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Modeling
Proportional Relationships

Learn to represent proportional relationships using tables, graphs, and equations, and translate fluently between all three forms.

Pre-Algebra

What Is a Proportional Relationship?

Two quantities are proportional when they increase or decrease at the same constant rate. If you double one quantity, the other doubles too. If you triple one, the other triples.

Example: If apples cost $2 each, then 3 apples cost $6, 5 apples cost $10, and the ratio \(\frac{\text{cost}}{\text{apples}}\) is always 2.

Three Ways to Model Proportions

Table
Organize \(x\) and \(y\) pairs. Check that \(\frac{y}{x} = k\) for every row.
Graph
Plot points on a coordinate plane. A proportional graph is a straight line through the origin.
Equation
Write \(y = kx\), where \(k\) is the constant of proportionality (unit rate).

Why Proportions Matter

Proportional relationships appear everywhere in real life: speed and distance, recipes and ingredients, currency exchange, map scales, and unit pricing. Recognizing them lets you predict, compare, and calculate efficiently.

Speed
If \(x\) is hours driven, distance \(y\) is \(y = 60x\).
Recipes
Double a recipe: double every ingredient
Currency
If \(x\) is dollars, euros \(y\) is \(y = 0.85x\).

The Core Rule: \(y = kx\)

x y 1 2 3 4 1 2 3 4 (0, 0) (1, 1.5) (2, 3) (3, 4.5)
\(y = 1.5x\)

Finding \(k\) from a Table

\(x\) \(y\) \(\frac{y}{x}\)
263
5153
8243
10303

Proportional vs Non-Proportional

Proportional
Equation: \(y = kx\)
Graph passes through (0, 0)
\(\frac{y}{x}\) is constant
No added constant
Non-Proportional
Equation: \(y = mx + b\), \(b \neq 0\)
Graph does not pass through (0, 0)
\(\frac{y}{x}\) changes row to row
Has a starting value \(b\)

Translating Between Representations

Table Graph Equation plot points read coords find slope make table
compute \(\frac{y}{x} = k\)

You can move in any direction between the three representations. The constant of proportionality \(k\) is the bridge: find it from one form, use it to build the others.

How to Apply It: Step by Step

1
Identify the two quantities that vary together (e.g., hours and miles).
2
Build a table of at least 3-4 \((x, y)\) pairs, including \((0, 0)\) if possible.
3
Compute \(\frac{y}{x}\) for each pair. If all ratios are equal, you have found \(k\).
4
Write the equation \(y = kx\) and graph the line through the origin with slope \(k\).
5
Use the model to predict: substitute any \(x\) to find \(y\), or any \(y\) to find \(x\).

Example 1: Table to Equation

A car uses gas at a steady rate. Use the table to find the equation.

Gallons (\(x\)) Miles (\(y\))
256
4112
7196

Example 2: Equation to Graph

A store sells notebooks for $4 each. The equation is \(y = 4x\). Graph it.

Notebooks Cost ($) 1 2 3 4 4 8 12 16 (0, 0) (1, 4) (2, 8) (3, 12) (4, 16)
\(y = 4x\)

Example 3: Graph to Equation

A line passes through the origin and the point (3, 9). Write the equation and build a table.

\(x\) \(y = 3x\)
00
13
39
515

Example 4: Real-World Translation

A faucet leaks at a constant rate. After 5 minutes, 2 cups of water have dripped. Model this relationship.

1
Identify quantities: \(x\) = minutes, \(y\) = cups.
2
Find \(k\): \(k = \frac{2}{5} = 0.4\) cups per minute.
3
Equation: \(y = 0.4x\).
4
Predict: After 30 min: \(y = 0.4 \times 30 = 12\) cups.

Spotting Non-Proportional Data

Hours (\(x\)) Earnings (\(y\)) \(\frac{y}{x}\)
11515
22512.5
33511.67
44511.25

Common Mistakes to Avoid

✕ Forgetting the Origin
A proportional graph must pass through \((0, 0)\). If you only check the slope but ignore the intercept, you may misclassify \(y = 2x + 3\) as proportional.
✕ Checking Only One Ratio
One matching \(\frac{y}{x}\) is not enough. Check all pairs. A single coincidence does not prove proportionality.
✕ Confusing \(k\) with \(x\) or \(y\)
\(k\) is the ratio \(\frac{y}{x}\), not either value alone. Writing \(k = y\) or \(k = x\) leads to wrong equations.

Summary: The Three Forms

Feature Table Graph Equation
What you see \((x, y)\) pairs Plotted points / line \(y = kx\)
How to find \(k\) Divide \(\frac{y}{x}\) Rise over run (slope) Coefficient of \(x\)
Proportional test All ratios equal Straight line through (0,0) No added constant
Best for Exact values Visual patterns Predictions / algebra

Question 1 of 5

Which equation represents a proportional relationship?

Question 2 of 5

A table shows: \((2, 10)\), \((4, 20)\), \((6, 30)\). What is the constant of proportionality \(k\)?

Question 3 of 5

A graph of a proportional relationship passes through \((4, 12)\). What is the equation?

Question 4 of 5

Which table does NOT show a proportional relationship?

Question 5 of 5

If \(y = 6x\), how many units of \(y\) correspond to \(x = 15\)?