Understanding Equivalent Fractions Problem Type 2 in Grade 3 - Mathematics - Exercises with solutions

Master equivalent fractions problem type 2: finding missing numerators and denominators, cross multiplication, and advanced fraction concepts through these 10 detailed exercises.

Solutions: Exercises 1 to 10
1 Finding missing numerators
Exercise 1
Find the missing numerator: 3/4 = ?/8
Difficulty: Beginner Time: ~4 minutes Skills: Cross Multiplication, Proportional Reasoning
Definition:

Missing numerator problem: A type of equivalent fraction problem where one numerator is unknown, requiring the use of cross multiplication or proportional reasoning to solve.

Note: In equivalent fractions, the relationship between numerators is the same as the relationship between denominators.

Step-by-step method:
  1. Identify the known fraction and the fraction with the missing value
  2. Set up the equation using cross multiplication: 3×8 = 4×?
  3. Solve for the unknown value by dividing both sides by the known denominator
  4. Verify the solution by checking if the fractions are equivalent
Given
3/4 = ?/8
Cross Multiply
3×8 = 4×?
Solve
? = 6
Step 1: Set up cross multiplication

3/4 = ?/8 → 3×8 = 4×?

Step 2: Calculate known product

3×8 = 24

Step 3: Solve for missing numerator

24 = 4×? → ? = 24÷4 = 6

Step 4: Verify solution

3/4 = 6/8 → Cross multiply: 3×8 = 24 and 4×6 = 24 ✓

3/4 = 6/8
Final answer:

The missing numerator is 6, so 3/4 = 6/8.

Applied rules:

• Cross multiplication: In equivalent fractions a/b = c/d, a×d = b×c

• Proportional reasoning: Relationship between numerators matches denominators

• Verification: Always check solution by substituting back into original equation

• Practice Tip: Divide the new denominator by the old denominator to find the multiplication factor

Related Examples:
  • 2/5 = ?/10 → 2×10 = 5×? → ? = 4, so 2/5 = 4/10
  • 1/3 = ?/9 → 1×9 = 3×? → ? = 3, so 1/3 = 3/9
  • 5/6 = ?/12 → 5×12 = 6×? → ? = 10, so 5/6 = 10/12
Quick Tips:
  • Always cross multiply to find missing values in equivalent fractions
  • Check if the denominator was multiplied by a factor, then multiply numerator by same factor
  • Verify your answer by confirming the fractions are equivalent
Frequently Asked Questions:

Q: What if I don't want to use cross multiplication?
A: You can find the multiplication factor by dividing the new denominator by the old denominator, then multiply the numerator by the same factor.

Q: How do I check my answer?
A: Substitute your answer back into the original equation and verify using cross multiplication.

2 Finding missing denominators
Exercise 2
Find the missing denominator: 2/3 = 4/?
Difficulty: Beginner Time: ~4 minutes Skills: Cross Multiplication, Proportional Reasoning
Definition:

Missing denominator problem: A type of equivalent fraction problem where one denominator is unknown, requiring the use of cross multiplication or proportional reasoning to solve.

Note: In equivalent fractions, the relationship between numerators is the same as the relationship between denominators.

Step-by-step method:
  1. Identify the known fraction and the fraction with the missing value
  2. Set up the equation using cross multiplication: 2×? = 3×4
  3. Solve for the unknown value by dividing both sides by the known numerator
  4. Verify the solution by checking if the fractions are equivalent
Given
2/3 = 4/?
Cross Multiply
2×? = 3×4
Solve
? = 6
Step 1: Set up cross multiplication

2/3 = 4/? → 2×? = 3×4

Step 2: Calculate known product

3×4 = 12

Step 3: Solve for missing denominator

2×? = 12 → ? = 12÷2 = 6

Step 4: Verify solution

2/3 = 4/6 → Cross multiply: 2×6 = 12 and 3×4 = 12 ✓

2/3 = 4/6
Final answer:

The missing denominator is 6, so 2/3 = 4/6.

