Definition of Joint Variation
Joint variation is the same as direct variation with two or more quantities.
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
Increase in one quantity causes an increase in the other quantity
Decrease in one quantity causes a decrease in the other quantity
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 1. Area of a triangle varies jointly with base 'b' and height 'h'
Area of a rectangle = L x M represents joint variation. Here the constant is 1. Area of a rectangle varies jointly with length 'l' and width 'w'.
Video Examples: Joint Variation
Solved Example on Joint Variation
Ques: 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.
- 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 x 1 x 6
⇒ 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 x 2 x 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.