Chapter 17: Ratios, Rates, Percentages, and Number Applications
Learn Grade 6 mathematics with very-beginner explanations, original worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Ratios, Rates, Percentages, and Number Applications in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.
Key Technical Terms
- Meaning of ratio (A ratio compares two quantities by division.)
- Meaning of rate (A rate compares two quantities measured in different units.)
- Connecting fractions ratios and percents (A fraction represents equal parts of a whole, set, or quantity.)
- Percent of a quantity (Percent means 'per hundred' and compares a quantity with 100 equal parts.)
- Multi-step number problems (Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
How to Learn This Chapter
Read the explanation first, follow each worked example slowly, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
17.1 Meaning of ratio
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 8:5.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:5
Worked Example 2
Problem: Simplify the ratio 10:4.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:2
Worked Example 3
Problem: Simplify the ratio 2:6.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:3
Worked Example 4
Problem: Simplify the ratio 5:14.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:14
Worked Example 5
Problem: Simplify the ratio 7:14.
- Find the greatest common factor, 7.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:2
Worked Example 6
Problem: Simplify the ratio 11:15.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:15
Worked Example 7
Problem: Simplify the ratio 4:8.
- Find the greatest common factor, 4.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:2
Worked Example 8
Problem: Simplify the ratio 5:13.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:13
Worked Example 9
Problem: Simplify the ratio 9:3.
- Find the greatest common factor, 3.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:1
Worked Example 10
Problem: Simplify the ratio 8:5.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:5
Practice Exercise
Create one new Grade 6 problem about Meaning of ratio. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.2 Part-to-part ratios
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 11:8.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:8
Worked Example 2
Problem: Simplify the ratio 12:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 12:7
Worked Example 3
Problem: Simplify the ratio 4:9.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 4:9
Worked Example 4
Problem: Simplify the ratio 11:3.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:3
Worked Example 5
Problem: Simplify the ratio 12:12.
- Find the greatest common factor, 12.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:1
Worked Example 6
Problem: Simplify the ratio 6:6.
- Find the greatest common factor, 6.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:1
Worked Example 7
Problem: Simplify the ratio 8:3.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:3
Worked Example 8
Problem: Simplify the ratio 8:8.
- Find the greatest common factor, 8.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:1
Worked Example 9
Problem: Simplify the ratio 10:4.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:2
Worked Example 10
Problem: Simplify the ratio 10:14.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:7
Practice Exercise
Create one new Grade 6 problem about Part-to-part ratios. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.3 Part-to-whole ratios
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 11:15.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:15
Worked Example 2
Problem: Simplify the ratio 2:6.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:3
Worked Example 3
Problem: Simplify the ratio 10:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 10:7
Worked Example 4
Problem: Simplify the ratio 7:8.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 7:8
Worked Example 5
Problem: Simplify the ratio 10:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 10:7
Worked Example 6
Problem: Simplify the ratio 10:5.
- Find the greatest common factor, 5.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:1
Worked Example 7
Problem: Simplify the ratio 2:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:7
Worked Example 8
Problem: Simplify the ratio 2:5.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:5
Worked Example 9
Problem: Simplify the ratio 3:8.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:8
Worked Example 10
Problem: Simplify the ratio 6:13.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 6:13
Practice Exercise
Create one new Grade 6 problem about Part-to-whole ratios. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.4 Writing ratios in different forms
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 8:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:7
Worked Example 2
Problem: Simplify the ratio 3:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:7
Worked Example 3
Problem: Simplify the ratio 9:8.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 9:8
Worked Example 4
Problem: Simplify the ratio 2:8.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:4
Worked Example 5
Problem: Simplify the ratio 11:10.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:10
Worked Example 6
Problem: Simplify the ratio 3:13.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:13
Worked Example 7
Problem: Simplify the ratio 2:14.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:7
Worked Example 8
Problem: Simplify the ratio 5:3.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:3
Worked Example 9
Problem: Simplify the ratio 10:4.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:2
Worked Example 10
Problem: Simplify the ratio 11:10.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 11:10
Practice Exercise
Create one new Grade 6 problem about Writing ratios in different forms. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.5 Simplifying ratios
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 10:14.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:7
Worked Example 2
Problem: Simplify the ratio 4:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 4:7
Worked Example 3
Problem: Simplify the ratio 9:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 9:7
Worked Example 4
Problem: Simplify the ratio 8:7.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:7
Worked Example 5
Problem: Simplify the ratio 7:12.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 7:12
Worked Example 6
Problem: Simplify the ratio 2:3.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:3
Worked Example 7
Problem: Simplify the ratio 10:12.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:6
Worked Example 8
Problem: Simplify the ratio 4:8.
