Comparing Fractions Exercises - Solved Problems for Grade 3

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

Solutions: Exercises 1 to 3
1 Exercise: Comparing fractions with same denominator
Exercise 1
Compare 2
5 and 3
5.
Definition:

Like denominators: Fractions with the same denominator that can be compared by their numerators.

Same denominator method:
  1. Check if denominators are the same
  2. If same, compare numerators directly
  3. Larger numerator means larger fraction
2
5
?
3
5
2/5
3/5
Step 1: Identify denominators

Both fractions have denominator 5, so we compare numerators: 2 vs 3

Step 2: Compare numerators

Since 3 > 2, then 3

5 > 2
5

Step 3: State the final answer

3

5 is greater than 2
5

3
5 > 2
5
Final answer:

3

5 is greater than 2
5

Applied rules:

• Same denominator rule: When denominators are equal, compare numerators directly

• Greater than symbol: Use > when first fraction is larger

• Visual representation: Pie charts help visualize the comparison

2/5
3/5
2 Exercise: Comparing fractions with same numerator
Exercise 2
Compare 1
3 and 1
4.
Definition:

Same numerator: Fractions with the same numerator that can be compared by their denominators.

1
3
?
1
4
1/3
1/4
Step 1: Identify numerators

Both fractions have numerator 1, so we compare denominators: 3 vs 4

Step 2: Apply same numerator rule

When numerators are equal, the fraction with the smaller denominator is larger

Step 3: Compare denominators

Since 3 < 4, then 1

3 > 1
4

Step 4: Explain why

One-third represents one part out of three equal parts (larger pieces) compared to one-fourth (one part out of four equal parts)

1
3 > 1
4
Final answer:

1

3 is greater than 1
4

Applied rules:

• Same numerator rule: When numerators are equal, smaller denominator means larger fraction

• Visual reasoning: Smaller denominator means fewer, larger parts

• Logical thinking: One piece of 3 is bigger than one piece of 4

1/3
1/4
3 Exercise: Comparing fractions with unlike denominators
Exercise 3
Compare 2
3 and 3
4 using cross multiplication.
Definition:

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

2
3
?
3
4
2/3
3/4
Step 1: Cross multiply the fractions

Multiply the numerator of the first fraction by the denominator of the second: 2 × 4 = 8

Multiply the numerator of the second fraction by the denominator of the first: 3 × 3 = 9

Step 2: Compare the products

Since 9 > 8, then 3

4 > 2
3

Step 3: Verification using common denominators

LCM of 3 and 4 is 12

2

3 = 8
12 and 3
4 = 9
12

Since 9 > 8, our cross multiplication was correct!

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:

• 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

2/3
3/4
Additional Practice Exercises

More Exercises with Complete Solutions

4

Exercise: Compare 3

4 and 5
6

Solution: Using cross multiplication:

3 × 6 = 18 and 5 × 4 = 20

Since 20 > 18, then 5

6 > 3
4

5

Exercise: Compare 1

2 and 2
5

Solution: Using common denominator:

LCM of 2 and 5 is 10

1

2 = 5
10 and 2
5 = 4
10

Since 5 > 4, then 1

2 > 2
5

6

Exercise: Compare 4

7 and 3
8

Solution: Using cross multiplication:

4 × 8 = 32 and 3 × 7 = 21

Since 32 > 21, then 4

7 > 3
8

a
c > b
c when a > b
Same Denominator Rule
Rule 1
a
c > a
d when c < d
Same Numerator Rule
Rule 2
a
b = a×n
b×n
Equivalent Fraction
Rule 3
a
b > c
d when ad > bc
Cross Multiplication
Key definitions:

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

Unlike denominators: Different denominators requiring special comparison methods

Equivalent fractions: Different fractions representing the same value

Common denominator: Same denominator allowing direct comparison

Cross multiplication: Method to compare fractions without finding common denominator

Complete methodology:
  1. Check denominators: If same, compare numerators directly
  2. Check numerators: If same, compare denominators (smaller denominator = larger fraction)
  3. Find common denominator: When neither numerator nor denominator match
  4. Convert to equivalent fractions: With common denominator
  5. Compare numerators: Of equivalent fractions
Tip 1: Visualize fractions as pie slices or bar segments to understand size.
Tip 2: Remember: Larger denominator means smaller pieces.
Tip 3: Cross multiplication: a/b > c/d if a×d > b×c.
Tip 4: Always find LCM when denominators are different.
Fraction Comparison Techniques and Strategies
2
3 vs 3
4 → 8
12 vs 9
12
Common Denominator Method
Comparison Rules: Same denominator → compare numerators; Same numerator → compare denominators (smaller wins); Different → find common denominator.
Visual Aids: Pie charts, bar models, and number lines help students understand fractional relationships.
Fraction Comparison Strategies:

• Strategy 1: Same denominator → Compare numerators directly

• Strategy 2: Same numerator → Compare denominators (smaller denominator = larger fraction)

• Strategy 3: Different numerators/denominators → Find common denominator

• Strategy 4: Cross multiplication: a/b vs c/d → Compare a×d vs b×c

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 numerators and denominators
  2. Choose method: Based on fraction characteristics
  3. Execute: Apply chosen comparison method
  4. Verify: Check with alternative method if possible
Key Notes and Memory Aids
1
2 = 2
4 = 3
6
Equivalent Fractions
Key definitions:

Comparing fractions: Determining which fraction represents a larger portion

Common denominator: Same bottom number allowing direct comparison

Cross multiplication: Method to compare fractions without finding common denominator

Comparing fractions exercises: Practice problems for mastering fraction comparison skills

Complete methodology:
  1. Analyze the fractions: Check if numerators or denominators match
  2. Choose the method: Same denominator, same numerator, or common denominator
  3. Apply the method: Follow the specific steps for chosen approach
  4. State the result: Use correct inequality symbols (> , < , =)
Memory Aid 1: "Same bottom, compare tops" (same denominator)
Memory Aid 2: "Same top, smaller bottom wins" (same numerator)
Memory Aid 3: "Cross multiply and compare" - for different numerators and denominators.
Memory Aid 4: "Visualize pizza slices" - helps understand relative sizes.
Common errors: Confusing numerator and denominator roles, forgetting to find common denominator, misapplying cross multiplication.
Practice strategies: Use visual models, practice with equivalent fractions, work with number lines.
Quick Reference Guide:

• Same denominator: Compare numerators directly

• Same numerator: Compare denominators (smaller wins)

• Different: Find common denominator or use cross multiplication

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

Questions & Answers

Question: I'm confused about comparing 2

3 and 4
6. Are they the same or different?

Answer: Great question! Let's explore this step by step:

  • First, let's check if 4
    6     is equivalent to 2
    3
  • We can simplify 4
    6     by dividing both numerator and denominator by 2: 4÷2
    6÷2     = 2
    3
  • This shows that 4
    6     and 2
    3     are equivalent fractions!

So: 2

3 = 4
6

When comparing fractions, always simplify first to see if they're equivalent!

Question: Why is 1

4 smaller than 1
2? Both have 1 on top.

Answer: Excellent observation! When fractions have the same numerator, the denominator determines the size:

  • 1
    4     means 1 part out of 4 equal parts
  • 1
    2     means 1 part out of 2 equal parts
  • Think of it as pizza: 1 slice from a pizza cut into 4 pieces vs 1 slice from a pizza cut into 2 pieces
  • The slice from the 2-piece pizza is bigger!

Rule: When numerators are the same, the fraction with the smaller denominator is larger.

So: 1

2 > 1
4

Question: What's the easiest way to compare 3

5 and 2
3? They have different tops and bottoms.

Answer: When fractions have different numerators AND denominators, you have two main options:

Method 1: Common Denominator

  • Find LCM of 5 and 3 = 15
  • Convert: 3
    5     = 9
    15     and 2
    3     = 10
    15
  • Compare: 10
    15     > 9
    15
  • So: 2
    3     > 3
    5

Method 2: Cross Multiplication

  • Multiply diagonally: 3×3 = 9 and 2×5 = 10
  • Compare: 10 > 9
  • Since 2×5 > 3×3, then 2
    3     > 3
    5

Both methods give the same result: 2

3 is greater!

