Comparing Fractions with Unlike Denominators - Solved Exercises for Grade 3

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

Solution: Exercises 1 to 3
1 Comparing fractions using common denominators
Exercise 1
Compare 2
3 and 3
4.
Definition:

Unlike denominators: Fractions with different denominators that require conversion to like denominators for comparison.

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 Comparing fractions using cross multiplication
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 Comparing fractions using visual models
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
Rules and methods, laws,...
a
b > c
d when ad > bc
Cross Multiplication Rule
Method 1
a
b = a×n
b×n
Equivalent Fraction
Method 2
a
b > c
d when ad > bc
Cross Multiplication
Method 3
LCM(b, d) → convert fractions
Common Denominator
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.
Comparison Techniques and Strategies
2
3 vs 3
4 → 8
12 vs 9
12
Common Denominator Method
Comparison Rules: When denominators differ, convert to same denominator or use cross multiplication.
Visual Aids: Pie charts and bar models help students understand fractional relationships.
Comparison Strategies:

• Strategy 1: Common denominator → Convert fractions to like denominators

• Strategy 2: Cross multiplication → Compare products of cross multiplication

• Strategy 3: Visual estimation → Use models to predict before calculating

• Strategy 4: Benchmark comparison → Compare to 0, 1/2, or 1 first

Fraction Concepts:

Proper fraction: Numerator < denominator (value < 1)

Improper fraction: Numerator ≥ denominator (value ≥ 1)

Unit fraction: Numerator = 1 (like 1/2, 1/3, 1/4)

Like fractions: Same denominator

Unlike fractions: Different denominators

Step-by-step Comparison Process:
  1. Analyze: Look at denominators to determine if they are alike or unlike
  2. Choose method: Based on complexity of denominators
  3. Execute: Apply chosen comparison method
  4. Verify: Check with alternative method if possible
Key Notes and Memory Aids
2
3 vs 3
4 → 2×4 vs 3×3 → 8 vs 9 → 8 < 9 → 2
3 < 3
4
Cross Multiplication Example
Key definitions:

Comparing fractions with unlike denominators: Method of comparing fractions when denominators are different.

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: Why can't I just compare the numerators when denominators are different? Why do I need to make them the same?

Answer: Great question! Think of it this way:

  • 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!

We need common denominators because they make the "size" of the pieces the same, so we can just count how many pieces we have.

Question: When should I use cross multiplication versus finding common denominators?

Answer: Here's how to choose:

  • Use cross multiplication when: Denominators are large or don't share obvious factors (like 7 and 11)
  • Use common denominators when: Denominators share factors or are small numbers (like 2, 3, 4, 5)
  • Cross multiplication is faster: Just two multiplications instead of multiple conversions
  • Common denominators are clearer: Easier to understand the concept visually

Example: For 3

4 vs 5
6, common denominators might be easier (LCM = 12). For 7
13 vs 9
17, cross multiplication is better.

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 with Unlike Denominators

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.
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.
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.
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.