Chapter 25: Rates and Unit Rates
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Rates and Unit Rates with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Meaning of a rate (a Grade 5 idea used in this chapter)
- Comparing two different units (a Grade 5 idea used in this chapter)
- Finding a unit rate (a Grade 5 idea used in this chapter)
- Cost per item (a Grade 5 idea used in this chapter)
- Distance per unit of time (a Grade 5 idea used in this chapter)
- Items per package (a Grade 5 idea used in this chapter)
- Equivalent rates (a Grade 5 idea used in this chapter)
- Rate tables (a Grade 5 idea used in this chapter)
- Rates in everyday contexts (a Grade 5 idea used in this chapter)
- Choosing the better rate (a Grade 5 idea used in this chapter)
How to Learn This Chapter
Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.
25.1 Meaning of a rate
Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Meaning of a rate?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Meaning of a rate?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Meaning of a rate.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Meaning of a rate problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Meaning of a rate.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Meaning of a rate can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Meaning of a rate. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.2 Comparing two different units
Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Comparing two different units?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Comparing two different units?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Comparing two different units.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Comparing two different units problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Comparing two different units.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Comparing two different units can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Comparing two different units in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing two different units becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Comparing two different units and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing two different units becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Comparing two different units using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing two different units becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Comparing two different units problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing two different units becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Comparing two different units could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Comparing two different units becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Comparing two different units. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.3 Finding a unit rate
Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Finding a unit rate?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Finding a unit rate?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Finding a unit rate.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Finding a unit rate problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Finding a unit rate.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Finding a unit rate can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Finding a unit rate. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.4 Cost per item
Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Cost per item?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Cost per item?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Cost per item.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Cost per item problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Cost per item.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Cost per item can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Cost per item in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Cost per item becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Cost per item and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Cost per item becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Cost per item using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Cost per item becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Cost per item problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Cost per item becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Cost per item could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Cost per item becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Cost per item. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.5 Distance per unit of time
Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Distance per unit of time?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Distance per unit of time?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Distance per unit of time.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Distance per unit of time problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Distance per unit of time.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Distance per unit of time can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Distance per unit of time in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Distance per unit of time becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Distance per unit of time and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Distance per unit of time becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Distance per unit of time using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Distance per unit of time becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Distance per unit of time problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Distance per unit of time becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Distance per unit of time could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Distance per unit of time becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Distance per unit of time. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.6 Items per package
Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Items per package?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Worked Example 2
Problem: What should you identify first before solving a problem about Items per package?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Items per package.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Items per package problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Items per package.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Answer: Items per package can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Explain Items per package in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Items per package becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Items per package and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Items per package becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Items per package using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Items per package becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Items per package problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Items per package becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Items per package could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Items per package becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Items per package. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.7 Equivalent rates
Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Equivalent rates?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Equivalent rates?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Equivalent rates.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Equivalent rates problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Equivalent rates.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Equivalent rates can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Equivalent rates. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.8 Rate tables
Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Rate tables?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Rate tables?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Rate tables.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Rate tables problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Rate tables.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Rate tables can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Rate tables. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.9 Rates in everyday contexts
Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Rates in everyday contexts?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Rates in everyday contexts?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Rates in everyday contexts.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Rates in everyday contexts problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Rates in everyday contexts.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Rates in everyday contexts can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Rates in everyday contexts. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.10 Choosing the better rate
Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Beginner Note
Begin with meaning and a visual or concrete example. Work one step at a time and check the answer before moving on.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: In your own words, what is the main idea of Choosing the better rate?
- Read the topic name carefully.
- Identify the quantity, relationship, shape, or process it describes.
- Explain the idea without using a memorized sentence.
Very beginner explanation: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Worked Example 2
Problem: What should you identify first before solving a problem about Choosing the better rate?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Choosing the better rate.
- Decide whether a model, number line, table, graph, diagram, equation, or labelled calculation best fits the idea.
- Label the representation clearly.
- Check that it matches the mathematical relationship.
Very beginner explanation: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Use the representation that makes the important relationship easiest to see and check.
Worked Example 4
Problem: A learner gets an answer in a Choosing the better rate problem. How can the learner check it?
- Estimate the expected size or direction of the answer.
- Check units, labels, signs, and place values.
