Chapter 12: Operations with Decimals
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Operations with Decimals with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Ratio (a comparison of two quantities)
- Estimate (a close approximation used to check whether an answer is reasonable)
- Solution (a value or result that satisfies the problem)
- Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
- Reasonableness (whether an answer makes sense in the context of the problem)
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 Adding decimals
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Calculate 3.47 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Answer: 4.72
Worked Example 2
Problem: Calculate 8.205 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Answer: 9.455
Worked Example 3
Problem: Calculate 0.96 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Answer: 2.21
Worked Example 4
Problem: Calculate 12.375 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Answer: 13.625
Worked Example 5
Problem: Calculate 5.004 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals .
Answer: 6.254
Worked Example 6
Problem: Calculate 3.75 + 1.2.
- Align decimal points.
- Add by place value.
Very beginner explanation: Aligning decimal points keeps tenths with tenths and hundredths with hundredths.
Answer: 4.95
Worked Example 7
Problem: Calculate 8.4 + 0.6.
- Align decimal points.
- Add by place value.
Very beginner explanation: Aligning decimal points keeps tenths with tenths and hundredths with hundredths.
Answer: 9
Worked Example 8
Problem: Calculate 12.05 + 2.5.
- Align decimal points.
- Add by place value.
Very beginner explanation: Aligning decimal points keeps tenths with tenths and hundredths with hundredths.
Answer: 14.55
Worked Example 9
Problem: Calculate 0.96 + 0.4.
- Align decimal points.
- Add by place value.
Very beginner explanation: Aligning decimal points keeps tenths with tenths and hundredths with hundredths.
Answer: 1.36
Worked Example 10
Problem: Calculate 5.125 + 1.25.
- Align decimal points.
- Add by place value.
Very beginner explanation: Aligning decimal points keeps tenths with tenths and hundredths with hundredths.
Answer: 6.375
Practice Exercise
Create one new question about Adding decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.2 Subtracting decimals
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Calculate 3.47 - 0.75.
- Align decimal points.
- Subtract by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Answer: 2.72
Worked Example 2
Problem: Calculate 8.205 - 0.75.
- Align decimal points.
- Subtract by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Answer: 7.455
Worked Example 3
Problem: Calculate 0.96 - 0.75.
- Align decimal points.
- Subtract by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Answer: 0.21
Worked Example 4
Problem: Calculate 12.375 - 0.75.
- Align decimal points.
- Subtract by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Answer: 11.625
Worked Example 5
Problem: Calculate 5.004 - 0.75.
- Align decimal points.
- Subtract by place value.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals .
Answer: 4.254
Worked Example 6
Problem: Calculate 3.75 - 1.2.
- Align decimal points.
- Add trailing zeros if helpful.
- Subtract by place value.
Very beginner explanation: Trailing zeros can be added without changing a decimal value.
Answer: 2.55
Worked Example 7
Problem: Calculate 8.4 - 0.6.
- Align decimal points.
- Add trailing zeros if helpful.
- Subtract by place value.
Very beginner explanation: Trailing zeros can be added without changing a decimal value.
Answer: 7.8
Worked Example 8
Problem: Calculate 12.05 - 2.5.
- Align decimal points.
- Add trailing zeros if helpful.
- Subtract by place value.
Very beginner explanation: Trailing zeros can be added without changing a decimal value.
Answer: 9.55
Worked Example 9
Problem: Calculate 0.96 - 0.4.
- Align decimal points.
- Add trailing zeros if helpful.
- Subtract by place value.
Very beginner explanation: Trailing zeros can be added without changing a decimal value.
Answer: 0.56
Worked Example 10
Problem: Calculate 5.125 - 1.25.
- Align decimal points.
- Add trailing zeros if helpful.
- Subtract by place value.
Very beginner explanation: Trailing zeros can be added without changing a decimal value.
Answer: 3.875
Practice Exercise
Create one new question about Subtracting decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.3 Multiplying decimals
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Calculate 3.47 × 0.4.
- Multiply as whole numbers.
