Proportional Relationship Worksheet- Practice Problems and Solutions

What Is a Proportional Relationship?

A proportional relationship between two quantities means they maintain a constant ratio. If one variable always changes by the same factor when the other changes, you're looking at a proportional relationship.

The standard form is y = kx, where k is the constant of proportionality. That's it. No extra terms, no tricks.

For example, if gas costs $4 per gallon, the relationship between gallons and total cost is proportional. Buy 1 gallon, pay $4. Buy 5 gallons, pay $20. The ratio stays locked at 4:1.

How to Spot a Proportional Relationship

Before you start solving problems, you need to recognize when a relationship is proportional. Here are the markers:

• The graph forms a straight line through the origin (0, 0)
• The ratio y/x is always the same value
• The equation can be written as y = kx with no added constant
• A table shows consistent products or quotients across all pairs

If any of these fail, you're not dealing with a proportional relationship. Move on.

Constant of Proportionality Explained

The constant of proportionality (k) is just the ratio between the two variables. Find it by dividing y by x for any point on the line.

Example

Given: 3 apples cost $9

k = 9/3 = 3

Each apple costs $3. The equation is y = 3x.

Practice Problems and Solutions

Work through these. Cover the solutions, try the problem, then check your answer.

Problem 1: Finding the Constant

A car travels 150 miles using 5 gallons of gas. Assuming the relationship is proportional, what is the constant of proportionality?

Solution:

k = miles/gallons = 150/5 = 30 miles per gallon

The equation is y = 30x.

Problem 2: Completing a Table

If y varies proportionally with x, and y = 24 when x = 8, complete the table:

Solution:

First find k: 24/8 = 3

• When x = 2: y = 3 × 2 = 6
• When x = 15: y = 3 × 15 = 45
• When y = 72: 72/3 = 24

Problem 3: Writing the Equation

A copy shop charges $0.15 per page. Express the total cost as a proportional relationship.

Solution:

Total cost = 0.15 × number of pages

C = 0.15p

If you print 200 pages, cost = 0.15 × 200 = $30

Problem 4: Graph Interpretation

A graph shows points (0,0), (2,6), (4,12), (6,18). Is this proportional? Find the equation.

Solution:

Yes, it's proportional. The line passes through the origin.

k = 6/2 = 3

y = 3x

Problem 5: Word Problem

Sarah earns $12 per hour babysitting. How much does she earn for 7 hours? For 20 hours?

Solution:

Equation: E = 12h

• 7 hours: 12 × 7 = $84
• 20 hours: 12 × 20 = $240

Common Mistakes to Avoid

• Ignoring the origin: Proportional graphs MUST pass through (0,0). If the line starts elsewhere, it's not proportional.
• Adding instead of multiplying: The relationship is multiplication, not addition. Don't confuse y = kx with y = kx + b.
• Finding k wrong: Always divide y by x, not x by y. Check which way the ratio goes.
• Assuming proportionality when it doesn't exist: Some relationships have a constant rate but aren't proportional if there's a starting fee or base amount.

Getting Started: How to Solve Any Proportional Relationship Problem

Follow this step-by-step process:

1. Identify the two variables. What are you comparing? (Distance and time, cost and quantity, etc.)
2. Find the constant k. Use any given pair: divide the dependent variable by the independent variable.
3. Write the equation. y = kx
4. Answer the question. Plug in the given value and solve.

That's the whole process. Practice it until it becomes automatic.

Proportional vs. Non-Proportional Relationships

Quick Reference Formulas

• Find k: k = y/x
• Find y: y = kx
• Find x: x = y/k

Keep these three formulas memorized. Every proportional relationship problem uses one of them.

When You're Done

If you can solve the five practice problems without looking at the solutions, you understand proportional relationships. If you can't, go back and work through each step again.

That's the only way to actually learn this. Not reading, not highlighting—doing.