Comparing Fractions Worksheets - Solved Exercises for Grade 3

Master comparing fractions: common denominators, cross multiplication, and visual comparison methods through these 3 detailed worksheet exercises.

Solution: Worksheet Exercises 1 to 3
1 Worksheet: Common Denominator Method
Exercise 1
Compare 2
3 and 3
4 using the common denominator method.
Definition:

Worksheet format: Practice problems designed to reinforce fraction comparison skills.

Common denominator method:
  1. Find the least common multiple (LCM) of the denominators
  2. Convert each fraction to an equivalent fraction with the common denominator
  3. Compare the numerators of the equivalent fractions
2
3
?
3
4
2/3
3/4
Step 1: Find the LCM of denominators 3 and 4

Factors of 3: 1, 3

Factors of 4: 1, 2, 4

LCM(3, 4) = 12

Step 2: Convert to equivalent fractions with denominator 12

2

3 = 2×4
3×4 = 8
12

3

4 = 3×3
4×3 = 9
12

Step 3: Compare the numerators of equivalent fractions

Since 9 > 8, then 9

12 > 8
12

Step 4: State the final answer

Therefore, 3

4 > 2
3

3
4 > 2
3
Final answer:

3

4 is greater than 2
3

Applied rules:

• LCM method: Find least common multiple of denominators

• Equivalent fraction conversion: Multiply numerator and denominator by the same number

• Same denominator comparison: Compare numerators once denominators match

2/3
3/4
2 Worksheet: Cross Multiplication Method
Exercise 2
Compare 5
6 and 7
8 using cross multiplication.
Definition:

Cross multiplication: Multiply the numerator of one fraction by the denominator of the other and compare the products.

5
6
?
7
8
5/6
7/8
Step 1: Cross multiply the fractions

Multiply the numerator of the first fraction by the denominator of the second: 5 × 8 = 40

Multiply the numerator of the second fraction by the denominator of the first: 7 × 6 = 42

Step 2: Compare the products

Since 42 > 40, then 7

8 > 5
6

Step 3: Verification using common denominators

LCM of 6 and 8 is 24

5

6 = 20
24 and 7
8 = 21
24

Since 21 > 20, our cross multiplication was correct!

7
8 > 5
6
Final answer:

7

8 is greater than 5
6

Applied rules:

• Cross multiplication rule: For fractions a/b and c/d, if a×d > b×c, then a/b > c/d

• Product comparison: Compare the results of cross multiplication

• Verification: Use another method to verify the result

5/6
7/8
3 Worksheet: Visual Model Method
Exercise 3
Compare 1
3 and 2
7 using visual models and equivalent fractions.
Definition:

Visual models: Using pie charts, bars, or other visual representations to understand fractional relationships.

1
3
?
2
7
1/3
2/7
Step 1: Use visual inspection to estimate

Looking at the pie charts, 1/3 appears larger than 2/7, but we need to verify mathematically.

Step 2: Find the LCM of denominators 3 and 7

Since 3 and 7 are both prime numbers, LCM(3, 7) = 3 × 7 = 21

Step 3: Convert to equivalent fractions with denominator 21

1

3 = 1×7
3×7 = 7
21

2

7 = 2×3
7×3 = 6
21

Step 4: Compare the numerators of equivalent fractions

Since 7 > 6, then 7

21 > 6
21

Step 5: State the final answer

Therefore, 1

3 > 2
7

1
3 > 2
7
Final answer:

1

3 is greater than 2
7

Applied rules:

• Prime denominators: When denominators are prime, LCM is their product

• Visual estimation: Use models to predict the outcome before calculating

• Equivalent fraction method: Convert to same denominator for direct comparison

1/3
2/7
Additional Worksheet Problems
Problem Set A: Like Denominators
Problem A1
Compare: 3
5 ___ 4
5
Answer: 4
5 > 3
5 (same denominator, compare numerators)
Problem A2
Compare: 7
8 ___ 5
8
Answer: 7
8 > 5
8 (same denominator, compare numerators)
Problem A3
Compare: 2
9 ___ 6
9
Answer: 6
9 > 2
9 (same denominator, compare numerators)
Problem Set B: Same Numerators
Problem B1
Compare: 1
4 ___ 1
6
Answer: 1
4 > 1
6 (same numerator, smaller denominator is larger)
Problem B2
Compare: 3
5 ___ 3
7
Answer: 3
5 > 3
7 (same numerator, smaller denominator is larger)
Problem B3
Compare: 2
3 ___ 2
8
Answer: 2
3 > 2
8 (same numerator, smaller denominator is larger)
Worksheet Solutions and Strategies
a
b > c
d when ad > bc
Cross Multiplication Rule
Problem Set C: Unlike Denominators
Problem C1
Compare: 2
3 ___ 3
5
Solution: LCM(3,5)=15 → 10
15 vs 9
15 → 2
3 > 3
5
Problem C2
Compare: 4
7 ___ 3
5
Solution: Cross multiply: 4×5=20, 3×7=21 → 21>20 → 3
5 > 4
7
Problem C3
Compare: 5
8 ___ 7
12
Solution: LCM(8,12)=24 → 15
24 vs 14
24 → 5
8 > 7
12
Key definitions:

Fraction: Represents a part of a whole (numerator/denominator)

Unlike denominators: Fractions with different denominators that require special methods for comparison

Equivalent fractions: Different fractions representing the same value

Least Common Multiple (LCM): The smallest number that is a multiple of both denominators

Common denominator: Same denominator for comparing fractions

Complete methodology:
  1. Identify denominators: Note that they are different
  2. Choose method: Common denominator, cross multiplication, or visual model
  3. Execute: Apply chosen method systematically
  4. Compare: Determine which fraction is larger
  5. Verify: Use alternative method if needed
Tip 1: Use cross multiplication for quick comparisons when denominators are large.
Tip 2: Use common denominators when denominators share factors.
Tip 3: Visual models help understand the concept before calculating.
Tip 4: Always verify your answer with an alternative method.
Practice Problems and Key Notes
Mixed Practice Problems
Problem D1
Compare: 3
4 ___ 5
6
Solution: Cross multiply: 3×6=18, 5×4=20 → 20>18 → 5
6 > 3
4
Problem D2
Compare: 7
10 ___ 4
5
Solution: LCM(10,5)=10 → 7
10 vs 8
10 → 4
5 > 7
10
Problem D3
Compare: 1
2 ___ 2
5
Solution: Cross multiply: 1×5=5, 2×2=4 → 5>4 → 1
2 > 2
5
Key definitions:

Comparing fractions worksheets: Practice materials for reinforcing fraction comparison skills.

Common denominator: Same bottom number allowing direct comparison.

Cross multiplication: Method to compare fractions without finding common denominator.

Equivalent fractions: Fractions that represent the same value but have different numerators and denominators.

Complete methodology:
  1. Analyze the fractions: Check if denominators are the same or different
  2. Choose the method: Common denominator or cross multiplication
  3. Apply the method: Follow the specific steps for chosen approach
  4. State the result: Use correct inequality symbols (> , < , =)
Memory Aid 1: "Cross multiply and compare" - multiply diagonally and compare products.
Memory Aid 2: "Find common ground" - convert to same denominator.
Memory Aid 3: "Multiply across, compare results" - for cross multiplication.
Memory Aid 4: "Visualize pizza slices" - helps understand relative sizes.
Common errors: Forgetting to find common denominator, mixing up cross multiplication, not simplifying fractions.
Practice strategies: Use visual models, practice with various denominators, work with number lines.
Quick Reference Guide:

• Common denominator: Convert fractions to same denominator, then compare numerators

• Cross multiplication: For a/b vs c/d, compare a×d vs b×c

• LCM method: Find least common multiple of denominators

• Equivalent fractions: Multiply/divide both parts by same number

Questions & Answers

Question: How do I know which method to use when comparing fractions on worksheets?

Answer: Here's how to choose the right method:

  • Same denominators: Just compare numerators directly
  • Same numerators: The fraction with the smaller denominator is larger
  • Different denominators: Use common denominators or cross multiplication
  • Cross multiplication: Better when denominators are large or don't share factors
  • Common denominators: Better when denominators share factors or are small

Example: For 3

4 vs 5
6, cross multiplication gives 3×6=18 vs 5×4=20, so 5
6 is larger.

Question: Why do I need to find common denominators when comparing fractions?

Answer: Think of it like comparing apples to oranges:

  • If I have 2 pieces of a pizza cut into 3 pieces, that's 2
    3
  • If you have 3 pieces of a pizza cut into 10 pieces, that's 3
    10
  • Even though 3 > 2, 3
    10     is actually smaller than 2
    3     because tenths are much smaller pieces than thirds!

Common denominators make all the "pieces" the same size, so we can just count how many pieces we have.

Question: How do I find the LCM of two numbers quickly?

