1. Slideshow 29, Mathematics Mr Richard Sasaki
Inverse Proportion
Slideshow 29, Mathematics
Mr Richard Sasaki

2. Objectives
Understand how to calculate 𝑘, the constant of proportionality
Building inversely proportional equations and using them
Note: Make sure you understand the meaning of “where 𝑦 is the subject” and “in terms of 𝑥”.
𝑦=7𝑥+5
𝑦 is the subject. It is on the left and on its own.
𝑥 is the only unknown on the right.

3. Rate of Change Consider a table for the following example. Example
A triangle has an area of 24 𝑐 𝑚 2 . Build a table where 𝑥 refers to the size of its base and 𝑦 refers to its height for integer values of 𝑥 where 1≤𝑥≤6.
𝒙− Base (𝑐𝑚) 1 2 3 4 5 6
𝒚− Height (𝑐𝑚) 48 24 16 12
9.6 8
We can see that the connection between 𝑥 and 𝑦 is not The rate of change
linear
changes

4. Relationship of 𝑥 and 𝑦 𝒙− Base (𝑐𝑚) 1 2 3 4 5 6 𝒚− Height (𝑐𝑚) 48 2416 12
9.6 8
The relationship between 𝑥 and 𝑦 is not linear so what is it?
If it were possible for the base (𝑥) to be 0 𝑐𝑚, what would the height (𝑦) be?
∞ 𝑐𝑚
If it were possible for the height (𝑦) to be 0 𝑐𝑚, what would the base (𝑥) be?
∞ 𝑐𝑚
For some relationship where when 𝑥=0, 𝑦 tends to ∞ and when 𝑥 tends to ∞, 𝑦=0, they are
inversely proportional

5. Inverse Proportion 𝒙− Base (𝑐𝑚) 1 2 3 4 5 6 𝒚− Height (𝑐𝑚) 48 24 16 12
9.6 8
With this example 𝑥∙𝑦= for all pairs. 48
In fact, when two variables, 𝑥 and 𝑦 are inversely proportional, they always have some product . 𝑘
So as 𝑥∙𝑦=𝑘, 𝑦= when 𝑥 and 𝑦 are inversely proportional.
𝑘 𝑥
𝑘 is known as the
constant of proportionality
The notation of 𝑦 is inversely proportional to 𝑥 is commonly shown as or
𝑦∝ 1 𝑥
𝑦∝ 𝑥 −1

6. Inverse Proportion Example
A car travels 200 𝑘𝑚 in 𝑥 hours at a constant speed of 𝑦 𝑘𝑚 ℎ −1 . Write an equation for 𝑦 in terms of 𝑥 regarding their relationship.
𝑦= 200 𝑥
As 𝑦∝ 1 𝑥 , 𝑦= 𝑘 𝑥 ⇒
(𝑘=200)
Why does 𝑘=200?
𝑘 is the constant value, the car always travels 200 𝑘𝑚 irrelevant of 𝑥 and 𝑦.
If the car travelled for 4 hours, state its average speed.
𝑦= =
50 𝑘𝑚 ℎ −1

7. 𝑦= 35 𝑥
𝑦= 10 𝑥
𝑦= −2 𝑥 7 2
−0.4
𝑦= 24 𝑥
𝑦= 24 6 =4 𝑐𝑚
Half a person makes no sense.
Natural numbers
The length of a piece.
𝑦= 200 𝑥
𝑦= =12.5 𝑐𝑚
25= 200 𝑥 ⇒8 pieces

8. 𝑦=−2.5
𝑥=−2.5
𝑦= 8 𝑥
𝑦 refers to the rate of the fuel being used per hour
𝑦= 8 𝑥
𝒙− Width (𝑐𝑚) 1 2 3 4 5 6
𝒚− Length (𝑐𝑚) 96 48 32 24
19.2 16
𝑦= 96 𝑥
The length and width (in either order).
They have many factors (easy to divide).

9. Finding 𝑘
As with direct proportion, we may need to calculate 𝑘 if we are given 𝑥 and 𝑦 at a point.
As 𝑦= 𝑘 𝑥 ,
𝑘=𝑥∙𝑦
We just multiply them together!
Example
If 𝑦∝ 1 𝑥 and at one point, 𝑥=5 and 𝑦=8, write down a formula for both 𝑦 and 𝑥 where 𝑦 is the subject.
𝑘=𝑥∙𝑦⇒
𝑦= 40 𝑥
𝑘=8∙5⇒

10. 𝑦= 24 𝑥
𝑘=24
𝑦= 45 𝑥
𝑘=45
𝑦= 66 𝑥
𝑘=66
𝑦= 84 𝑥
𝑘=84
𝑦= 12 𝑥
3𝑚 𝑠 −1
𝑦= 32 𝑥
None
40 days

11. 𝑘=2
𝑦= 2 𝑥
𝑦= 6 𝑥
𝑘=6
𝑘=−33
𝑦=− 33 𝑥
𝑦=− 4 𝑥
𝑘=−4
20∙60=1200𝑚𝑙, 𝑘
𝑦= 1200 𝑥
100 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
𝑦=−1 or anything negative
The number of minutes