© T Madas
© T Madas
What do we mean when we say two quantities are
in proportion?
It means that if:
one of them doubles, the other one also doubles.
one of them trebles, the other one also trebles.
one of them x4, the other one also x4.
one of them halves, the other one also halves.
one of them ÷4, the other one also ÷4.
Can you give examples of directly proportional
quantities from every day life?
© T Madas
Directly proportional quantities:
They increase or decrease at the same rate
More formally:
Two variables are directly proportional
if the ratio between them remains constant.
© T Madas
© T Madas
Two variables v and t are directly proportional.
When t = 8, v =18.
Write a formula which links v and t, in the form v = …
v
t
Proportional
© T Madas
Two variables v and t are directly proportional.
When t = 8, v =18.
Write a formula which links v and t, in the form v = …
v
t
v = kt
This will be the formula when
we find the value of k
Proportionality
Constant
© T Madas
Two variables v and t are directly proportional.
When t = 8, v =18.
Write a formula which links v and t, in the form v = …
v
t
v = kt
9
So: v = 4 t
or: v
9t
= 4
v = kt
18 = k x 8
8k = 18
k =
18
9
= 2.25
=
8
4
or: v = 2.25t
© T Madas
© T Madas
In a chemistry experiment, the reaction time t is directly
proportional to the mass m of the compound present.
When the mass is 3 grams the reaction time is 0.2 seconds.
1.
Write a formula which links t and m, in the form t = …
2.
What is the reaction time when the mass is 8 grams?
t
m
Proportional
© T Madas
In a chemistry experiment, the reaction time t is directly
proportional to the mass m of the compound present.
When the mass is 3 grams the reaction time is 0.2 seconds.
1.
Write a formula which links t and m, in the form t = …
2.
What is the reaction time when the mass is 8 grams?
t
m
t = km
This will be the formula when
we find the value of k
Proportionality
Constant
© T Madas
In a chemistry experiment, the reaction time t is directly
proportional to the mass m of the compound present.
When the mass is 3 grams the reaction time is 0.2 seconds.
1.
Write a formula which links t and m, in the form t = …
2.
What is the reaction time when the mass is 8 grams?
t
m
t = km
1
So: t = 15 m
m
or: t = 15
or: t ≈ 0.067m
t = km
0.2 = k x 3
3k = 0.2
k =
2
1
0.2
=
=
30 15 ≈0.067
3
m
using: t = 15
t =
8
15 ≈ 0.53 s
© T Madas
© T Madas
What do we mean when we say two quantities are
inversely proportional ?
It means that if:
one of them doubles, the other one halves.
one of them x3, the other one ÷3.
one of them x4, the other one ÷4.
one of them ÷2, the other one x2.
one of them ÷10, the other one x10.
Can you give an example of inversely proportional
quantities from every day life?
© T Madas
The Civic Centre is to be painted, so they call a firm
of decorators.
If this firm provide:
1 decorator will take 60 days for the job
2 decorators will take 30 days for the job
3 decorators will take 20 days for the job
4 decorators will take 15 days for the job
5 decorators will take 12 days for the job
6 decorators will take 10 days for the job
10 decorators will take 6 days for the job
12 decorators will take 5 days for the job
15 decorators will take 4 days for the job
20 decorators will take 3 days for the job
30 decorators will take 2 days for the job
60 decorators will take 1 day for the job
120 decorators will take ½ day for the job
1 x 60
2 x 30
3 x 20
4 x 15
5 x 12
6 x 10
10 x 6
12 x 5
15 x 4
20 x 3
30 x 2
60 x 1
120 x ½
© T Madas
INVERSELY PROPORTIONAL QUANTITIES
One increases at the same rate as the other
one decreases.
More formally:
Two variables are inversely proportional
if their product remains constant.
© T Madas
© T Madas
A variable P is inversely proportional to a variable A.
When A = 2, P = 36.
1.
Write a formula which links P and A, in the form P = …
2.
Find the value of P when A is 2.5.
P
1
A
Inversely
Proportional
© T Madas
A variable P is inversely proportional to a variable A.
When A = 2, P = 36.
1.
Write a formula which links P and A, in the form P = …
2.
Find the value of P when A is 2.5.
P
1
A
1
P =kxA
Proportionality
Constant
© T Madas
A variable P is inversely proportional to a variable A.
