Chapter 19: Proportional Reasoning
Learn Grade 7 mathematics from very beginner explanations through worked examples, practice, reasoning, and review.
Chapter Overview
This chapter teaches Proportional Reasoning with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.
Key Technical Terms
- Percent (a ratio out of 100)
- Ratio (a comparison of two quantities)
- Unit Rate (a rate per one unit)
- Proportion (an equation showing two equal ratios)
- Circle Graph (a graph using sectors of a circle to show parts of a whole)
- Exchange Rate (the value of one currency expressed in another currency)
- Interest Rate (percentage used to calculate interest)
- 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)
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.
19.1 Meaning of proportion
A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
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 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion .
Answer: x=18
Worked Example 6
Problem: Solve 4/6 = x/15.
- Use equivalent ratios or cross multiplication.
- 6x = 60.
- x = 60 ÷ 6.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 10
Worked Example 7
Problem: Solve 8/12 = x/20.
- Use equivalent ratios or cross multiplication.
- 12x = 160.
- x = 160 ÷ 12.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 13.3333
Worked Example 8
Problem: Solve 15/25 = x/25.
- Use equivalent ratios or cross multiplication.
- 25x = 375.
- x = 375 ÷ 25.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 15
Worked Example 9
Problem: Solve 18/30 = x/30.
- Use equivalent ratios or cross multiplication.
- 30x = 540.
- x = 540 ÷ 30.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 18
Worked Example 10
Problem: Solve 21/28 = x/35.
- Use equivalent ratios or cross multiplication.
- 28x = 735.
- x = 735 ÷ 28.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 26.25
Practice Exercise
Create one new question about Meaning of proportion. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.2 Equivalent ratios
A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
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: Simplify 4:6.
- GCF(4,6)=2.
- Divide both terms by 2.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
Answer: 2:3
Worked Example 2
Problem: Simplify 8:12.
- GCF(8,12)=4.
- Divide both terms by 4.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
Answer: 2:3
Worked Example 3
Problem: Simplify 15:25.
- GCF(15,25)=5.
- Divide both terms by 5.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
Answer: 3:5
Worked Example 4
Problem: Simplify 21:28.
- GCF(21,28)=7.
- Divide both terms by 7.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
Answer: 3:4
Worked Example 5
Problem: Simplify 18:30.
- GCF(18,30)=6.
- Divide both terms by 6.
Very beginner explanation: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios .
Answer: 3:5
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 Equivalent ratios. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.3 Solving missing values
Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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: Solving missing values: Identify a correct example.
- State the definition and show why the example fits.
- Show all important reasoning.
Very beginner explanation: Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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: Solving missing values: Identify a non-example.
- Explain which requirement is missing.
- Show all important reasoning.
Very beginner explanation: Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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: Solving missing values: Compare two cases.
- State one similarity and one difference.
- Show all important reasoning.
Very beginner explanation: Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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: Solving missing values: Apply the idea in a real situation.
- Translate the situation into mathematical language.
- Show all important reasoning.
Very beginner explanation: Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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: Solving missing values: Create and check your own example.
- Use the definition, then verify each condition.
- Show all important reasoning.
Very beginner explanation: Solving missing values is an important Grade 7 idea in Proportional Reasoning. 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 Solving missing values 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: Solving missing values 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 Solving missing values 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: Solving missing values 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 Solving missing values 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: Solving missing values 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 Solving missing values 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: Solving missing values 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 Solving missing values 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: Solving missing values 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 Solving missing values. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.4 Unit-rate method
A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
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: 120 km in 2 h. Find the unit rate.
- Use rate = distance ÷ time.
- 120÷2=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
Answer: 60 km/h
Worked Example 2
Problem: 180 km in 3 h. Find the unit rate.
- Use rate = distance ÷ time.
- 180÷3=60.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
Answer: 60 km/h
Worked Example 3
Problem: 250 km in 5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 250÷5=50.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
Answer: 50 km/h
Worked Example 4
Problem: 72 km in 1.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 72÷1.5=48.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
Answer: 48 km/h
Worked Example 5
Problem: 315 km in 4.5 h. Find the unit rate.
- Use rate = distance ÷ time.
- 315÷4.5=70.
Very beginner explanation: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method .
