middle
( unit Finding equivalent fractions son would get 3 of the camels, and
Using fractions to describe parts of a whole
the youngest son would get § of the
Simplifying fractions
camels.
Converting between Improper fractions
As it was not possible to divide 17 in
and mixed numbers
half, or by three, or by nine, the sons
Expressing remainders In fractions
started fighting with one another.
Comparing and ordering fractions
How will the sons divide the camels?
Adding and subtracting fractions
Finding fractions of a quantity






5
.A



















li

.The sons visited a wise woman in the neighbouring village to try and settle their dispute. The woman "
agreed that it was.a difficult problem and that'she would reflect on if and advise them the next day.
The riext morning, the woman told the sons she could not solve the problem, but would give them her
own camel, in the hope that It would resolve their problem and end the feud.
The sons were puzzled, but pleased to have an additional camel, and began to walk home. On the
long walk home, they calculated how a herd of 18 camels could be divided. Half (or nine camels) would
go to the eldest son, the middle would get a third (six camels), and after the youngest son received his
ninth (two camels), there was still one camel left over which they could return to the wise woman.

Can you work out how this trick works?
Think about the solution. We will come back to this at the end of Chapter 8.6.



Fractions
I

CHAPTER 8.1
In this chapter

Pupils should be able to:
Understanding Fractions • understand and use

fraction notation
recognising that
8.1.1 Parts of 0 whole fractions are several
parts of one whole

^ and ^ are examples of fractions. They are parts of a whole.
A fraction has a numerator and a denominator.
O Spotlight
The number above is called the numerator.
The number below is called the denominator.
The word fraction
comes from the Latin
word "frangere" which
The denominator shows the
parts of a whole means "to break". The
number of equal parts a whole is
total number of equal parts history of fractions goes
divided into. The numerator shows
back to the Babylonians
the number of parts we have.
(who lived where Iraq is
today) and the ancient
Egyptians.
Example 1


The figure is divided into equal parts.
a) What fraction of the figure is coloured?
b) What fraction of the figure is not coloured?

Solution
The figure Is divided into 9 equal parts.
a) 4 parts are coloured.
So, ^ of the figure is coloured.

b) 5 parts are not coloured.
So, I of the figure is not coloured.


The figure is divided into equal parts.
How many more parts must be coloured to show^
?
Solution
The figure is divided into 12 equal parts.
5 parts are coloured.
To show . 11 parts must be coloured.
11-6 = 5
So, 5 more parts must be coloured to show .

Spotlight


Fractions can be used to O Divide 3 pizzas equally among 5 people.
describe © How can you share them equally so that
• a part of a whole, for each person gets a fair share?
example ^ of a cake

• a part of a group of
items, for example | Fraction of a quantity
of the girls in the class
Let us learn to find a fraction of a group of objects.
the positions of
fractions on a number The diagram below show 3 red soccer balls and 1 blue soccer ball.
line, for example 1 on
a number line




balls
of
the
We say that | and T of the soccer balls are blue.
red
are
^ RECALL
Example 2
Find the value of each of
the following.
O There are 50 people at a health workshop. 30 of them are men and the rest
a) 1 of 15 = are women. What fraction of the people at the workshop are men? Express
the answer in its simplest form.
b) 1 of 20 =
4
30 men
Solution
c) 1 Of 35 =
d) 1 Of 32 =
8 50 people
Fraction of people who are men = Number of men—
Total number of people

50-
= 3
3 5
5 of the people at the workshop are men.

0 Mimie baked 160 cookies. She gave 120 of the cookies to her neighbours.
What fraction of the cookies did she have left?

Solution
Number of cookies left = 160-120
= 40
Number of cookies left
Fraction of cookies left =
Total number number of cookies
Jm:

1^4
= 1
4
Mimie had J of the cookies left.


