We can locate a point on a coordinate grid using its distance from the ^-axis and Think and Share
y-axis. Look at the example below.
Can you write the
Point A is located on the coordinates (1, 2) on the grid below in the first quadrant.
coordinates of some
The point B{-3, 3) lies in the second quadrant. In the second quadrant, the points in the third and
.Y-coordinate is a negative number and the v-coordinate is a positive number. fourth quadrants?
Point C has the coordinates (-5,-2) lies in the third quadrant. Both the
How are they different
.Y-coordinate and v-coordinate are negative.
or similar to the
Point D lies in the fourth quadrant. coordinates of the
points in the other two
quadrants?
This is also known as a
B(-:> 3)
Cartesian plane. It consists
of two axes: the .v-axis and
the _v-axis.
When naming a point, we
write the .v-coordinate first,
2 -
-6 -? -4 -3 -
then the v-coordinate. The
coordinates are written
C{-!, -2 within a pair of brackets and
separated by a comma. The
e ^
pair of coordinates (0, 0) is
D(3.-4)
called the origin. I
Example 1
Mark and label the following points on the coordinate grid below. Then draw
straight lines to join the points in order. Name the polygon that is formed.
P{3, 3) Q(3, 1) R{1,-2) S(-2,-3) T(-4,3) U(-2,4)
Solution
PQRSTU is a hexagon.
Check My
Understanding
O Mark and label the following coordinates on a coordinate grid.
A(1,6) B(3, 7) C(5,6) D(6,4) E(5,2) F(3,2) G(1,4)
Draw straight lines to join all the points together from A to G, and back to A again.
What is the name of the shape formed?
@ Mark and label the following points on a coordinate grid.
Then draw straight lines to join the points in alphabetical order such that they form a polygon.
Name the polygon that is formed.
D(1,4) E(4, 0) F(-2, -2) G(-3, -1) H{0, 2)
Write down the
coordinates of the
points shown on the
grid to the right. rj
r;
->
H
I 1
6 - 5 - 4 - 3 -2 -\ 0 ) ; \ . *-.v
F T n
1 ?
K 1 , 1
-4 f\/\
-S
-6
O Match the letter used to label each of the
following points shown on the grid to the
coordinates given. ■Aa- f-fr-
M r
a) 0.3) b) (0, 2)
c) (3, 2) d) (0, -6)
e) (-4,2) f) (-3,4) p
g) (-5, -1) h) (-3, -4) H -) p
i) (-3, 6) j) (-6,5) 1
k) (-2, 0) 1) (-6,-6) N
0
m)(2, -4) n) (-5, -2) ■6 - 5 - 4 - 3 2 -[ » \ ^ .
-
o) (2, -5) P) (0,-5) n 1
q) (4,-4) r) (5,-6) , p ?
-4 1. ,
-S n s
-A ► F
24^1 UNIT 11 I Coordinate Geometry and Graphs
State which quadrant each of the following points lies in.
a) (14,-10) b) (8,4) c) (-^,-19) d) (-15, 2)
e) (10,-8) f) (6,-7) g) (23,9) h) (-1,-1)
i) (-16,6) j) (-2,2) k) (-35,-^1) I) (9,9)
m) (6,-24) n) (-7,-9) o) (-3, 16) P) (5, -5)
O The diagram below shows the map of a theme park drawn on a coordinate grid.
Mega-Maze
Ice-cream Entrance
Stall
riar Thunder
Bumper
Boat Pond
Go-Kart I
Burger Bar
*• X
9 10 11 12 13 14 15 16 17 18 19 20
a) Write down the coordinates of the
I) Burger Bar, ii) start of the Thunder Ride,
iii) centre of the Bumper Boat Pond, iv) ice-cream stall and
v) Mega-Maze exit.
b) What location would you be near to if you were at
i) point (5, 8)? ii) point (16, 10)?
Try and Apply!
O Draw a set of axes on a grid.
¨ Amazing
IV/ia+ham:
Mathematician
The coordinate system we
commonly use is called
the Cartesian system, after
the French mathematician
Rene Descartes (1596-
1650), who developed it in
the 17'^ century.
1
Legend has it that
Descartes, who liked to
6 - 5 - 4 -3 -2 - 1 0 1 \ ' . (
\
stay in bed until late, was "1
watching a fly on the
-?
ceiling from his bed. He
wondered how to best
describe the fly's location -A
and decided that one
-S
of the corners of the
-6
ceiling could be used as a
reference point.
Q Plot the following points. Join each set of points using straight lines.
a) (2, 6), {3, 6), {3, 7), (2, 7), (2, 6) STOP
b) (2, 2), {3, 2), {3, 5), (2, 5), (2, 2) STOP
c) (-5, 7), (-5, -3), (-3, -3), (-3, 1), (-1, 1), {-1, -3), (1, -3),
(1, 7), (-1, 7), (-1, 3), (-3, 3), (-3, 7), (-5, 7) STOP
@ What do you think you have drawn?
- Think and Share
O Are (2, 3) and (3, 2) coordinates of the same point?
Is the order in which the two numbers are written important?
© The pairs of numbers (-2, 0), (2, 3) and (8, 10) are examples of
ordered pairs. Why are they called ordered pairs?
UNIT 11 Coordinate Geometry and Graphs
CHAPTER 11.2
In this chapter
Linear Graphs Using Pupils should be able to:
• generate coordinate
pairs that satisfy a linear
Coordinate Pairs equation where, y is
given explicitly in terms
of .V
A function, or an equation, is a written relationship or pattern between two
• plot the corresponding
quantities. Coordinate geometry helps us to display the patterns we observe
graphs
between two quantities visually. A coordinate graph is like a picture of their
• recognise straight-line
relationship. We use coordinate grids to plot the points of an equation to help us
graphs parallel to the
draw the line graphs.
.V- or y-axis
To draw the graph, we plot coordinate points and linear equations (where _v is
given in terms of x) on a set of axes. A linear graph is a straight line.
Example 1
O On a piece of graph paper, draw an x-axis and a y-axis from -5 to 5.
a) Plot the points (4, 4), (3, 3), (2, 2), (1, 1) and (0, 0).
b) Use a ruler to draw a line though the points.
c) Extend the line into the third quadrant. Write down the coordinates of
any three points on the line that lies in the third quadrant.
d) Write down what you observe about the coordinates of the points you
have marked. Hence, write down the equation of the line.
Solution
a) We plot the five points on the coordinate b) Draw a straight line through the five
grid. points using a ruler and extend the line
into the third quadrant.
•>
1
U
5 -4 -3 - 2 - 1 ; 1 5 - 4 - 3 - 2 - 1 M >
n n - -
1 1
/ 7
/
7, .3
/
rA 4
/
-S -S
c) Mark any three points on the line that lie in the third quadrant.
The points shown are {-1, -1), (-3, -3) and {-4, -4).
2
■\
-♦•A-
5 4 - 3 2 1 M ; 1
-
-
-
(1
7
/\
s
d) The y-values of each point is the same as the A-vaiue of each point.
So, the equation of the line is = a-.
Check My
Understanding
O a) Complete the table of values for the equation _y = a + 1. X
3
X -3 -2 -1 0 1 2 3 2
y = A+ 1 -1 3 4 1
0
-1
b) Plot the points.
-2
Draw a line through the points with a ruler and label -3
the liney = A+ 1.
O a) Complete the table of values for the equation y = x + 2. X
3
A -3 -2 -1 0 1 2 3 2
y = A + 2 -1 3 5 1
0 + 2
b) Plot the points. -1
Draw a line through the points with a ruler and label -2
the line y = a + 2. -3
O Look at the straight lines you have drawn in Example 1 and Questions 1 to 2.
a) Write down two facts about the three lines y = a, y = a + 1 and y = A+ 2.
What are the similarities and differences between the lines?
b) Use these facts to draw the line v = a + 3.
Coordinate Geometry and Graphs
O a) Complete the table for the equation 1.
X -3 -2 -1 0 1 2 3
-4 2
II
b) Plot the points.
H
Draw a line through the points with a ruler and label the liney = x- 1.
1
O Draw the graph ofy = x-2 using a table of values to find the points that lie on the line.
Try and Apply!
Match the equations on the left-hand side to the correct scenarios on the right-hand side,
X 1 -3
a) y = 3;i: + 1 •
T 5 1
X V
-2 -11
-1 -1
b) y = X — 3
0 -3
1 1
2 5
c) y = 4x-3 X 3 +1
d) V = 1 + 4x When X = 0, y = 6, and when y = 0, x = |.
