Chapter 25: Equations with Fractions and Decimals
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Equations with Fractions and 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)
- Coefficient (the number multiplying a variable)
- Equation (a statement that two expressions are equal)
- 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.
25.1 Equations with decimal constants
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Equations with decimal constants .
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 Equations with decimal constants .
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 Equations with decimal constants .
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 Equations with decimal constants .
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 Equations with decimal constants .
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 Equations with decimal constants .
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 Equations with decimal constants. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.2 Equations with decimal coefficients
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Equations with decimal coefficients .
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 Equations with decimal coefficients .
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 Equations with decimal coefficients .
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 Equations with decimal coefficients .
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 Equations with decimal coefficients .
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 Equations with decimal coefficients .
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 Equations with decimal coefficients. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.3 Equations with fraction constants
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
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: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Equations with fraction constants. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.4 Equations with fraction coefficients
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
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: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Equations with fraction coefficients. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.5 Clearing simple fractions
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
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: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Clearing simple fractions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.6 Using inverse operations
A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
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 12 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 96 ÷ 8 = 12.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
Answer: 96
Worked Example 2
Problem: Calculate 9 × 7.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 63 ÷ 7 = 9.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
Answer: 63
Worked Example 3
Problem: Calculate 144 × 12.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1728 ÷ 12 = 144.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
Answer: 1728
Worked Example 4
Problem: Calculate 325 × 6.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 1950 ÷ 6 = 325.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
Answer: 1950
Worked Example 5
Problem: Calculate 728 × 8.
- Use a known fact, distributive strategy, or standard multiplication.
- Check using division: 5824 ÷ 8 = 728.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations .
Answer: 5824
Worked Example 6
Problem: Simplify the ratio 4:6.
- Find the greatest common factor, 2.
- Divide both terms by 2.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 2:3
Worked Example 7
Problem: Simplify the ratio 8:12.
- Find the greatest common factor, 4.
- Divide both terms by 4.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 2:3
Worked Example 8
Problem: Simplify the ratio 15:25.
- Find the greatest common factor, 5.
- Divide both terms by 5.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:5
Worked Example 9
Problem: Simplify the ratio 18:30.
- Find the greatest common factor, 6.
- Divide both terms by 6.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:5
Worked Example 10
Problem: Simplify the ratio 21:28.
- Find the greatest common factor, 7.
- Divide both terms by 7.
Very beginner explanation: Simplifying a ratio is like simplifying a fraction: divide both quantities by the same common factor.
Answer: 3:4
Practice Exercise
Create one new question about Using inverse operations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.7 Multi-step equations
An equation says that two expressions are equal. This topic focuses on Multi-step equations .
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: Solve 2x+5=3x-1.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Multi-step equations .
Answer: x=6
Worked Example 2
Problem: Solve 4x-7=13.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Multi-step equations .
Answer: x=5
Worked Example 3
Problem: Solve 3x+2=2x+9.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Multi-step equations .
Answer: x=7
Worked Example 4
Problem: Solve 5x+4=2x+19.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Multi-step equations .
Answer: x=5
Worked Example 5
Problem: Solve 2(x+3)=18.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Multi-step equations .
Answer: x=6
Worked Example 6
Problem: Solve 2x + 5 = 17.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 6
Worked Example 7
Problem: Solve 3x - 4 = 11.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 8
Problem: Solve 5x + 7 = 2x + 22.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 9
Problem: Solve 4(x + 2) = 24.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 4
Worked Example 10
Problem: Solve 7x - 3 = 4x + 12.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Practice Exercise
Create one new question about Multi-step equations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.8 Checking decimal solutions
A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal solutions .
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 solutions .
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 solutions .
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 solutions .
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 solutions .
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 solutions .
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 solutions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.9 Checking fraction solutions
A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
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: Write 1/2 as a decimal.
- Divide 1 by 2.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
Answer: 0.5
Worked Example 2
Problem: Write 2/3 as a decimal.
- Divide 2 by 3.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
Answer: 0.6667
Worked Example 3
Problem: Write 3/4 as a decimal.
- Divide 3 by 4.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
Answer: 0.75
Worked Example 4
Problem: Write 5/6 as a decimal.
- Divide 5 by 6.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
Answer: 0.8333
Worked Example 5
Problem: Write 7/8 as a decimal.
- Divide 7 by 8.
Very beginner explanation: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions .
Answer: 0.875
Worked Example 6
Problem: Write 1/3 as a decimal.
- A fraction bar means division.
- Divide 1 by 3.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.3333
Worked Example 7
Problem: Write 2/5 as a decimal.
- A fraction bar means division.
- Divide 2 by 5.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4
Worked Example 8
Problem: Write 3/8 as a decimal.
- A fraction bar means division.
- Divide 3 by 8.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.375
Worked Example 9
Problem: Write 5/12 as a decimal.
- A fraction bar means division.
- Divide 5 by 12.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.4167
Worked Example 10
Problem: Write 7/9 as a decimal.
- A fraction bar means division.
- Divide 7 by 9.
Very beginner explanation: Fractions and decimals can represent the same rational number.
Answer: 0.7778
Practice Exercise
Create one new question about Checking fraction solutions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.10 Measurement equations
An equation says that two expressions are equal. This topic focuses on Measurement equations .
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: Solve 2x+5=3x-1.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Measurement equations .
Answer: x=6
Worked Example 2
Problem: Solve 4x-7=13.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Measurement equations .
Answer: x=5
Worked Example 3
Problem: Solve 3x+2=2x+9.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Measurement equations .
