Comparing Fractions with Different Numerators and Denominators: Visual Exercises for Grade 4

Master fraction comparison using cross multiplication, common denominators, and visual methods through 5 detailed exercises with step-by-step solutions.

Solution: Exercises 1 to 3
1 Cross multiplication method
Exercise 1
Compare 2/3 and 3/4 using cross multiplication.
Definition:

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

Cross multiplication method:
  1. Multiply numerator of first fraction by denominator of second: a × d
  2. Multiply numerator of second fraction by denominator of first: b × c
  3. Compare results: if ad > bc, then a/b > c/d
First fraction
2/3
Second fraction
3/4
Compare
?
Step 1: Cross multiply

2/3 vs 3/4
First cross: 2 × 4 = 8
Second cross: 3 × 3 = 9

Step 2: Compare cross products

8 vs 9 → Since 8 < 9, then 2/3 < 3/4

Step 3: Verify with common denominator

2/3 = 8/12 and 3/4 = 9/12
Since 8 < 9, then 8/12 < 9/12

8/12
4/12
9/12
3/12
2/3 < 3/4
Final answer:

2/3 is less than 3/4 because when cross multiplied, 8 < 9.

Applied rules:

• Cross multiplication: a/b > c/d if ad > bc

• Verification: Convert to common denominator to confirm result

• Comparison principle: When denominators are equal, compare numerators

2/3
3/4
2 Common denominator method
Exercise 2
Compare 5/6 and 3/4 using the common denominator method.
Definition:

Common denominator: Converting fractions to have the same denominator to enable direct comparison.

First fraction
5/6
Second fraction
3/4
Compare
?
Step 1: Find least common denominator (LCD)

Denominators: 6 and 4
Multiples of 6: 6, 12, 18, 24...
Multiples of 4: 4, 8, 12, 16...
LCD = 12

Step 2: Convert to equivalent fractions

5/6 = (5×2)/(6×2) = 10/12
3/4 = (3×3)/(4×3) = 9/12

Step 3: Compare numerators

10/12 vs 9/12 → Compare numerators: 10 vs 9
Since 10 > 9, then 10/12 > 9/12

10/12
2/12
9/12
3/12
5/6 > 3/4
Final answer:

5/6 is greater than 3/4 because when converted to twelfths, 10/12 > 9/12.

Applied rules:

• Least common denominator: Find smallest number divisible by both denominators

• Equivalent fractions: Multiply numerator and denominator by same number

• Direct comparison: When denominators are equal, compare numerators

5/6
3/4
3 Equivalent fractions
Exercise 3
Are 2/5 and 4/10 equal? Use the cross multiplication method.
Definition:

Equivalent fractions: Different fractions that represent the same value.

First fraction
2/5
Second fraction
4/10
Relationship
?
Step 1: Cross multiply

2/5 vs 4/10
First cross: 2 × 10 = 20
Second cross: 5 × 4 = 20

Step 2: Compare cross products

20 vs 20 → Since 20 = 20, then 2/5 = 4/10

Step 3: Verify by simplifying

4/10 = (4÷2)/(10÷2) = 2/5
This confirms they are equivalent

Step 4: Visual confirmation

Both fractions represent the same portion of their respective wholes

2/5
3/5
4/10
6/10
2/5 = 4/10
Final answer:

2/5 and 4/10 are equal because their cross products are equal (20 = 20).

Applied rules:

• Equivalence test: a/b = c/d if ad = bc

• Simplification: Divide numerator and denominator by GCD

• Verification: Multiple methods confirm the same result

2/5
4/10
Comparison Methods and Key Concepts
a/b > c/d when ad > bc
Cross Multiplication Rule
Cross Multiply
a/b vs c/d → compare ad vs bc
Quick comparison method
Common Denominator
a/b vs c/d → convert to same denominator
Direct comparison
Equivalent Check
a/b = c/d when ad = bc
Test equality
Key definitions:

Unlike fractions: Fractions with different denominators (e.g., 2/3 and 3/4)

Like fractions: Fractions with the same denominator (e.g., 2/5 and 3/5)

Least Common Denominator (LCD): Smallest number divisible by all denominators being compared