—fr
/
/
1
e) y = X + 4
6 -5 - 4 -3 - 2 ^1 u 1 y \
1
/
n ■?
/
-.3
/
/
/ -6
/
Mr Singh gives his students 1 homework problem during the first week
f) y = -5x +
6
of school. He gives 4 problems every week thereafter. The total number of
problems pupils have done thus far is?
CHAPTER 11.3
-t'K-A
In this chapter
Drawing Lines Parallel to
Pupils should be able to:
• generate coordinate
pairs that satisfy a linear the y-axis or the x-axis
equation, where y is
given explicitly in terms
of .V
• plot the corresponding
t
graphs Investigate!
• recognise straight-line
graphs parallel to the Two straight line graphs are drawn on the coordinate grid shown below.
X- or y-axis
What equation
->A
-5 -4 -3 -2 -
Check My
..=4
Understanding
O 3) On a graph paper, a) Use the graphs to help you complete the following tables.
draw a set of axes
from -5 to 5. Use Horizontal Line
the tables above X -A -3 -2 -1 0 1 2 3 4
to plot each linear
y
graph. Label
each line with its Vertical Line
equation.
X
b) On the same set -4 -3 -2
y 0 1 2 3 4
of axes, draw and
b) What do you notice about the y-coordinates of the points on the horizontal
label the following
line?
straight lines.
c) What do you notice about the x-coordinates of the points on the
1) A = -3
vertical line?
ii) a = 4
d) Write down the equation of the horizontal line and the vertical line on the
iii)y = 2
coordinate grid.
iv) y = -5
25^ mMITII I Coordinate Geometry and Graphs
In this chapter
Pupils should be able to:
• draw and interpret
graphs in real life
contexts involving
more than one stage
Conversion graphs
We can use a conversion graph to show the relationship between two units of
measurement. It can be used to help us convert from one unit to another.
Example 1
This conversion graph shows the approximate conversion between miles and
kilometres.
Kilometres and Miles
1 k
/]
40
/
36
/
32
b. ?R
1/> 74
/
a> 20
F /
o 16
/
12
/
4
/
0
t
1 ) 1 ! 2) 2 I 3)
Miles (ml)
O Convert 5 mi to kilometres. O Convert 20 mi to kilometres,
O Convert 16 km to miles. o Convert 28 km to miles.
Solution
o Use the graph to find 5 miles on the horizontal axis.
Follow the blue dotted line to go up to a point on the straight-line graph.
Then draw a red dotted line to the left to find the corresponding value on the
vertical axis.
5 mi = 8 km
@ Use the graph to find 20 miles on the horizontal axis.
Follow the blue dotted line to go up to a point on the straight-line graph.
Then draw a red dotted line to the left to find the corresponding value on the
vertical axis.
20 mi = 32 km
O Use the graph to find 16 km on the vertical axis.
Follow the red dotted line to the right to go to the straight-line graph.
Then draw a blue dotted line to go down to find the corresponding value on
the horizontal axis.
16 km = 10 mi.
O Use the graph to find 28 km on the vertical axis.
Follow the red dotted line to the right to go to the straight-line graph.
Then draw a blue dotted line to go down to find the corresponding value on
the horizontal axis.
28 km = 17.5 mi.
.4.2 Wages chart
Straight line graphs are useful to track one quantity against another quantity that
changes proportionally to each other.
Example 2
Trevor has a part-time job as a waiter. He earns $3 per hour. The table below
shows how much he can earn.
Number of hours of work 1 2 3 4 5
Wages {$) 3 6 9 12 15
We see that wages (w) = $3 x number of hours worked (h)
We can write this as: u- = 3 x h
= 3/7
Trevor's wages
24
Trevor draws a graph to represent his work-wage
equation.
21
O How much does he earn if he works 7 hours?
@ How much does he earn if he works 3.5 hours?
0
18
0
Solution
0
15 From the graph we can see that
/
if
O he works 7 hours, he earns 7 x $3 = $21.
a>
I 12- / © if he works 3.5 hours, he earns 3.5 x $3 = $10.50.
0
0
0
3
Hours (h)
Coordinate Geometry and Graphs
Check My
Understanding
A few Singaporean students are visiting Kuala Lumpur. They want to buy
some souvenirs. To convert the prices from Malaysian Ringgit to Singapore
dollar, they use a conversion graph to check the prices.
The line graph below shows the conversion between Malaysian Ringgit
(RM) and Singapore dollars {S$) on a particular day.
Ringgit and Singapore Dollar
4
1
r
r 3
o
0
1 2
Q.
C
If-
4 5 6 7 8 9 10 11
Malaysian Ringgit (RM)
a) Ann is buying a model of the Petronas Twin Towers.
It costs RM 12. Use the graph to work out the cost in
Singapore dollars (S$).
b) John is buying a postcard of Kuala Lumpur.
It costs RM 9. Use the graph to work out the cost in
Singapore dollars {S$).
c) Mary is buying 2 pens. They cost RM 3 each.
What is the cost of 1 pen in Singapore dollars?
How much do 2 pens cost in Singapore dollars?
/! I/'
d) Paul wants to buy 3 bars of chocolate showing the
Petronas Twin Towers. Each chocolate bar costs RM 6.
What is the cost of 1 bar of chocolate in Singapore dollars?
How many do 3 bars of chocolate cost in Singapore dollars?
Think and Share
Let's refer to Question 1. 1 bar of chocolate costs RM 6 and 1 pen costs RM 3.
Jane has RM 20 and wants to buy as many chocolate bars and pens as she can.
She wants to buy at least 1 chocolate bar and 1 pen.
How many combinations of chocolate bars and pens can she buy with RM 20?
© Anita works in the same restaurant as Trevor. She has been there longer and earns $4 per hour,
a) Express Anita's wages (14/) in terms of the number of hours she works (h).
b) Complete the table below.
Anita's wages
Number of
24
1 2 3 4 5
hours of work
Wages ($) 4
21
Use the set of axes shown and draw a
graph to show Anita's wages. 18
d) Use your graph to work out how much
Anita will earn if she works
15
i) 6 hours.
ii) 4^ hours.
12
01
e) Use your graph to work out how many CTl
(5
hours Anita needs to work to earn:
i) $10.
ii) $21.
Joanne and Laura are jogging together. Laura
soon gets ahead of Joanne.
Laura jogs at 4 m per second (4 m/s) and
Joanne jogs at 3 m per second (3 m/s).
a) Write down the equation for the
distance (D) covered by Laura in terms 1 2 3 4 5 6 7 8
Hours (h)
of the time she took (t).
':-S
b) Complete the table below for Laura.
Time (s) 10 20 30 40 50
Distance (m)
Use the set of axes Distance that Joanne and Laura jog
shown and draw a graph
to show the distance 200
that Laura jogs on a
piece of graph paper.
d) Express the distance (D)
150
Joanne jogs in terms of
the time she took (t).
e) Make a table of values to
plot Joanne's graph. 100
f) On the same set of axes,
draw a graph to show the
distance that Joanne jogs.
g) Who covered more
distance in 50 seconds?
1 1 1 1 2r) 1 3b 40 50
Time (s)
UNIT 11 I Coordinate Geometry and Graphs
straight line distance-time graphs
A distance-time graph gives us information about a journey. It tells us the
distance from where the journey started, how long the journey took, or how fast
one travelled.
Example 5
Peter recorded the distance he cycled from his house to his office in a table.
Time (min)
0 5 10 15 20 25 30 35
Distance (km) 3.5
0 1.5 2 3 3.5 4 5
a) Plot a distance-time graph to show the distance Peter cycled.
b) Describe Peter's journey.
Solution
a) Distance Alex cycled
-
/.
^40- -A 5... 1 S- •3 S-
Time (mtn)
b) He cycled faster from the 10"' to 15^^ minute since he covered the most
distance in this interval than in other time intervals. He covered 1 km in the
first 5 minutes compared to 0.5 km between the and 10"^ minute, and
0.8 km between the 30^^ and 35^^ minute. He stopped to rest from the 20^^ to
25**^ minute.
Investigate!
The distance-time graph below shows Paul's journey from home to school.
