Proportional relationships
The idea
Two quantities are proportional when one is always the same multiple of the other: double one and the other doubles, triple one and the other triples. You already know how to multiply, divide, and scale fractions, so the new idea is just naming that constant multiplier — the constant of proportionality, usually written k. If pages printed and minutes are proportional with k = 15, then pages = 15 × minutes, every single time, with no exceptions and no leftover amount.
Think of a proportional relationship as a perfectly fair vending machine: zero in, zero out, and every extra unit in buys exactly the same amount out. On a graph that means a straight line through the origin. The common trap is calling every increasing relationship proportional. A gym that charges a $20 signup fee plus $5 per visit is linear but not proportional — at zero visits you still owe $20, so doubling visits does not double cost. Always check the zero point and the constant multiple.
Worked example
A school printer prints 45 pages in 3 minutes at a steady rate. Write an equation relating pages p to minutes t, and find how long it takes to print 120 pages.
- Find the constant of proportionality first: 45 pages ÷ 3 minutes = 15 pages per minute, so k = 15.
- Write the relationship as an equation: p = 15t, meaning every minute adds exactly 15 pages starting from zero.
- To find the time for 120 pages, solve 120 = 15t by dividing both sides by 15: t = 120 ÷ 15 = 8 minutes.
- Check that the answer fits the original rate: in 8 minutes at 15 pages per minute the printer produces 8 × 15 = 120 pages, exactly what was asked.
Answer. The equation is p = 15t, and printing 120 pages takes 8 minutes.
Check your understanding
- How can you tell from a table of values whether a relationship is proportional without graphing it?
- Why does a flat starting fee break proportionality even though the cost still rises at a steady rate?
- What does the constant of proportionality mean in everyday words for a situation you choose, like recipes or downloads?
- If a graph is a straight line that misses the origin, what kind of relationship is it, and what real situations behave that way?
Build the foundations first
Proportional relationships builds on these concepts. If any feel shaky, start there.