Answer: 70 km/h
Worked Example 6
Problem: 120 km are travelled in 2 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 120 ÷ 2 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 7
Problem: 180 km are travelled in 3 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 180 ÷ 3 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 8
Problem: 240 km are travelled in 4 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 240 ÷ 4 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 9
Problem: 300 km are travelled in 5 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 300 ÷ 5 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Worked Example 10
Problem: 360 km are travelled in 6 hours. Find the unit rate.
- Use rate = distance ÷ time.
- 360 ÷ 6 = 60.
Very beginner explanation: A unit rate compares a quantity with exactly one unit of another quantity.
Answer: 60 km/h
Practice Exercise
Create one new question about Unit-rate method. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.5 Scale-up method
Scale-up method is an important Grade 7 idea in Proportional Reasoning. 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: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 10 cm
Worked Example 6
Problem: Explain Scale-up method 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: Scale-up method 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 Scale-up method 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: Scale-up method 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 Scale-up method 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: Scale-up method 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 Scale-up method 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: Scale-up method 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 Scale-up method 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: Scale-up method 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 Scale-up method. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.6 Scale-down method
Scale-down method is an important Grade 7 idea in Proportional Reasoning. 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: Scale factor 2; original side 6 cm. Find image side.
- Multiply 6 by 2.
Very beginner explanation: Scale-down method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 2
Problem: Scale factor 1.5; original side 8 cm. Find image side.
- Multiply 8 by 1.5.
Very beginner explanation: Scale-down method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 12 cm
Worked Example 3
Problem: Scale factor 0.5; original side 14 cm. Find image side.
- Multiply 14 by 0.5.
Very beginner explanation: Scale-down method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 7 cm
Worked Example 4
Problem: Scale factor 3; original side 5 cm. Find image side.
- Multiply 5 by 3.
Very beginner explanation: Scale-down method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 15 cm
Worked Example 5
Problem: Scale factor 4; original side 2.5 cm. Find image side.
- Multiply 2.5 by 4.
Very beginner explanation: Scale-down method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: 10 cm
Worked Example 6
Problem: Explain Scale-down method 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: Scale-down method 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 Scale-down method 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: Scale-down method 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 Scale-down method 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: Scale-down method 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 Scale-down method 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: Scale-down method 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 Scale-down method 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: Scale-down method 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 Scale-down method. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.7 Cross multiplication introduction
A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Cross multiplication introduction .
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 Cross multiplication introduction .
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 Cross multiplication introduction .
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 Cross multiplication introduction .
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 Cross multiplication introduction .
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 Cross multiplication introduction .
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 Cross multiplication introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.8 Proportion tables
A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
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 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables .
Answer: x=18
Worked Example 6
Problem: Solve 4/6 = x/15.
- Use equivalent ratios or cross multiplication.
- 6x = 60.
- x = 60 ÷ 6.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 10
Worked Example 7
Problem: Solve 8/12 = x/20.
- Use equivalent ratios or cross multiplication.
- 12x = 160.
- x = 160 ÷ 12.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 13.3333
Worked Example 8
Problem: Solve 15/25 = x/25.
- Use equivalent ratios or cross multiplication.
- 25x = 375.
- x = 375 ÷ 25.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 15
Worked Example 9
Problem: Solve 18/30 = x/30.
- Use equivalent ratios or cross multiplication.
- 30x = 540.
- x = 540 ÷ 30.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 18
Worked Example 10
Problem: Solve 21/28 = x/35.
- Use equivalent ratios or cross multiplication.
- 28x = 735.
- x = 735 ÷ 28.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 26.25
Practice Exercise
Create one new question about Proportion tables. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.9 Proportional graphs
A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
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 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs .
Answer: x=18
Worked Example 6
Problem: Solve 4/6 = x/15.
- Use equivalent ratios or cross multiplication.
- 6x = 60.
- x = 60 ÷ 6.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 10
Worked Example 7
Problem: Solve 8/12 = x/20.
- Use equivalent ratios or cross multiplication.
- 12x = 160.
- x = 160 ÷ 12.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 13.3333
Worked Example 8
Problem: Solve 15/25 = x/25.
- Use equivalent ratios or cross multiplication.
- 25x = 375.