- Find the greatest common factor, 4.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:2
Worked Example 9
Problem: Simplify the ratio 5:14.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:14
Worked Example 10
Problem: Simplify the ratio 3:3.
- Find the greatest common factor, 3.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 1:1
Practice Exercise
Create one new Grade 6 problem about Simplifying ratios. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.6 Equivalent ratios
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Simplify the ratio 3:14.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:14
Worked Example 2
Problem: Simplify the ratio 9:10.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 9:10
Worked Example 3
Problem: Simplify the ratio 2:13.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:13
Worked Example 4
Problem: Simplify the ratio 5:6.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 5:6
Worked Example 5
Problem: Simplify the ratio 12:4.
- Find the greatest common factor, 4.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 3:1
Worked Example 6
Problem: Simplify the ratio 2:3.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:3
Worked Example 7
Problem: Simplify the ratio 4:14.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 2:7
Worked Example 8
Problem: Simplify the ratio 4:5.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 4:5
Worked Example 9
Problem: Simplify the ratio 8:14.
- Find the greatest common factor, 2.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 4:7
Worked Example 10
Problem: Simplify the ratio 8:9.
- Find the greatest common factor, 1.
- Divide both parts by the same factor.
Very beginner explanation: A ratio is simplified the same way as a fraction.
Answer: 8:9
Practice Exercise
Create one new Grade 6 problem about Equivalent ratios. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.7 Meaning of rate
A rate compares two quantities measured in different units. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: 91 units are used in 13 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 91 ÷ 13 = 7.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 7 per group
Worked Example 2
Problem: 55 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 55 ÷ 11 = 5.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 5 per group
Worked Example 3
Problem: 132 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 132 ÷ 11 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Worked Example 4
Problem: 10 units are used in 5 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 10 ÷ 5 = 2.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 2 per group
Worked Example 5
Problem: 24 units are used in 8 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 24 ÷ 8 = 3.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 3 per group
Worked Example 6
Problem: 154 units are used in 14 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 154 ÷ 14 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 7
Problem: 121 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 121 ÷ 11 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 8
Problem: 90 units are used in 15 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 90 ÷ 15 = 6.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 6 per group
Worked Example 9
Problem: 110 units are used in 10 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 110 ÷ 10 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 10
Problem: 16 units are used in 4 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 16 ÷ 4 = 4.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 4 per group
Practice Exercise
Create one new Grade 6 problem about Meaning of rate. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.8 Unit rates
A rate compares two quantities measured in different units. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: 20 units are used in 4 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 20 ÷ 4 = 5.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 5 per group
Worked Example 2
Problem: 20 units are used in 5 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 20 ÷ 5 = 4.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 4 per group
Worked Example 3
Problem: 36 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 36 ÷ 3 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Worked Example 4
Problem: 12 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 12 ÷ 3 = 4.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 4 per group
Worked Example 5
Problem: 77 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 77 ÷ 11 = 7.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 7 per group
Worked Example 6
Problem: 90 units are used in 9 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 90 ÷ 9 = 10.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 10 per group
Worked Example 7
Problem: 16 units are used in 8 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 16 ÷ 8 = 2.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 2 per group
Worked Example 8
Problem: 42 units are used in 7 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 42 ÷ 7 = 6.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 6 per group
Worked Example 9
Problem: 14 units are used in 7 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 14 ÷ 7 = 2.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 2 per group
Worked Example 10
Problem: 120 units are used in 10 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 120 ÷ 10 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Practice Exercise
Create one new Grade 6 problem about Unit rates. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.9 Connecting fractions ratios and percents
A fraction represents equal parts of a whole, set, or quantity. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: The fraction bar means division.
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: The fraction bar means division.
Answer: 0.667
Worked Example 3
Problem: Write 3/5 as a decimal.
- Divide 3 by 5.
Very beginner explanation: The fraction bar means division.
Answer: 0.6
Worked Example 4
Problem: Write 5/8 as a decimal.
