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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.

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.

Answer: 12

Practice problems

1. Solve 2/3 = x/12.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 2/3 = x/12.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. The denominator 3 is multiplied by 4 to get 12.
  4. Multiply 2 by 4 to get 8.
  5. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: A recipe uses 5 cups of rice for 10 servings. How many cups for 2 servings?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 2 servings is 1/5 of 10 servings.
  4. 5 cups divided by 5 equals 1 cup.
  5. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

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. Warm-up: First identify exactly what the question is asking: Which table is proportional?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Each y-value is 4 times the x-value.
  4. The constant of proportionality is 4.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

4. Solve 4/7 = 20/x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 4/7 = 20/x.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. 4 is multiplied by 5 to get 20.
  4. Multiply 7 by 5 to get 35.
  5. Check the result by substituting or estimating: the response should match 35 and make sense in the original problem.

Answer: 35

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: If y = 6x, what is y when x = 9?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Multiply x by 6.
  4. 6 x 9 = 54.
  5. Check the result by substituting or estimating: the response should match 54 and make sense in the original problem.

Answer: 54

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