- x = 375 ÷ 25.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 15
Worked Example 9
Problem: Solve 18/30 = x/30.
- Use equivalent ratios or cross multiplication.
- 30x = 540.
- x = 540 ÷ 30.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 18
Worked Example 10
Problem: Solve 21/28 = x/35.
- Use equivalent ratios or cross multiplication.
- 28x = 735.
- x = 735 ÷ 28.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 26.25
Practice Exercise
Create one new question about Proportional graphs. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.10 Constant of proportionality introduction
A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
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 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction .
Answer: x=18
Worked Example 6
Problem: Solve 4/6 = x/15.
- Use equivalent ratios or cross multiplication.
- 6x = 60.
- x = 60 ÷ 6.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 10
Worked Example 7
Problem: Solve 8/12 = x/20.
- Use equivalent ratios or cross multiplication.
- 12x = 160.
- x = 160 ÷ 12.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 13.3333
Worked Example 8
Problem: Solve 15/25 = x/25.
- Use equivalent ratios or cross multiplication.
- 25x = 375.
- x = 375 ÷ 25.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 15
Worked Example 9
Problem: Solve 18/30 = x/30.
- Use equivalent ratios or cross multiplication.
- 30x = 540.
- x = 540 ÷ 30.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 18
Worked Example 10
Problem: Solve 21/28 = x/35.
- Use equivalent ratios or cross multiplication.
- 28x = 735.
- x = 735 ÷ 28.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 26.25
Practice Exercise
Create one new question about Constant of proportionality introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.11 Direct variation introduction
Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. 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: Solve 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Answer: x=18
Worked Example 6
Problem: Explain Direct variation introduction 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: Direct variation introduction 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 Direct variation introduction 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: Direct variation introduction 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 Direct variation introduction 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: Direct variation introduction 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 Direct variation introduction 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: Direct variation introduction 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 Direct variation introduction 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: Direct variation introduction 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 Direct variation introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.12 Percent proportions
Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
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.
- 10% = 0.1.
- 0.1×80=8.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
Answer: 8
Worked Example 2
Problem: Find 25% of 60.
- 25% = 0.25.
- 0.25×60=15.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
Answer: 15
Worked Example 3
Problem: Find 15% of 120.
- 15% = 0.15.
- 0.15×120=18.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
Answer: 18
Worked Example 4
Problem: Find 5% of 250.
- 5% = 0.05.
- 0.05×250=12.5.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
Answer: 12.5
Worked Example 5
Problem: Find 13% of 75.
- 13% = 0.13.
- 0.13×75=9.75.
Very beginner explanation: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions .
Answer: 9.75
Worked Example 6
Problem: Find 10% of 90.
- 10% = 0.1.
- 0.1 × 90 = 9.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 9
Worked Example 7
Problem: Find 25% of 64.
- 25% = 0.25.
- 0.25 × 64 = 16.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 16
Worked Example 8
Problem: Find 15% of 140.
- 15% = 0.15.
- 0.15 × 140 = 21.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 21
Worked Example 9
Problem: Find 5% of 260.
- 5% = 0.05.
- 0.05 × 260 = 13.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 13
Worked Example 10
Problem: Find 20% of 75.
- 20% = 0.2.
- 0.2 × 75 = 15.
Very beginner explanation: Percent means per hundred, so converting the percent to a decimal makes multiplication straightforward.
Answer: 15
Practice Exercise
Create one new question about Percent proportions. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.
19.13 Real-life proportion problems
A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion 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: Solve 2/5 = x/10.
- Use equivalent ratios or cross multiplication.
- x=2×10÷5.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems .
Answer: x=4
Worked Example 2
Problem: Solve 3/4 = x/20.
- Use equivalent ratios or cross multiplication.
- x=3×20÷4.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems .
Answer: x=15
Worked Example 3
Problem: Solve 5/8 = x/24.
- Use equivalent ratios or cross multiplication.
- x=5×24÷8.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems .
Answer: x=15
Worked Example 4
Problem: Solve 4/7 = x/21.
- Use equivalent ratios or cross multiplication.
- x=4×21÷7.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems .
Answer: x=12
Worked Example 5
Problem: Solve 6/9 = x/27.