UNITS Fractions

O Fon's father gave her $700. Her mother gave her $500. Fon bought a
dress and had $200 left. What fraction of her money did she spend on the
dress?
Solution
Amount of money Fon had = $700 + $500
Try and Apply!
= $1200

Amount Fon spent on the dress = $1200 - $200
O A square drawn on
= $1000
a square dot paper
is divided into three
Fraction of money spent on the dress
sections.
= 5 ^
6
She spent 5 of her money on the dress.
6

Check My
Understanding


O What fraction of each shape is shaded?
What fraction of
the square is each
sections?
'
^ Look at the square.



f) g)








How many different
e A box of eggs is divided equally among 4 people. What fraction of the ways can you join
box of eggs will each person get?
the dots to divide the
e Sally shares her 2 sandwiches with her 2 friends. square into two equal
What fraction of the sandwiches will Sandy get? parts?
o There are 10 students from the same school on a bus. 2 of them are Use only straight
boys. What fraction of the students are girls? You may draw a bar lines. All lines must
diagram to help you. join from dot to dot.
o Jenni gets an allowance of $60. She spends $15 of her allowance. What For example:
fraction of her allowance did she spend?
e There are 44 pupils in a class. 28 of the pupils are girls. What fraction of
the pupils are girls?

o Ice has 200 beads. 50 beads are red, 70 beads are blue and the rest of
the beads are pink. What fraction of the beads are pink?
© There are 24 pupils in a class. One-sixth of the pupils play tennis and
half of the class play soccer. How many pupils do not play tennis and
soccer?

CHAPTER 8.2


In this chapter
Equivalent Fractions and
Pupils should be able to:
• simplify fractions by
cancelling common Simplifying Fractions
factors and identify
equivalent fractions
8.2.1 Finding equivalent fractions I



The rectangle is divided into 10 equal parts and
8 parts are shaded.
Look at the shapes below
So, of the rectangle is shaded.
Is there another way we can use fractions to
describe the shaded parts?
If we group 2 units into 1 part, we see that the rectangle has
of
the
5 equal parts and 4 parts are shaded. We say |
rectangle is shaded.
Shape A We say that is equivalent to We write ^ = f.
Q A
and 5 are called equivalent fractions.


We can find equivalent fractions by multiplying or dividing the numerator and
denominator of a fraction by a common factor.


Example 1
Shape B
O Find two fractions that are equivalent to
What fraction of
each shape has been Solution
the
we
both
multiply
shaded? To find an equivalent fraction to | numerator and
denominator by the same number.
Are the two fractions
equivalent? Discuss Here we multiply by 2 and 3 to get two equivalent fractions.
your answers.
X 2 X 3
3 6 3 _9_
4 V^8 4 12
X 2 X 3
Shape A is an octagon.
The words octo, meaning I and ^ are equivalent to
eight, and gonia, meaning
There are many other fractions that are equivalent to |.
angle, come from ancient
Greek. Can you name We find them by multiplying both the numerator and denominator by 4, 5, 6
Shape B?
and so on.



Fractions

.
© Find two fractions that are equivalent to ^

Solution Equivalent fractions are
Q
To find an equivalent fraction to ^ we can also divide both the numerator fractions that are equal
and denominator by the same number. in value even though
Here we divide by 2, or we divide twice by 2 (this is the same as dividing by 4). their numerators or
denominators are not the
■r2 -r2
same.
_8_ _4 2
A A

20 10 20 10 5
-5-2 ^2 4-2
^ and I are equivalent to ^ .




© Fill in the blank.

7 _ □
8 " 40
Solution
X 5
We multiply the denominator 8 by 5 to get 40.
2^"^ 35
To find an equivalent fraction of we should also 8V_>40
multiply the numerator by 5. X 5

So ^ 8 - 40 ■
50, ^



8.2.2 Simplifying fractions



Look at the fraction wall below.

