What is direct proportion?
Direct proportion describes a relationship between two variables where they always change together in the same direction, by the same factor. If one doubles, the other doubles. If one triples, the other triples.
The Core Idea
Imagine you're paid £5 per hour of work. If you work 1 hour, you earn £5. If you work 2 hours, you earn £10. If you work 10 hours, you earn £50. Your earnings are directly proportional to the hours you work — the ratio stays constant (5:1).
Key Facts About Direct Proportion
✓ The symbol ∝ means "is proportional to"
✓ If y is directly proportional to x, we write: y ∝ x
✓ This means there's always a constant multiplier k such that: y = kx
✓ The ratio y : x is always the same
✓ The graph is a straight line through the origin with gradient k
y = kx
What is k (the constant of proportionality)?
It's the fixed number that multiplies your variable. In the wage example, k = 5 (pounds per hour). Once you know k, you can find y for any value of x instantly.
Direct proportion with powers and roots
Direct proportion doesn't always mean y = kx. Sometimes y is proportional to a power or root of x, and the formula changes but the method stays the same.
The Four Types
Square: y ∝ x² means y = kx²
The graph curves upward like a parabola.
Square root: y ∝ √x means y = k√x
The graph curves more gently.
Cube: y ∝ x³ means y = kx³
The graph curves steeply.
Cube root: y ∝ ³√x means y = k ³√x
The graph is similar to square root but smoother.
How to find the equation between two directly proportional variables
Every direct proportion question follows the same 4-step process. Once you master this, you can solve any variation.
The Universal 4-Step Method
Step 1: Identify the two variables and decide if you're dealing with x, x², √x, etc.
Step 2: Write the formula involving k (y = kx, y = kx², etc.)
Step 3: Substitute the given values to find k, then solve
Step 4: Rewrite your formula with the k value, then use it to answer the question
Worked Example: y is directly proportional to x²
Given: When x = 3, y = 18. Find: y when x = 4.
Write the formula with k
y = kx²