Paul's journey to school
3.0 ' c
/
2.8
f
2.6
/
2.4
2.2
/
2.0
f
1.8 B
E /
1.6
/ c
1.4
/
1.2
/
1.0
0.8
/
0.6
/
0.7
.... /
0.6 —
0.4
I
> 1 10. -15- n 7S- io--4s-. -50-
Time (min)
a) Paul started his journey by walking. How far did he walk?
b) Paul stopped at a shop to buy a pen.
i) Which segment on the graph does this correspond to?
ii) How long was Paul stop at the shop?
c) Paul was late and started to run.
i) How long did he run?
ii) How far did Paul run?
d) How long did Paul's journey take altogether?
e) If Paul left home at 07:05, what time did Paul arrive at school?
f) What is the distance Paul has to travel to school?
Paul's classmate, Annabelle, takes a bus to school.
a) On the same axes above, plot a graph of Annabelle's journey
from home to school.
i) Annabelle walks 1 km in 15 minutes from her home to the bus
stop.
ii) Annabelle then waits 10 minutes for the bus.
iii) The bus travels 2 km in 15 minutes to get to Annabelle's school.
b) How far has Annabelle travelled in total?
c) How many minutes does Annabelle take to get from her home to
school?
2601 UNIT 11 Coordinate Geometry and Graphs
Check My
Understanding
O a) An express train travels at an average speed of 200 km/h.
i) Complete the table of values below.
Time taken (h)
0 1 2 3 4 5 6
Distance travelled (km) 0 200
11) Plot the points on the set of axes provided. Draw a line through
Express tram
the points. Label the line 'Express train'.
b) A goods train travels at an average speed of 100 km/h.
i) Complete the table of values below.
Time taken (h) 0 1 2 3 4 5 6
Distance travelled (km) 0 100
ii) Plot the points on the set of axes provided. Draw a line through
Goods train
the points. Label the line 'Goods train'.
A cross-country train travels at an average speed of 150 km/h.
i) Describe how the line for this train will look like on your graph
compared to the other lines.
ii) Complete the table of values below.
Time taken (h)
0 1 2 3 4 5 6
Distance travelled (km) 0 150 Cross-country train
Hi) Plot the points on the set of axes. Draw a line through the points.
Label the line 'Cross-country train'.
Distance travelled
600
500
400
E
O)
^ 300
(Q
200
100
1 > 1
Time (h)
d) Study the graph after you have drawn the straight-line graphs for the three trains
on the same grid.
i) Describe how the speed of a train affects its graph.
ii) Think about the graph of a train that is not moving. How will its distance-time
graph look like?
@ Joseph travelled from his home
Joseph's journey
to his friend's house 15 km away.
He stayed at his friend's house
15 / \
for some time and then returned > s
home. ■D / \
a) At what time did Joseph f 10 / \
/ \
/
leave his home? S.
s
b) How far from his home was
/ \
Joseph at 14:30? f \
/ S
c) How long did Joseph spend / s
at his friend's house? / \
14 00J,..l 4-: n 1^■46 5-nn 1 i-7 0 15 4r 1 in
d) How far did Joseph
Time (h)
travel altogether?
Beth goes to a shopping centre on a Saturday morning. She leaves her home at 10 o'clock.
Draw the axes shown on a piece of graph paper,
a) Beth starts cycling at a speed of 20 km/h.
i) How far does she cycle in
Beth's journey
30 min?
ii) Plot a point to show this
on your graph.
20
iii) Draw a line for this part of "D
0)
Beth's Journey on your graph.
b) Beth makes a stop at the library 10
while on the way to the shopping
centre. She spends 20 min at
the library.
Plot a point and draw a line
10 00 10 30 n 00 11 :3n
segment to represent this on
Time (h)
your graph.
c) Beth meets her friend in the library.
They walk together for 2 km for 20 min with Beth pushing her bike.
Plot a point to show this on your graph. Then draw a line segment to represent Beth's walk
on your graph.
d) Beth then cycles 8 km to the shopping centre at the same starting speed.
Plot a point to show this, and then draw a line segment to represent Beth's ride to the shopping
centre on your graph.
e) How far does Beth travel in total?
f) At what time does Beth arrive at the shopping centre?
Coordinate Geometry and Graphs
o Journal Writing
A driver travels from Town A to Town B.
Another driver travels from Town B to Town A.
Two journeys between Town A and Town B
—f~
40
ro\vn B /
\ >
\
/
\ /
E 30 \ /
/
/
/
20
/
/
/
/
/
10 -
/
/I
3 ■) 4 s 6 )
1 >
Time (min)
a) Write a scenario that fits the two graphs above.
b) Come up with questions that you can ask based on the scenarios
you have written that can be solved using the graphs.
d
Mathematics Connect
Coordinates control air flights
The Speed of cars on the road can be monitored using 35Q DID
traffic lights and cameras. Unlike cars, airplanes in flight are
difficult to track and monitor. There are no traffic lights in
air space. How do air traffic controllers manage the planes 200..^
from their control towers? How do they inform the pilots
when they are safe to take off or land? In real life, air traffic
is managed and regulated using coordinate geometry.
An air traffic controller tracks the location of every aircraft jP
at any particular moment in time using coordinates. Every
aircraft's movement updates the coordinates in the system.
The coordinate system is one of the most important tools in
airtransport.
nevision
O Use the set of axes to plot each
set of given points on a piece of graph
paper. Draw a straight line through each
set of points and name each geometrical
shape that you get.
a) (-5,-1), (-3, 3), (-1,-1), (-5,-1)
b) (-1,3), (7, 3), (7,1), (-1,1), (-1,3) ■5
c) (-1,-3), (3,0), (5,0), (5,-2), (-1,-3) ,?
d) (-5, 8), (4, 7), (5, 5), (-3, 4), (-5, 8)
0 Given the equation 3'= 2v +5, 5 - 4 - 3 - 2 1 0 1 ( !
-
-1
find the value of v when
?
a) A = 1,
b) A = 6,
c) A = -2.
0 Given the equationy 3' = -5jc-2, find the value of y when
X-
a) X = -3, b) X 0, c) x= 15.
0 Given the equation >' = 120 - 3x, find the value of
a) y when x = 20 and b) X when v = 30.
0 This conversion graph shows the
Conversion from Euros to Baht
approximate conversion from
Euros (€) to Thai Baht (B) 200 --
a) Convert €2 to Thai Baht. 1
! 3
b) Convert €4.50 to Thai Baht.
150 /
c) Convert 8100 to Euros. /
d) Convert B200 to Euros.
100
/
50 /
-
1 \
- -
-»
Euros(€)
UNIT 11 Coordinate Geometry and Graphs
0 Liana earns $5 per hour working as a waitress,
a) Complete the table of values below.
Number of hours of work 1 2 3 4 5
Wages ($) 5 10
b) Use the set of axes Liana's wages
to plot a graph to show
25
Liana's wages on a piece of
graph paper. 20
15
4^
o> 10
2 3
Time (h)
0 Use the set of axes to draw a graph showing Patel family's journey
the Patel family's journey on a piece of graph
paper. 200
The Patel family are travelling on the
highway In their car.
a) They travel at 100 km/h. How far do they
150
travel In 1 h?
Plot a point to show this on your graph.
b) They continue at the same speed for
100
another ^ h.
i) How far do they travel In j h?
II) How far have they travelled In total?
50
III) Plot a point to show this on the
graph.
Iv) Draw a line segment to represent this
part of their journey on your graph.
Time (h)
c) The Patels stop for coffee for 20 min.
Draw a line segment to represent this part
of their journey on your graph.
d) They get back Into their car and travel at a different speed for another hour.
They are now 200 km from home.
i) Draw a line segment to represent this part of their journey on your graph,
il) What speed are they travelling at after having coffee?
Help
Sheet
Coordinate grid and ordered pairs
The coordinates of a point in the
coordinate grid are (.v, y), where .v is the
.v-coordinate and _v is the v-coordinate. (n
■, 3)
The axes intersect at right angles at the ' \ 1 •i)
origin (0, 0). -)
The -v-coordinates are written first, 1
(-3, 0) (2 0)
followed by the v-coordinates.
-
6 5 - 4 - 3 2 1 0 1 > 1 ■
-
-
The-V-coordinate tells you how far right -1
(■!■) or left (-) a point is. _■?
(2, -2)
The y-coordinate tells you how far up 1-2 -3
(+) or down {-) a point is. -A
-S
-6
Lines parallel to the x-axis or y-axis
• Some line graphs can be vertical, like
the green graph shown here. i
It is parallel to the v-axis.
All the coordinates have the same value P irall ;l to the v-ax s
fory, so the line is named after its
y value.
1
• Some line graphs can be horizontal, like
the red graph shown here. 6 5 4 3 - 2 i 0 > ; (
-
-
-
-
It is parallel to the.v-axis. -1
pa alle to -?
