joint variation


Definition of Joint Variation

Joint variation is the same as direct variation with two or more quantities.

That is:

Joint variation is a variation where a quantity varies directly as the product of two or more other quantities.

 
Let’s first understand direct variation.

Direct variation occurs when two quantities change in the same manner.

That is:

Increase in one quantity causes an increase in the other quantity.

Decrease in one quantity causes a decrease in the other quantity.

For example:

The cost of a pencil and the number of pencils you buy.

Buy more pay more….Buy less pay less.

 
Direct variation between variables x and y can be expressed as:

y = kx, where ‘k’ is the constant of variation and k ≠ 0.

 
y = kxz represents joint variation. Here, y varies jointly as x and z.


More Examples on Joint Variation

  • y = 7xz, here y varies jointly as x and z. 
  • y = 7x2z3, here y varies jointly as x2 and z3.
  • Area of a triangle = is an example of joint variation. Here the constant is ½. Area of a triangle varies jointly with base ‘b’ and height ‘h’. 
  • Area of a rectangle = l × w represents joint variation. Here the constant is 1. Area of a rectangle varies jointly with length ‘l’ and width ‘w’.


Solved Example on Joint Variation

 

Assume a varies jointly with b and c. If b = 2 and c = 3, find the value of a. Given that a = 12 when b = 1 and c = 6.

 

Solution:

 

Step 1: First set up the equation. a varies jointly with b and c

           

            a = kbc

                       

Step 2: Find the value of the constant, k.

           

Given that a = 12 when b = 1 and c = 6

 

            a = kbc

 

            12 = k × 1 × 6

 

Þ         k = 2   

 

Step 3: Rewrite the equation using the value of the constant ‘k’.

 

a = 2bc

 

Step 4: Using the new equation, find the missing value.

 

If b = 2 and c = 3, then a = 2 × 2 × 3 = 12.  

 
Step 5:
So, when a varies jointly with b and c and If b = 2 and c = 3, then the value of a is 12.

 
 
Real-world Connections for Joint Variation

Force = mass × acceleration. The force exerted on an object varies jointly as the mass of the object and the acceleration produced.

 
Related Terms for Joint Variation
 
  • Variable
  • Constant
  • Direct Variation