I and ^ are equivalent

fractions.
We say that ^ is the simplest form

of

4' 8'^ ie ai'eequivalent
fractions.
1 1 1 1 1
7 7 7 7 7 7 We say that is the simplest
1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 8 form of .
1 1 1 1 1 1 1 1
9 9 9 9 9 9 9 9 9
A fraction that cannot be
JL J_ ± 1 1 1 1 1 j_ simplified further is said to be in
10 10 10 10 10 10 10 10 10 10
1 1 1 1 1 1 1 1 1 1 its simplest form or lowest terms.
11 11 11 11 11 11 11 11 11 11 11
J. 1. 1 ± 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12

O Write the fraction ^ Its simplest form.
in
Solution
18 and 24 are multiples of several factors, like 2, 3 and 6. In fact, 6 is the
highest common factor (HCF) of 18 and 24.
What happens when you
To get the simplest form, we divide the numerator and denominator by 6.
divide the numerator and
by
denominator of ^ ~6
2 or 3? 18 3
24 V_>4
-r6
18 :
.
g in its simplest form is f


© Express each fraction in its simplest form.
IS
a) ^ 15 c) ^ d) 64
28 45 W "7^ 80
70
Solution
Method 1 Method 2

a) The greatest number that divides 18 and 28 is 2.
o Journal Writing 28 28 -r 2 14 28 2r 14
1^^ 18-^2 _ _9_
18 =
±
James lists some fractions 14
that are equivalent to i: b) The greatest number that divides 15 and 45 is 15.

A 16 32_ 1
12 24 48 96
15-= 15 -r 15 ^ 1 15 ^ 4^^ 1
a) What is James doing 45 45 T 15 3 45 45- 3
to work out the 3
equivalent fractions?

b) James says that he
c) The greatest number that divides 42 and 70 is 14.
will eventually list all
3
the fractions that are
equivalent to^ if he 42-= 42 4-14 _ 5 42 ^ 4r^ 3
5
70 70 T 14 " 10 jer S
keeps on going in this
-KT
way. Show how James 5
may not be correct.
c) Suggest a way that
d) The greatest number that divides 64 and 80 is 16.
could be used to find
all the fractions that 4
are equivalent to^. 6A= 64 T 16 ^ 4 M = M'- 4
80 80 -r 16 5 80 88-"" 5

5



Fractions

8.23 Using a calculator to simplify fractions



You will need a scientific calculator with a fraction key either or


Example 3


Use a calculator to express the fraction in its simplest form. Calculator display will show:

Solution Q Utlh A
8
Use the key sequence GDOQQED- 10
So ^ ^
10 ~ 5


Check My
Understanding


O Find the missing number In each pair of equivalent fractions.
O 5 □
a) 5 = 9 b) 24 c) 6 30

7_
□
d) ^ - □ e) 32 f) 12 o
^
a; 4
60
Write each fraction in its simplest form.
a) I b) i c) ^ 10 e) 15
20
33
VS g) t2 h) ^ j) 20
30
What fraction of an hour is each of the following? Write each fraction in its simplest form.
a) ISmin b) 30 min c) 45 min d) 20 min e) 40 min
f) 10 min g) 50 min h) 5 min i) 55 min j) 25 min

o Some of the following fractions can be simplified and some cannot.
Use your calculator to try to write each fraction in its simplest form.

a; 21 b) ^ 0 H d) i e) ^
48
30 17 i) i
f) 48 g) 34 h) i i) ^

Hexominoes are figures you can form using 6 squares that
are of the same size.
i
This rectangle is divided into 4 hexominoes.
-
Give your answers in its simplest form.
a) What fraction of the total area is blue? r- 1
b) What fraction of the total area is red? J
c) What fraction of the total area is green?

CHAPTER 8.3

In this chapter
onverting Between
Pupils should be able to:
• change an improper
fraction to a mixed
number, and vice versa Improper Fractions and
• know that in any
division where the
Mixed Numbers
dividend is not a
multiple of the
divisor there will be a
remainder and it can be
Proper fractions
expressed as a fraction
and
of the divisor In Stage 3, you learnt about fractions. In fractions such as |
the numerator is smaller than the denominator.
These are called proper fractions.

Mixed numbers and improper fractions
1
The whole number part is 1. 1 The proper fraction part is ^.