All the coordinates have the same value thi ?y-a (is
for.v, so the line is named after its -3
-V value. -4
a- = 4
-6
UNIT 11 I Coordinate Geometry and Graphs
Linear Line Graphs
Given the equation of a linear graph, we work out the values for by substituting fixed values for x.
Example
Apples are sold at $3 per kilogram.
Cost of apples iy) = $3 x mass of apples in kilograms U).
We write the equation of the line as y = 3x
We can work out ordered pairs or coordinates on the graph using a table.
Total cost of apples
Mass (kg) 0 1 2 3 4
Total cost ($) 0 3 6 9 12 20
18 /
/
Plan the scales of the axes to fit all the coordinate 16 /
14
points on the graph paper. /
12
Plot the points on the graph. 10
8
Draw a straight line through the points.
6
4
/
2
0
—<>—
Total cost
Distance-time graphs
The vertical axis represents distance. The horizontal axis represents time.
The blue line shows constant speed. The object is moving away from the starting point at a constant speed.
The red line shows a journey with a stop from the 4*^ to 8'^ second. Then the object slows down.
Different types of movement
100
stf ad) /
90
fastspeed /
80
/
70
/
60
/
>tat oni ry
50 1
/ / \
40
/ / re urn to
30
/ ^St(ad) st< rtin 9
20 ml
10 / spjed C
/
\
0
1 i > i 0
Time (s)
•'—I
o r . '
m
I i Ol
UNIT 12 I Percentages
If! In this chapter
CHAPTER 12.1
■" -r=-*.r
II
,ij
.
Understanding Pupils should be able to:
• understand percentage
Percentages as the number of parts
in every 100
• use fractions and
percentages to
^ RECALL describe parts of
shapes, quantities and
measures
O What percentage of each figure is shaded?
a) b)
•v K"
■ fti
1
Mm m !f5
m
%
%
O Convert these fractions into percentages.
a) ^ = % b) g = % c) -^ = %
200
'
O Convert these fractions into percentages. Round your answers to
1 decimal place.
' ' 8
a) % b) 15 % c) 121 = %
180
Percentages as parts of a whole A percentage is
a fraction with a
The large square is divided into 100 equal parts. denominator of 100.
1 part is coloured.
gp of the large square is coloured. 'Cent' is a word that
is commonly used in
We say that 1 percent or 1 % of the large square systems of measure
is coloured. The word 'percent' comes from the where there are one
Latin phrase, per centum, which means 'per hundred'. hundred smaller
units to one larger
We use the symbol, %, for percent. unit. Examples are
We say the percentage of the large square that cents in a dollar and
is coloured is 1 % or one percent. centimetres in a
Percentages can be used to describe part of a whole metre.
in the same way as fractions.
Example 1
The large square is divided Into equal parts. What percentage of the square is
coloured? Write in numerals and in words.
a)
d)
inBOBD
m
iriBrjBn
B
Solution
There are 100 equal parts in each large square,
a) Number of coloured parts = 60
60% or sixty percent of the large square is coloured.
b) Number of coloured parts = 20
20% or twenty percent of the large square is coloured.
c) Number of coloured parts = 3
3% or three percent of the large square is coloured.
d) Number of coloured parts = 25
25% or twenty-five percent of the large square is coloured.
e) Number of coloured parts = 8
-S. _ 8x10 _ 80 _ Qrto/
10 10 X 10 " 100
80% or eighty percent of the large square is coloured.
Example 2
The label on the tin lists the amount of protein, fats and fibre in every 100 g of
beans.
100 g
contains:
What percentage of the beans is:
Protetn: 7g a) protein? b) fats?
Fats: 1g
Fibre: 6.2g c) fibre? d) not protein, fat or fibre?
Solution
a) 7 g out of 100 g are protein. So, = 70% of the beans is protein.
b) 1 g out of 100 g is fat. So, = 1 % of the beans is fat.
c) 6.2 g out of 100 g is fibre. So, = 6.2% of the beans is fibre.
i UNIT 12 Percentages
d) Mass of protein, fats and fibre in 100 g of beans = 7g + 1g + 6.2g = 14.2 g
14.2
= 14.2% of the beans is protein, fats or fibre.
100
Mass of beans that is not protein, fats or fibre = 100 g - 14.2 g = 85.8 g
So, = 85.8% of the beans is not protein, fats or fibre.
100
Check My
Understanding
O What percentage of each large square is shaded?
a) d)
■ ■ *1 J
■[ ■ ■ ■
•
If
i BX ! IJ
u r* n
■
r i M i i w 1
_ ■ ■ ■ Ji y i H m
O 87% of the seats at a football match are occupied. What percentage of the seats are not occupied?
@ Sally spent 25% of her weekly pocket money during the week. She spent 45% of her weekly pocket
money over the weekend and saved the rest. What percentage of her weekly pocket money does she
save?
O 75 of the 100 balls in a box are red and the rest are blue. What percentage of the balls is red?
12.1.2 Converting between
percentages and fractions
You have seen how a percentage can be easily expressed as a fraction with a
denominator of 100. Let us convert percentages to fractions in its simplest form.
Example 3
Express 40% as a fraction in its simplest form.
Solution
40% = ^
A percentage is a fraction
_ 40-r20
" 100T20 with a denominator of
- 2 100.
5
On a calculator, use the % key and the equal sign. Enter:
The display gives the answer [Ti [Ti [iii^ rn R
5'
To express a fraction as a percentage, multiply the fraction by 100%.
Example 4
Express ^ as a percentage.
100% = j^ =
1
Solution
So, multiplying by 100%
^ = ^ X 100%
is the same as multiplying
by 1. = 12%
On a calculator, enter the fraction, press Enter: (T) [T] fT] fT] fTI
the % key and then press the equal sign.
or (I]@(I][I]@II](=)
The display gives the answer 12 or 12%.
Check My
Understanding
Express each percentage as a fraction in its simplest form,
To express .v% as a
a) 29% b) 67% c) 80%
fraction, divide a- by
d) 65% e) 12% f) 4%
100.
To express a fraction as Express each fraction as a percentage.
a percentage, multiply 73 24
100 b) 100 50
by 100%.
18 16
d) e) I
25 f) 20
12.1.3 Converting between
percentages and decimals
To convert between percentages and decimals, write the percentage as a fraction
with a denominator of 100 and then divide.
Example 5
Express the percentages as decimals,
a) 64% b) 7%
Solution
a)
= Wo
= 0.64
On a calculator enter the percentage using the Enter:
% key, and then press the equal sign to the
= S«D
fraction to a decimal.
The display gives the answer 0.64.
b)
= 0.07
B UNIT 12 I Percentages
To express a decimal as a percentage, multiply the decimal by 100%.
Example 6
To express .v% as a
Express the decimals as percentages, decimal, divide .v by
100.
a) 0.85 b) 0.3 c) 0.02
To express a decimal as
Solution a percentage, multiply
a) 0.85 = 0.85 x 100% by 100%.
= 85%
On a calculator enter the decimal and multiply it by 100. Enter:
The display gives the answer 85. CD ® © ® QD ® ® ®
b) 0.3 = 0.3x100% c) 0.02 = 0.02 X 100%
= 30% = 2%
Check My
Understanding
Express these decimals as percentages,
a) 0.25 b) 0.45 c) 0.5 d) 0.7
e) 0.07 f) 0.09 g) 0.4 h) 0.375
Express these percentages as decimal.
a) 30% b) 42% c) 80% d) 3%
e) 8% f) 12% g) 67% h) 17.5%
Complete the table. The first one has been done for you.
Fraction Decimal Percentage
1
0.05 5%
20
1
10
20%
0.25
35%
2
5
0.5
3
4
12.1.4 Comparing two quantities in
percentages
Sometimes you have to compare different amounts which can be a percentage, a
fraction or a decimal.
Example 7
O Which is greater, ^ or 16%?
Solution
Express ^ as a percentage:
^ X 100% = 15%
16% > 15%, so 16% is greater than ^
.
0 Three friends are at a basketball game stall. They each have the same number
of basketballs to throw into the hoop. Friend A gets 72% of her basketballs
Into the hoop. Friend B gets ^ of his basketballs into the hoop and Friend C
gets 0.67 of her basketballs in. Who got the least number of basketballs into
the hoop?
Solution
Friend A got 72% of her shots in.
Express ^ as a percentage: ^ x 100% = 76%. So, Friend C got 76% of her
19
shots in.
