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Proportional vs.
Non-Proportional

Learn to identify, compare, and justify whether a relationship is proportional or non-proportional using tables, graphs, and equations.

Pre-Algebra

What Is a Proportional Relationship?

A proportional relationship exists between two quantities when they always have the same ratio. As one quantity changes, the other changes by a constant factor.

★ Constant Ratio
Every \( \frac{y}{x} \) gives the same value \( k \).
★ Through Origin
The graph is a straight line through \( (0, 0) \).

What Is a Non-Proportional Relationship?

A non-proportional relationship exists when the ratio \( \frac{y}{x} \) is not constant. The equation has an added or subtracted value:

★ Changing Ratio
The value of \( \frac{y}{x} \) differs for each data point.
★ \( y \)-intercept \( \neq 0 \)
The line crosses the \( y \)-axis at \( b \), not the origin.

Seeing the Difference on a Graph

0 2 4 6 8 1 2 3 4
\( y = 2x \)
\( y = 2x + 1 \)
\( x \)
\( y \)

The Core Rule

Proportional
Equation: \( y = kx \)
Ratio \( \frac{y}{x} = k \) is constant
Graph passes through \( (0, 0) \)
Non-Proportional
Equation: \( y = mx + b,\; b \neq 0 \)
Ratio \( \frac{y}{x} \) changes
Graph does not pass through \( (0, 0) \)

Testing with a Table of Values

Compute \( \frac{y}{x} \) for every row. If every ratio equals the same constant, it is proportional.

Proportional
\( x \)
\( y \)
\( \frac{y}{x} \)
 1 3 3 2 6 3 4 12 3 5 15 3 ✓ Constant! Non-Proportional
\( x \)
\( y \)
\( \frac{y}{x} \)
 1 5 5 2 7 3.5 4 11 2.75 5 13 2.6 ✕ Not constant!

Constant Rate vs. Constant Ratio

Both proportional and non-proportional linear relationships have a constant rate of change (slope). The key distinction is the ratio test.

Constant Rate of Change
Both \( y = 3x \) and \( y = 3x + 2 \) increase by 3 for every +1 in \( x \). This makes them both linear.
Constant Ratio
Only \( y = 3x \) has \( \frac{y}{x} = 3 \) for every point. The "+2" in \( y = 3x + 2 \) breaks the constant ratio.

How to Decide and Justify

1
Check the origin. Does \( (0, 0) \) belong to the data? If not, it is non-proportional.
2
Compute all ratios. Calculate \( \frac{y}{x} \) for every data point where \( x \neq 0 \).
3
Compare. If every ratio equals the same number \( k \), the model is proportional: \( y = kx \).
4
Justify. State the constant \( k \) or explain why the ratios differ. Cite the origin test or the ratio test.

Example 1: Buying Apples

Apples cost $2 each. The table shows total cost \( y \) for \( x \) apples.

Apples \( x \) 1 2 3 5
Cost \( y \) $2 $4 $6 $10
Ratio \( \frac{y}{x} \) 2 2 2 2

Example 2: Taxi Fare

A taxi charges a $3 flat fee plus $2 per mile. The table shows total fare \( y \) for \( x \) miles.

Miles \( x \) 0 1 2 4
Fare \( y \) $3 $5 $7 $11
Ratio \( \frac{y}{x} \) -- 5 3.5 2.75

Example 3: Reading the Graph

Two students earn money. Decide which relationship is proportional.

0 10 20 30 40 1 2 3 4 Hours Worked Dollars ($) Student A Student B

Example 4: Classify from an Equation

Determine if each equation represents a proportional or non-proportional relationship:

\( y = 7x \)
Proportional: the form is \( y = kx \) with \( k = 7 \) and no added constant.
\( y = 4x - 1 \)
Non-proportional: the \( -1 \) means \( b \neq 0 \). When \( x = 0 \), \( y = -1 \).
\( y = \frac{1}{2}x \)
Proportional: \( k = \frac{1}{2} \), no additive constant, passes through origin.

Real-World Context Clues

Word problems often give context clues that signal whether a relationship is proportional or non-proportional:

Proportional Clues
"$5 per hour"
"3 miles for every gallon"
"at a constant rate"
No starting fee or fixed cost
Non-Proportional Clues
"$5 per hour plus a $10 fee"
"starts with 20 already done"
"initial deposit" or "flat fee"
Any fixed starting amount

Common Mistakes

✕ Confusing Slope with Proportionality
Having a constant slope does not guarantee proportionality. \( y = 5x + 3 \) has a constant slope but is not proportional.
✕ Ignoring the Origin
Forgetting to check if \( (0, 0) \) is on the graph. A line can look "proportional" but miss the origin by a small amount.
✕ Checking Only One Ratio
Computing \( \frac{y}{x} \) for just one point is not enough. You must verify the ratio is the same for all data points.

Summary Comparison

Feature Proportional Non-Proportional
Equation \( y = kx \) \( y = mx + b,\; b \neq 0 \)
Origin Passes through \( (0,0) \) Does not pass through \( (0,0) \)
Ratio \( \frac{y}{x} \) Constant Changes
Graph Straight line through origin Straight line, shifted up or down
Example \( y = 4x \) \( y = 4x + 5 \)

Question 1 of 5

A table shows: \( (1, 4),\; (2, 8),\; (3, 12),\; (4, 16) \). Is this relationship proportional or non-proportional?

Question 2 of 5

Which equation represents a non-proportional relationship?

Question 3 of 5

A gym charges a $20 membership fee plus $5 per visit. The total cost is modeled by \( y = 5x + 20 \). What is the best justification that this is non-proportional?

Question 4 of 5

A graph shows a straight line passing through \( (0, 0) \) and \( (3, 9) \). What is the constant of proportionality \( k \)?

Question 5 of 5

A table shows: \( (1, 5),\; (2, 8),\; (3, 11),\; (4, 14) \). Which model fits this data, and why?