Key takeaway: In a proportional relationship, two quantities keep one unchanging ratio, so scaling one by any factor scales the other by the same factor and the graph runs from the origin.
Grade 77.RP.A.2aRatios, Rates and Percents
Definition of Proportional Relationship
A proportional relationship ties two quantities together so that one is always the same multiple of the other. Triple the first and the second triples; cut the first to a fifth and the second shrinks to a fifth.
Three checks reveal it. A table must give the same quotient \(y : x\) in every row. A graph must be a line starting at (0, 0). An equation must look like \(y = kx\), with nothing added. A fee charged up front, such as a delivery charge, spoils the pattern: the relationship can still be linear without being proportional.
Formula and notation
Every proportional relationship can be written as
\[y = kx\]
where \(x\) and \(y\) are the two quantities and \(k\) is a fixed nonzero number, the constant of proportionality. Equivalently, \(\frac{y}{x} = k\) for every pair.
Key facts
- Zero of one quantity pairs with zero of the other, so the origin lies on the line.
- Equal steps in x produce equal steps in y.
- The quotient y divided by x is identical in every row of the table.
- Rules like \(y = 2x + 3\) are linear but not proportional.
Proportional Relationship example
A sub shop’s catering order costs \$22.50 for 3 sandwiches, \$37.50 for 5 and \$60 for 8. Is the cost proportional to the number of sandwiches? Write the rule.
-
Compute the quotient cost \(: \) sandwiches for each pair: \(22.5 : 3 = 7.5\), \(37.5 : 5 = 7.5\), \(60 : 8 = 7.5\).
-
The quotient is the same every time, and the graph passes through (0, 0).
-
The rule is \(y = 7.5x\), with \(y\) the cost in dollars and \(x\) the number of sandwiches.
Yes, the cost is proportional to the number of sandwiches: \(y = 7.5x\).
Common mistake
Watch out
Gia says \(y = 2x + 3\) shows a proportional relationship “because its graph is a straight line.”
Why it is wrong
Being straight is not enough; the line must also start at the origin. At \(x = 0\) this rule gives \(y = 3\), not 0, and the quotients \(\frac{5}{1}\), \(\frac{7}{2}\), \(\frac{9}{3}\) are all different. The added 3 breaks the proportion.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Ratios, Rates and Percents