The lesson
This lesson teaches Proportional Relationships. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
What makes a relationship proportional
Two quantities are proportional when they always change by the same rate. If you earn $12 every hour, dollars and hours are proportional - the rate stays 12 no matter how many hours you work.
When you study what makes a relationship proportional, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
Solving a proportion
Set two equal ratios next to each other and cross-multiply to find the missing value.
A ratio compares two quantities with the same units. Order matters: a ratio of cats to dogs is not the same as dogs to cats unless the problem says they are equivalent.
Write ratios in three ways: with a colon (3:4), as a phrase (3 to 4), or as a fraction (3/4) when it fits the context.
Solve 3/5 = x/20
- 1Cross-multiply: 3 × 20 = 5 × x.
- 260 = 5x.
- 3Divide by 5: x = 12.
Why this matters
Proportional Relationships shows up constantly in two quantities that grow together at one steady rate. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- What makes a relationship proportional
- Solving a proportion
Video walkthrough
Proportional Relationships
Spot when two quantities grow together at a steady rate.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfSolve the proportion: 2/7 = x/21.
Step-by-step solution
- 12 × 21 = 7 × x → 42 = 7x.
- 2Divide by 7: x = 6.
Exercise 2
Try it yourselfIf 4 pencils cost $3, how much do 10 pencils cost?
Step-by-step solution
- 1Unit price: $3 ÷ 4 = $0.75 per pencil.
- 210 × $0.75 = $7.50.
Exercise 3
Try it yourselfA map scale is 1 inch = 15 miles. Two towns are 4.5 inches apart on the map. How many miles apart are they?
Step-by-step solution
- 1Multiply: 4.5 × 15 = 67.5 miles.
Exercise 4
Try it yourselfSolve: 5/8 = 15/x.
Step-by-step solution
- 1Cross-multiply: 5x = 8 × 15 = 120.
- 2x = 120 ÷ 5 = 24.
Exercise 5
Try it yourselfA recipe for 6 servings uses 2 cups of rice. How many cups are needed for 15 servings?
Step-by-step solution
- 1Set up 2/6 = x/15. Cross-multiply: 6x = 30.
- 2x = 5 cups of rice.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Proportional Relationships
Question 1 of 5
If 5 notebooks cost $7.50, how much do 8 notebooks cost?