Express 0.67 as a percentage: 0.67 x 100% = 67%. So, Friend C got 67% of
her shots in.
67% < 72% < 76%
So, Friend C got the least number of basketballs into the hoop.
O Journal Writing
I O Can 25% of something be greater than 50% of something else?
L Explain.
n © A big company says that its earnings for this year is 120% of last
I year's earnings.
I Explain how the percentage can be greater than 100%.
UNIT 12 Percentages
Check My
Understanding
O which number is greater?
a) 0.9 or 95% b) |^or37% c) 0.12 or 1.2%
0 Order these numbers from least to greatest.
a) 38%, ^,0.41 b) 84%, 0.91, c) 2.62,2|,2.26, 271%, 26.8%
@ Which is larger, 0.37 of a cup of flour or ^ of a cup of flour?
or
75
60
80%
of
of
m
O Which is larger, | m?
@ Josh is aiming to score 80% or better in a test. He answered 21 out of 25 questions correctly.
Did he achieve his goal? Explain.
O The table shows the portions of a day during which various animals sleep.
Animal Portion of day sleeping
Dolphin 0.433
Lion 56.3%
19
Rabbit
40
31
Squirrel 50
Tiger 65.8%
a) Order the animals by sleep time from least to greatest.
b) Estimate the portion of a day that you spend sleeping.
c) Where do you fit on the ordered list?
0* Challenge!
Will has a pack of 24 sweets.
He gives 25% of the sweets to his brother.
He eats ^ of the rest himself.
He then gives the remaining sweets to his sister.
What percentage of the pack of sweets does he give to his sister?
0* Challenge!
Sanjay sowed 240 flower seeds.
60% of the seeds germinated. 75% of those that germinated grew into plants.
^ of the plants grew yellow flowers and the rest grew blue flowers.
How many plants grew blue flowers?
©* Challenge!
In maths class, 52% of the students are girls.
What percentage of the students are boys? Are there more girls or boys?
CHAPTER 12.2
In this chapter
Percentages in
Pupils should be able to:
• calculate simple
percentages of
quantities (whole Quantities
number answers)
• express a smaller
quantity as a fraction or
Investigate!
percentage of a larger
one
• use percentages to Look at the notice that was on display at a local shop.
represent and compare
different quantities
^ RECALL
O Find the value of each
of the following.
a) 27% of 450
b) 81% of 96
c) 36% of
1250 m m
What is wrong with the notice?
d) 74% of
65^
12.2.1 Finding the percentage of
0 Mrs Singh had $2400.
She spent ^ of her 0 quantity
money on a printer
and 28% of her money To find a percentage of an amount, first write the percentage as a fraction.
on a wardrobe. How Then multiply the fraction by that amount.
much money does she
have left? Example 1
Find the following values,
a) 3% of 200 b) 24% of 400 c) 5% of 20 d) 70% of 285
Solution
Take the whole
Method 1
number as 100%.
a) 100% 200 b) 100%-^400
200 ^
1% Find 1% 1% ^ =4 Find 1%
^
100 "
3%' 2x3 = 6 Multiply by 3 24% -► 4 X 24 = 96 Multiply by 24
3% of 200 = 6 24% of 400 = 96
UNIT 12 I Percentages
100% 20 d) 100%-►285
20
1% Find 1% Find 1%
100
57 7
5% Multiply by 5 70% ^ Multiply by 70
^ 1
= 1 2
57 X 7
5% of 20 = 1 = 2
_ 399
2
= 199
70% of 285 = 199^
Method 2
3% of 200= ^ X 20& b) 24% of 400= ^ X4e0
1 1
= 3X2 = 24x4
= 6 = 96
7 57 The word 'of has
c) 5%of20=3|^x^ d) 70% of 285 = ^x 285 the same meaning
70
as multiply as seen
40^
in Method 2. The
- 5
~ 5 percentage is expressed
_ 7 X 57
= 1 2 as a fraction with a
_ 399 denominator of 100,
- 2 and multiplied by the
= 1991 whole number.
Example 2
We can also use a calculator to find the percentage of a number.
Find 35% of 1800.
Solution
On a calculator, enter the percentage Enter:
and multiply it by 1800. (T)|T|(s^|13@0]® BB(=]
The display gives the answer 630.
35% of 1800= :^x 1800
= 630
Example 3
Find the following values.
a) 6% of 25 m b) 15% of 280 kg c) 28% of 300 I d) 54% of 420 g
Solution
3 1 3 14
a) 6% of 25 m b) 15% of 280 kg
.Mn)
2 1
_ 3 = 3 X 14
2
= 42 kg
= 1r m
21
X
_ ^
c) 28% of 300/ = X 30e d) 54% of 420 9 = ^ 42e
iecr
= 28 X 3 _ 54 X 21
5
= 84/ _ 1134
5
= 226fg
Example 4
There are 64 seats in a restaurant. 75% of the seats are occupied. How many seats
are not occupied?
Solution
Method 1 Method 2
Find the number of seats that are occupied. Find the percentage of seats that are not occupied.
"I 100%-75% = 25%
3 16
75% of 64 =^XM' Find 25% of the seats in the restaurant.
^ 1 16
25% of 64= ^x^
1
= 3 X16
= 48
1
Find the number of seats that are not occupied. = 16
% 64-48= 16
So, 16 seats are not occupied.
So, 16 seats are not occupied.
UNIT 12 I Percentages
Example 5
There are 36 000 spectators at a football match. 85% of the spectators are adults.
How many children are there at the football match?
Solution
85% of the 36 000 spectators are adult.
360
1
= 85 X 360
= 30 600
There are 30 600 adults at the football match.
There are 36 000 - 30 600 = 5400 children at the football match.
Check My
Understanding
O Find the following values without using a calculator,
a) 50% of $30 b) 30% of $60 c) 40% of $50
d) 60% of $80 e) 25% of $72 f) 10% of $80
O Find the following values without using a calculator.
a) 35% of 400 cm b) 25% of 12 kg c) 15% of 240 g
d) 5% of 20 i e) 80% of 320 m f) 75% of 1 h
O 25% of the 40 trees in our school compound are non-flowering. How many non-flowering trees
are there?
O 40% of the competitors in a cycling race were women. If 15 000 people competed in the race,
how many women were there?
O Gravel loses 10% of its mass when it dries. A load of wet gravel has a mass of 450 kg.
What will the mass of the gravel be when it Is dry?
O 240 people are watching a netball match. 25% of the people watching are children.
a) How many children are watching the netball match?
b) How many people watching the match are not children?
O 90% of the machines in a factory are working properly. There are 800 machines in the factory.
How many machines are not working properly?
O Harry is going on holiday. They will cover a distance of 640 km. His mother says they have travelled
30% of the distance. Find the distance they still have to travel.
O A test has 24 multiple choice questions. How many questions do you need to answer correctly to
score at least 80%?
12.2.2 Express one quantity as a
percentage of another
To express Quantity A as percentage of Quantity B, write Quantity A as a fraction
of Quantity B. Then express the fraction as a percentage.
Example 6
28 of the 40 pupils in a class are boys. What percentage of the pupils are boys?
Solution
Number of boys in the class = 28.
Percentage of the pupils who are boys = N^^^befofpu^ ^ ""OOyo
7 10
1
= 7 X 10%
= 70%
70% of the pupils are boys.
Example 7
May has 54 stickers and ice has 90 stickers. Express the number of stickers May has
as a percentage of the number of stickers Ice has.
Solution
1^X100% = 60%
The number of stickers May has is 60% of the number of stickers Ice has.
Jim has a mass of 60 kg and Ong has a mass of 56 kg. Express Ong's mass as a
percentage of Jim's mass.
Solution
X 100% = 93^ %
To express a quantity as
a percentage of another
Ong's mass is 93i % of Jim's mass.
quantity, make sure that
they are expressed in the
same units.
Express 90 minutes as a percentage of 2 hours.
Solution
2 h = 120 min
90
X 100% = 75%
120
90 minutes is 75% of 2 hours.
Percentages
Express 180 cm as a percentage of 3 m.
Solution
3 m = 300 cm
X
^ 100% = 60%
180 cm is 60% of 3 m.
Example 11
Em scored 62 marks out of a maximum of 75 marks in a test. Express Em's score as
a percentage of the full marks. Round your answer to 1 decimal place.
Solution
X 100% = 82.66...%
= 82.7% (rounded to 1 decimal place)
Em's score was about 82.7% of the full marks (rounded to 1 decimal place).