- Use equivalent ratios or cross multiplication.
- x=6×27÷9.
Very beginner explanation: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems .
Answer: x=18
Worked Example 6
Problem: Solve 4/6 = x/15.
- Use equivalent ratios or cross multiplication.
- 6x = 60.
- x = 60 ÷ 6.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 10
Worked Example 7
Problem: Solve 8/12 = x/20.
- Use equivalent ratios or cross multiplication.
- 12x = 160.
- x = 160 ÷ 12.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 13.3333
Worked Example 8
Problem: Solve 15/25 = x/25.
- Use equivalent ratios or cross multiplication.
- 25x = 375.
- x = 375 ÷ 25.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 15
Worked Example 9
Problem: Solve 18/30 = x/30.
- Use equivalent ratios or cross multiplication.
- 30x = 540.
- x = 540 ÷ 30.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 18
Worked Example 10
Problem: Solve 21/28 = x/35.
- Use equivalent ratios or cross multiplication.
- 28x = 735.
- x = 735 ÷ 28.
Very beginner explanation: A proportion states that two ratios have equal value.
Answer: 26.25
Practice Exercise
Create one new question about Real-life proportion 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 Meaning of proportion . Show your reasoning and check your answer.
- Create and solve a new question about Equivalent ratios . Show your reasoning and check your answer.
- Create and solve a new question about Solving missing values . Show your reasoning and check your answer.
- Create and solve a new question about Unit-rate method . Show your reasoning and check your answer.
- Create and solve a new question about Scale-up method . Show your reasoning and check your answer.
- Create and solve a new question about Scale-down method . Show your reasoning and check your answer.
- Create and solve a new question about Cross multiplication introduction . Show your reasoning and check your answer.
- Create and solve a new question about Proportion tables . Show your reasoning and check your answer.
- Create and solve a new question about Proportional graphs . Show your reasoning and check your answer.
- Create and solve a new question about Constant of proportionality introduction . Show your reasoning and check your answer.
- Create and solve a new question about Direct variation introduction . Show your reasoning and check your answer.
- Create and solve a new question about Percent proportions . 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 Meaning of proportion?
Answer: A proportion is an equation showing that two ratios are equal. This topic focuses on Meaning of proportion.
Q2. What is important to remember about Equivalent ratios?
Answer: A ratio compares two quantities in a fixed order. This topic focuses on Equivalent ratios.
Q3. What is important to remember about Solving missing values?
Answer: Solving missing values is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q4. What is important to remember about Unit-rate method?
Answer: A rate compares two quantities measured in different units. This topic focuses on Unit-rate method.
Q5. What is important to remember about Scale-up method?
Answer: Scale-up method is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q6. What is important to remember about Scale-down method?
Answer: Scale-down method is an important Grade 7 idea in Proportional Reasoning. 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 Cross multiplication introduction?
Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Cross multiplication introduction.
Q8. What is important to remember about Proportion tables?
Answer: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportion tables.
Q9. What is important to remember about Proportional graphs?
Answer: A proportion is an equation showing that two ratios are equal. This topic focuses on Proportional graphs.
Q10. What is important to remember about Constant of proportionality introduction?
Answer: A proportion is an equation showing that two ratios are equal. This topic focuses on Constant of proportionality introduction.
Q11. What is important to remember about Direct variation introduction?
Answer: Direct variation introduction is an important Grade 7 idea in Proportional Reasoning. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.
Q12. What is important to remember about Percent proportions?
Answer: Percent means per hundred, so 35% means 35 out of 100. This topic focuses on Percent proportions.
Q13. What is important to remember about Real-life proportion problems?
Answer: A proportion is an equation showing that two ratios are equal. This topic focuses on Real-life proportion problems.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.
Q16. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q17. When should you round?
Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.
Q18. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the relationship connecting them.
Q20. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q21. What should you do if an answer seems unreasonable?
Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer you obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.
Q26. How should you study this chapter?
Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.
Q27. 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.
Q28. Why compare more than one strategy?
Answer: Different strategies can make a problem easier and provide a way to verify the result.
Q29. What is a mathematical model?
Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.
Q30. Why should assumptions be stated?
Answer: Assumptions show what conditions the solution depends on and help readers judge whether the model is reasonable.