Proportional Relationships

Two quantities are in a proportional relationship when their ratio stays exactly the same no matter how large or small the amounts get. If 2 items cost $6, and 4 items cost $12, and 10 items cost $30, the cost-per-item ratio (y/x) is always 3 – that constant value is called the constant of proportionality, usually written k. This idea takes everything learned so far about ratios and formalises it into a single, powerful equation: y = kx.

Testing for Proportionality

To check if y is proportional to x, divide y by x for each pair of values. If the result (y/x) is the same every time, the relationship is proportional.

Is y proportional to x for (x=3, y=12) and (x=5, y=20)?

12 ÷ 3 = 4, and 20 ÷ 5 = 4. Both give the same value, so yes, y is proportional to x, with k = 4.

Finding the Constant of Proportionality

y is proportional to x. Given (x=6, y=42), find k.

k = y ÷ x = 42 ÷ 6 = 7.

Real-Life Application

Shopping: total cost is proportional to the number of items bought (at a fixed price).
Travel at constant speed: distance travelled is proportional to time.
Currency exchange: the converted amount is proportional to the original amount.

Key Takeaways

A proportional relationship has a constant ratio y/x = k for every pair of values.
The equation y = kx describes any proportional relationship.
Direct proportion, covered next, is exactly this relationship in its most common everyday form.

Practice: Proportional Relationships

Is It a Proportional Relationship?