Check My
Understanding
O Express the first quantity as a percentage of the second quantity,
a) 7 out of 10 b) 42 out of 50 c) 2 out of 3 d) 16 out of 3 000
O Express the first quantity as a percentage of the second quantity,
a) 40 min, 50 min b) 45 min, 1 h c) 27 s, 3.6 min
d) 3 mth, 1 yr e) $2.40, 72 c f) 1 kg, 800 g
Q Draw 8 cylinders.
a) Label 5 of them'C for cola.
b) Write down the percentage of cans that are cola.
c) Write down the percentage of cans that are not cola. Label them 'O' for orange.
d) Write down the fraction of cans that are orange.
O There are 90 teachers in a school. 40 of them are male. Find the percentage of
a) male teachers and
b) female teachers in the school.
@ List the numbers 1 to 10.
a) What percentage of these numbers are odd numbers?
b) What percentage are even numbers?
c) What percentage are numbers greater than 7?
d) What percentage are numbers less than 5?
e) What percentage are prime numbers?
Revision
For each of the following shapes, write down
a) the fraction that is shaded.
b) the percentage that is shaded.
i) ii) iv)
O Find the missing numbers. Show your working.
a) 60% = -D b) 75% =n 0 70% = □ d) 72% = n
© Express each percentage as a fraction in the simplest form.
a) 20% b) 11% c) 66%
d) 24% e) 80% f) 84%
g) 105% h) 125% i) 164%
j) 276% k) 265% 1) 342%
o Express each percentage as a decimal,
a) 30% b) 68% c) 209% d) 3012%
e Express each fraction as a percentage.
a) 9 b)g c) 2 d) 17
10 5 20
e) 3 f) ¥o 21 h) 48
4 9) 30 60
32
i) i) ^ k) 7
80 J' 90 8
4 m)2 I
1) n)
7
© Express each decimal as a percentage,
a) 0.05 b) 0.67 c) 1.46 d) 3.3
© 32 of the 80 apples in a basket are green and the rest are red. What percentage of the apples is green?
® A shop has 400 music albums — 120 are pop, 140 are rock, 80 are dance and 60 are classical.
What percentage of the albums are
a) rock music?
b) classical music?
c) pop and dance music?
d) not pop music?
Percentages
o Lucy got back her results for the following tests:
• 38 out of 50 marks in biology
• 77% in history
• 16 out of 20 marks in mathematics
a) Which subject did Lucy score the highest?
b) What was her percentage score in this subject?
<E) Work out the following values,
a) 50% of 20 m b) 20% of 15 kg c) 15%of120min d) 60% of 1200 kg
In a class of 30 pupils, 60% are girls.
a) What percentage of the pupils are boys?
b) How many of the pupils are boys?
Which is greater, ^ of 49 or 30% of 50?
0) At an international conference, 24 of the 200 women are from Thailand.
What percentage of the women are from Thailand?
Find the following values.
a) 3% of 80 km b) 17% of 200 g c) 65% of 580 /
d) 82% of 360 kg e) 15% of 90 f) 48% of 350
® There are 225 children at a concert. 56% of them are girls. How many boys are there?
© At a party, 260 guests were served either coffee or tea. 35% of them drank coffee.
How many guests drank tea?
The usual price of an oven is 8500 Baht. May bought it at a discount of 8%.
How much did she pay for the oven?
A pack contains 50 seeds and costs $50.
Erica sows all the seeds.
40% of the seeds grow into plants.
She sells the plants for $1 each.
How much profit does she make?
© A tree is 650 cm tall at the start of summer.
The tree grows during summer and its height increases by 8%.
Calculate the increase in the height of the tree.
Help
Sheet Percent and Percentage
'Percent' means 'per hundred'. We use the symbol, %, for percent.
We say 'the percentage of a quantity.
Converting percentages to fractions and decimals
Examples
175
a) 32% b) 175% =
100
32 _i 100 =
1
100
" - ' 100 100
= ^ Simplify = Simplify
1
0 44%=^ d) 155%= if
= 0.44 = 1.55
Converting fractions and decimals to percentages
Examples
a) 3 _ 3 X 20 b) 14 5
5 " 5 X 20
60=^ X :WtJ'% Multiply by 100%
60
•" 100
1
= 60%
= 14X5% Simplify
= 70%
Comparing fractions, decimals and percentages
We can convert all the terms to percentages.
Percentage of a quantity
Example:
Method 1 Method 2
100%
15% of 60= ^ x^
Find 1%
.sr
1
15% ^ x-fS- Multiply by 15
= 3X3
= 9
1
15% of 60 = 9
= 9
15% of 60 = 9
UNIT 12 Percentages
Finding a quantity given a percentage of the quantity
Example
15% of the fish in an aquarium are goldfish. There are 3 goldfish.
The total number of fish in the aquarium is 20.
Check: 15% of 20 = x 20 = 3
Finding the percentage of one quantity of another quantity
To express Quantity A as a percentage of Quantity B, write Quantity A as a fraction of Quantity B.
Then express the fraction as a percentage by multiplying by 100%.
Examples
a) There are 60 basketball players and 75 football players in Stage 7. Percentage of basketball players
expressed as a percentage of the number of football players in Stage 7 = ^ x 100%
= 80%
b) 13 g expressed as a percentage of 1 kg = x 100% = 1.3%
(The quantities have to be expressed in the same units.)
Mathematics Connect
.1 llitHlllU t t ii V
- :l L •! WL:! , '
.
1
Percentages work for you
Whilst doing research work, you may come across data in the form of tables, imsnuuiiHiB- a, i 1
graphs or reports. This can be difficult to read and compare if the data is recorded
in different formats.
t'' llMll i.::!!!!!
For example, a bookstore owner may be researching book sales figures. The
following table shows the sales of different genres of books of a bookstore in
2017 and 2018.
Year Horror Non-fiction Romance Crime
2017 2.4 46.2 159.7 78.3
2018 2.5 36.6 158.8 87.0
We can convert the number of books sold over the years to percentages to
make it easier to compare the sales across different companies or platforms. For
example, the table below shows that in 2017, 16.1% of the books sold were non-
fiction books, but in 2018, this quantity had fallen to 12.8%.
Year Horror Non-fiction Romance Crime
2017 0.8% 16.1% 55.7% 27.4%
2018 0.9% 12.8% 55.7% 30.6%
The bookstore owner may use this information to decide what books to purchase
or whether to mark up the price of future bestsellers by a percentage.
Percentages are useful for comparing information where the sample sizes or
totals are different. By converting different data to percentages, you can readily
compare them.
The aspect ratio of a television is the ratio of its width to its height.
The most common aspect ratio today is 16 : 9, which means that if
the width is divided into 16 equal parts, the height of the TV screen
or picture should be 9 parts.
Find the ratio of the width of a mobile phone to its height. Compare
the length of your tv screen to the length of a projector screen.
You will learn about:
Simplifying ratios
Dividing quantities using ratio
Proportion
i
U*
Ji
FiCG2a I -J
UNIT 13 I Ratio, Rate and Proportion
CHAPTER 13.1 In this chapter
Pupils should be able to:
• recognise the
Ratio relationship between
ratio and proportion
^
13.1.1 Writing ratio • use ratio notation
• simplify ratios
• divide a quantity into
A ratio compares quantities of the same kind or of the same unit. The quantities
two parts in a given ratio
may be numbers, measurements or sums of money. The ratio of one quantity to
another quantity tells us how much the first quantity is as compared to the second
quantity. Ratios can also be used to compare more than two quantities.
There are 2 oranges and 3 pears. We can express the ratio of the number of
oranges to the number of pears in two ways:
The colon (:) in a ratio
• Number of oranges: Number of pears = 2:3
means 'is to'.
• Number of oranges _ 2
Number of pears " Hi 4 : 3 is read as
3
"4 is to 3".
OOO
There are 5 nuts and 3 bolts. ' mrnt m I
OO
The ratio of the number of nuts to the number of bolts = 5:3
The order in which
_ 5
" the ratio is written is
3
The ratio of the number of bolts to the number of nuts = 3:5 important.
.
3
"
5
^ RECALL
O Ruby uses 3 parts flour and 2 parts egg to make pasta,
a) Complete the table.