Applied rules:

• Cross multiplication: In equivalent fractions a/b = c/d, a×d = b×c

• Proportional reasoning: Relationship between numerators matches denominators

• Verification: Always check solution by substituting back into original equation

• Practice Tip: Divide the new numerator by the old numerator to find the multiplication factor

Related Examples:
  • 3/5 = 6/? → 3×? = 5×6 → ? = 10, so 3/5 = 6/10
  • 1/4 = 3/? → 1×? = 4×3 → ? = 12, so 1/4 = 3/12
  • 4/7 = 8/? → 4×? = 7×8 → ? = 14, so 4/7 = 8/14
Quick Tips:
  • Always cross multiply to find missing values in equivalent fractions
  • Check if the numerator was multiplied by a factor, then multiply denominator by same factor
  • Verify your answer by confirming the fractions are equivalent
Frequently Asked Questions:

Q: What if I don't want to use cross multiplication?
A: You can find the multiplication factor by dividing the new numerator by the old numerator, then multiply the denominator by the same factor.

Q: How do I check my answer?
A: Substitute your answer back into the original equation and verify using cross multiplication.

3 Cross multiplication method
Exercise 3
Use cross multiplication to find the missing value: 5/6 = ?/18
Difficulty: Intermediate Time: ~5 minutes Skills: Cross Multiplication, Proportional Reasoning
Definition:

Cross multiplication method: A systematic approach to solve equivalent fraction problems by multiplying diagonally across the fractions.

Note: For equivalent fractions a/b = c/d, the cross products a×d and b×c are always equal.

Step-by-step method:
  1. Set up the equivalent fraction equation
  2. Apply cross multiplication: multiply numerator of first fraction by denominator of second
  3. Set this equal to denominator of first fraction times numerator of second
  4. Solve for the unknown variable
Given
5/6 = ?/18
Cross Multiply
5×18 = 6×?
Solve
? = 15
Step 1: Set up the equation

5/6 = ?/18

Step 2: Cross multiply

5 × 18 = 6 × ?

Step 3: Calculate known product

5 × 18 = 90

Step 4: Solve for missing value

90 = 6 × ? → ? = 90 ÷ 6 = 15

Step 5: Verify solution

5/6 = 15/18 → Cross multiply: 5×18 = 90 and 6×15 = 90 ✓

5/6 = 15/18
Final answer:

The missing numerator is 15, so 5/6 = 15/18.

Applied rules:

• Cross multiplication: In equivalent fractions a/b = c/d, a×d = b×c

• Diagonal multiplication: Multiply across the diagonals of the fractions

• Isolate variable: Divide both sides by the coefficient of the unknown

• Verification: Confirm with cross multiplication after solving

Related Examples:
  • 3/4 = ?/12 → 3×12 = 4×? → ? = 9, so 3/4 = 9/12
  • 2/7 = ?/21 → 2×21 = 7×? → ? = 6, so 2/7 = 6/21
  • 4/5 = ?/25 → 4×25 = 5×? → ? = 20, so 4/5 = 20/25
Quick Tips:
  • Cross multiplication is the most reliable method for finding missing values
  • Always multiply numerator of first fraction by denominator of second
  • Then multiply denominator of first fraction by numerator of second
Frequently Asked Questions:

Q: Does cross multiplication work for all equivalent fraction problems?
A: Yes, it works for any equivalent fraction problem with one unknown value.

Q: Can I use cross multiplication to compare fractions?
A: Yes, cross multiplication is also used to compare fractions to see which is larger.

Solutions: Exercises 4 to 5
4 Advanced missing values
Exercise 4
Find the missing value: 7/? = 21/27
Definition:

Advanced missing value problem: A type of equivalent fraction problem where the unknown value requires careful analysis of the relationship between numerators and denominators.

Note: When solving for a missing denominator, analyze how the numerator changed to determine the denominator relationship.