Equivalent fractions: Fractions representing the same value (e.g., 1/2 = 2/4)

Complete methodology:
  1. Identify the method: Choose cross multiplication or common denominator based on situation
  2. Apply the method: Either cross multiply or find LCD and convert fractions
  3. Compare results: Compare numerators or cross products
  4. State the relationship: Use <, >, or = to show the comparison
Tip 1: Cross multiplication is faster for comparing two fractions.
Tip 2: Use common denominator when comparing more than two fractions.
Tip 3: Always verify your result with a different method when possible.
Tip 4: For LCD, find multiples of larger denominator until you find one divisible by smaller.
Common errors: Mixing up numerators with denominators, forgetting to multiply both numerator and denominator when finding equivalent fractions.
Key insight: Unlike fractions must be converted to like fractions before direct comparison.
Rules to know by heart:

• Cross multiplication: a/b > c/d if ad > bc

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

• Equivalence: a/b = c/d if ad = bc

• Order preservation: Equivalent transformations preserve inequality relationships

Solution: Exercises 4 to 5
4 Ordering multiple fractions
Exercise 4
Order these fractions from least to greatest: 1/2, 2/3, 3/4, 5/6.
Definition:

Ordering fractions: Arranging fractions in ascending or descending order by comparing their values.

Fractions to order
1/2, 2/3, 3/4, 5/6
Method
Common denominator
Result
?
Step 1: Find least common denominator

Denominators: 2, 3, 4, 6
Multiples of 6: 6, 12, 18, 24...
12 is divisible by 2, 3, 4, and 6
LCD = 12

Step 2: Convert all fractions to twelfths

1/2 = (1×6)/(2×6) = 6/12
2/3 = (2×4)/(3×4) = 8/12
3/4 = (3×3)/(4×3) = 9/12
5/6 = (5×2)/(6×2) = 10/12

Step 3: Compare numerators

6/12, 8/12, 9/12, 10/12
Numerators: 6, 8, 9, 10
Ascending order: 6 < 8 < 9 < 10

Step 4: Write original fractions in order

From least to greatest: 1/2, 2/3, 3/4, 5/6

6/12
6/12
8/12
4/12
9/12
3/12
10/12
2/12
1/2 < 2/3 < 3/4 < 5/6
Final answer:

From least to greatest: 1/2, 2/3, 3/4, 5/6

Applied rules:

• Multiple comparison: Convert all fractions to same denominator

• Systematic ordering: Arrange by comparing numerators

• Verification: Check that order makes logical sense

1/2
2/3
3/4
5/6
5 Complex comparison
Exercise 5
Which is greater: 7/12 or 5/9? Use both cross multiplication and common denominator methods.
Definition:

Method verification: Using multiple approaches to confirm the same result increases confidence in the answer.

First fraction
7/12
Second fraction
5/9
Compare
?
Method 1: Cross multiplication

7/12 vs 5/9
First cross: 7 × 9 = 63
Second cross: 12 × 5 = 60
Since 63 > 60, then 7/12 > 5/9

Method 2: Common denominator

Denominators: 12 and 9
Multiples of 12: 12, 24, 36, 48...
Multiples of 9: 9, 18, 27, 36...
LCD = 36

Step 3: Convert to equivalent fractions

7/12 = (7×3)/(12×3) = 21/36
5/9 = (5×4)/(9×4) = 20/36

Step 4: Compare numerators

21/36 vs 20/36 → Since 21 > 20, then 21/36 > 20/36

21/36
15/36
20/36
16/36
7/12 > 5/9
Final answer:

7/12 is greater than 5/9. Both methods confirm this result.

Applied rules:

• Cross multiplication: a/b > c/d if ad > bc

• Common denominator: Convert to same denominator and compare numerators

• Method verification: Different approaches yielding same result increase confidence

7/12
5/9
Detailed Summary: Comparing Fractions with Different Numerators and Denominators
a/b > c/d when ad > bc (Cross Multiplication)
Core Comparison Principle
Key definitions:

Fraction: A number representing part of a whole, written as a/b where a is the numerator and b is the denominator.

Like fractions: Fractions with the same denominator (e.g., 2/7 and 5/7).

