Comparing Fractions 4th Grade - Solved Exercises for Grade 4

Master comparing fractions: common denominators, cross multiplication, and mixed number comparisons through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing fractions with unlike denominators
Exercise 1
Compare 5
6 and 7
8 using cross multiplication.
Definition:

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

Cross multiplication method:
  1. Multiply the numerator of the first fraction by the denominator of the second
  2. Multiply the numerator of the second fraction by the denominator of the first
  3. 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
2 Comparing mixed numbers
Exercise 2
Compare 23
4 and 25
6.
Definition:

Mixed numbers: Numbers that combine a whole number and a fraction.

2
3
4
?
2
5
6
2 whole
3/4
5/6
Step 1: Compare whole numbers

Both mixed numbers have the same whole number part: 2

Step 2: Compare fractional parts

Now we need to compare 3

4 and 5
6

Step 3: Use cross multiplication for fractional parts

3 × 6 = 18 and 5 × 4 = 20

Since 20 > 18, then 5

6 > 3
4

Step 4: State the final answer

Therefore, 25

6 > 23
4

25
6 > 23
4
Final answer:

25

6 is greater than 23
4

Applied rules:

• Mixed number comparison: Compare whole numbers first, then fractional parts

• Same whole numbers: When whole numbers are equal, compare fractional parts

• Equivalent fractions: Convert to same denominator if needed

2 3/4
2 5/6
3 Comparing improper fractions
Exercise 3
Compare 11
4 and 13
5 using common denominators.
Definition:

Improper fractions: Fractions where the numerator is greater than or equal to the denominator.

11
4
?
13
5
11/4
13/5
Step 1: Find the LCM of denominators 4 and 5

Since 4 and 5 share no common factors, LCM(4, 5) = 4 × 5 = 20

Step 2: Convert to equivalent fractions with denominator 20

11

4 = 11×5
4×5 = 55
20

13

5 = 13×4
5×4 = 52
20

Step 3: Compare the numerators of equivalent fractions

Since 55 > 52, then 55

20 > 52
20

Step 4: State the final answer

Therefore, 11

4 > 13
5

11
4 > 13
5
Final answer:

11

4 is greater than 13
5

Applied rules:

• Improper fraction comparison: Use same methods as proper fractions

• Common denominator method: Find LCM of denominators

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

11/4
13/5
Rules and methods, laws,...
a
b > c
d when ad > bc
Cross Multiplication Rule
Rule 1
a
c > b
c when a > b
Same Denominator
Rule 2
a
c > a
d when c < d
Same Numerator
Rule 3
a
b = a×n
b×n
Equivalent Fraction
Key definitions:

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

Equivalent fractions: Different fractions representing the same value

Common denominator: Same denominator for comparing fractions

Numerator: Top number representing how many parts we have

Denominator: Bottom number representing total number of equal parts

Improper fraction: Numerator ≥ denominator

Mixed number: Whole number + fraction

Complete methodology:
  1. Check denominators: If same, compare numerators
  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

Mixed number: Whole number + proper fraction

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

Improper fractions: Fractions where numerator ≥ denominator

Mixed numbers: Combination of whole number and fraction

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: "More pieces, smaller pieces" - larger denominator means smaller parts.
Memory Aid 2: "Same top, smaller bottom wins" - same numerator, smaller denominator is larger.
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

• Mixed numbers: Compare whole numbers first, then fractional parts

Questions & Answers

Question: I'm confused about comparing 3

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

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

  • First, let's check if 6
    8     is equivalent to 3
    4
  • We can simplify 6
    8     by dividing both numerator and denominator by 2: 6÷2
    8÷2     = 3
    4
  • This shows that 6
    8     and 3
    4     are equivalent fractions!

So: 3

4 = 6
8

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 7

12 and 5
8? 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 12 and 8 = 24
  • Convert: 7
    12     = 14
    24     and 5
    8     = 15
    24
  • Compare: 15
    24     > 14
    24
  • So: 5
    8     > 7
    12

Method 2: Cross Multiplication

  • Multiply diagonally: 7×8 = 56 and 5×12 = 60
  • Compare: 60 > 56
  • Since 5×12 > 7×8, then 5
    8     > 7
    12

Both methods give the same result: 5

8 is greater!

Detailed Summary: Comparing Fractions 4th Grade

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).
Mixed Number: A combination of a whole number and a proper fraction (e.g., 2 1/3).
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.
Mixed Number Comparison: Compare whole numbers first, then fractional parts if whole numbers are equal.
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: Mixed Number Comparison
• Compare whole number parts first
• If whole numbers are equal, compare fractional parts
• Use any of the above methods for fractional parts
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: 32
5 vs 33
7 → Compare fractional parts: 2
5 vs 3
7 → 2×7=14, 3×5=15 → 15>14 → 33
7 > 32
5
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: For mixed numbers, compare whole numbers first, then fractional parts
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: When comparing mixed numbers, forgetting to compare fractional parts if whole numbers are equal
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: "Whole numbers first in mixed numbers" (compare whole numbers before fractions)
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