1. Direct Variation

2. Consider the amount required to buy a certain number of cinema tickets.
No. of tickets bought (x) 1 2 3 4 5 …
Amount of money required ($y) 80
160
240
320
400
How does y change when x increases?

3. Consider the amount required to buy a certain number of cinema tickets.
No. of tickets bought (x) 1 2 3 4 5 …
Amount of money required ($y) 80
160
240
320
400
When x increases, y increases.

4. Consider the amount required to buy a certain number of cinema tickets.
No. of tickets bought (x) 1 2 3 4 5 …
Amount of money required ($y) 80
160
240
320
400
Do you have any further observations about the relation between x and y?

5. It is observed that the value of is a constant. x y
No. of tickets bought (x) 1 2 3 4 5 …
Amount of money required ($y) 80
160
240
320
400 x y 1 80
= 80 2
160
= 80 3
240
= 80 4
320
= 80 5
400
= 80 …
It is observed that the value of is a constant. x y
That is, (or y = 80x). 80 = x y
We say that the amount of money required ($y) varies directly as the number of tickets bought (x).
Such a relation is called a direct variation (or direct proportion).

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

7. Consider any two quantities x and y which are in direct variation.
i.e. y = kx, where k
The graph of y against x is a straight line
(i) passing through the origin
(ii) with its slope equal to the variation constant k. y x
y = kx
Slope = k O

8. Follow-up question If P µ w, and P = 12 when w = 2, find
an equation connecting P and w,
the value of w when P = 78.
(a)
∵ P µ w
k is the variation constant.
∴ P = kw , where k ¹ 0
By substituting P = 12 and w = 2 into the equation, we have
12 = k(2)
k = 6
∴ P = 6w

9. Follow-up question If P µ w, and P = 12 when w = 2, find
an equation connecting P and w,
the value of w when P = 78.
(b)
When P = 78,
78 = 6w
w = 13