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Chapter 13: Understanding Percent and Proportional Reasoning

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

13.1 Meaning of percent

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Meaning of percent.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Meaning of percent. Show the important steps, include units when needed, and explain how you checked the answer.

13.2 Percent as per hundred

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Percent as per hundred.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Percent as per hundred. Show the important steps, include units when needed, and explain how you checked the answer.

13.3 Benchmark percents

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Benchmark percents.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Benchmark percents. Show the important steps, include units when needed, and explain how you checked the answer.

13.4 Finding a percent of a number

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Finding a percent of a number.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Finding a percent of a number. Show the important steps, include units when needed, and explain how you checked the answer.

13.5 Finding the percent

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Finding the percent.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: 8 is what percent of 80?

  1. Divide part by whole: 8 ÷ 80.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 10%

Worked Example 2

Problem: 15 is what percent of 60?

  1. Divide part by whole: 15 ÷ 60.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 25%

Worked Example 3

Problem: 18 is what percent of 120?

  1. Divide part by whole: 18 ÷ 120.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 15%

Worked Example 4

Problem: 12.5 is what percent of 250?

  1. Divide part by whole: 12.5 ÷ 250.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 5%

Worked Example 5

Problem: 9.75 is what percent of 75?

  1. Divide part by whole: 9.75 ÷ 75.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 13%

Worked Example 6

Problem: 49 is what percent of 140?

  1. Divide part by whole: 49 ÷ 140.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 35%

Worked Example 7

Problem: 40 is what percent of 500?

  1. Divide part by whole: 40 ÷ 500.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 8%

Worked Example 8

Problem: 27 is what percent of 45?

  1. Divide part by whole: 27 ÷ 45.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 60%

Worked Example 9

Problem: 8 is what percent of 320?

  1. Divide part by whole: 8 ÷ 320.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 2.5%

Worked Example 10

Problem: 80 is what percent of 64?

  1. Divide part by whole: 80 ÷ 64.
  2. Multiply by 100%.

Very beginner explanation: Percent tells what fraction of the whole would correspond to 100 equal parts.

Answer: 125%

Practice exercise

Create one new Grade 8 problem involving Finding the percent. Show the important steps, include units when needed, and explain how you checked the answer.

13.6 Finding the whole

Finding the whole is an important Grade 8 concept in Understanding Percent and Proportional Reasoning. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Finding the whole: Explain Finding the whole in one simple sentence.

  1. Look at the words in “Finding the whole”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Finding the whole is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Finding the whole: A student says, “I can use Finding the whole without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Finding the whole: What is the first step when solving a problem about Finding the whole?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Finding the whole: After solving a Finding the whole problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Finding the whole: Give one way Finding the whole could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Finding the whole can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Finding the whole: Which representation could help explain Finding the whole: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Finding the whole: A student gets an answer for Finding the whole but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Finding the whole: Why can estimation help before a detailed Finding the whole calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Finding the whole: How can you test whether your rule for Finding the whole works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Finding the whole: How would you teach Finding the whole to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Finding the whole. Show the important steps, include units when needed, and explain how you checked the answer.

13.7 Percent proportions

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Percent proportions.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Percent proportions. Show the important steps, include units when needed, and explain how you checked the answer.

13.8 Equivalent ratios

A ratio compares two quantities in a fixed order. In this section, the focus is Equivalent ratios.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Simplify the ratio 4:6.

  1. Find the GCF: 2.
  2. Divide both terms by 2.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 2

Problem: Simplify the ratio 8:12.

  1. Find the GCF: 4.
  2. Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 3

Problem: Simplify the ratio 15:25.

  1. Find the GCF: 5.
  2. Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 4

Problem: Simplify the ratio 21:28.

  1. Find the GCF: 7.
  2. Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:4

Worked Example 5

Problem: Simplify the ratio 18:30.

  1. Find the GCF: 6.
  2. Divide both terms by 6.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 6

Problem: Simplify the ratio 12:20.

  1. Find the GCF: 4.
  2. Divide both terms by 4.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 7

Problem: Simplify the ratio 9:15.

  1. Find the GCF: 3.
  2. Divide both terms by 3.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 3:5

Worked Example 8

Problem: Simplify the ratio 16:24.

  1. Find the GCF: 8.
  2. Divide both terms by 8.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:3

Worked Example 9

Problem: Simplify the ratio 25:35.

  1. Find the GCF: 5.
  2. Divide both terms by 5.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 5:7

Worked Example 10

Problem: Simplify the ratio 14:49.

  1. Find the GCF: 7.
  2. Divide both terms by 7.

Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both parts by the same common factor.

Answer: 2:7

Practice exercise

Create one new Grade 8 problem involving Equivalent ratios. Show the important steps, include units when needed, and explain how you checked the answer.

13.9 Proportional reasoning

A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportional reasoning.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve 4/6 = x/15.