A mixed number is a number that has a whole number and a proper fraction.
An Improper fraction is a fraction where the numerator is greater than or equal to
the denominator.




A group of pupils is split 8.3.1 Converting mixed numbers tp
into four groups.
improper fractions I
They shared some
chocolate bars as follows:
We learnt that an improper fraction is a fraction where the numerator is greater
1" group: 4 learners
than the denominator. Let us learn how to convert mixed numbers to improper
shared 3 chocolate bars.
fractions.
2"'' group: 5 learners
Example 1
shared 4 chocolate bars.
3'" group: 8 learners
Write 2| as an improper fraction.
shared 7 chocolate bars.
4'^ group: 5 learners Solution
shared 3 chocolate bars. Method 1 We can represent 2| as a diagram.

Is this fair? Did each pupil 1 - - I
' 4 1
get the same amount of I
2^ -2 + ^ ■H
^
4
chocolate? ^4 " ^
=1+1+1
-4 4 3
- 4 ^4 ^4
11
- 4

UNIT 8 I Fractions

Method 2 We can also do it this way without the help of a diagram.
2^-2+^
^4-^+4
First, work out the number of quarters in the whole number part.
1
2 whole ones =2x4 quarters
= 8 quarters
8
- 4
8 ^ 3 _ 11
Add the total number of quarters. 4+4-4
So, 23=^.


O Convert each mixed number to an improper fractions.
To get the numerator of
a) 3f b) 10^-
the improper fraction,
multiply the number of
Solution
wholes by the number
-4 (3 X 7) + 4
a) of equal parts it is
- 1 '^10 10
divided into. Then add
_ 21+4 _ 100 + 9
" 7 10 the numerator in the
_ 1M fractional part.
- 25
10
~ 7 "
To convert a mixed number to an improper fraction,

• multiply the whole number by the denominator of the proper fraction and

• add this product to the numerator of the proper fraction.
The result is the numerator of the improper fraction. The denominator
remains unchanged.


Check My

Understanding

Convert the mixed numbers to improper fractions.


a) 1 b) 3I c) 4 d) e) 2|

9) 2I h) 2| i) 2| j) si

83.2 Converting improper fractions


to mixed numbers



Example 2


O Convert each improper fraction to a mixed number.
Express your answer in the simplest form.

a) ^ b) f c) ^ d) 24
8 10
Solution

=
b) ^





=
1

d) ^ 30 + 4
-
O) 10 -
10
30 , A
10 10
= 3 + §



We can also use division to help us convert improper fractions to mixed numbers.

O Write 4 as a mixed number.

The numerator {the Solution
M
number being divided)
+
=\,\
=1
a) Since |
^+1
is the dividend.
The denominator (the =1+1+1
number dividing the
= 2i
dividend) is the divisor.
The quotient (the b) 9 -r 4 = 2 R 1 2 R 1
Write the remainder 1 as 419
answer to the division)
= 2 a fraction of the divisor 4. 8
is the whole number
1
part of the mixed
number.
To convert an improper fraction to a mixed number, divide the numerator by the
denominator.
• If the dividend is not a multiple of the divisor, then there will be a remainder.
• The remainder can be written as a fraction of the divisor. This is the fractional
part of the mixed number. quotient
^ remainder
157 ^ 25 = 6^
divisor
t t
dividend divisor


UNIT 8 I Fractions

8.3 3 Using a calculator to convert


between mixed numbers and

improper fractions



We can use a scientific calculator to convert from a mixed number to an improper
fraction, and from an improper fraction to a mixed number.