- Place the decimal using the total number of decimal places.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Answer: 1.388
Worked Example 2
Problem: Calculate 8.205 × 0.4.
- Multiply as whole numbers.
- Place the decimal using the total number of decimal places.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Answer: 3.282
Worked Example 3
Problem: Calculate 0.96 × 0.4.
- Multiply as whole numbers.
- Place the decimal using the total number of decimal places.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Answer: 0.384
Worked Example 4
Problem: Calculate 12.375 × 0.4.
- Multiply as whole numbers.
- Place the decimal using the total number of decimal places.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Answer: 4.95
Worked Example 5
Problem: Calculate 5.004 × 0.4.
- Multiply as whole numbers.
- Place the decimal using the total number of decimal places.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals .
Answer: 2.0016
Worked Example 6
Problem: Calculate 3.75 × 1.2.
- Multiply as whole numbers.
- Count total decimal places in the factors.
- Place the decimal in the product.
Very beginner explanation: The decimal position in the product comes from the total decimal places in both factors.
Answer: 4.5
Worked Example 7
Problem: Calculate 8.4 × 0.6.
- Multiply as whole numbers.
- Count total decimal places in the factors.
- Place the decimal in the product.
Very beginner explanation: The decimal position in the product comes from the total decimal places in both factors.
Answer: 5.04
Worked Example 8
Problem: Calculate 12.05 × 2.5.
- Multiply as whole numbers.
- Count total decimal places in the factors.
- Place the decimal in the product.
Very beginner explanation: The decimal position in the product comes from the total decimal places in both factors.
Answer: 30.125
Worked Example 9
Problem: Calculate 0.96 × 0.4.
- Multiply as whole numbers.
- Count total decimal places in the factors.
- Place the decimal in the product.
Very beginner explanation: The decimal position in the product comes from the total decimal places in both factors.
Answer: 0.384
Worked Example 10
Problem: Calculate 5.125 × 1.25.
- Multiply as whole numbers.
- Count total decimal places in the factors.
- Place the decimal in the product.
Very beginner explanation: The decimal position in the product comes from the total decimal places in both factors.
Answer: 6.4062
Practice Exercise
Create one new question about Multiplying decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.4 Dividing decimals
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Calculate 3.47 ÷ 2.
- Divide normally.
- Keep the decimal aligned in the quotient.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Answer: 1.735
Worked Example 2
Problem: Calculate 8.205 ÷ 2.
- Divide normally.
- Keep the decimal aligned in the quotient.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Answer: 4.1025
Worked Example 3
Problem: Calculate 0.96 ÷ 2.
- Divide normally.
- Keep the decimal aligned in the quotient.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Answer: 0.48
Worked Example 4
Problem: Calculate 12.375 ÷ 2.
- Divide normally.
- Keep the decimal aligned in the quotient.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Answer: 6.1875
Worked Example 5
Problem: Calculate 5.004 ÷ 2.
- Divide normally.
- Keep the decimal aligned in the quotient.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals .
Answer: 2.502
Worked Example 6
Problem: Calculate 3.75 ÷ 1.2.
- Make the divisor a whole number by shifting its decimal if needed.
- Shift the dividend by the same amount.
- Divide.
Very beginner explanation: Multiplying dividend and divisor by the same power of 10 keeps the quotient unchanged.
Answer: 3.125
Worked Example 7
Problem: Calculate 8.4 ÷ 0.6.
- Make the divisor a whole number by shifting its decimal if needed.
- Shift the dividend by the same amount.
- Divide.
Very beginner explanation: Multiplying dividend and divisor by the same power of 10 keeps the quotient unchanged.
Answer: 14
Worked Example 8
Problem: Calculate 12.05 ÷ 2.5.
- Make the divisor a whole number by shifting its decimal if needed.
- Shift the dividend by the same amount.
- Divide.
Very beginner explanation: Multiplying dividend and divisor by the same power of 10 keeps the quotient unchanged.
