## Related Words

# 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 = 7x^{2}z^{3}, here y varies jointly as x^{2} and z^{3}

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.

#### 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 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.