Answer: x=7
Worked Example 4
Problem: Solve 5x+4=2x+19.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Measurement equations .
Answer: x=5
Worked Example 5
Problem: Solve 2(x+3)=18.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Measurement equations .
Answer: x=6
Worked Example 6
Problem: Solve 2x + 5 = 17.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 6
Worked Example 7
Problem: Solve 3x - 4 = 11.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 8
Problem: Solve 5x + 7 = 2x + 22.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 9
Problem: Solve 4(x + 2) = 24.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 4
Worked Example 10
Problem: Solve 7x - 3 = 4x + 12.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Practice Exercise
Create one new question about Measurement equations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.11 Money equations
An equation says that two expressions are equal. This topic focuses on Money equations .
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: Solve 2x+5=3x-1.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Money equations .
Answer: x=6
Worked Example 2
Problem: Solve 4x-7=13.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Money equations .
Answer: x=5
Worked Example 3
Problem: Solve 3x+2=2x+9.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Money equations .
Answer: x=7
Worked Example 4
Problem: Solve 5x+4=2x+19.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Money equations .
Answer: x=5
Worked Example 5
Problem: Solve 2(x+3)=18.
- Simplify each side if needed.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide to isolate the variable.
- Check by substitution.
Very beginner explanation: An equation says that two expressions are equal. This topic focuses on Money equations .
Answer: x=6
Worked Example 6
Problem: Solve 2x + 5 = 17.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 6
Worked Example 7
Problem: Solve 3x - 4 = 11.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 8
Problem: Solve 5x + 7 = 2x + 22.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Worked Example 9
Problem: Solve 4(x + 2) = 24.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 4
Worked Example 10
Problem: Solve 7x - 3 = 4x + 12.
- Simplify each side if needed.
- Use inverse operations to collect variable terms and constants.
- Keep both sides balanced.
- Check by substitution.
Very beginner explanation: An equation stays balanced when equivalent operations are applied correctly.
Answer: x = 5
Practice Exercise
Create one new question about Money equations. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
25.12 Word problems
Word problems is an important Grade 7 idea in Equations with Fractions and 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: Word problems: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Word problems is an important Grade 7 idea in Equations with Fractions and 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: Word problems: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Word problems is an important Grade 7 idea in Equations with Fractions and 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: Word problems: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Word problems is an important Grade 7 idea in Equations with Fractions and 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: Word problems: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Word problems is an important Grade 7 idea in Equations with Fractions and 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: Word problems: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Word problems is an important Grade 7 idea in Equations with Fractions and 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 Word problems 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: Word problems 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 Word problems 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: Word problems 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 Word problems 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: Word problems 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 Word problems 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: Word problems 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 Word problems 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: Word problems 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 Word problems. 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 Equations with decimal constants . Show your reasoning and check your answer.
- Create and solve a new question about Equations with decimal coefficients . Show your reasoning and check your answer.
- Create and solve a new question about Equations with fraction constants . Show your reasoning and check your answer.
- Create and solve a new question about Equations with fraction coefficients . Show your reasoning and check your answer.
- Create and solve a new question about Clearing simple fractions . Show your reasoning and check your answer.
- Create and solve a new question about Using inverse operations . Show your reasoning and check your answer.
- Create and solve a new question about Multi-step equations . Show your reasoning and check your answer.
- Create and solve a new question about Checking decimal solutions . Show your reasoning and check your answer.
- Create and solve a new question about Checking fraction solutions . Show your reasoning and check your answer.
- Create and solve a new question about Measurement equations . Show your reasoning and check your answer.
- Create and solve a new question about Money equations . Show your reasoning and check your answer.
- Create and solve a new question about Word problems . Show your reasoning and check your answer.
Common Mistakes
- Combining unlike terms.
- Changing only one side of an equation.
- Forgetting to distribute to every term.
- Using a pattern rule that works only for the first few terms.
- Writing code without tracing variable values.
- Using a mathematical model without stating assumptions.
30 Review Questions and Answers
Q1. What is important to remember about Equations with decimal constants?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Equations with decimal constants.
Q2. What is important to remember about Equations with decimal coefficients?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Equations with decimal coefficients.
Q3. What is important to remember about Equations with fraction constants?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction constants.
Q4. What is important to remember about Equations with fraction coefficients?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Equations with fraction coefficients.
Q5. What is important to remember about Clearing simple fractions?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Clearing simple fractions.
Q6. What is important to remember about Using inverse operations?
Answer: A ratio compares two quantities in a fixed order. This topic focuses on Using inverse operations.
Q7. What is important to remember about Multi-step equations?
Answer: An equation says that two expressions are equal. This topic focuses on Multi-step equations.
Q8. What is important to remember about Checking decimal solutions?
Answer: A decimal uses place value to represent parts of one, such as tenths, hundredths, and thousandths. This topic focuses on Checking decimal solutions.
Q9. What is important to remember about Checking fraction solutions?
Answer: A fraction (a numerator over a denominator) can represent part of a whole, a quotient, or a ratio. This topic focuses on Checking fraction solutions.
Q10. What is important to remember about Measurement equations?
Answer: An equation says that two expressions are equal. This topic focuses on Measurement equations.
Q11. What is important to remember about Money equations?
Answer: An equation says that two expressions are equal. This topic focuses on Money equations.
Q12. What is important to remember about Word problems?
Answer: Word problems is an important Grade 7 idea in Equations with Fractions and Decimals. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
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.