Use the Is*J key to enter the improper fraction
SHIFT
or the key to enter the mixed number. Press the key and
the at the same
To convert an improper fraction to a mixed
time to get the mixed
number, press the key. M-B-j format.
number

Example 3


O Use the calculator to convert

a) 2^ to an improper fraction.

b) 3 to a mixed number.
Calculator display will show:
doiuiion
a MlU A
a) Use the kev sequence SHIFT □ H] CE) E ® d] d] 2^
So.2l = |. 7

b) Use the key sequence 5 |^| [3] ® ShD ■ Hltt A
5
So,f =l|. 3



Check My
Understanding

O Change the following improper fractions to mixed numbers.

a) I b) I T d) t e) I

h) I
f) ^ 9) 2 "' 3 " 5 j) ¥
i)
6
J' 5
y 2
O Use your calculator to convert the mixed numbers to improper fractions.
a) b) c) 2| d) 3l e) 5| f) 32

© Use your calculator to convert the improper fractions to mixed numbers.

a) b) I I


d) f e) 10 f) ^
7

CHAPTER 8.4

In this chapter

Pupils should be able to: Comparing and Ordering
• compare two fractions
by using diagrams, or
Fractions
by using a calculator to
convert the fractions to
decimals
8.4.1 Comparing and ordering


^ RECALL unlike fractions


You have learnt to compare and arrange like fractions which have the same
O Which is greater,
denominators. Let us now learn to compare and arrange unlike fractions. These
are fractions with different numerators and denominators.

O Which is greater,
Example 1
lorl7

@ Arrange and
Which of the fractions, ^ and^ Is the smallest? Which is the greatest?
in ascending order.
Solution






Mathematician


Thomas
Harriot
(1560- 1621)
The signs for
greater than The circles are of the same size.
P
(>) and less than (<) were Circle P is divided into 8 equal parts.
introduced in 1631 in his Circle Q is divided into 11 equal parts. Q
book "Artis Analyticae
Circle R is divided into 5 equal parts. R
Praxis ad Aequationes
The part coloured in Circle Q is the smallest.
Algebraicas Resolvendas."
It was only published 10 is the smallest fraction.
years after his death! The
The part coloured in Circle R is the greatest.
symbols were actually
invented by the book's ^ is the greatest fraction.
editor. Harriot initially
used triangular symbols
which the editor later
modified to resemble the
modern less/greater than
symbols.





UNIT 8 I Fractions

When comparing fractions that have the same numerator but different
denominators, we compare the denominators.

• The smaller the denominator, the greater the fraction.
• The greater the denominator, the smaller the fraction.



Example 2


ascending
in
Arrange ^ and | bar diagrams to help you.
Use
order.
'ascending' means 'from
Solution smallest to biggest'.
Write each fraction as an equivalent fraction with 12 as the denominator. 'descending' means 'from
biggest to smallest'.
A
12
3_
12
_2_
12

From the diagram, we can see that

So, writing the fractions in ascending order, we get; 5.4.3-



To compare unlike fractions, find the lowest common multiple (LCM) of the two
denominators to find equivalent fractions for each fraction with the LCM as the
denominator.

Example 3


Which of the fractions, ^ and 11 is the smallest? Which is the greatest?

Solution

Write the fractions as like fractions.
^ _3_ 3X2 _ _6_ 4 _ 4X4 16 The LCM of 10 and 5
10~10x2~ 20 5 "4X4 20
is 20.

2- We compare ^ ^ and

^ has the smallest numerator.

So, is the smallest fraction.

^ has the greatest numerator.

So, I is the greatest fraction.

Example 4


Which of the fractions, ^ and |, is the smallest? Which is the greatest?

Solution
Write the fractions as like fractions.
1 2 2 X 10 X 5 100
3 " 3 X 10 X 5 ~ 150
The LCM of 3, 10
7 7x3x5 105
10 " 10 X 3 X 5 ~ 150 and 5 is 150.
3 3 X 3 X 10 90
5 " 5 X 3 X 10 ~ 150


We compare, ]iand:^.


has the smallest numerator.
So, I is the smallest fraction.

has the greatest numerator.

So, is the greatest fraction.


When comparing unlike fractions:
• convert all the fractions to like fractions and
• then compare their numerators.



Example 5


Arrange the fractions, |, f :j^, in increasing order.

