1. Partial Variation

2. Consider a trip to Maldives which lasts n days.
Air ticket fare
$5000
Expense for each day $2000
Total expenditure $E:
E = n

3. Does E varies directly as n?
Total expenditure $E:
E = n
When n changes,
the value of E changes.
Does E varies directly as n?
When n = 2,
E = 9000
When n = 3,
E =
When n = 4,
E =
is not a constant. n E
∴ E does not vary directly as n.

4. constant part variable part
E = n
Consider
constant part
variable part
E is the sum of two parts.
We say:
E is partly constant and partly varies directly as n.

5. Consider the situation of a basketball match.
A basketball player made x two-point and y three-point field goals in the match.
What is his total score?

6. x two-point field goals
y three-point field goals
Points scored 2x 3y
Total score S:
S = 2x + 3y
When x and/or y change,
the value of S changes.

7. This part varies directly as x. This part varies directly as y.
S = 2x + 3y
Consider
This part varies directly as x.
This part varies directly as y.
S is the sum of two parts.
We say:
S partly varies directly as x and partly varies directly as y.
In each of the above situations, the relation is called a partial variation.

8. Partial Variation
In partial variation, a quantity is the sum of two or more parts, where some of the parts may be constants, while other part(s) vary with other quantity / quantities.

9. Here are three examples:
(1) z is partly constant and partly varies inversely as x.
z = k , where k1, k2 ≠ 0 k2 x
k2 is called a variation constant.
Note that k1 is not necessarily equal to k2.

10.  Here are three examples: k2 is called a variation constant.
(1) z is partly constant and partly varies inversely as x.
z = k , where k1, k2 ≠ 0 k2 x
k2 is called a variation constant.
Therefore, it is wrong to put  k
z = k + . x

11.  k1 and k2 are called the variation constants.
(2) z partly varies directly as x and
partly varies directly as
z = k1x + k , where k1, k2 ≠ 0 x
k1 and k2 are called the variation constants.
Similarly, it is wrong to put 
z = kx + k x .

12.  k1 and k2 are called the variation constants.
(3) z partly varies directly as x and partly varies inversely as y.
z = k1x , where k1, k2 ≠ 0 k2 y
k1 and k2 are called the variation constants.
Similarly, it is wrong to put  k
z = kx + . y

13. Follow-up question
It is given that z is partly constant and partly varies inversely as y2. When y = 2, z = 4. When y = 1, z = 7.
Find an equation connecting y and z.
Find the value of z when y = –1.
(a)
∵ z is partly constant and partly varies inversely as y2.
∴ , where k1, k2 ¹ 0 2 1 y k z + =
By substituting y = 2 and z = 4 into the equation, we have ) 2 ( 4 1 + = k 16 4 2 1 = + k )
......(

14. Follow-up question
It is given that z is partly constant and partly varies inversely as y2. When y = 2, z = 4. When y = 1, z = 7.
Find an equation connecting y and z.
Find the value of z when y = –1.
By substituting y = 1 and z = 7 into the equation, we have 7 2 1 + = k 7 2 1 = + k )
......( 9 3 : ) 2 ( 1 = - k 3 1 = k

15. Follow-up question
It is given that z is partly constant and partly varies inversely as y2. When y = 2, z = 4. When y = 1, z = 7.
Find an equation connecting y and z.
Find the value of z when y = –1.
By substituting k1 = 3 into (2), we have
3 + k2 = 7
k2 = 4 ∴ 2 4 y 3 z + =

16. Follow-up question
It is given that z is partly constant and partly varies inversely as y2. When y = 2, z = 4. When y = 1, z = 7.
Find an equation connecting y and z.
Find the value of z when y = –1.
(b)
When y = –1,
(–1) 2 4 3 z + =
= 7