Detailed Summary: Comparing Fractions Exercises

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.
Equivalent Fractions: Different fractions that represent the same amount (e.g., 1/2 = 2/4 = 3/6).
Comparing Fractions: Determining which fraction is greater, lesser, or if they are equal.
Like Fractions: Fractions with the same denominator.
Unlike Fractions: Fractions with different denominators.
Unit Fraction: A fraction with numerator 1 (e.g., 1/2, 1/3, 1/4).
Proper Fraction: A fraction where the numerator is less than the denominator (value less than 1).
Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (value greater than or equal to 1).
Comparing fractions exercises: Practice problems designed to reinforce fraction comparison skills.
Common denominator: Same denominator for comparing unlike fractions.
Cross multiplication: Method of comparing fractions by multiplying diagonally.
Core Rules and Principles:
Same Denominator Rule: When denominators are equal, the fraction with the larger numerator is greater.
Same Numerator Rule: When numerators are equal, the fraction with the smaller denominator is greater.
Common Denominator Method: Convert unlike fractions to like fractions by finding the least common multiple (LCM) of denominators.
Cross Multiplication: For fractions a/b and c/d, if a×d > b×c, then a/b > c/d.
Unit Fraction Rule: A unit fraction has numerator 1, and among unit fractions, the one with the smallest denominator is largest.
Zero and One Rule: Any proper fraction is between 0 and 1, and improper fractions are equal to or greater than 1.
Transitive Property: If a > b and b > c, then a > c.
Reflexive Property: Any fraction is equal to itself (a = a).
Symmetric Property: If a = b, then b = a.
Step-by-Step Methods:
Method 1: Same Denominator
• Check if denominators are equal
• Compare numerators directly
• Larger numerator means larger fraction
Method 2: Same Numerator
• Check if numerators are equal
• Compare denominators
• Smaller denominator means larger fraction
Method 3: Different Numerators and Denominators
• Find LCM of denominators
• Convert fractions to equivalent forms with common denominator
• Compare numerators of new fractions
Method 4: Cross Multiplication
• Multiply numerator of first fraction by denominator of second fraction
• Multiply numerator of second fraction by denominator of first fraction
• Compare the products
Method 5: Decimal Conversion
• Divide numerator by denominator for each fraction
• Compare decimal results
• Use calculator if available
Examples (Simple to Advanced):
Simple: 2
5 vs 3
5 → Same denominator, 3 > 2, so 3
5 > 2
5
Intermediate: 1
4 vs 1
3 → Same numerator, 3 < 4, so 1
3 > 1
4
Advanced: 2
3 vs 3
4 → Convert to 8
12 vs 9
12 → 9
12 > 8
12
Complex: 5
12 vs 7
18 → Cross multiply: 5×18=90, 7×12=84 → 90 > 84, so 5
12 > 7
18
Most Complex: 11
15 vs 13
20 → LCM of 15 and 20 is 60 → 44
60 vs 39
60 → 44
60 > 39
60
Tips, Tricks, and Common Pitfalls:
Tip 1: Always visualize fractions as parts of a whole (pies, bars, etc.)
Tip 2: Remember: larger denominator means smaller pieces
Tip 3: Cross multiplication is faster than finding common denominators for large numbers
Tip 4: Simplify fractions before comparing to make calculations easier
Tip 5: Use benchmarks (0, 1/2, 1) to estimate before calculating
Pitfall 1: Don't compare numerators when denominators are different without converting
Pitfall 2: Don't assume larger denominators mean larger fractions
Pitfall 3: Forgetting to simplify fractions before comparing
Pitfall 4: Mixing up which product corresponds to which fraction when cross multiplying
Pitfall 5: Not verifying answers with alternative methods
Key Notes for Memorization:
Memory Aid 1: "Same bottom, compare tops" (same denominator)
Memory Aid 2: "Same top, smaller bottom wins" (same numerator)
Memory Aid 3: "Different tops and bottoms → Make same bottom first"
Memory Aid 4: "More pieces = smaller pieces" (larger denominator = smaller fraction)
Memory Aid 5: "Cross multiply and compare products" (for quick comparisons)
Memory Aid 6: "Equal parts, more parts win" (when denominators are same)
Memory Aid 7: "Visualize pizza slices" (helps understand relative sizes)
Memory Aid 8: "Calculator for verification" (check manual work)
Application in Real Life:
Sharing Food: Understanding who gets more when dividing pizza or cake
Time Management: Comparing portions of time spent on different activities
Measurements: Comparing lengths, weights, or volumes expressed as fractions
Grades: Understanding fractional scores and comparing performance
Budgeting: Comparing fractional portions of money spent on different items
Recipes: Adjusting ingredient amounts when scaling recipes up or down
Games: Comparing fractional progress or scores in educational games