Answer: Here are quick methods to find LCM:

Method 1: Prime Factorization

  • Break each number into prime factors
  • Take the highest power of each prime that appears
  • Multiply those together

Method 2: For small numbers

  • List multiples of the larger number until you find one divisible by the smaller
  • Example: LCM of 4 and 6 → multiples of 6: 6, 12... 12 ÷ 4 = 3, so LCM = 12

Special cases:

  • If one number divides the other, LCM is the larger number
  • If numbers are both prime, LCM is their product
  • If numbers share no common factors, LCM is their product

Example: LCM of 8 and 12 → 8 = 2³, 12 = 2² × 3 → LCM = 2³ × 3 = 24

Detailed Summary: Comparing Fractions Worksheets

Definitions and Key Concepts:
Fraction: A number representing a part of a whole, written as numerator/denominator.
Numerator: The top number in a fraction, showing how many parts we have.
Denominator: The bottom number in a fraction, showing how many equal parts the whole is divided into.
Unlike denominators: Fractions with different denominators that require special methods for comparison.
Like denominators: Fractions with the same denominator that can be compared directly.
Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.
Equivalent fractions: Different fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6).
Cross multiplication: A method of comparing fractions by multiplying diagonally.
Worksheets: Practice materials designed to reinforce fraction comparison skills through repeated exercises.
Core Rules and Principles:
Common Denominator Rule: Convert unlike fractions to like fractions by finding the LCM of denominators.
Cross Multiplication Rule: For fractions a/b and c/d, if a×d > b×c, then a/b > c/d.
Same Denominator Rule: When denominators are equal, compare numerators directly.
Same Numerator Rule: When numerators are equal, the fraction with the smaller denominator is larger.
Equivalent Fraction Rule: Multiply both numerator and denominator by the same number to get an equivalent fraction.
Comparison Principle: Only compare fractions when they refer to the same-sized whole.
Order Property: If a > b and b > c, then a > c (transitivity).
Step-by-Step Methods:
Method 1: Common Denominator
• Find the LCM of the denominators
• Convert each fraction to an equivalent fraction with the common denominator
• Compare the numerators of the equivalent fractions
• The fraction with the larger numerator is greater
Method 2: Cross Multiplication
• Multiply the numerator of the first fraction by the denominator of the second
• Multiply the numerator of the second fraction by the denominator of the first
• Compare the two products
• The fraction associated with the larger product is greater
Method 3: Visual Model
• Draw pie charts or bar models for each fraction
• Estimate which appears larger
• Verify with mathematical method
Examples (Simple to Advanced):
Simple: Compare 1
2 and 2
3 → LCM(2,3)=6 → 3
6 vs 4
6 → 4
6 > 3
6 → 2
3 > 1
2
Intermediate: Compare 3
5 and 4
7 → Cross multiply: 3×7=21, 4×5=20 → 21>20 → 3
5 > 4
7
Advanced: Compare 5
12 and 7
18 → LCM(12,18)=36 → 15
36 vs 14
36 → 5
12 > 7
18
Complex: Compare 11
15 and 13
18 → Cross multiply: 11×18=198, 13×15=195 → 198>195 → 11
15 > 13
18
Tips, Tricks, and Common Pitfalls:
Tip 1: Use cross multiplication for quick comparisons when denominators are large or complex.
Tip 2: Use common denominators when denominators share obvious factors.
Tip 3: Visual models help understand the concept before calculating.
Tip 4: Always verify your answer with an alternative method.
Tip 5: Practice regularly with worksheets to build fluency.
Pitfall 1: Comparing numerators directly without considering denominators.
Pitfall 2: Forgetting to convert both fractions when using common denominators.
Pitfall 3: Mixing up which product corresponds to which fraction in cross multiplication.
Pitfall 4: Not simplifying fractions before comparison.
Key Notes for Memorization:
Memory Aid 1: "Cross multiply and compare" - multiply diagonally and compare products.
Memory Aid 2: "Find common ground" - convert to same denominator.
Memory Aid 3: "Multiply across, compare results" - for cross multiplication.
Memory Aid 4: "Make denominators the same, then compare tops" - for common denominator method.
Memory Aid 5: "Bigger denominator means smaller pieces" - helps remember why conversion is needed.
Memory Aid 6: "Pizza slices" - visualize different sized pieces to understand the concept.
Application in Real Life:
Sharing Food: Determining who got more when different numbers of people shared different amounts.
Test Scores: Comparing grades with different total points possible.
Time Management: Comparing portions of time spent on different activities.
Shopping: Comparing discount rates or sale prices.
Measurements: Understanding fractional measurements relative to each other.