Variation in Mathematics

Variation describes the relationship between two or more quantities. In the JAMB Mathematics syllabus, you are expected to master how variables change in relation to one another and how to find the 'constant of proportionality' (k).

1. Direct Variation

In direct variation, as one quantity increases, the other increases in a constant ratio. It is represented as:

• Mathematical form: y ∝ x
• Equation form: y = kx (where k is the constant)

Example: If y varies directly as x, and y = 10 when x = 2, find the constant k.
Solution: y = kx ⇒ 10 = k(2) ⇒ k = 5.

2. Inverse Variation

In inverse variation, as one quantity increases, the other decreases. It is represented as:

• Mathematical form: y ∝ 1/x
• Equation form: y = k/x or xy = k

Example: If y is inversely proportional to x, and y = 4 when x = 3, find k.
Solution: y = k/x ⇒ 4 = k/3 ⇒ k = 12.

3. Joint Variation

Joint variation involves more than two variables. One variable varies directly and/or inversely as the product of other variables.

• Example: z ∝ xy ⇒ z = kxy
• Example: z ∝ x/y ⇒ z = kx/y

4. Partial Variation

In partial variation, a variable is the sum of two or more parts. Usually, one part is a constant and the other varies with a variable, or both parts vary with different variables.

• General form: y = a + bx (where a and b are constants)
• General form: y = kx + cz

To solve partial variation problems, you typically use simultaneous equations to find the two constants.

5. Percentage Increase and Decrease

JAMB often asks how a certain percentage change in one variable affects another. For example, if y ∝ x² and x increases by 10%, y increases by 21%.

• Method: Set initial values to 100 or 1. Calculate the new value and find the difference.