Partial Variation

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

Does E varies directly as n? Total expenditure $E: E = 5000 + 2000n When n changes, the value of E changes. Does E varies directly as n? When n = 2, E = 9000 When n = 3, E = 11 000 When n = 4, E = 13 000 is not a constant. n E ∴ E does not vary directly as n.

constant part variable part E = 5000 + 2000n Consider constant part variable part E is the sum of two parts. We say: E is partly constant and partly varies directly as n.

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?

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.

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.

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.

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

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

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

 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

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 ) ......(

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

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 + =

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