When A = 2, P = 36.
1.
Write a formula which links P and A, in the form P = …
2.
Find the value of P when A is 2.5.
P
1
A
1
P =kxA
k
P =A
This will be the formula when
we find the value of k
© T Madas
A variable P is inversely proportional to a variable A.
When A = 2, P = 36.
1.
Write a formula which links P and A, in the form P = …
2.
Find the value of P when A is 2.5.
P
1
A
1
P =kxA
k
P =A
k
P =A
k
36 =
2
k = 72
72
So: P =
A
72
using P =
A
72
144 288
P = 2.5 = 5 = 10 = 28.8
© T Madas
© T Madas
A variable F is inversely proportional to a variable t.
When t = 3, F = 12.
Find the value of t when F is 48.
F
1
t
Inversely
Proportional
© T Madas
A variable F is inversely proportional to a variable t.
When t = 3, F = 12.
Find the value of t when F is 48.
1
F
t
F =k
x
1
t
Proportionality
Constant
© T Madas
A variable F is inversely proportional to a variable t.
When t = 3, F = 12.
Find the value of t when F is 48.
F
F =k
k
F = t
1
t
x
1
t
This will be the formula when
we find the value of k
© T Madas
A variable F is inversely proportional to a variable t.
When t = 3, F = 12.
Find the value of t when F is 48.
F
F =k
1
t
x
k
F = t
36
So: F =
t
36
1
t
using F =
t
36
48 =
t
48t = 36
k
F = t
k
12 =
3
k = 36
36
3
t = 48 = 4
© T Madas
A variable F is inversely proportional to a variable t.
When t = 3, F = 12.
Find the value of t when F is 48.
Since we do not require a formula in this example we could
also have worked as follows:
The product of inversely proportional
quantities remains constant
F x t = constant
12 x 3 = 36
48 x t = 36
36 ÷ 48 = 0.75
© T Madas
© T Madas
Sometimes we may be asked to set and
solve problems involving direct or inverse
proportion to the:
• square of a variable
• cube of a variable
• square root of a variable
or simply combine 3 variables with direct
and inverse proportion in the same problem.
© T Madas
© T Madas
A variable A is directly proportional to the square of another
variable r .
When r = 3, A = 36.
Find the value of A, when r = 2.5
A
r2
A = kr 2
So: A = 4r 2
using: A = 4r 2
A = kr 2
36 = k x 32
9k = 36
k =4
A = 4 x 2.52
5 2
A =4x 2
25
A =4x 4
A = 25
© T Madas
© T Madas
A variable y is directly proportional to the SQUARE ROOT of
another variable x .
When x = 25, y = 3.
Find the value of x, when y = 1.2
y
x
y = k x
3
So: y = 5 x
3
using: y = 5 x
3
1.2 = 5 x
6
3
5x
x
=
5
5
6= 3 x
x5
y = k x
3 = k x 25
5k = 3
3
k = 5 = 0.6
x =2
x =4
© T Madas
© T Madas
A variable W is directly proportional to a variable m and
inversely proportional to another variable t.
When m = 2 and t = 8, W = 15.
Find the value of W when m = 6 and t = 4.
W
m
W
m
t
x
1
t
W=kx m
t
W = km
t
60m
So: W =
t
using: W =
60m
t
W=
60 x 6
4
W=
360
4
W = 90
W = km
t
kx2
15 =
8
15 = 2k
8
2k = 120
k = 60
© T Madas
© T Madas
A variable F is directly proportional to a variable m and
inversely proportional to the square of another variable r.
When m = 10 and r = 2, F = 15.
Find the value of F when m = 24 and r = 3.
F
m
F
m
r2
x
1
r2
F =kx m
r2
F = km
r2
6m
So: F = 2
r
using: F =
6m
r2
F =
6 x 24
32
F =
144
9
F = 16
F = km
r2
k x 10
15 =
22
15 = 10k
4
10k = 60
k = 6
© T Madas
© T Madas
Cost of packets of pens
3 pens cost £2
What does the graph of two
directly proportional quantities
looks like?