Answer: 4.82
Worked Example 9
Problem: Calculate 0.96 ÷ 0.4.
- Make the divisor a whole number by shifting its decimal if needed.
- Shift the dividend by the same amount.
- Divide.
Very beginner explanation: Multiplying dividend and divisor by the same power of 10 keeps the quotient unchanged.
Answer: 2.4
Worked Example 10
Problem: Calculate 5.125 ÷ 1.25.
- Make the divisor a whole number by shifting its decimal if needed.
- Shift the dividend by the same amount.
- Divide.
Very beginner explanation: Multiplying dividend and divisor by the same power of 10 keeps the quotient unchanged.
Answer: 4.1
Practice Exercise
Create one new question about Dividing decimals. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.5 Multiplying by 10, 100, and 1000
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Find 10% of $80.
- Convert 10% to 0.1.
- Multiply by 80.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Answer: $8.00
Worked Example 2
Problem: Find 25% of $60.
- Convert 25% to 0.25.
- Multiply by 60.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Answer: $15.00
Worked Example 3
Problem: Find 15% of $120.
- Convert 15% to 0.15.
- Multiply by 120.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Answer: $18.00
Worked Example 4
Problem: Find 5% of $250.
- Convert 5% to 0.05.
- Multiply by 250.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Answer: $12.50
Worked Example 5
Problem: Find 13% of $75.
- Convert 13% to 0.13.
- Multiply by 75.
Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000 .
Answer: $9.75
Worked Example 6
Problem: Find 10% of $90.
- Convert 10% to 0.1.
- Multiply by 90.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $9.00
Worked Example 7
Problem: Find 25% of $64.
- Convert 25% to 0.25.
- Multiply by 64.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $16.00
Worked Example 8
Problem: Find 15% of $140.
- Convert 15% to 0.15.
- Multiply by 140.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $21.00
Worked Example 9
Problem: Find 5% of $260.
- Convert 5% to 0.05.
- Multiply by 260.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $13.00
Worked Example 10
Problem: Find 20% of $75.
- Convert 20% to 0.2.
- Multiply by 75.
Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.
Answer: $15.00
Practice Exercise
Create one new question about Multiplying by 10, 100, and 1000. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.6 Dividing by 10, 100, and 1000
Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Dividing by 10, 100, and 1000: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Dividing by 10, 100, and 1000: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Dividing by 10, 100, and 1000: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Dividing by 10, 100, and 1000: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Dividing by 10, 100, and 1000: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 6
Problem: Explain Dividing by 10, 100, and 1000 in one simple sentence.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Dividing by 10, 100, and 1000 becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: State the meaning, then give one small example.
Worked Example 7
Problem: Give one example that does not satisfy Dividing by 10, 100, and 1000 and explain why.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Dividing by 10, 100, and 1000 becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Identify the rule or condition that fails.
Worked Example 8
Problem: Show Dividing by 10, 100, and 1000 using a table, diagram, number line, graph, or equation.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Dividing by 10, 100, and 1000 becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Choose the representation that makes the relationship easiest to see.
Worked Example 9
Problem: After solving a Dividing by 10, 100, and 1000 problem, how can you check the answer?
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Dividing by 10, 100, and 1000 becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Use estimation, an inverse operation, substitution, or a second representation.
Worked Example 10
Problem: Give one real-life situation where Dividing by 10, 100, and 1000 could be useful.
- Read the definition of the topic.
- Use the definition to build the response.
- Check that the response matches every important condition.
Very beginner explanation: Dividing by 10, 100, and 1000 becomes easier to understand when you connect its definition to examples, non-examples, and representations.
Answer: Connect the idea to money, measurement, data, design, or technology.
Practice Exercise
Create one new question about Dividing by 10, 100, and 1000. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.7 Aligning decimal places
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places .
Answer: 0
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 Aligning decimal places. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.8 Estimating decimal results
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results .
