Proportionality Formula

The Formula

y = kx where k = \frac{y}{x} is the constant of proportionality

When to use: If you double one, you double the other. Triple one, triple the other.

Quick Example

Cost is proportional to quantity: 3 apples cost 6, so 6 apples cost 12.

Notation

y \propto x means 'y is proportional to x'

What This Formula Means

A relationship where two quantities maintain a constant ratio: doubling one always doubles the other, giving y = kx.

If you double one, you double the other. Triple one, triple the other.

Formal View

y \propto x \iff \exists\, k \in \mathbb{R},\; k \neq 0,\; \text{such that } y = kx. Equivalently, \frac{y}{x} = k is constant for all (x, y) with x \neq 0. The graph passes through the origin.

Worked Examples

Example 1

easy
A car travels 150 miles in 3 hours at constant speed. How far will it travel in 5 hours?

Solution

  1. 1
    Find the unit rate (speed): \dfrac{150 \text{ miles}}{3 \text{ hours}} = 50 mph.
  2. 2
    Distance in 5 hours: 50 \times 5 = 250 miles.
  3. 3
    Alternatively, set up a proportion: \dfrac{150}{3} = \dfrac{d}{5}, so d = \dfrac{150 \times 5}{3} = 250 miles.

Answer

The car travels 250 miles in 5 hours.
Two quantities are proportional when their ratio is constant. Here, distance and time have a constant ratio (speed). Setting up a proportion or multiplying by the unit rate both give the same result.

Example 2

medium
The table shows: x = 2, y = 8; x = 5, y = 20; x = 9, y = 36. Determine whether y is proportional to x, and if so write the proportionality equation.

Common Mistakes

  • Assuming any linear equation is proportional — y = 3x + 5 is linear but not proportional because it does not pass through the origin
  • Setting up the ratio upside down — if 3 apples cost 6, the unit rate is \frac{6}{3} = \2 per apple, not \frac{3}{6}
  • Cross-multiplying incorrectly — in \frac{a}{b} = \frac{c}{d}, students write ab = cd instead of ad = bc

Why This Formula Matters

Foundation for linear relationships, similar figures, and rate problems.

Frequently Asked Questions

What is the Proportionality formula?

A relationship where two quantities maintain a constant ratio: doubling one always doubles the other, giving y = kx.

How do you use the Proportionality formula?

If you double one, you double the other. Triple one, triple the other.

What do the symbols mean in the Proportionality formula?

y \propto x means 'y is proportional to x'

Why is the Proportionality formula important in Math?

Foundation for linear relationships, similar figures, and rate problems.

What do students get wrong about Proportionality?

Not all linear relationships are proportional (y = 2x + 3 is not).

What should I learn before the Proportionality formula?

Before studying the Proportionality formula, you should understand: ratios, multiplication.

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