© T Madas
3 pens cost £2
Cost of packets of pens
Number of pens
3
6
9
12
15
18
Cost (£)
2
4
6
8
10
12
Let us plot the information of this table in a graph
© T Madas
3 pens cost £2
Cost of packets of pens
Number of pens
3
6
9
12
15
18
Cost (£)
2
4
6
8
10
12
£
12
10
8
6
4
2
0
4
8
12
16
20
24
pens
© T Madas
when graphed the points of Directly Proportional Quantities:
1.
always form a straight line through the origin
2.
always form the corners of similar rectangles whose
opposite corner is at the origin.
3.
the line is a diagonal of every rectangle
£
12
10
8
6
4
2
0
4
8
12
16
20
24
pens
© T Madas
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
The data above has been obtained from a chemistry
experiment and concerns two quantities, u and v.
Are u and v directly proportional quantities?
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
v
20
15
10
5
0
the quantities u and v
are directly proportional
5
10
15
20
25
u
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
u
v
What is the gradient of the line?
gradient =
20
the ratio between directly
proportional quantities
remains constant.
15
Work the ratio v : u from the
table and compare it with the
gradient of this line.
25
10
What would have happened if
we plotted the data with the
axes the other way round?
5
0
25
diff in y
=
≈ 1.47
diff in x
17
5
17
10
15
20
25
u
v
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
u
What is the gradient of the line?
gradient =
20
the ratio between directly
proportional quantities
remains constant.
15
10
17
5
0
17
25
diff in y
= 1.47
0.68
=
≈
25
diff in x
17
5
10
25
15
20
25
Work the ratio u
v : vu from the
table and compare it with the
gradient of this line.
What would have happened if
we plotted the data with the
axes the other way round?
v
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
We could obtain a formula linking u and v
u
v
u = kv
So: u = 0.68v
u = kv
3.4 = k x 5
5k = 3.4
k =
3.4 6.8
=0.68
5 = 10
The proportionality constant is the
gradient of the line in the graph
1
u
0.68
v ≈ 1.47 u
or v =
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
v
v ≈ 1.47 u
20
15
10
5
0
5
10
15
20
25
u
© T Madas
u
3.4
5.44
6.8
8.16
10.88
12.92
14.28
17
v
5
8
10
12
16
19
21
25
u
20
u = 0.68 v
15
10
5
0
5
10
15
20
25
v
© T Madas
© T Madas
1 decorator takes 24 days to finish a job
What does the graph of two
inversely proportional quantities
looks like?
© T Madas
1 decorator takes 24 days to finish a job
No of decorators
Days
1
2
24 12
3
4
6
8
8
6
4
3
12 24
2
1
20
days
15
10
5
0
5
10
15
decorators
20
25
© T Madas
The graphed points of Inversely Proportional Quantities:
1.
always lie on a curve like the one shown below.
2.
always form the corners of rectangles of constant area
whose opposite corner is at the origin.
20
days
15
10
5
0
5
10
15
decorators
20
25
© T Madas
© T Madas
P
A
5
6
8
9
12
15
20
24
18
15
11.25
10
7.5
6
4.5
3.75
The data above has been obtained from the physics
department and concerns two quantities, P and A.
Are P and A inversely proportional quantities?
© T Madas
P
A
5
6
8
10
12
15
20
24
18
15
11.25
9
7.5
6
4.5
3.75
A
20
15
10
5
0
P
5
10
15
20
25
© T Madas
When plotted, Inversely Proportional quantities,
always show as Hyperbolas.
4.5
4
3.5
3
2.5
2
1.5
1
0.5
1
2
3
4
5
6
7
© T Madas
© T Madas
Suppose we have a formula which
contains 2 or more variables.
The data which produced this
formula is not available.
Is it possible to establish if variables are
directly proportional or inversely proportional?
This is how this is done.
© T Madas
The variable for which the formula is solved for is:
Directly proportional to Inversely proportional to
variables which appear in the variables which appear in the
numerator of the R.H.S
denominator of the R.H.S
v =s
t
c
s = vt
v
v
s
s
v and s are directly proportional
1
v and t are inversely proportional
t
s and t are directly proportional
t
© T Madas
The variable for which the formula is solved for is:
Directly proportional to Inversely proportional to
variables which appear in the variables which appear in the
numerator of the R.H.S
denominator of the R.H.S
V = mgh
c
V
m=
gh
c
h = V
mg
V
V
V
m V and m are directly proportional
g V and g are directly proportional
h V and h are directly proportional
m
1
m and g are inversely proportional
m
1
m and h are inversely proportional
h
1
h and g are inversely proportional
g
h
g
© T Madas
The variable for which the formula is solved for is:
Directly proportional to Inversely proportional to
variables which appear in the variables which appear in the
numerator of the R.H.S
denominator of the R.H.S
F = GMm
r2
F
F
F
G
M
m
F
1
r2
F and G are directly proportional
F and M are directly proportional
F and m are directly proportional
F is inversely proportional to the square of r
To get relationships between any other 2 variables
we appropriately rearrange the formula.
