Proportional Relationships

Grade 7 · Math · Study Guide

Explore how two quantities relate proportionally and learn to identify, write, and graph proportional relationships.

A proportional relationship is a relationship between two quantities where the ratio between them always stays the same. This constant ratio is called the unit rate or constant of proportionality, often written as k.

When you write a proportional relationship as an equation, it looks like y = kx, where k is the constant of proportionality. For example, if a car travels 60 miles every hour, the relationship between miles (y) and hours (x) is y = 60x.

You can identify a proportional relationship in a table by checking that the ratio y/x is the same for every row. On a graph, a proportional relationship always forms a straight line that passes through the origin (0, 0).

Proportional relationships appear everywhere in real life: recipes, map scales, unit pricing, and currency conversion all use proportional reasoning. Understanding them helps you solve problems like finding the cost of multiple items when you know the cost of one, or scaling a recipe up or down.

To find the constant of proportionality from a table, divide any y-value by its corresponding x-value. Always double-check with another pair to confirm the relationship is truly proportional.

Key terms

Proportional relationship
A relationship where two quantities always have the same ratio
Constant of proportionality (k)
The fixed ratio between two proportional quantities; the value k in y = kx
Unit rate
The rate for one unit of a given quantity, such as miles per one hour
Origin
The point (0, 0) on a coordinate plane where proportional relationship graphs always pass through
Ratio
A comparison of two quantities by division

Flashcards

y = kx
The equation form of a proportional relationship, where k is the constant of proportionality
How to find k from a table
Divide any y-value by its corresponding x-value: k = y ÷ x
Graph of a proportional relationship
A straight line that passes through the origin (0, 0)
Unit rate
The value of y when x = 1; equivalent to the constant of proportionality k
Non-proportional relationship
A relationship where the ratio y/x is NOT constant, or the graph does not pass through the origin
Cross multiplication
A method to solve proportions: if a/b = c/d, then ad = bc
Equivalent ratios
Two ratios that represent the same proportional relationship, e.g., 1:2 and 3:6
Scale factor
The constant of proportionality used when scaling measurements up or down

Questions

  1. A recipe calls for 2 cups of flour for every 3 cups of oats. Which equation represents this proportional relationship if y = cups of flour and x = cups of oats?

    1. y = 3x
    2. y = (2/3)x
    3. y = (3/2)x
    4. y = 2x
    Show answer

    Answer: y = (2/3)x

    There are 2 cups of flour for every 3 cups of oats, so flour = (2/3) × oats, meaning y = (2/3)x.

  2. A graph showing a proportional relationship must pass through the origin (0, 0).

    Show answer

    Answer: True

    Proportional relationships always pass through (0, 0) because when one quantity is 0, the other must also be 0.

  3. In the equation y = kx, the letter k represents the _______ of proportionality.

    Show answer

    Answer: constant

    k is the constant of proportionality — the fixed ratio between the two quantities.

  4. A store sells 5 apples for $2.00. What is the unit rate (cost per apple)?

    1. $0.50
    2. $0.40
    3. $0.25
    4. $0.30
    Show answer

    Answer: $0.40

    $2.00 ÷ 5 apples = $0.40 per apple.

  5. If a table shows x-values of 2, 4, 6 and y-values of 5, 10, 16, the relationship is proportional.

    Show answer

    Answer: False

    Check the ratios: 5/2 = 2.5, 10/4 = 2.5, but 16/6 ≈ 2.67. The ratios are not equal, so it is not proportional.

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