Step-by-step method:
  1. Identify how the known numerator changed to become the unknown numerator
  2. Apply the same relationship to the known denominator
  3. Alternatively, use cross multiplication to solve algebraically
  4. Verify the solution by confirming the fractions are equivalent
Given
7/? = 21/27
Cross Multiply
7×27 = ?×21
Solve
? = 9
Step 1: Set up cross multiplication

7/? = 21/27 → 7×27 = ?×21

Step 2: Calculate known product

7×27 = 189

Step 3: Solve for missing denominator

189 = ?×21 → ? = 189÷21 = 9

Step 4: Verify solution

7/9 = 21/27 → Cross multiply: 7×27 = 189 and 9×21 = 189 ✓

7/9 = 21/27
Final answer:

The missing denominator is 9, so 7/9 = 21/27.

Applied rules:

• Cross multiplication: In equivalent fractions a/b = c/d, a×d = b×c

• Proportional reasoning: Relationship between numerators matches denominators

• Verification: Always check solution by substituting back into original equation

• Alternative method: 21÷7=3, so 27÷3=9 (divide by same factor)

Related Examples:
  • 5/? = 15/24 → 5×24 = ?×15 → ? = 8, so 5/8 = 15/24
  • 2/? = 10/35 → 2×35 = ?×10 → ? = 7, so 2/7 = 10/35
  • 3/? = 18/42 → 3×42 = ?×18 → ? = 7, so 3/7 = 18/42
Quick Tips:
  • When the unknown is in the denominator, cross multiply and divide by the numerator
  • Alternatively, find the multiplication factor from numerators and apply to denominators
  • Check your work by ensuring the fractions are equivalent
Frequently Asked Questions:

Q: What if the numbers are large?
A: Cross multiplication still works, but you may need to simplify fractions first.

Q: Can I solve this by looking at patterns?
A: Yes, if you notice that 21 = 7×3, then the denominator must be 27÷3 = 9.

5 Multiple missing values
Exercise 5
Find the missing values: 4/6 = ?/9 = 8/?
Definition:

Multiple missing values problem: A type of equivalent fraction problem where more than one value is unknown in a chain of equivalent fractions.

Note: When solving for multiple missing values, solve one at a time using the known equivalent fraction as a reference.

Step-by-step method:
  1. Identify the complete fraction to use as a reference
  2. Solve for the first missing value using cross multiplication
  3. Use the newly found equivalent fraction to solve for the next missing value
  4. Verify all fractions are equivalent
Given
4/6 = ?/9 = 8/?
First Missing
4/6 = ?/9 → ? = 6
Second Missing
4/6 = 8/? → ? = 12
Step 1: Solve 4/6 = ?/9

4×9 = 6×? → 36 = 6×? → ? = 6

Step 2: Solve 4/6 = 8/?

4×? = 6×8 → 4×? = 48 → ? = 12

Step 3: Verify all fractions

4/6 = 6/9 = 8/12 → All simplify to 2/3 ✓

4/6 = 6/9 = 8/12
Final answer:

The missing values are 6 and 12, so 4/6 = 6/9 = 8/12.

Applied rules:

• Chain of equivalence: All fractions in the chain are equal

• Step-by-step solving: Solve one missing value at a time

• Reference fraction: Use a complete fraction to solve for missing values

• Verification: All fractions should simplify to the same value

Related Examples:
  • 2/3 = ?/6 = 4/? → ? = 4 and ? = 6, so 2/3 = 4/6 = 4/6
  • 3/5 = ?/10 = 9/? → ? = 6 and ? = 15, so 3/5 = 6/10 = 9/15
  • 1/4 = ?/8 = 3/? → ? = 2 and ? = 12, so 1/4 = 2/8 = 3/12
Quick Tips:
  • Work with one pair of fractions at a time
  • Use the fraction with both values known as your reference
  • Verify that all fractions in the chain are equivalent
Frequently Asked Questions:

Q: Can I solve multiple missing values at once?
A: It's better to solve one at a time to avoid confusion.

Q: What if I get different answers for the same fraction?
A: Check your work - all fractions in the chain must be equivalent.

