Chapter 12: Mental Math with Decimals
Learn Grade 5 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Mental Math with Decimals with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Multiplying whole numbers by 0.1 (a Grade 5 idea used in this chapter)
- Multiplying whole numbers by 0.01 (a Grade 5 idea used in this chapter)
- Understanding decimal shifts using place value (a Grade 5 idea used in this chapter)
- Estimating decimal sums (a Grade 5 idea used in this chapter)
- Estimating decimal differences (a Grade 5 idea used in this chapter)
- Using rounding for decimal estimates (a Grade 5 idea used in this chapter)
- Using benchmarks for decimal estimates (a Grade 5 idea used in this chapter)
- Front-end estimation with decimals (a Grade 5 idea used in this chapter)
- Explaining a mental-math strategy (a Grade 5 idea used in this chapter)
- Checking whether a decimal estimate is reasonable (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.
12.1 Multiplying whole numbers by 0.1
Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1?
- 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: Multiplying whole numbers by 0.1 helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1.
- 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: Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1 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: Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1.
- 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: Multiplying whole numbers by 0.1 helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Multiplying whole numbers by 0.1 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 Multiplying whole numbers by 0.1. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.2 Multiplying whole numbers by 0.01
Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01?
- 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: Multiplying whole numbers by 0.01 helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01.
- 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: Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01 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: Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01.
- 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: Multiplying whole numbers by 0.01 helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Multiplying whole numbers by 0.01 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 Multiplying whole numbers by 0.01. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.3 Understanding decimal shifts using place value
Understanding decimal shifts using place value 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 Understanding decimal shifts using place value?
- 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: Understanding decimal shifts using place value helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Understanding decimal shifts using place value 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 Understanding decimal shifts using place value?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Understanding decimal shifts using place value 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 Understanding decimal shifts using place value.
- 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: Understanding decimal shifts using place value 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 Understanding decimal shifts using place value 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: Understanding decimal shifts using place value 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 Understanding decimal shifts using place value.
- 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: Understanding decimal shifts using place value helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Answer: Understanding decimal shifts using place value 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 Understanding decimal shifts using place value. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.4 Estimating decimal sums
Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal sums?
- 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: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Estimating decimal sums?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Estimating decimal sums.
- 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: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal sums 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: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal sums.
- 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: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Estimating decimal sums can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Estimating decimal sums. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.5 Estimating decimal differences
Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal differences?
- 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: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Estimating decimal differences?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Estimating decimal differences.
- 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: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal differences 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: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Estimating decimal differences.
- 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: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Estimating decimal differences can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Estimating decimal differences. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.6 Using rounding for decimal estimates
Using rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 rounding for decimal estimates?
- 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 rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Using rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Using rounding for decimal estimates?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Using rounding for decimal estimates.
- 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 rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 rounding for decimal estimates 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 rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 rounding for decimal estimates.
- 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 rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Using rounding for decimal estimates can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 638,420,715 to the nearest 10.
- Locate the 10 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 638,420,720
Worked Example 7
Problem: Round 92,305,004 to the nearest 100.
- Locate the 100 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 92,305,000
Worked Example 8
Problem: Round 704,090,650 to the nearest 1,000.
- Locate the 1,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 704,091,000
Worked Example 9
Problem: Round 18,765,432 to the nearest 10,000.
- Locate the 10,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 18,770,000
Worked Example 10
Problem: Round 999,500,001 to the nearest 100,000.
- Locate the 100,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 999,500,000
Practice Exercise
Create one new question about Using rounding for decimal estimates. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.7 Using benchmarks for decimal estimates
Using benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 benchmarks for decimal estimates?
- 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 benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Using benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Using benchmarks for decimal estimates?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Using benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Using benchmarks for decimal estimates.
- 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 benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 benchmarks for decimal estimates 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 benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 benchmarks for decimal estimates.
- 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 benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Using benchmarks for decimal estimates can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Using benchmarks for decimal estimates. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.8 Front-end estimation with decimals
Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Front-end estimation with decimals?
- 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: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Front-end estimation with decimals?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Front-end estimation with decimals.
- 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: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Front-end estimation with decimals 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: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Front-end estimation with decimals.
- 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: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Front-end estimation with decimals can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 638,420,715 to the nearest 10.
- Locate the 10 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 638,420,720
Worked Example 7
Problem: Round 92,305,004 to the nearest 100.
- Locate the 100 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 92,305,000
Worked Example 8
Problem: Round 704,090,650 to the nearest 1,000.