Unlike fractions: Fractions with different denominators (e.g., 3/4 and 2/5).

Least Common Denominator (LCD): The smallest number that is a multiple of all denominators in a set.

Equivalent fractions: Different fractions that represent the same value (e.g., 1/2 = 2/4 = 4/8).

Core methods and procedures:
  1. Cross Multiplication Method:
    • Multiply numerator of first fraction by denominator of second: a × d
    • Multiply numerator of second fraction by denominator of first: b × c
    • Compare results: if ad > bc, then a/b > c/d
  2. Common Denominator Method:
    • Find the least common denominator of all fractions
    • Convert each fraction to an equivalent fraction with the common denominator
    • Compare numerators: the fraction with the larger numerator is greater
  3. Visual/Model Method:
    • Draw area models or number lines to represent each fraction
    • Compare the visual representations to determine which is larger
Tip 1: Cross multiplication is the fastest method for comparing two fractions.
Tip 2: For ordering multiple fractions, the common denominator method is usually more efficient.
Tip 3: When denominators are small, try listing multiples to find the LCD quickly.
Tip 4: Always verify your results by using a different method when possible.
Common mistakes: Forgetting to multiply both numerator and denominator when converting fractions, mixing up numerators with denominators during cross multiplication, not finding the true LCD.
Memory aids: "Cross multiply and compare" for cross multiplication method; "Find same bottom number, then compare tops" for common denominator method.
Essential rules and formulas:

• Cross multiplication: a/b > c/d if and only if ad > bc

• Equality test: a/b = c/d if and only if ad = bc

• Common denominator: a/b > c/b if and only if a > c (same denominator)

• Transitive property: If a/b > c/d and c/d > e/f, then a/b > e/f

• Addition property: If a/b > c/d, then a/b + e/f > c/d + e/f

Examples: 2/3 vs 3/4 → 2×4 vs 3×3 → 8 vs 9 → 2/3 < 3/4
Practical Example
Cross Multiply
2/3 vs 3/4 → 8 vs 9 → 2/3 < 3/4
Quick comparison
Common Denominator
2/3 vs 3/4 → 8/12 vs 9/12 → 2/3 < 3/4
Direct comparison
Equivalent Test
2/3 = 4/6 → 2×6 = 3×4 → 12 = 12 ✓
Equality verification

Questions & Answers

Question: Why do we need to convert fractions to the same denominator when comparing them? Can't we just compare the numerators directly?

Answer: Great question! We need the same denominator because the denominator tells us the size of the pieces we're comparing. Think of it this way:

  • If you have 3 slices of pizza cut into 4 pieces (3/4) vs 2 slices of pizza cut into 8 pieces (2/8), you can't just compare 3 vs 2
  • The pieces in the first pizza are larger (1/4) than the pieces in the second pizza (1/8)
  • So 3/4 is actually much larger than 2/8, even though 3 > 2

When denominators are the same, we're comparing pieces of the same size, so we can directly compare the number of pieces (numerators).

Question: How do I find the least common denominator quickly? Sometimes the numbers are big and I get confused.

Answer: Here are some strategies to find the LCD quickly:

  1. Start with the larger denominator: List its multiples until you find one divisible by the smaller denominator
  2. Example: For 1/6 and 1/8, start with multiples of 8: 8, 16, 24. 24 is divisible by 6, so LCD = 24
  3. Look for factors: If one denominator is a factor of the other, use the larger one as LCD
  4. For prime numbers: Multiply them together to get LCD

Remember, any common multiple will work, but using the smallest one (LCD) makes calculations easier.

Question: What's the most effective method to teach fraction comparison to struggling students?

Answer: For struggling students, I recommend starting with visual models before moving to abstract methods:

  1. Area models: Draw circles or rectangles divided into equal parts and shade to represent fractions
  2. Number lines: Plot fractions on a number line to see their relative positions
  3. Concrete manipulatives: Use fraction tiles or strips to physically compare sizes
  4. Gradual progression: Move from like fractions to unlike fractions once students understand the concept

Once students grasp the concept visually, introduce the cross multiplication method as a shortcut. The key is ensuring students understand WHY the method works, not just HOW to apply it.