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Daily Math Minute

Ratios & Proportional Relationships

Identifying Proportional Relationships

Deciding whether two quantities are in a proportional relationship from a table or graph.

Foundational15 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Unit Rates with Fractions

Same Rate Every Time, or Not?

A rideshare service charges by the mile. Table A shows 2 miles for $6, 4 miles for $12, and 6 miles for $18. Table B shows 2 miles for $6, 4 miles for $11, and 6 miles for $15. Before reading on — what do you notice that's different between these two tables?

Definition — Proportional Relationship

A relationship between two quantities where their ratio stays exactly constant — every pair of corresponding values divides to the same number.

In Table A, every price divided by its miles gives the same result: 6÷2 = 3, 12÷4 = 3, 18÷6 = 3 — a genuinely constant rate of $3 per mile. In Table B, the ratios drift: 6÷2 = 3, but 11÷4 = 2.75, and 15÷6 = 2.5. Table A represents a proportional relationship; Table B doesn't, even though the two tables start out looking almost identical.

Worked Example — Testing a Table for Proportionality

A table shows: 3 hours worked for $45, 5 hours for $75, 8 hours for $120. Check every ratio: 45÷3 = 15, 75÷5 = 15, 120÷8 = 15. Every ratio matches, so this is a proportional relationship, with a constant rate of $15 per hour.

Worked Example — Testing a Graph for Proportionality

A graph of cost versus number of items passes through (0, 0) and forms a perfectly straight line through every plotted point. Does this represent a proportional relationship? Yes — a straight line through the origin is exactly the graphical signature of a proportional relationship, since it means the ratio y/x stays the same at every point.

Graph Visualizer

Domain & range
2
Evaluate a point
  • x^2 = 0

Tip

Checking just two pairs of values isn't enough to confirm a proportional relationship — Table B's first ratio also happened to equal 3. Always check at least three pairs before concluding a relationship is truly proportional.

Common Mistakes

  • Concluding a relationship is proportional after checking only one or two pairs of values, missing that later ratios don't match.

    Check every pair of values available, or at least three, since a coincidental match on the first pair doesn't guarantee the relationship holds throughout.

  • Mistaking any straight-line graph for a proportional relationship, without checking that it passes through the origin.

    A proportional relationship's graph must pass through (0, 0) — a straight line that starts anywhere else represents a relationship with a fixed starting amount, not a pure proportional one.

Key Takeaways

  • A proportional relationship has a constant ratio between every pair of corresponding values.
  • Checking several pairs of values, not just one or two, confirms whether a relationship is truly proportional.
  • A proportional relationship graphs as a straight line that passes through the origin.

Summary

Identifying a proportional relationship means checking that the ratio between two quantities never changes. The next lesson names that constant ratio directly and uses it to connect a proportional relationship's table, graph, and equation.

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