Amount of
2 6
eggs used
Amount of
3 6 12
flour used
b) The ratio of the amount of flour used to the amount of egg used is
c) The amount of eggs used is j=j of the amount of flour used.
d) The amount of flour used is I times as much as the amount of eggs used.
a) The ratio of the number of ants to the number of butterflies is
b) The ratio of the number of butterflies to the number of ants is
Check My
Understanding
O There are 6 sparrows and 9 woodpeckers on a tree.
a) What is the ratio of the number of woodpeckers to the number of sparrows?
b) Express the number of sparrows as a fraction of the number of woodpeckers.
© The number of blue beads in a bag is | of beads in the bag.
total
the
number
of
a) What is the ratio of the number of blue beads to the number of beads that are not blue in the bag?
b) What is the ratio of the number of beads that are not blue to the total number of beads in the bag?
the
number
of
of
© The number of girls at a concert is | children at the concert.
a) What is the ratio of the number of girls to the number of boys at the concert?
b) What is the ratio of the number of boys to the total number of children at the concert?
O Yan has 9 hairbands, May has 6 hairbands and Fiona has 6 hairbands.
a) What is the ratio of the number of hairbands May has to the number of hairbands
Yan has to the number of hairbands Fiona has?
b) Express the number of hairbands Yan has as a fraction of the total number of hairbands that
the girls have.
© There are 40 girls, 32 boys and 16 adults at a party.
a) What is the ratio of the number of girls to the number of boys to the number of adults in the simplest form?
b) Express the number of adults as a fraction of the total number of people at the party.
o I of the children in a class are boys.
8
a) What fraction of the class are girls?
b) What is the ratio of the number of boys to girls?
o I of the hibiscus plants in a garden are red. The rest are yellow.
What is the ratio of the number of red hibiscus plants to the number of yellow hibiscus plants?
o I of the audience in a cinema are women.
What is the ratio of the number of men to the number of women in the cinema?
13.1.1 Equivalent ratios and simplifying
ratios
You learnt about equivalent fractions and expressing fractions in the simplest form.
Similarly, we can simplify ratios or find equivalent ratios of any given ratio.
Finding equivalent ratios
Example 1
Molly used the information in the table below to prepare some lemonade.
Number of glasses of blended lemon 1 2
Number of glasses of water
3 6
Blended lemon : Water (simplest form) 1 :3 1 :3
UNIT 13 I Ratio, Rate and Proportion
Molly used 4 glasses of blended lemon to prepare the lemonade. How many glasses of
water did she use?
Solution
To find the number of glasses of water, we use the ratio 1 : 3.
1 ;3 = 4:? 1 : 3 and 4 :12 are
equivalent ratios.
1 : 3 j
X 4 X 4
c;;j[
So, Molly used 12 glasses of water to prepare the lemonade.
Simplifying ratios
Example 2
long bought 4 bookshelves and 6 chairs for his new apartment.
a) What is the ratio of the number of chairs to the number of bookshelves?
b) Express the number of bookshelves as a fraction of the number of chairs.
Solution
bookshelves 1
\ k
To simplify a fraction or
chairs
1 1 1
ratio, find the highest
1 1 1
common factor (HCF) of
a) Number of chairs: Number of bookshelves
both numbers in the ratio.
= 6:4 Divide each number by their HCF, 2 Then divide each number
= 3:2 Simplify by the HCF.
The ratio of the number of chairs to the number of bookshelves is 3 : 2.
A ratio Is In its simplest
Number of bookshelves _ 4
b)
Number of chairs 6 form when 1 is the only
_2
whole number that
3
divides exactly into each
of
the
chairs.
of
number
The number of bookshelves is |
part.
Example 3
The ingredients used to make some shortbread biscuits are as follows:
• 360 g flour • 120 g sugar • 240 g butter
Work out the following, giving your answers in simplest form.
a) What is the ratio of the amount of sugar used to the amount of flour used?
b) What is the ratio of the amount of flour used to the amount of butter used?
Solution
a) The recipe needs 120 g sugar and 240 g of flour.
It is important that you
The ratio of the amount of sugar used to the amount of flour used is 120 : 240
write the numbers in the
The highest common factor of 120 and 240 is 120.
correct order.
120:240 The ratio of the amount
•r 1201 ^120 of sugar used to the
amount of flour used is
1 : 2, not 2 : 1.
So, the simplified ratio is 1 : 2.
b) The recipe needs 360 g of flour and 240 g of butter.
The ratio of the amount of flour used to the amount of butter used is 360 : 240.
The highest common factor of 240 and 360 is 120.
TheHCF of 120 and 240 360 : 240
is 120. t120 -r120
The HCF of 360 and 240 c...)
3:2
is 120.
So, the simplified ratio is 3 : 2.
Example 4
Simplify the ratio I • g-
Solution
Express the fractions Find the equivalent fractions for each fraction in the ratio such that they have the
as like fractions for same denominator.
comparison. Then we
The lowest common multiple of 4 and 8 is 8.
only need to look at the
3 _ 6
number of parts in their 4 8
numerators. 3 . 5.6 . 5
4 " 8 ~ 8 n 8
So, the ratio in its simplest form is 6 : 5.
Check My
Understanding
O Ong mixed some nuts using the ratio given in the following table.
Number of cups of peanuts
1 2
Number of cups of almonds
4 8
Peanuts : Almonds (simplest form) 1 :4 1 :4
Ong used 4 cups of peanuts. How many cups of almonds did he use?
o At a cafe, a cup of latte Is made by mixing 2 portions of coffee and 3 portions of milk. How many portions
of coffee and portions of milk are needed to make 5 cups of latte?
o May mixes 2 glasses of syrup and 5 glasses of fruit juice to make some fruit punch. Ying wants to make
5 times the amount of fruit punch that May makes. How many glasses of syrup and glasses of fruit juice
does she need?
o Freshly-squeezed lime juice is added to water to make a drink for a party. The ratio of the amount of
lime juice used to the amount of water used is 2 : 7. If 600 ml of lime juice is used, what is the volume
of water used?
The ratio of the number of chickens to the number of ducks in a farm is 5 : 3. The ratio of the number of
ducks to the number of geese is 6 : 1. Find the ratio of the number of chickens to the number of ducks to
the number of geese in the simplest form.
Simplify each of these ratios.
a) 6:4 b) 8:16 c) 2 : 32 d) 12:10
e) 15:20 f) 35:15 g) 16:16 h) 70 : 50
i) 24:12 j) 21 :12 k) 18:27 I) 15:33
m) 16:12 n) 8 : 5 o) 8 : 6 p) 5 : 8
Ratio, Rate and Proportion
O There are 21 girls and 15 boys in the Stage 7 class.
a) What is the ratio of the number of girls to the number of boys?
b) What is the ratio of the number of boys to the number of girls?
Give both answers in their simplest form.
©* Challenge! Simplify each of these ratios,
C) 2 . 7 _5_
X
a) b) i d) I
10 12
o Journal Writing <^^eese Scones
The ingredients needed to make cheese scones are listed.
^'^Ogpour
a) Write down the ratio of the amount of flour needed to the amount
of butter needed. Simplify your answer.
b) Write down the ratio of the amount of cheese needed to the amount
of butter needed. Simplify your answer.
c) Abdul says that the ratio of the amount of cheese needed to the
amount of milk needed is 1 : 5. Is he correct? Explain your answer.
d) As Liana has only 20 g of cheese, she decides to adjust the amount of
the ingredients used to make a smaller batch of scones.
i) How many grams of butter does she need?
ii) How many grams of flour does she need?
13.1.1 Sharing quantities in a given ratio
Work backwards.
Example 5
Check that 250: 100
Yi Ling and Grace share a sum of money in the ratio 5 : 2. simplifies to 5 : 2 as given
Yi Ling receives $250. in the problem.
a) How much does Grace receive? 250:100
c .)
b) Find the sum of money that the two girls are sharing.
Solution ^ 5:2 ^ /
$250
Yi Ling
Alternative solution
Grace using fractions
Yi Ling receives | of the
a) 5 parts—►$250 b) 7 parts—^7 X $50 = $350 sum of money.
1 part—►$250v5 = $50 or Grace receives y of the
money.
2 parts —► 2 x $50 = $100 Sum of money = $250 + $100 = $350
Amount of money Grace
Grace receives $100. The sum of money that the two girls receives
are sharing is $350. = $250 V f X f
= $250 X ^ X 2
= $250 X I
= $100
Check My
Understanding
o Ker Jin and Asyraff share 42 sweets in the ratio 3 :4.
Who gets more sweets, Ker Jin or Asyraff?
e Lily and Chloe share some money in the ratio 1 : 3.