Solution
Write the fractions as like fractions.
^
and
We see that |, | 3x3 _9 5x4 20 7_
1
7X2
can be expressed with 8x3 24 6X4 24 12 12 X 2 14
24
denominators of 24. Use
Weget^,^,g and
their equivalent fractions. 24 n
Compare their numerators.
9<14<19<20

So ^< 19 20 14
24 ^ 24 ^ 24 ^ 24 •
So, the fractions arranged in increasing order are:

3 19 5
8' 12' 24' 6







UNIT 8 I Fractions

Comparing and ordering unlike


fractions and improper fractions







and
in
Arrange the fractions, | order.
decreasing
Solution
Write the fractions as like fractions.
1 5 5x4 _ 20 3 _ 3X8 24 5.1 16
4~4x4"16 2"2x8~16 5 - ' - 16
\A/p npt 20 15 24 j lb
16
vveget^g, .,g, .,5 ana ^g.

Compare their numerators.
^ 24>20>16>15
Weseethat|, I and|
So 24 >20 16 .15 can be expressed with
16 ^ 16 > 16 ^ 16 •
denominators of 16. Use j
So, the fractions arranged in decreasing order are: 1
their equivalent fractions.
3 5 5 15
2' 4' 5' 16

To compare unlike fractions and improper fractions easily, we make the
denominators of the fractions the same.
O Find the lowest common multiple (LCM) of the denominators.
Find equivalent fractions for each fraction with the LCM as the denominator.



Check My
Understanding



O Which fraction in each pair is greater?
^
and
e)
^
f)
and
and
\
and
a) I and f b) | | | | | | |
c)
d)
and
O Fill in the blanks with <, > or =.
100
a) ini b) ini 1000 e) 2_\ I _21L
I 1000
10 I
fl 2 □ I 9) i □ ^ b) f i i) ^ ^ i) 1 □ f

O Redraw the number line below and then place
correct
positions
the fractions ^ and |
in
the
++
on the number line.
O Which of the fractions, |and is the smallest? Which is the greatest?
© Which of the fractions, |, | Which is the greatest?
smallest?
is
the
and
© Arrange the fractions, ^.3.5 and in increasing order.
,
decreasing
and
© Arrange the fractions, | order.
in
® Arrange the following numbers in ascending order.
a) |,iand| b) 1 and J
© Arrange the following numbers In descending order.

a) 3,f andg b) f.^and^

® Write the fractions in ascending order.

a) Z ^ 4 1 and^ M 280 543 28 3_d 150 17 500 13 790 34 820 and 9222
9' 10 ' 6' 3 ' 12 ' 400'800'50 200 18 000'20 000'45 000 15 000
11
© Write in descending order.

a) 7, 1,:^, f,|andO b) 115 60 103 482 11 500 2500 6000 and 7650
.^and
125' 250' 500 1250 22 950' 3825' 11 475 15 300
of
answers
Amy got 13 out of 25 answers correct in a Mathematics test while Victor got | correct.
the
3
5
Who did better in this Mathematics test?
race.
the
of
0) Adam and his father took part in a bicycle race. Adam only managed to finish |
His father managed to finish of the race. Who rode the furthest?
® A can of cola is divided between three friends. Jack drinks ^ of the cola, Jane drinks ^ of the cola and
Sheila drinks ^ of the cola. Who gets the biggest share of the drink?





























Fractions
I

i CHAPTER 8.5



Adding and Subtracting In this chapter

Pupils should be able to:
• add and subtract two
Fractions simple fractions
• find fractions of
___
quantities {whole
8.5.1 Adding and subtracting like number answers)


fractions


Fractions can be added and subtracted easily if they have the same denominators.


Example 1


Add ^ and |

Solution
1
5

m



1 fifth + 3 fifths = 4 fifths The bar diagram has
5 equal parts. We add
1 +3
i ^
5 ^ 5 5 1 part and 3 parts.

Example 2


Subtract |
and
|
Solution
take away |











We add or subtract like
fractions by adding
7 eighths - 3 eighths = 4 eighths
or subtracting their
Z _ 3 _ 7^ _ 4 _ 1
8 8 ~ 8 "8"2 numerators.

