1. Inverse Variation

2. Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people.
No. of people (x) 1 2 3 4 5 …
Amount (y mL)
1500
750
500
375
300
How does y change when x increases?

3. Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people.
No. of people (x) 1 2 3 4 5 …
Amount (y mL)
1500
750
500
375
300
When x increases,
y decreases.

4. Consider the amount of soft drink each person gets when 1500 mL of soft drink is shared equally among different number of people.
No. of people (x) 1 2 3 4 5 …
Amount (y mL)
1500
750
500
375
300
Do you have any further observations about the relation between x and y?

5. It is observed that the value of xy is a constant.
No. of people (x) 1 2 3 4 5 …
Amount (y mL)
1500
750
500
375
300 xy
1500
1500
1500
1500
1500 …
It is observed that the value of xy is a constant.
That is, xy = (or ). y = x
1500
We say that the amount of soft drink (y mL) obtained by each person varies inversely as the number of people (x).
Such a relation is called an inverse variation (or inverse proportion).

6. Inverse Variation x k y = then xy = k or ,
In general, if y varies inversely as x,
 ‘y varies inversely as x’ can also be written as ‘y is inversely proportional to x’. x k
y =
then xy = k or ,
where k is a non-zero constant.
k is called the variation constant.
In symbols, we write x y 1

7. Consider any two quantities x and y which are in inverse variation.
i.e , where k x y =
The graph of y against x is a curve like the one shown below.
As x tends to 0, the curve gets closer to the y-axis. y x O k =
The curve does not intersect the axes.
As x increases, the curve gets closer to the x-axis.

8. Follow-up question
If d varies inversely as n, and d = when n = 4, find
(a) the equation connecting d and n,
(b) the value of n when d = 8. 2 1
(a)
∵ d varies inversely as n.
∴ , where k ¹ 0 n k d =
k is the variation constant.
By substituting n = 4 and d = into the equation,
we have 2 1 4 2 1 = k 2 = k

9. Follow-up question
If d varies inversely as n, and d = when n = 4, find
(a) the equation connecting d and n,
(b) the value of n when d = 8. 2 1 ∴ 2 n d =
(b)
When d = 8, 2 8 = n 4 1 = n