- Divide 5 by 8.
Very beginner explanation: The fraction bar means division.
Answer: 0.625
Worked Example 5
Problem: Write 7/10 as a decimal.
- Divide 7 by 10.
Very beginner explanation: The fraction bar means division.
Answer: 0.7
Worked Example 6
Problem: Write 4/9 as a decimal.
- Divide 4 by 9.
Very beginner explanation: The fraction bar means division.
Answer: 0.444
Worked Example 7
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: The fraction bar means division.
Answer: 0.833
Worked Example 8
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: The fraction bar means division.
Answer: 0.75
Worked Example 9
Problem: Write 2/5 as a decimal.
- Divide 2 by 5.
Very beginner explanation: The fraction bar means division.
Answer: 0.4
Worked Example 10
Problem: Write 7/12 as a decimal.
- Divide 7 by 12.
Very beginner explanation: The fraction bar means division.
Answer: 0.583
Practice Exercise
Create one new Grade 6 problem about Connecting fractions ratios and percents. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is combining numerators and denominators without first applying the correct fraction rule.
17.10 Percent of a quantity
Percent means 'per hundred' and compares a quantity with 100 equal parts. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find 1% of 80.
- Think of 1% as 1/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 0.8
Worked Example 2
Problem: Find 5% of 120.
- Think of 5% as 5/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 6
Worked Example 3
Problem: Find 10% of 64.
- Think of 10% as 10/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 6.4
Worked Example 4
Problem: Find 20% of 250.
- Think of 20% as 20/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 50
Worked Example 5
Problem: Find 25% of 96.
- Think of 25% as 25/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 24
Worked Example 6
Problem: Find 50% of 40.
- Think of 50% as 50/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 20
Worked Example 7
Problem: Find 75% of 200.
- Think of 75% as 75/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 150
Worked Example 8
Problem: Find 15% of 60.
- Think of 15% as 15/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 9
Worked Example 9
Problem: Find 30% of 150.
- Think of 30% as 30/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 45
Worked Example 10
Problem: Find 40% of 90.
- Think of 40% as 40/100.
- Multiply the quantity by the percent as a decimal or fraction.
Very beginner explanation: Percent means per hundred, so benchmark percents can often be found mentally.
Answer: 36
Practice Exercise
Create one new Grade 6 problem about Percent of a quantity. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is treating a percent as a whole number instead of converting it to a fraction or decimal when needed.
17.11 Ratio and rate tables
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: 70 units are used in 7 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 70 ÷ 7 = 10.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 10 per group
Worked Example 2
Problem: 165 units are used in 15 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 165 ÷ 15 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 3
Problem: 180 units are used in 15 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 180 ÷ 15 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Worked Example 4
Problem: 54 units are used in 9 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 54 ÷ 9 = 6.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 6 per group
Worked Example 5
Problem: 63 units are used in 9 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 63 ÷ 9 = 7.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 7 per group
Worked Example 6
Problem: 56 units are used in 7 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 56 ÷ 7 = 8.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 8 per group
Worked Example 7
Problem: 36 units are used in 4 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 36 ÷ 4 = 9.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 9 per group
Worked Example 8
Problem: 33 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 33 ÷ 3 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 9
Problem: 15 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 15 ÷ 3 = 5.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 5 per group
Worked Example 10
Problem: 60 units are used in 5 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 60 ÷ 5 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Practice Exercise
Create one new Grade 6 problem about Ratio and rate tables. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.12 Real-life ratio and rate problems
A ratio compares two quantities by division. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: 55 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 55 ÷ 11 = 5.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 5 per group
Worked Example 2
Problem: 44 units are used in 4 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 44 ÷ 4 = 11.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 11 per group
Worked Example 3
Problem: 72 units are used in 9 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 72 ÷ 9 = 8.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 8 per group
Worked Example 4
Problem: 18 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 18 ÷ 3 = 6.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 6 per group
Worked Example 5
Problem: 42 units are used in 6 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 42 ÷ 6 = 7.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 7 per group
Worked Example 6
Problem: 72 units are used in 6 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 72 ÷ 6 = 12.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 12 per group
Worked Example 7
Problem: 6 units are used in 3 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 6 ÷ 3 = 2.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 2 per group
Worked Example 8
Problem: 36 units are used in 6 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 36 ÷ 6 = 6.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 6 per group
Worked Example 9
Problem: 90 units are used in 10 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 90 ÷ 10 = 9.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 9 per group
Worked Example 10
Problem: 110 units are used in 11 equal groups. Find the unit rate.