Example 3


Add 2j and 1

Solution

^7 I








Jt
4 wholes
2^lf=3f

= 42



Example 4

Find the difference between 2 ^ and 11

Solution






2l_i4_i6_44
I
z 5 I 5 - 5 I 5


take away 1 |



Check My

Understanding


O Add.
a) 1 and ^ b) and ^

© Subtract.
a) ^andi b) 1 and |

© Find the sum.
a) 1 § + ^ b) 2U1I


O Find the difference,
a) 3f-1§ b) 2

8.5.2 Adding and subtracting unlike

fractions



Example 5

Add ^ and


Solution







1 . 1 _ 1 . 2
6
6 ^ 3 " ^ 6
_ 3
"
5
_ 1
"
2

If fractions do not have the same denominator, we express them as equivalent
fractions with the same denominators by finding the LCM of their denominators.


Example 6


O Work the following. Express your answers in the simplest form,
a) 1 + I b) _L
12
Solution
1+5^2 5
a) I and I do not have the same denominator.
3^6 6^6
_ 7
~ 6 The LCM of 3 and 6 is 6.
-6 1
"6^6
= 1+1




s_ 5_
b) I and do not have the same denominator.
12 12 12
-4
12 The LCM of 4 and 12 is 12.
1
3

Example 7


.
O Find the sum of ^ and ^ Q Find the sum of :j^ and |.
Give your answer in the simplest form. Express your answer as a mixed
number in the simplest form.
Solution
1 + = 2. i Solution
6 ^ 12 12 ^ 12
= ^ +-S-
_ S_ 10^ 5 10 ^ 10
"
12
_ 15
10
_ 3 "
4
"
= 1-^
' 10
= li
and
Express
^
.
your
O Find the sum of ^| answer as a mixed number in the
simplest form.
Solution
The LCM of 2, 4 and 8 is 8.
4 . 6 Z
2 ^ 4^8" 8 8^8
1x4 = 4
17 2x4 = 8
3X2 = 6
= 2^ 4x2 =
8



O Subtract J from ^
0 Find the difference between |
and
Solution
Give your answer in the simplest form.
Solution
j_ 6_ J_
10 10 10
_5.
10
1
2
Example 9
What is the value of | | answer in the simplest form.
Give
^
?
your
+
Solution
3^2 J_
4 5 20 ~ 20 20 20
_ 22
~ 20
=
1
20
- 10






UNIT 8 I Fractions

Example 10


O Work out the following. Express your answers in the simplest form.

a) 2^+1^ b) 8-l§
6
^ 3 ^ '
Solution
a; ^3+«6-3 + 6 Write each mixed number as an improper fraction

_ 14^ n
The LCM of 3 and 6 is 6
- 6 6
- 25 Add the two fractions by adding the numerators
6
Write as a mixed number



b) 8-l| = f-| Write each number as an improper fraction

3 8x3 5
The LCM of 1 and 3 is 3
1x3 3
_ 24
You have two fractions with the same denominator
~ 3
_ IS Subtract the two fractions by subtracting their numerators
~ 3
= 6i Write as a mixed number


Example 11

Find the sum of 21 and 31.


Solution
■y2 . :,2 _,-2 . 2 Add the whole numbers
25 +33 -55 + 3
= 5-^ + ^ Convert the proper fractions to like fractions
^ 15 ^ 15
Add the proper fractions

= 5 + 1^ Convert the improper fraction to a mixed number
= 6i Simplify


Example 12


Subtract 1 ^ from 3 ^ .

Solution

-2^ -
2
3^ 2 7 -•^14 " 7 Subtract the whole numbers
-2^ - 4 Convert the proper fractions to like fractions
-4 14
14
-4 14 Simplify
_ 2 —

Example 13


.
Subtract 2 ^ and 5 ^
Solution
5I _ 2I -3I - 1 Subtract the whole numbers
^3 ^2-^3 2
= 3^ - ^ Convert the proper fractions to like fractions
^6 6
We cannot subtract |
from
o b
So, convert 1 whole to f.
b
= 2l Simplify


Example 14


What is the value of 3^-1:^ + 2^?

