Chapter 22: Multiplying Whole Numbers by Unit Fractions
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Multiplying Whole Numbers by Unit Fractions with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Meaning of a unit fraction (a Grade 5 idea used in this chapter)
- One-half of a whole number (a Grade 5 idea used in this chapter)
- One-third of a whole number (a Grade 5 idea used in this chapter)
- One-fourth of a whole number (a Grade 5 idea used in this chapter)
- Using number lines (a Grade 5 idea used in this chapter)
- Using area models (a Grade 5 idea used in this chapter)
- Whole number times a unit fraction (a Grade 5 idea used in this chapter)
- Unit fraction times a whole number (a Grade 5 idea used in this chapter)
- Word problems with unit fractions (a Grade 5 idea used in this chapter)
- Checking fraction products with models (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.
22.1 Meaning of a unit fraction
Meaning of a unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 unit fraction?
- 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 unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Meaning of a unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Meaning of a unit fraction?
- 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 unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 unit fraction.
- 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 unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 unit fraction 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 unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 unit fraction.
- 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 unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Meaning of a unit fraction can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Meaning of a unit fraction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.2 One-half of a whole number
One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-half of a whole number?
- 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: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about One-half of a whole number?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for One-half of a whole number.
- 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: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-half of a whole number 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: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-half of a whole number.
- 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: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-half of a whole number can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about One-half of a whole number. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.3 One-third of a whole number
One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-third of a whole number?
- 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: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about One-third of a whole number?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for One-third of a whole number.
- 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: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-third of a whole number 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: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-third of a whole number.
- 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: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-third of a whole number can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about One-third of a whole number. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.4 One-fourth of a whole number
One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-fourth of a whole number?
- 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: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about One-fourth of a whole number?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for One-fourth of a whole number.
- 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: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-fourth of a whole number 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: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 One-fourth of a whole number.
- 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: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: One-fourth of a whole number can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about One-fourth of a whole number. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.5 Using number lines
Using number lines 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 Using number lines?
- 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: Using number lines 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: Using number lines 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 Using number lines?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using number lines 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 Using number lines.
- 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: Using number lines 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 Using number lines 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: Using number lines 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 Using number lines.
- 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: Using number lines 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: Using number lines can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Which is farther right on a number line: 3,000 or 3,750?
- Numbers get larger as you move right.
- 3,750 is greater than 3,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 3,750
Worked Example 7
Problem: Which is farther right on a number line: 4,000 or 4,750?
- Numbers get larger as you move right.
- 4,750 is greater than 4,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 4,750
Worked Example 8
Problem: Which is farther right on a number line: 5,000 or 5,750?
- Numbers get larger as you move right.
- 5,750 is greater than 5,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 5,750
Worked Example 9
Problem: Which is farther right on a number line: 6,000 or 6,750?
- Numbers get larger as you move right.
- 6,750 is greater than 6,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 6,750
Worked Example 10
Problem: Which is farther right on a number line: 7,000 or 7,750?
- Numbers get larger as you move right.
- 7,750 is greater than 7,000.
Very beginner explanation: On a standard number line, the greater number is farther to the right.
Answer: 7,750
Practice Exercise
Create one new question about Using number lines. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.6 Using area models
Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
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 Using area models?
- 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: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
Answer: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
Worked Example 2
Problem: What should you identify first before solving a problem about Using area models?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Using area models.
- 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: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
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 Using area models 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: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
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 Using area models.
- 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: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
Answer: Using area models can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Tile a floor
- Write a simple rule: area = length × width.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Estimate tile quantity and include a small waste allowance.
Worked Example 7
Problem: Plan monthly savings
- Write a simple rule: total = starting amount + monthly deposit × months.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Predict when the savings goal is reached.
Worked Example 8
Problem: Simulate a die
- Write a simple rule: generate a random whole number from 1 to 6.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Repeat many times and compare frequencies.
Worked Example 9
Problem: Translate a point
- Write a simple rule: new x = x + 3; new y = y - 2.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Apply the same rule to every point.
Worked Example 10
Problem: Estimate paint needed
- Write a simple rule: paint = wall area ÷ coverage per can.
- State any assumptions.
- Use the rule on sample values.
- Check whether the result is realistic.