- Locate the 1,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 704,091,000
Worked Example 9
Problem: Round 18,765,432 to the nearest 10,000.
- Locate the 10,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 18,770,000
Worked Example 10
Problem: Round 999,500,001 to the nearest 100,000.
- Locate the 100,000 place.
- Look at the next digit to the right.
- 0–4: keep the target digit; 5–9: increase it by 1.
- Replace lower places with zeros.
Very beginner explanation: Rounding gives a nearby value that is easier to use while staying close to the original number.
Answer: 999,500,000
Practice Exercise
Create one new question about Front-end estimation with decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.9 Explaining a mental-math strategy
Explaining a mental-math strategy 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 Explaining a mental-math strategy?
- 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: Explaining a mental-math strategy compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Explaining a mental-math strategy 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 Explaining a mental-math strategy?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Explaining a mental-math strategy 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 Explaining a mental-math strategy.
- 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: Explaining a mental-math strategy 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 Explaining a mental-math strategy 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: Explaining a mental-math strategy 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 Explaining a mental-math strategy.
- 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: Explaining a mental-math strategy compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Answer: Explaining a mental-math strategy 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 Explaining a mental-math strategy. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.10 Checking whether a decimal estimate is reasonable
Checking whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 whether a decimal estimate is reasonable?
- 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 whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Checking whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Worked Example 2
Problem: What should you identify first before solving a problem about Checking whether a decimal estimate is reasonable?
- Read the complete problem.
- Mark the information that is given.
- Identify exactly what the question asks you to find.
Very beginner explanation: Checking whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Identify the known information and the unknown before choosing an operation or rule.
Worked Example 3
Problem: Choose a useful representation for Checking whether a decimal estimate is reasonable.
- 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 whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 whether a decimal estimate is reasonable 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 whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 whether a decimal estimate is reasonable.
- 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 whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Answer: Checking whether a decimal estimate is reasonable can be used in a real situation where mathematics helps make a clear calculation, comparison, measurement, prediction, or decision.
Worked Example 6
Problem: Round 3.75 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 3.8
Worked Example 7
Problem: Round 8.4 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 8.4
Worked Example 8
Problem: Round 12.05 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 12.1
Worked Example 9
Problem: Round 0.96 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 1
Worked Example 10
Problem: Round 5.125 to the nearest tenth.
- Find the tenths digit.
- Look at the hundredths digit.
- Round up if the hundredths digit is 5 or more.
Very beginner explanation: Rounding uses the next digit to decide whether the target digit stays or increases.
Answer: 5.1
Practice Exercise
Create one new question about Checking whether a decimal estimate is reasonable. 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 Multiplying whole numbers by 0.1.
- Create and solve one original problem about Multiplying whole numbers by 0.01.
- Create and solve one original problem about Understanding decimal shifts using place value.
- Create and solve one original problem about Estimating decimal sums.
- Create and solve one original problem about Estimating decimal differences.
- Create and solve one original problem about Using rounding for decimal estimates.
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 Multiplying whole numbers by 0.1?
Answer: Multiplying whole numbers by 0.1 helps learners understand how digits represent quantities and how whole numbers can be written, compared, or used in everyday situations.
Q2. What is the key idea in Multiplying whole numbers by 0.01?
Answer: Multiplying whole numbers by 0.01 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 Understanding decimal shifts using place value?
Answer: Understanding decimal shifts using place value 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 Estimating decimal sums?
Answer: Estimating decimal sums uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q5. What is the key idea in Estimating decimal differences?
Answer: Estimating decimal differences uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q6. What is the key idea in Using rounding for decimal estimates?
Answer: Using rounding for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q7. What is the key idea in Using benchmarks for decimal estimates?
Answer: Using benchmarks for decimal estimates uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q8. What is the key idea in Front-end estimation with decimals?
Answer: Front-end estimation with decimals uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
Q9. What is the key idea in Explaining a mental-math strategy?
Answer: Explaining a mental-math strategy compares quantities. Equivalent ratios and unit rates help learners scale situations and compare choices fairly.
Q10. What is the key idea in Checking whether a decimal estimate is reasonable?
Answer: Checking whether a decimal estimate is reasonable uses place value to describe parts smaller than one. Keeping decimal places aligned makes comparisons and calculations easier to understand.
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 Mental Math with Decimals 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 Mental Math with Decimals 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 Mental Math with Decimals 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 Mental Math with Decimals 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 Mental Math with Decimals effectively?
Answer: Work a small example, explain every step, check the answer, then try a similar problem without looking at the first solution.