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Proportions and Scale Factors

A free Pre-Algebra lesson from the “Proportional Reasoning” unit, with a worked example and practice problems including step-by-step solutions.

A proportion states that two ratios are equal. When two quantities are proportional, multiplying one quantity by a constant scale factor gives the matching quantity. Tables, equations, and diagrams can all show proportional relationships. In Proportional Reasoning, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

What you'll learn

Why it matters: Map distances, model cars, recipe doublings, and architectural blueprints all use proportions and scale factors. The same ratio holds at every size, which is why you can solve for the missing measurement.

Worked example

Problem. Solve 3/5 = x/20.

  1. The denominator changed from 5 to 20.
  2. That is a scale factor of 4.
  3. Multiply 3 by 4, so x = 12.
  4. Connect the calculation back to Proportions and Scale Factors so the method, not just the arithmetic, is clear.

Answer: 12

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Solve 2/3 = x/12.

Show solution
  1. The denominator 3 is multiplied by 4 to get 12.
  2. Multiply 2 by 4 to get 8.

Answer: 8

2. A recipe uses 5 cups of rice for 10 servings. How many cups for 2 servings?

Show solution
  1. 2 servings is 1/5 of 10 servings.
  2. 5 cups divided by 5 equals 1 cup.

Answer: 1

3. Which table is proportional?

Choices: x: 1, 2, 3; y: 4, 8, 12 · x: 1, 2, 3; y: 4, 7, 12 · x: 1, 2, 3; y: 5, 8, 11 · x: 1, 2, 3; y: 2, 4, 7

Show solution
  1. Each y-value is 4 times the x-value.
  2. The constant of proportionality is 4.

Answer: x: 1, 2, 3; y: 4, 8, 12

4. Solve 4/7 = 20/x.

Show solution
  1. 4 is multiplied by 5 to get 20.
  2. Multiply 7 by 5 to get 35.

Answer: 35

5. If y = 6x, what is y when x = 9?

Show solution
  1. Multiply x by 6.
  2. 6 x 9 = 54.

Answer: 54

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