  1. Cross multiply or use equivalent ratios.
  2. 6x = 4 × 15.
  3. x = 60 ÷ 6.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 10

Worked Example 2

Problem: Solve 8/12 = x/20.

  1. Cross multiply or use equivalent ratios.
  2. 12x = 8 × 20.
  3. x = 160 ÷ 12.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 13.3333

Worked Example 3

Problem: Solve 15/25 = x/25.

  1. Cross multiply or use equivalent ratios.
  2. 25x = 15 × 25.
  3. x = 375 ÷ 25.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 15

Worked Example 4

Problem: Solve 21/28 = x/30.

  1. Cross multiply or use equivalent ratios.
  2. 28x = 21 × 30.
  3. x = 630 ÷ 28.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 22.5

Worked Example 5

Problem: Solve 18/30 = x/35.

  1. Cross multiply or use equivalent ratios.
  2. 30x = 18 × 35.
  3. x = 630 ÷ 30.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 21

Worked Example 6

Problem: Solve 12/20 = x/40.

  1. Cross multiply or use equivalent ratios.
  2. 20x = 12 × 40.
  3. x = 480 ÷ 20.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 24

Worked Example 7

Problem: Solve 9/15 = x/45.

  1. Cross multiply or use equivalent ratios.
  2. 15x = 9 × 45.
  3. x = 405 ÷ 15.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 27

Worked Example 8

Problem: Solve 16/24 = x/50.

  1. Cross multiply or use equivalent ratios.
  2. 24x = 16 × 50.
  3. x = 800 ÷ 24.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 33.3333

Worked Example 9

Problem: Solve 25/35 = x/55.

  1. Cross multiply or use equivalent ratios.
  2. 35x = 25 × 55.
  3. x = 1375 ÷ 35.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 39.2857

Worked Example 10

Problem: Solve 14/49 = x/60.

  1. Cross multiply or use equivalent ratios.
  2. 49x = 14 × 60.
  3. x = 840 ÷ 49.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 17.1429

Practice exercise

Create one new Grade 8 problem involving Proportional reasoning. Show the important steps, include units when needed, and explain how you checked the answer.

13.10 Proportion tables

A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportion tables.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Solve 4/6 = x/15.

  1. Cross multiply or use equivalent ratios.
  2. 6x = 4 × 15.
  3. x = 60 ÷ 6.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 10

Worked Example 2

Problem: Solve 8/12 = x/20.

  1. Cross multiply or use equivalent ratios.
  2. 12x = 8 × 20.
  3. x = 160 ÷ 12.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 13.3333

Worked Example 3

Problem: Solve 15/25 = x/25.

  1. Cross multiply or use equivalent ratios.
  2. 25x = 15 × 25.
  3. x = 375 ÷ 25.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 15

Worked Example 4

Problem: Solve 21/28 = x/30.

  1. Cross multiply or use equivalent ratios.
  2. 28x = 21 × 30.
  3. x = 630 ÷ 28.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 22.5

Worked Example 5

Problem: Solve 18/30 = x/35.

  1. Cross multiply or use equivalent ratios.
  2. 30x = 18 × 35.
  3. x = 630 ÷ 30.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 21

Worked Example 6

Problem: Solve 12/20 = x/40.

  1. Cross multiply or use equivalent ratios.
  2. 20x = 12 × 40.
  3. x = 480 ÷ 20.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 24

Worked Example 7

Problem: Solve 9/15 = x/45.

  1. Cross multiply or use equivalent ratios.
  2. 15x = 9 × 45.
  3. x = 405 ÷ 15.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 27

Worked Example 8

Problem: Solve 16/24 = x/50.

  1. Cross multiply or use equivalent ratios.
  2. 24x = 16 × 50.
  3. x = 800 ÷ 24.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 33.3333

Worked Example 9

Problem: Solve 25/35 = x/55.

  1. Cross multiply or use equivalent ratios.
  2. 35x = 25 × 55.
  3. x = 1375 ÷ 35.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 39.2857

Worked Example 10

Problem: Solve 14/49 = x/60.

  1. Cross multiply or use equivalent ratios.
  2. 49x = 14 × 60.
  3. x = 840 ÷ 49.

Very beginner explanation: A proportion states that two ratios have the same value.

Answer: 17.1429

Practice exercise

Create one new Grade 8 problem involving Proportion tables. Show the important steps, include units when needed, and explain how you checked the answer.

13.11 Cross multiplication

Cross multiplication is an important Grade 8 concept in Understanding Percent and Proportional Reasoning. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: An item costs $50.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 50.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $6.50

Worked Example 2

Problem: An item costs $100.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 100.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $13.00

Worked Example 3

Problem: An item costs $75.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 75.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $9.75

Worked Example 4

Problem: An item costs $120.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 120.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $15.60

Worked Example 5

Problem: An item costs $200.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 200.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $26.00

Worked Example 6

Problem: An item costs $150.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 150.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $19.50

Worked Example 7

Problem: An item costs $250.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 250.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $32.50

Worked Example 8

Problem: An item costs $80.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 80.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $10.40

Worked Example 9

Problem: An item costs $300.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 300.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $39.00

Worked Example 10

Problem: An item costs $60.00. Find 13% sales tax.