Solution
j. i_ 2
^ 2- Work out the whole numbers
4.
10 20 5 10 20
8 6 4. 7
= 4 20 "*■ 20 Convert to like fractions
20
9
= 4
20
Check My
Understanding


O Work out the following. Write your answer in simplest form.

a) i + 5 d)i-i
8 8 6 ' 6 6
O Work out the following. Write your answer in simplest form.

d) i
10 2
p) ^ J. ^ —L f
2 ^ 3 ^ 12 f) 2 4 3 g) 1 - 12
6

Determine the following. Give your answers as mixed numbers in simplest form.
a) i + 2 b) f+ ^ c) ^
3
8
5 . 2
5 . 3
d) 1^ 4 e) ^ + f f) 5 + i
6 ^ 12
O Determine the following. Give your answers as mixed numbers in simplest form.
a) 21 + 31 b)4l-2U 0 3^+ 7§ d) 2 — - 1 ' 100
20

e) 4 + 3^ f) 12| -9 g) 10-5| h) 7-6|







Fractions

8.5.3 Solving word problems



Let us solve some word problems involving addition and subtraction of fractions.

Example 15



O Theresa sold 14 out of 30 papayas at the Fresh Produce Market and John sold
I of his papayas. Who sold more papayas?

the
of
O Jill and Peter sold oranges together at the same market. Jill sold |
oranges and Peter sold | fraction of the oranges was left at
of
them.
What
the end of the day?
Solution
O Theresa sold 14 of 30 papayas: ^ 15

John sold I of his papayas: f ^


15 ^ 15
So, John sold more papayas.

O Jill and Peter sold oranges together, so we add the fractions together to find
the fraction of oranges sold.
1 + 3 _ 4 3
8
2 ^ 8 " ^ 8
_ 7
8
"
1-^=1
' 8 8
i of the oranges were left at the end of the day.

Example 16


flour
kg
to
make
of
Yim had ^ kg of flour. She used | some buns. She then
bought another ^ kg of flour. How many kilograms of flour did she have in the
end?
Solution
To find the mass of flour Yim had in the end, first subtract the mass of flour she
used and then add the mass of flour she bought.

2_3 . 1 _7_6 . 4
8 4 '^ 2 ~8 8 '^ 8
_ 5
- 8
of
kg
Yim had |
flour
end.
the
in

Gino had 42 ^ / of petrol in the petrol tank of his car at the start of his journey.
At the end of his journey, the volume of petrol left was 261 /. How many litres of
petrol did Gino use for the journey?

Solution
To find the volume of petrol Gino used, subtract the volume of petrol left from
the volume of petrol at the start of the journey.
42i-26|=16l-



= 15 + I +


= 15
Gino used 151 / of petrol for the journey.



Check My
Understanding

o On Monday, Keith spends ^ of his day at school and ^ of his day practising soccer.
How many hours does he have left to do his homework, play and sleep?
e Ying had ^ kg of grapes. She gave j kg of grapes to her neighbour. Then, she bought

another ^ kg of grapes. How many kilograms of grapes did she have in the end?
e Mimie mixed 8 ^ / of water and ^ / of orange squash to make a drink.
How many litres of drink did she make?

O A baker had 25 ^ kg of flour at the start of the day. At the end of the day, the mass of flour
he had left was 9 kg. How many kilograms of flour did baker use?
his
money
of
© Duncan spends ^ of his money on bread, | and ^ of his money on chocolate.
milk
on
a) What fraction of his money does Duncan spend at the shop?
b) What fraction of his money does he have left?



Try and Apply!


Fill in the missing numbers
a) b)
in the magic squares. The 1 3^ 2 21
numbers in the rows, columns
and diagonals should add up
to the same total — the magic
1 2 3
number!






Fractions