Very beginner explanation: Coding and modelling organize a real situation into clear steps, variables, rules, and checks.
Answer: Round up because a partial can may not be enough.
Practice Exercise
Create one new question about Using area models. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.7 Whole number times a unit fraction
Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Whole number times a unit fraction?
- 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: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about Whole number times a unit fraction?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Whole number times a unit fraction.
- 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: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Whole number times a unit fraction 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: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Whole number times a unit fraction.
- 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: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Whole number times a unit fraction can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about Whole number times a unit fraction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.8 Unit fraction times a whole number
Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Unit fraction times a whole number?
- 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: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Worked Example 2
Problem: What should you identify first before solving a problem about Unit fraction times a whole number?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Unit fraction times a whole number.
- 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: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Unit fraction times a whole number 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: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
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 Unit fraction times a whole number.
- 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: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Unit fraction times a whole number can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: What is the value of the first digit in 638,420,715?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 600,000,000
Worked Example 7
Problem: What is the value of the first digit in 92,305,004?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 90,000,000
Worked Example 8
Problem: What is the value of the first digit in 704,090,650?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 700,000,000
Worked Example 9
Problem: What is the value of the first digit in 18,765,432?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 10,000,000
Worked Example 10
Problem: What is the value of the first digit in 999,500,001?
- Find the position of the first digit.
- Multiply the digit by its place value.
Very beginner explanation: Place value tells the value of a digit because of where it is located.
Answer: 900,000,000
Practice Exercise
Create one new question about Unit fraction times a whole number. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.9 Word problems with unit fractions
Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Word problems with unit fractions?
- 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: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Word problems with unit fractions?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Word problems with unit fractions.
- 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: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Word problems with unit fractions 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: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Word problems with unit fractions.
- 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: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Word problems with unit fractions can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Word problems with unit fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
22.10 Checking fraction products with models
Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Checking fraction products with models?
- 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: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Worked Example 2
Problem: What should you identify first before solving a problem about Checking fraction products with models?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Checking fraction products with models.
- 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: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Checking fraction products with models 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: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Checking fraction products with models.
- 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: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Answer: Checking fraction products with models can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Calculate 84 × 7.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 588
Worked Example 7
Problem: Calculate 125 × 8.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1000
Worked Example 8
Problem: Calculate 936 × 9.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 8424
Worked Example 9
Problem: Calculate 48 × 12.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 576
Worked Example 10
Problem: Calculate 315 × 5.
- Break one factor into easier parts if helpful.
- Multiply carefully.
- Check with division.
Very beginner explanation: Multiplication combines equal groups and can be checked with the inverse operation, division.
Answer: 1575
Practice Exercise
Create one new question about Checking fraction products with models. 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 unit fraction.
- Create and solve one original problem about One-half of a whole number.
- Create and solve one original problem about One-third of a whole number.
- Create and solve one original problem about One-fourth of a whole number.
- Create and solve one original problem about Using number lines.
- Create and solve one original problem about Using area models.
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 unit fraction?
Answer: Meaning of a unit fraction focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q2. What is the key idea in One-half of a whole number?
Answer: One-half of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q3. What is the key idea in One-third of a whole number?
Answer: One-third of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q4. What is the key idea in One-fourth of a whole number?
Answer: One-fourth of a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q5. What is the key idea in Using number lines?
Answer: Using number lines 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 Using area models?
Answer: Using area models uses mathematics to simplify a real situation, test possible solutions, and explain a reasonable recommendation.
Q7. What is the key idea in Whole number times a unit fraction?
Answer: Whole number times a unit fraction helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q8. What is the key idea in Unit fraction times a whole number?
Answer: Unit fraction times a whole number helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q9. What is the key idea in Word problems with unit fractions?
Answer: Word problems with unit fractions focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
Q10. What is the key idea in Checking fraction products with models?
Answer: Checking fraction products with models focuses on equal parts of a whole or set. Learners use models, number lines, and equivalent forms so the fraction has a clear meaning before using a rule.
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 Multiplying Whole Numbers by Unit Fractions 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 Multiplying Whole Numbers by Unit Fractions 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 Multiplying Whole Numbers by Unit Fractions 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 Multiplying Whole Numbers by Unit Fractions 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 Multiplying Whole Numbers by Unit Fractions effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.