Answer: 0
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 results. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.9 Money calculations
Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Money calculations: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Money calculations: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Money calculations: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Money calculations: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Money calculations: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
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 Money calculations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.10 Measurement calculations
Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Measurement calculations: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 2
Problem: Measurement calculations: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 3
Problem: Measurement calculations: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 4
Problem: Measurement calculations: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
Worked Example 5
Problem: Measurement calculations: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: A correct response must satisfy the definition or rule and include a check.
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 Measurement calculations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.11 Multi-step decimal problems
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems .
Answer: 0
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 Multi-step decimal problems. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
12.12 Checking decimal answers
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Beginner Note
Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Identify the tenths digit in 3.47.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Answer: 4
Worked Example 2
Problem: Identify the tenths digit in 8.205.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Answer: 2
Worked Example 3
Problem: Identify the tenths digit in 0.96.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Answer: 9
Worked Example 4
Problem: Identify the tenths digit in 12.375.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Answer: 3
Worked Example 5
Problem: Identify the tenths digit in 5.004.
- The tenths digit is the first digit to the right of the decimal point.
Very beginner explanation: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers .
Answer: 0
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 decimal answers. 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 complete question before choosing an operation, formula, graph, or model.
- Write technical words together with their meaning until you are comfortable using them.
- Show all important steps so another student can follow your reasoning.
- Keep units, labels, signs, axes, variables, and mathematical symbols clear.
- Estimate before or after calculating when an estimate can help check reasonableness.
- For real-life problems, explain what the final number means in the situation.
Extra Practice
- Create and solve a new question about Adding decimals . Show your reasoning and check your answer.
- Create and solve a new question about Subtracting decimals . Show your reasoning and check your answer.
- Create and solve a new question about Multiplying decimals . Show your reasoning and check your answer.
- Create and solve a new question about Dividing decimals . Show your reasoning and check your answer.
- Create and solve a new question about Multiplying by 10, 100, and 1000 . Show your reasoning and check your answer.
- Create and solve a new question about Dividing by 10, 100, and 1000 . Show your reasoning and check your answer.
- Create and solve a new question about Aligning decimal places . Show your reasoning and check your answer.
- Create and solve a new question about Estimating decimal results . Show your reasoning and check your answer.
- Create and solve a new question about Money calculations . Show your reasoning and check your answer.
- Create and solve a new question about Measurement calculations . Show your reasoning and check your answer.
- Create and solve a new question about Multi-step decimal problems . Show your reasoning and check your answer.
- Create and solve a new question about Checking decimal answers . Show your reasoning and check your answer.
Common Mistakes
- Changing only one part of an equivalent fraction or ratio.
- Moving a decimal point without a mathematical reason.
- Using the new amount instead of the original amount for percent change.
- Mixing units when comparing rates or scale.
30 Review Questions and Answers
Q1. What is important to remember about Adding decimals?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Adding decimals.
Q2. What is important to remember about Subtracting decimals?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Subtracting decimals.
Q3. What is important to remember about Multiplying decimals?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multiplying decimals.
Q4. What is important to remember about Dividing decimals?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Dividing decimals.
Q5. What is important to remember about Multiplying by 10, 100, and 1000?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplying by 10, 100, and 1000.
Q6. What is important to remember about Dividing by 10, 100, and 1000?
Answer: Dividing by 10, 100, and 1000 is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q7. What is important to remember about Aligning decimal places?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Aligning decimal places.
Q8. What is important to remember about Estimating decimal results?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Estimating decimal results.
Q9. What is important to remember about Money calculations?
Answer: Money calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q10. What is important to remember about Measurement calculations?
Answer: Measurement calculations is an important Grade 7 idea in Operations with Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q11. What is important to remember about Multi-step decimal problems?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Multi-step decimal problems.
Q12. What is important to remember about Checking decimal answers?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal answers.
Q13. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q14. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q15. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q16. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q17. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q18. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q19. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q20. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q21. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q22. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q23. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q24. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q25. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q26. When is a calculator useful?
Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.
Q27. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q28. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q29. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.
Q30. Why should sources be checked in data problems?
Answer: Reliable sources and fair collection methods make conclusions more trustworthy.