© T Madas
The variable for which the formula is solved for is:
Directly proportional to Inversely proportional to
variables which appear in the variables which appear in the
numerator of the R.H.S
denominator of the R.H.S
V=4
3
π
r3
Rearranging the formula for r gives: r = 3V
4π
3
V
r3
V is directly proportional to the cube of r
V
π
ariable
Because π is not a v_______;
onstant n______
umber
π is a c_______
CHALLENGE
V
oot of __
r is directly proportional to the c___
ube r___
© T Madas
The variable for which the formula is solved for is:
Directly proportional to Inversely proportional to
variables which appear in the variables which appear in the
numerator of the R.H.S
denominator of the R.H.S
S =u+ v
t
S
1
S
S
u
v
S
u + v S is directly proportional to
the sum of u and v
t
S and t are inversely proportional
Because u and v are
not in a product
© T Madas
Now a harder, worded
proportionality problem
© T Madas
60 workers, working a 9 hour day produce 720 toys a day.
1.
Find a formula which relates the number of workers w, the
number of hours they work h and the number of toys T they
produce.
2.
How many hours a day, do 90 workers need to work if they
are to produce 1020 toys?
The formula must contain the 3 variables w, h and T
Suppose that:
the workers work a constant number of hours per day
Then:
the toys produced will also double
If we double the workers, ___________________________
Toys and workers are directly proportional quantities
T
w
Suppose that:
we keep the number of workers constant
Then:
toys produced will also double
doubling the hours they work, the
___________________________
Toys and hours are directly proportional quantities
T
h
© T Madas
60 workers, working a 9 hour day produce 720 toys a day.
1.
Find a formula which relates the number of workers w, the
number of hours they work h and the number of toys T they
produce.
2.
How many hours a day, do 90 workers need to work if they
are to produce 1020 toys?
The formula must contain the 3 variables w, h and T
T
w
T
h
T
h
© T Madas
60 workers, working a 9 hour day produce 720 toys a day.
1.
Find a formula which relates the number of workers w, the
number of hours they work h and the number of toys T they
produce.
2.
How many hours a day, do 90 workers need to work if they
are to produce 1020 toys?
The formula must contain the 3 variables w, h and T
T
w
T
h
T
wh
T = kwh
T = 4 wh
3
Check that
it works
T = kwh
720 = k x 60
540k = 720
k = 720
540 =
x9
4
3
© T Madas
60 workers, working a 9 hour day produce 720 toys a day.
1.
Find a formula which relates the number of workers w, the
number of hours they work h and the number of toys T they
produce.
2.
How many hours a day, do 90 workers need to work if they
are to produce 1020 toys?
The formula must contain the 3 variables w, h and T
T
w
T
h
T
wh
T = kwh
T = 4 wh
3
T = 4 wh
3
1020 = 4 x 90 x h
3
1020 = 120h
1020
h = 120
h = 8.5 hours
© T Madas
Come down…
There is more…
© T Madas
Three variables u, v and w are related by a formula.
The following table gives some of the values that these three
variables can take:
u
v
w
4
8
12
16
120 120 120 120
2
4
6
8
30
20
15
12
1
1
1
1
2
3
4
5
Obtain the formula linking these variables, solved for u.
u
v
© T Madas
Three variables u, v and w are related by a formula.
The following table gives some of the values that these three
variables can take:
u
v
w
4
8
12
16
120 120 120 120
2
4
6
8
30
20
15
12
1
1
1
1
2
3
4
5
Obtain the formula linking these variables, solved for u.
u
v
v
1
w
v
u
v
u
w
x
1
w
v =kx u
w
v = ku
w
v = ku
w
kx4
2 =
1
4k = 2
So: v =
u
w
u = vw
u = 2v w
1
k =2
© T Madas
© T Madas