5.3 Joint Variation

5.3 Joint Variation

5.3a Representing a Joint Variation using the symbol ‘α’.
1. If one quantity is proportional to two or more other quantities, this relationship is known as joint variation.
2.y varies directly as x and z’ is written as y α xz.
3. y varies directly as x and inversely z’ is written as y α  x z .
4. y varies inversely as x and z’ is written as y α  1 xz .

Example 1:
State the relationship of each of the following variations using the symbol 'α'.
(a) x varies jointly as y and z.
(b) x varies inversely as y and z .  
(c) x varies directly as r3 and inversely as y.

Solution:
(a) x α yz (b) x α  1 y z (c) x α  r 3 y


5.3b Solving Problems involving Joint Variation
1. If  y α  x n z n , then y=k x n z n , where k is a constant and n = 2, 3 and ½.

2. If y α  1 x n z n , then y= 1 k x n z n , where k is a constant and n = 2, 3 and ½.

3. If y α  x n z n , then y= k x n z n , where k is a constant and n = 2, 3 and ½.

Example 2:
Given that p α  1 q 2 r  when p = 4, q = 2 and r = 16, calculate the value of r when p = 9 and q = 4.

Solution:
Given that p α  1 q 2 r , p =  k q 2 r When p=4q=2 and r=16, 4 =  k 2 2 16 4= k 16 k=64 p =  64 q 2 r When p=9 and q=4, 9 =  64 4 2 r 9 =  4 r r = 4 9 r= ( 4 9 ) 2 = 16 81