Solutions: Exercises 6 to 10
6 Word problems with missing values
Exercise 6
If Sarah ate 3/4 of her pizza and this is equivalent to 6/8 of another pizza, how much of the second pizza did she eat?
Definition:

Word problems with missing values: Real-world problems that involve equivalent fractions where one value needs to be determined using cross multiplication or proportional reasoning.

Note: These problems connect mathematical concepts to practical situations, helping students understand the relevance of equivalent fractions.

Final answer:

Sarah ate 6/8 of the second pizza, which is equivalent to 3/4 of the first pizza.

7 Visual models with missing values
Exercise 7
A rectangle is divided into 5 equal parts with 2 parts shaded. Another rectangle of the same size is divided into 10 equal parts. How many parts should be shaded to represent an equivalent fraction?
Definition:

Visual models with missing values: Problems that use shapes or diagrams to represent fractions, where one aspect of the visual model is unknown.

Note: Visual models help students understand the concept of equivalent fractions by showing that different divisions can represent the same portion of a whole.

Final answer:

4 parts should be shaded in the second rectangle, since 2/5 = 4/10.

8 Complex fraction patterns
Exercise 8
Complete the pattern: 1/2, 2/4, 3/6, ?, 5/10, ?
Definition:

Complex fraction patterns: Sequences of equivalent fractions that follow a specific rule or pattern, requiring students to identify the relationship between terms.

Note: These problems help develop pattern recognition skills and deepen understanding of equivalent fractions.

Final answer:

The pattern is: 1/2, 2/4, 3/6, 4/8, 5/10, 6/12. Each fraction equals 1/2.

9 Error analysis
Exercise 9
A student wrote: 3/5 = 6/9. Is this correct? If not, what is the correct equivalent fraction with a numerator of 6?
Definition:

Error analysis: Problems that require identifying and correcting mistakes in fraction equivalence, helping students develop critical thinking and verification skills.

Note: Analyzing errors helps students understand the importance of following mathematical rules correctly.

Final answer:

The student is incorrect. 3/5 ≠ 6/9. The correct equivalent fraction with numerator 6 is 6/10, since 3/5 = 6/10.

10 Mixed problem types
Exercise 10
Solve: 2/3 = ?/12 = 8/? = ?/21
Definition:

Mixed problem types: Complex problems that combine multiple equivalent fraction concepts, requiring students to apply various techniques to solve.

Note: These problems test comprehensive understanding of equivalent fractions and problem-solving skills.

Final answer:

The solution is: 2/3 = 8/12 = 8/12 = 14/21. Wait, let me recalculate: 2/3 = 8/12 = 8/12 is redundant. Actually: 2/3 = 8/12 = 8/12 = 14/21. Let me fix: 2/3 = 8/12 = 8/? = ?/21. First, 2/3 = 8/?: 2×? = 3×8 → ? = 12. So 2/3 = 8/12. Second, 2/3 = ?/21: 2×21 = 3×? → ? = 14. So 2/3 = 14/21. Final answer: 2/3 = 8/12 = 14/21.

Detailed Summary of Equivalent Fractions Problem Type 2
Key Definitions:

Equivalent fractions: Fractions that represent the same value or occupy the same position on a number line, even though they have different numerators and denominators.

Missing value problems: Problems where one part of an equivalent fraction equation is unknown and needs to be determined using mathematical relationships.

Cross multiplication: A method to check if two fractions are equivalent or to solve for unknown values by multiplying diagonally across the fractions.

Proportional reasoning: Understanding that in equivalent fractions, the relationship between numerators is the same as the relationship between denominators.

Problem type 2: Specifically refers to finding missing values in equivalent fraction equations.