  1. Convert 13% to 0.13.
  2. Tax = 60.00×0.13.

Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.

Answer: $7.80

Practice exercise

Create one new Grade 8 problem involving Cross multiplication. Show the important steps, include units when needed, and explain how you checked the answer.

13.12 Unit-rate method

A rate compares quantities measured in different units. In this section, the focus is Unit-rate method.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A quantity of 80 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 80 ÷ 2 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 2

Problem: A quantity of 120 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 120 ÷ 3 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 3

Problem: A quantity of 190 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 190 ÷ 4 = 47.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 47.5 per unit

Worked Example 4

Problem: A quantity of 250 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 250 ÷ 5 = 50.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 50 per unit

Worked Example 5

Problem: A quantity of 220 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 220 ÷ 2 = 110.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 110 per unit

Worked Example 6

Problem: A quantity of 160 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 160 ÷ 3 = 53.3333.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 53.3333 per unit

Worked Example 7

Problem: A quantity of 130 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 130 ÷ 4 = 32.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 32.5 per unit

Worked Example 8

Problem: A quantity of 200 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 200 ÷ 5 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 9

Problem: A quantity of 290 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 290 ÷ 2 = 145.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 145 per unit

Worked Example 10

Problem: A quantity of 180 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 180 ÷ 3 = 60.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 60 per unit

Practice exercise

Create one new Grade 8 problem involving Unit-rate method. Show the important steps, include units when needed, and explain how you checked the answer.

13.13 Real-life percent problems

Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Real-life percent problems.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of 80.

  1. Convert 10% to decimal 0.1.
  2. Multiply 0.1 × 80.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 2

Problem: Find 25% of 60.

  1. Convert 25% to decimal 0.25.
  2. Multiply 0.25 × 60.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 15

Worked Example 3

Problem: Find 15% of 120.

  1. Convert 15% to decimal 0.15.
  2. Multiply 0.15 × 120.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 18

Worked Example 4

Problem: Find 5% of 250.

  1. Convert 5% to decimal 0.05.
  2. Multiply 0.05 × 250.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 12.5

Worked Example 5

Problem: Find 13% of 75.

  1. Convert 13% to decimal 0.13.
  2. Multiply 0.13 × 75.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 9.75

Worked Example 6

Problem: Find 35% of 140.

  1. Convert 35% to decimal 0.35.
  2. Multiply 0.35 × 140.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 49

Worked Example 7

Problem: Find 8% of 500.

  1. Convert 8% to decimal 0.08.
  2. Multiply 0.08 × 500.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 40

Worked Example 8

Problem: Find 60% of 45.

  1. Convert 60% to decimal 0.6.
  2. Multiply 0.6 × 45.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 27

Worked Example 9

Problem: Find 2.5% of 320.

  1. Convert 2.5% to decimal 0.025.
  2. Multiply 0.025 × 320.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 8

Worked Example 10

Problem: Find 125% of 64.

  1. Convert 125% to decimal 1.25.
  2. Multiply 1.25 × 64.

Very beginner explanation: “Of” usually means multiply in a percent-of-a-number problem.

Answer: 80

Practice exercise

Create one new Grade 8 problem involving Real-life percent problems. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 13 Review Questions and Answers

Q1. What is important to remember about Meaning of percent?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Meaning of percent.

Q2. What is important to remember about Percent as per hundred?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Percent as per hundred.

Q3. What is important to remember about Benchmark percents?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Benchmark percents.

Q4. What is important to remember about Finding a percent of a number?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Finding a percent of a number.

Q5. What is important to remember about Finding the percent?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Finding the percent.

Q6. What is important to remember about Finding the whole?

Answer: Finding the whole is an important Grade 8 concept in Understanding Percent and Proportional Reasoning. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Percent proportions?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Percent proportions.

Q8. What is important to remember about Equivalent ratios?

Answer: A ratio compares two quantities in a fixed order. In this section, the focus is Equivalent ratios.

Q9. What is important to remember about Proportional reasoning?

Answer: A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportional reasoning.

Q10. What is important to remember about Proportion tables?

Answer: A proportion is an equation showing that two ratios are equal. In this section, the focus is Proportion tables.

Q11. What is important to remember about Cross multiplication?

Answer: Cross multiplication is an important Grade 8 concept in Understanding Percent and Proportional Reasoning. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Unit-rate method?

Answer: A rate compares quantities measured in different units. In this section, the focus is Unit-rate method.

Q13. What is important to remember about Real-life percent problems?

Answer: Percent means per hundred, so 35% means 35 out of 100. In this section, the focus is Real-life percent problems.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.