- Use an inverse operation or a second method when possible.
Very beginner explanation: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: The answer should agree with an estimate, the problem conditions, and a second check.
Worked Example 5
Problem: Give a real-life use for Choosing the better rate.
- Think about money, measurement, data, patterns, design, or everyday quantities.
- Choose a situation where the topic helps compare, calculate, measure, predict, or organize.
- Explain the connection in one sentence.
Very beginner explanation: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Choosing the better rate can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Choosing the better rate. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
Chapter Notes, Practice, and Common Mistakes
Chapter Notes
- Read the question before calculating.
- Represent the mathematics with a model, table, graph, diagram, number line, or equation when useful.
- Show the reasoning and check the final answer with a second method when possible.
Extra Practice
- Create and solve one original problem about Meaning of a rate.
- Create and solve one original problem about Comparing two different units.
- Create and solve one original problem about Finding a unit rate.
- Create and solve one original problem about Cost per item.
- Create and solve one original problem about Distance per unit of time.
- Create and solve one original problem about Items per package.
Common Mistakes
- Skipping the meaning and trying to memorize a rule only.
- Using the wrong operation because the question was not read completely.
- Ignoring units, labels, place values, or the context of the problem.
- Not estimating or checking whether the final answer is reasonable.
30 Review Questions and Answers
Q1. What is the key idea in Meaning of a rate?
Answer: Meaning of a rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q2. What is the key idea in Comparing two different units?
Answer: Comparing two different units is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q3. What is the key idea in Finding a unit rate?
Answer: Finding a unit rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q4. What is the key idea in Cost per item?
Answer: Cost per item is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q5. What is the key idea in Distance per unit of time?
Answer: Distance per unit of time is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q6. What is the key idea in Items per package?
Answer: Items per package is an important Grade 5 mathematics idea. Learn its meaning first, then connect it to a representation, a worked example, and a real-life use.
Q7. What is the key idea in Equivalent rates?
Answer: Equivalent rates compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q8. What is the key idea in Rate tables?
Answer: Rate tables compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q9. What is the key idea in Rates in everyday contexts?
Answer: Rates in everyday contexts compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q10. What is the key idea in Choosing the better rate?
Answer: Choosing the better rate compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q11. Why should you estimate before or after a calculation?
Answer: Estimation gives a reasonable range and can reveal place-value or operation mistakes.
Q12. Why are labels and units important?
Answer: They show what a number represents and help prevent mixing unlike quantities.
Q13. How can a diagram help solve a problem?
Answer: A diagram makes quantities and relationships visible before calculation.
Q14. How can inverse operations check an answer?
Answer: An inverse operation reverses the original operation and should recover the starting value when the work is correct.
Q15. Why should you show steps?
Answer: Showing steps makes reasoning clear and helps find where an error happened.
Q16. What should you do after making a mistake?
Answer: Find the step where the reasoning changed, correct it, and try a similar problem again.
Q17. How can a number line support reasoning?
Answer: It shows order, distance, relative size, fractions, decimals, and inequality solutions visually.
Q18. When is a table useful?
Answer: A table organizes related values so patterns and comparisons are easier to see.
Q19. When is a graph useful?
Answer: A graph makes trends, comparisons, locations, or data patterns visible.
Q20. How do you decide which operation to use?
Answer: Use the meaning of the situation: combine, compare, find a difference, form equal groups, or find how many groups fit.
Q21. Why should you check place value?
Answer: A digit or decimal has a different value depending on its position.
Q22. What makes an answer reasonable?
Answer: It fits the context, has the expected size and units, and agrees with an estimate or second method.
Q23. How can you explain mathematical reasoning clearly?
Answer: Name the information, the rule or operation, the steps, and why the final answer fits the problem.
Q24. Why can more than one strategy be correct?
Answer: Different valid strategies can represent the same mathematical relationship and lead to the same result.
Q25. How should you approach a difficult Grade 5 problem?
Answer: Break it into smaller parts, solve one part at a time, and check each result before continuing.
Q26. How can you practise Rates and Unit Rates effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q27. How can you practise Rates and Unit Rates effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q28. How can you practise Rates and Unit Rates effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q29. How can you practise Rates and Unit Rates effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.
Q30. How can you practise Rates and Unit Rates effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.