Core Rules and Principles:
  1. Cross Multiplication Rule: For equivalent fractions a/b = c/d, a×d = b×c
  2. Proportional Relationship: The factor by which the numerator changes is the same as the factor by which the denominator changes
  3. Verification Principle: Always check your solution by substituting back into the original equation
  4. Step-by-Step Solving: When multiple values are missing, solve one at a time using known equivalents
Basic Example: 2/3 = ?/6 → 2×6 = 3×? → ? = 4, so 2/3 = 4/6
Advanced Example: 3/4 = ?/12 → 3×12 = 4×? → ? = 9, so 3/4 = 9/12
Real-World Application: If 1/2 of a pizza is the same as 2/4 of another pizza, both represent the same amount
Memory Tip: "Cross multiply to find missing values in equivalent fractions"
Common Mistakes: Forgetting to verify the solution, misapplying cross multiplication, not maintaining the same multiplication factor for numerator and denominator.
Memorization Tips: Practice with visual models, remember the cross multiplication rule, and check your work by verifying the fractions are equivalent.
Key Procedures:

• Finding Missing Numerator: Use cross multiplication: a/b = ?/d → a×d = b×?, then solve for ?

• Finding Missing Denominator: Use cross multiplication: a/b = c/? → a×? = b×c, then solve for ?

• Multiple Missing Values: Solve one at a time using known equivalent fractions

• Verification: Always substitute your answer back to ensure fractions are equivalent

Rules and Methods for Equivalent Fractions Problem Type 2
\(\frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c\)
Cross Multiplication Rule
Method 1
Cross Multiplication
Find missing values
Method 2
Proportional Reasoning
Find multiplication factors
Method 3
Verification
Check equivalence

Key Takeaways

  • Equivalent fractions have equal cross products
  • Use cross multiplication to find missing values
  • Verify your solution by checking equivalence
  • When multiple values are missing, solve one at a time
  • The relationship between numerators matches the relationship between denominators
  • Always check that your answer makes sense in the context of the problem

Questions & Answers

Question: I'm confused about finding missing numerators. How do I know what number to put?

Answer: To find a missing numerator, you can use cross multiplication:

  • Set up the equivalent fraction equation (like 2/3 = ?/6)
  • Multiply the known numerator by the opposite denominator (2×6 = 12)
  • Divide by the known denominator (12÷3 = 4)
  • So 2/3 = 4/6

The missing numerator is 4. You can verify: 2×6 = 12 and 3×4 = 12, so the cross products are equal.

Example: If 3/5 = ?/10, then 3×10 = 5×?, so ? = 30÷5 = 6. Therefore 3/5 = 6/10.

Question: How can I help my child master equivalent fraction problem type 2?

Answer: Here are effective strategies to master equivalent fraction problem type 2:

  • Start with visual models: Use fraction circles or rectangles to show equivalent fractions
  • Practice cross multiplication: This is the most reliable method for finding missing values
  • Work step-by-step: Solve one missing value at a time in complex problems
  • Verify solutions: Always check that the fractions are equivalent after finding missing values

Begin with simple problems like 1/2 = ?/4, then gradually increase difficulty.

Encourage your child to explain their thinking and verify each answer.

Question: What if I get stuck on a problem with multiple missing values?

Answer: When facing problems with multiple missing values:

  • Find a complete fraction first: Use the fraction that has both numerator and denominator
  • Solve one at a time: Focus on finding just one missing value before moving to the next
  • Use cross multiplication: This works for each pair of equivalent fractions
  • Check your work: Verify each step before proceeding to the next missing value

For example, in 2/3 = ?/6 = 4/? start with 2/3 = ?/6 to find the first missing value.

Once you have that, use your new equivalent fraction to find the next missing value.

Fraction Glossary

Equivalent Fractions
Fractions that represent the same value or portion of a whole, such as 1/2 and 2/4.
Cross Multiplication
A method to check if two fractions are equivalent by multiplying diagonally: a/b = c/d if a×d = b×c.
Proportional Reasoning
Understanding that in equivalent fractions, the relationship between numerators matches the relationship between denominators.
Missing Value Problem
A problem where one part of an equivalent fraction equation is unknown and needs to be determined.
Problem Type 2
Finding missing values in equivalent fraction equations.

Math Foundations Educational Team

Elementary Mathematics Specialists & Curriculum Developers

Our team of experienced elementary math teachers and curriculum specialists creates research-based, student-friendly fraction resources. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness for 3rd grade learners tackling equivalent fraction problem type 2.