CONSTANT OF
PROPORTIONALITY!
WHAT IS IT?
• A constant ratio in any proportional
relationship
• Really just another name for unit rate!
Remember, to be constant
means it never changes!
WITH RATIO TABLES
Find the constant of proportionality between the gallons
and the miles.
Miles
50
100
150
200
𝟏𝟎𝟎 ÷ 𝟒 = 𝟐𝟓
𝟏𝟓𝟎 ÷ 𝟔 = 𝟐𝟓
Gallons 2
4
÷2
50
?
=
2
1
÷2
25 miles
per gallon
6
8
𝟐𝟎𝟎 ÷ 𝟖 = 𝟐𝟓
This means our constant
of proportionality is 25,
so if we divide the miles
by gallons we should
always get 25.
Let’s check!
TRY THIS ONE!
Find the constant of proportionality between the gallons
and the miles.
Number of
Apples
9
27
36
Cost
$3.00
$9.00
$12.00
÷3
9
?
=
3
1
÷3
Constant of
proportionality = 3
Let’s Check!
𝟐𝟕 ÷ 𝟗 = 𝟑
𝟑𝟔 ÷ 𝟏𝟐 = 𝟑
WITH GRAPHS
Miles
Find the constant of proportionality.
200
180
160
140
120
(2,90)
100
80
(1,45)
60
40
(0,0)
20
0
0
1
2
(4,180)
Y
45
90
135
180
X
1
2
3
4
(3,135)
To find our constant of
proportionality we have to
divide!
3
Hours
4
5
y
x
45
1
90 135 180
3
2
4
45 =
1
135 =
3
45
45
90 =
2
180 =
4
Y
45
90
135
180
X
1
2
3
4
45
45
So, our constant of proportionality is
45.
Do you see that our x value
multiplied by our constant of
proportionality is our y
value?
We could write this
like: y=45(x)
We will ALWAYS be able
to write our constant of
proportionality as an
equation that looks like
this: y=kx.
And k will always be
our constant of
proportionality!
Just like in our
last example!
y=45x.
WHAT CAN WE DO WITH IT?
• Population Density: Finding the constant of
proportionality by dividing the population by the
area.
• Finding how many per 1 square mile
POPULATION DENSITY
Wildlife conservationists are concerned that the deer population
might not be constant across the National Forest. The scientists found
that there were 144 deer in a 16 square mile area of the forest. In
another part of the forest, conservationists counted 117 in a 13 square
mile area. Then a third conservationist counted 216 deer in a 24
square mile area of the forest. Do the conservationists need to be
worried?
Deer
Square miles
Do we have a constant
of proportionality?
144
16
117
13
216
24
144
=9
16
117
=9
13
216
=9
24
POPULATION DENSITY
So do the conservationists need to be worried?
NOPE! The deer population is
constant in the National Forest.
Let’s write an equation!
If we multiply our square miles by
9 we will have the number of
deer. So our equation will look
like:
Deer = 9 x Miles OR
y=9x
Deer
Square miles
Constant of
proportionality
144
16
9
117
13
9
216
24
9