- Divide the first quantity by the second.
- 110 ÷ 11 = 10.
Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.
Answer: 10 per group
Practice Exercise
Create one new Grade 6 problem about Real-life ratio and rate problems. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
17.13 Multi-step number problems
Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.
Beginner Note
Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Use Multi-step number problems to analyze the values 13 and 2. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 2
Problem: Use Multi-step number problems to analyze the values 29 and 15. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 3
Problem: Use Multi-step number problems to analyze the values 14 and 13. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 4
Problem: Use Multi-step number problems to analyze the values 11 and 18. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 5
Problem: Use Multi-step number problems to analyze the values 29 and 6. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 6
Problem: Use Multi-step number problems to analyze the values 3 and 10. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 7
Problem: Use Multi-step number problems to analyze the values 6 and 10. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 8
Problem: Use Multi-step number problems to analyze the values 19 and 19. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 9
Problem: Use Multi-step number problems to analyze the values 23 and 11. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Worked Example 10
Problem: Use Multi-step number problems to analyze the values 15 and 12. What should be checked first?
- Identify what each value represents.
- Choose the rule, representation, or comparison that matches the topic.
- Show the reasoning before accepting the result.
Very beginner explanation: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
Answer: Check the meaning of the values and the rule required by the problem.
Practice Exercise
Create one new Grade 6 problem about Multi-step number problems. Solve it step by step, label any units, and explain how you checked your answer.
Common Mistake
A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.
30 Review Questions and Answers
Q1. What is important to understand about Meaning of ratio?
Answer: A ratio compares two quantities by division.
Q2. What is important to understand about Part-to-part ratios?
Answer: A ratio compares two quantities by division.
Q3. What is important to understand about Part-to-whole ratios?
Answer: A ratio compares two quantities by division.
Q4. What is important to understand about Writing ratios in different forms?
Answer: A ratio compares two quantities by division.
Q5. What is important to understand about Simplifying ratios?
Answer: A ratio compares two quantities by division.
Q6. What is important to understand about Equivalent ratios?
Answer: A ratio compares two quantities by division.
Q7. What is important to understand about Meaning of rate?
Answer: A rate compares two quantities measured in different units.
Q8. What is important to understand about Unit rates?
Answer: A rate compares two quantities measured in different units.
Q9. What is important to understand about Connecting fractions ratios and percents?
Answer: A fraction represents equal parts of a whole, set, or quantity.
Q10. What is important to understand about Percent of a quantity?
Answer: Percent means 'per hundred' and compares a quantity with 100 equal parts.
Q11. What is important to understand about Ratio and rate tables?
Answer: A ratio compares two quantities by division.
Q12. What is important to understand about Real-life ratio and rate problems?
Answer: A ratio compares two quantities by division.
Q13. What is important to understand about Multi-step number problems?
Answer: Multi-step number problems is a Grade 6 mathematics idea in Ratios, Rates, Percentages, and Number Applications. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.
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 final answer is reasonable before accepting it.
Q16. Why do units matter?
Answer: Units tell what a value measures and help 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, a table, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the 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, operations, signs, units, rules, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships that are harder to see in words alone.
Q23. Why are tables useful?
Answer: Tables organize values so patterns, rates, comparisons, and missing information are easier to see.
Q24. Why are graphs useful?
Answer: Graphs make relationships, changes, trends, and comparisons visible.
Q25. Why should you explain your reasoning?
Answer: An explanation shows why a method works instead of only giving a final number.
Q26. Why can more than one strategy be correct?
Answer: Different correct strategies can represent the same mathematics in different but equivalent ways.
Q27. What should you do before using a formula or rule?
Answer: Check what each quantity means and whether the rule matches the situation.
Q28. How does practice help in mathematics?
Answer: Practice builds accuracy and fluency while helping you recognize which strategies fit different problems.
Q29. What is a good way to learn from a mistake?
Answer: Find the exact step where the reasoning changed from correct to incorrect, then solve a similar problem again.
Q30. Why should you compare an exact answer with an estimate?
Answer: The estimate gives a quick reasonableness check for the exact calculation.