What fraction of the money does each girl receive?
e Divide the following quantities in the given ratios,
a) $85 in the ratio 2:3 b) $50 in the ratio 4 :1 c) $210 in the ratio 5 : 2
d) 72 kg in the ratio 2:7 e) 42 sweets in the ratio 6 :1 f) 52 plants in the ratio 4: 9
g) $99 in the ratio 10:1 h) $400 in the ratio 3 : 7 i) $24 in the ratio 3 : 2
O Ang grows orchids as a hobby. The ratio of the number of pink orchids to the number of white orchids
that he has in his greenhouse is 4 : 1. How many pink orchids and white orchids doe he have if he has
225 orchids in total?
© In a packet of red and blue sweets, the ratio of the number of red sweets to the number of blue
sweets is 5 : 3. 75 of the sweets in the packet are red. How many blue sweets are there in the packet?
© Iris and Andrew want to share 2 ^ of milk in the ratio 7 : 3.
How many millilitres of milk does each of them receive?
O Ryan and Benjamin share a sum of money in the ratio 3 : 5. Benjamin receives $320 more than Ryan.
a) How much money does each boy receive?
b) How much money do they have altogether?
@* Challenge! The ratio of the number of cookies in Jar A to the number of cookies in Jar B is 3 : 8. Half
of the cookies in Jar A are transferred to Jar B. What is the new ratio of the number of cookies in Jar A
to the number of cookies in Jar B in the end?
©* Challenge! The ratio of the number of blue discs to the number of red discs in a box is 2 : 1. Ice puts
another 25 red discs into the box and the new ratio of the number of blue discs to the number of red
discs becomes 4 : 7. How many red discs are there in the box in the end?
UNIT 13 Ratio, Rate and Proportion
CHAPTER 13.2 In this chapter
Pupils should be able to:
Proportion • use direct proportion in
context
• solve simple problems
involving ratio and direct
proportion
^ RECALL
To make some bread dough, Mrs Tham uses 3 parts water and 5 parts flour,
a) Complete the table.
Think and Share
Amount of water used 3 9
Amount of flour used 5 10 20 To bake some cakes, a baker
used 40 g of strawberries for
b) The ratio of the amount of water used to the amount of flour used is every 220 g of flour.
c) The amount of flour used Is of the amount of water used.
d) How many mlllilitres of water is used in 3600 g of bread dough? ml
a) The proportion of
strawberries used
13.2.1 Proportion as port of a whole was of the total
amount of strawberries
and flour used.
Proportion is a comparison of the quantity of a part to the quantity of a whole.
b) If the total amount of
A bookstore sells pens in packs of four. There is 1 red pen and 3 blue pens in each strawberries and flour
pack.
that the baker used
was 1040 g, what was
the amount of flour the
baker used?
If the baker used 440 g
of flour to bake 1 cake,
how many such cakes
We can compare the number of red pens to the total number of pens in the
bookstore using ratio. can he bake with 1760 g
of flour?
red pens blue pens
The ratio of the number of red pens to the total number of pens is 1 : 4.
Since 1 in every 4 pens is a red pen, we say that the proportion of red pens is
i of the total number of pens.
We can say that 1 of
of
The ratio of the number of blue pens to the total number of pens is 3 : 4. the pens are red and |
the pens are blue.
of
Since 3 in every 4 pens are blue pens, the proportion of blue pens is |
total
the
number of pens.
Investigate!
Jaslyn measures the length of her foot and finds
that it is 21 cm long.
She then measures the length of her mother's foot
and finds that it is 24 cm long.
She decides to measure both their heights as well,
and records her measurements in a table like this:
Measurement Jaslyn jaslyn's Mother
Foot length (cm) 21 24
Height (cm) 133 152
Jaslyn says, "I must have big feet. My feet are nearly as long as my mother's!"
a) Calculate the ratio of the length of Jaslyn's foot to her height in simplest
form.
b) Calculate the ratio of the length of Jaslyn's mother's foot to her height in
simplest form.
c) What do you notice about the two ratios?
d) Does Jaslyn have bigger feet than normal for her height?
Xin mixes 3 ^ of red paint with 2 ^ of white paint.
Damien mixes 750 ml of red paint with 500 ml of white paint
Are the mixtures in proportion?
Solution
To compare the mixtures, we simplify the two ratios.
Xin's mixture is in the ratio of 3 : 2.
Damien's mixture Is in the ratio of 750 : 500 = (750 -r 250): (500 250)
o Journal Writing = 3:2.
They both mixed red and white paint in the ratio 3 : 2, so the mixtures are in
There were 15 girls and 10
proportion.
boys in Xin's maths class.
Xin said the ratio of the
Check My
number of girls to the
number of boys was 3 : 2. Understanding
Two new pupils, a girl
and a boy, joined the class
Sarah mixes 500 g flour with 200 g of sugar to make a cake.
halfway through the year.
Valerie mixes 360 g flour with 120 g of sugar to make another cake.
Xin says the ratio of the
a) For Sarah's cake mix, calculate the ratio of the amount of flour used
number of girls to the
to the amount of sugar used. Simplify your answer.
number of boys is still
b) For Valerie's cake mix, calculate the ratio of the amount of flour used
3:2.
to the amount of sugar used. Simplify your answer.
Is she correct? Why or why
c) Have the two girls mixed the flour and the sugar in the same
not?
proportion? Explain your answer.
UNIT 13 Ratio, Rate and Proportion
13.2.2 Direct proportion
When the ratio of 2 pairs of quantities are the same, we say the quantities are
proportional to each other.
To cook enough rice for 3 people, you need 2 cups of water and 1 cup of
i
dry rice. How many cups of water and cups of dry rice will you need to serve
rice to 12 people?
If two quantities are in the same ratio, they are said to be in direct proportion.
They increase or decrease at the same rate.
• If one quantity is doubled, then the other will double too.
• If one quantity is tripled, then the other will triple too.
• If one quantity is halved, then the other will be halved too, and so on.
Example 2
Emily is paid $5.30 per hour at work.
How much does she earn for working 4 hours?
Solution
1 h—►$5.30.
4h—►AX $5.30 = $21.20
Emily earns $21.20 for working 4 hours.
A food supplier for a restaraunt packs 300 pieces of tuna puffs in 12 cartons. The
restaurant owner orders 2100 pieces of tuna puffs to be delivered the next day.
How many cartons must the food supplier prepare?
Solution
12 cartons ► 300 puffs
X 7 X 7
? cartons ■>2100 puffs
2100 is 7 times of 300. So, we need 7 times the number of cartons.
12 cartons x 7 = 84 cartons
As the number of cartons
B ► 2100 increases, the number of
12 " ► 300 tuna puffs increases. This
CI]= 2100 X 12 is direct proportion.
300
= 7 X12
= 84
The supplier must prepare 84 cartons to pack 300 pieces of tuna puffs.
Check My
Understanding
O A machine packs 40 000 eggs in 5 h. How many eggs can it pack in 8 hours?
O Shane is paid $6.50 for an hour's work. How much is he paid if he works for
a) 4 hours? b) 11 hours?
O Bottles of juice cost 89 cents each. How much would
a) 5 bottles
b) 12 bottles
c) 25 bottles
of the juice cost?
O 12 exercise books have a total mass of 504 g. Work out the mass of
a) 6 exercise books, b) 3 exercise books and c) 24 exercise books.
O Felicia is paid $23.36 for 4 hours of work. How much is she paid for 12 hours' work?
O 5 identical cricket balls have a total mass of 700 g.
a) What is the mass of 1 cricket ball?
b) What is the mass of 8 cricket balls?
O The instruction on a bottle of fruit squash says, "Mix 1 part squash with 6 parts of water".
a) Write this as a ratio.
b) If you use 80 ml of squash, how many millilitres of water is needed?
© An American tourist changes some American dollars (US$) into Philippine peso (f).
She receives ^54.10 for each American dollar.
How many peso does she get for US$160?
O Three tourists from South Africa arrive in Australia for a holiday.
They exchange South African rand (R) into Australian dollars ($).
Xoli changes R500 and receives A$49.
a) Jannie changes R800. How many Australian dollars does he receive?
b) Charlene receives A$64.68 for the amount of rands she changes. Work out the amount of
rands she changes.
Try and Apply!
This is a recipe for making an apple pie. It makes
enough pie for 4 people. butter
a) How many apples are needed to make an apple
pie for 8 people?
b) What is the mass of butter needed to make an
self-raising pour
apple pie for 2 people?
c) Write down the ingredients needed for an
apple pie for 12 people.
UMIT 13 Ratio, Rate and Proportion
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