Comparing Fractions with Unlike Denominators: Complete Worksheet PDF

Master comparing fractions with unlike denominators through 5 detailed exercises with step-by-step solutions, rules, and visual aids.

Worksheet: Exercises 1 to 3

Worksheet Instructions

Compare the fractions using the LCD or cross multiplication method. Show your work in the space provided.

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1 LCD Method Exercise
Problem 1
Compare: \( \frac{2}{3} \) and \( \frac{3}{4} \)
Definition:

Unlike denominators: Fractions with different bottom numbers (denominators)

Least Common Denominator (LCD): The smallest number that both denominators divide into evenly

LCD Comparison Method:
  1. Find the LCD of the denominators
  2. Convert each fraction to an equivalent fraction with the LCD
  3. Compare the numerators
Original Fractions
\( \frac{2}{3}, \frac{3}{4} \)
LCD
12
Comparison
\( \frac{8}{12} < \frac{9}{12} \)
Step 1: Find the LCD

Denominators: 3 and 4

Multiples of 3: 3, 6, 9, 12, 15, 18...

Multiples of 4: 4, 8, 12, 16, 20...

LCD = 12

Step 2: Convert to equivalent fractions

\( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)

\( \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)

Step 3: Compare numerators

Since 8 < 9, then \( \frac{8}{12} < \frac{9}{12} \)

Solution: \( \frac{2}{3} < \frac{3}{4} \)
Final answer:

\( \frac{2}{3} < \frac{3}{4} \)

Applied rules:

• LCD Method: Convert fractions to have the same denominator

• Comparison: When denominators are equal, compare numerators

• Equivalent Fractions: Multiply numerator and denominator by the same number

2 Cross Multiplication Exercise
Problem 2
Compare: \( \frac{5}{6} \) and \( \frac{7}{8} \)
Definition:

Cross Multiplication: Multiply the numerator of one fraction by the denominator of the other

Original Fractions
\( \frac{5}{6}, \frac{7}{8} \)
Cross Products
\(5 \times 8 = 40, 7 \times 6 = 42\)
Comparison
\(40 < 42\)
Step 1: Apply cross multiplication

Numerator of first × Denominator of second: \(5 \times 8 = 40\)

Numerator of second × Denominator of first: \(7 \times 6 = 42\)

Step 2: Compare the products

Since 40 < 42, then \( \frac{5}{6} < \frac{7}{8} \)

Step 3: Verify with LCD method

LCD of 6 and 8 is 24

\( \frac{5}{6} = \frac{20}{24}, \frac{7}{8} = \frac{21}{24} \)

Since 20 < 21, then \( \frac{5}{6} < \frac{7}{8} \) ✓

Solution: \( \frac{5}{6} < \frac{7}{8} \)
Final answer:

\( \frac{5}{6} < \frac{7}{8} \)

Applied rules:

• Cross Multiplication Rule: \( \frac{a}{b} > \frac{c}{d} \) if \( a \times d > b \times c \)

• Comparison: Larger cross product indicates larger fraction

• Verification: Cross multiply and compare results

3 Visual Comparison Exercise
Problem 3
Compare: \( \frac{3}{5} \) and \( \frac{2}{3} \)
Definition:

Visual Method: Draw models or use number lines to represent fractions

Original Fractions
\( \frac{3}{5}, \frac{2}{3} \)
LCD
15
Result
\( \frac{9}{15} < \frac{10}{15} \)
Step 1: Find the LCD

Denominators: 5 and 3

Multiples of 5: 5, 10, 15, 20...

Multiples of 3: 3, 6, 9, 12, 15...

LCD = 15

Step 2: Convert to equivalent fractions

\( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)

\( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)

Step 3: Compare numerators

Since 9 < 10, then \( \frac{9}{15} < \frac{10}{15} \)

Step 4: Visual representation

Draw circles divided into 15 parts

\( \frac{3}{5} \) = 9 out of 15 parts shaded

\( \frac{2}{3} \) = 10 out of 15 parts shaded

Solution: \( \frac{3}{5} < \frac{2}{3} \)
Final answer:

\( \frac{3}{5} < \frac{2}{3} \)

Applied rules:

• Visual Verification: Draw models to confirm comparison

• Equivalent Fractions: Same value, different representation

• Common Denominator: Allows direct comparison

Worksheet Rules and Methods
\( \frac{a}{b} \text{ vs } \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \text{ vs } \frac{c \times b}{d \times b} \)
Cross Multiplication Formula
Method 1
LCD Method
Find common denominator
Method 2
Cross Multiply
Multiply diagonally
Method 3
Visual Model
Draw representations
Key definitions:

Like Denominators: Fractions with the same denominator (bottom number)

Unlike Denominators: Fractions with different denominators

Numerator: The top number in a fraction (how many parts)

Denominator: The bottom number in a fraction (how many total parts)

Equivalent Fractions: Different fractions with the same value

Least Common Denominator (LCD): The smallest number that both denominators divide into evenly

Complete methodology:
  1. Identify denominators: Check if all fractions have the same denominator
  2. Choose comparison method: LCD, cross multiplication, or visual based on efficiency
  3. Apply the method: Follow steps for chosen technique
  4. Compare: Determine which fraction is greater
  5. Verify: Check with alternative method if possible
Tip 1: Always verify denominators are different before comparing numerators.
Tip 2: Use LCD method when denominators are small numbers (≤ 12).
Tip 3: Use cross multiplication when denominators are large or don't share common factors.
Tip 4: Always multiply both numerator and denominator by the same number.
Common errors: Forgetting to multiply both numerator and denominator by the same number, confusing numerators with denominators, making calculation mistakes.
Memory aid: "Different bottom, same bottom - then compare tops!"
Essential formulas and rules:

• LCD Rule: \( \frac{a}{b} = \frac{a \times n}{b \times n} \) for equivalent fractions

• Cross Multiply: \( \frac{a}{b} > \frac{c}{d} \) if \( a \times d > b \times c \)

• Comparison: When denominators are equal, compare numerators directly

• Conversion: Always multiply both numerator and denominator by the same number

Download Complete Worksheet PDF
Worksheet: Exercises 4 to 5
4 Mixed Operations Exercise
Problem 4
Compare: \( \frac{7}{12} \) and \( \frac{5}{9} \)
Definition:

Mixed operations: Problems involving multiple mathematical operations

Original Fractions
\( \frac{7}{12}, \frac{5}{9} \)
Cross Products
\(7 \times 9 = 63, 5 \times 12 = 60\)
Comparison
\(63 > 60\)
Step 1: Use cross multiplication (efficient for these denominators)

\( \frac{7}{12} \) vs \( \frac{5}{9} \)

\( 7 \times 9 = 63 \)

\( 5 \times 12 = 60 \)

Step 2: Compare cross products

Since 63 > 60, then \( \frac{7}{12} > \frac{5}{9} \)

Step 3: Verify with LCD method

LCD of 12 and 9 is 36

\( \frac{7}{12} = \frac{21}{36}, \frac{5}{9} = \frac{20}{36} \)

Since 21 > 20, then \( \frac{7}{12} > \frac{5}{9} \) ✓

Solution: \( \frac{7}{12} > \frac{5}{9} \)
Final answer:

\( \frac{7}{12} > \frac{5}{9} \)

Applied rules:

• Efficiency: Cross multiplication is faster for larger denominators

• Multiple Verification: Use different methods to confirm results

• Pattern Recognition: Look for common factors to simplify

5 Advanced Comparison Exercise
Problem 5
Compare: \( \frac{11}{15} \) and \( \frac{13}{18} \)
Definition:

Advanced comparison: Using multiple methods efficiently

Original Fractions
\( \frac{11}{15}, \frac{13}{18} \)
Cross Product
11×18=198, 13×15=195
Result
198 > 195
Step 1: Use cross multiplication (most efficient for these denominators)

\( \frac{11}{15} \) vs \( \frac{13}{18} \)

\( 11 \times 18 = 198 \)

\( 13 \times 15 = 195 \)

Step 2: Compare cross products

Since 198 > 195, then \( \frac{11}{15} > \frac{13}{18} \)

Step 3: Verify with LCD method

LCD of 15 and 18 is 90

\( \frac{11}{15} = \frac{66}{90}, \frac{13}{18} = \frac{65}{90} \)

Since 66 > 65, then \( \frac{11}{15} > \frac{13}{18} \) ✓

Step 4: Final verification

Convert to decimals: \( \frac{11}{15} = 0.733... \), \( \frac{13}{18} = 0.722... \)

Since 0.733... > 0.722..., then \( \frac{11}{15} > \frac{13}{18} \) ✓

Solution: \( \frac{11}{15} > \frac{13}{18} \)
Final answer:

\( \frac{11}{15} > \frac{13}{18} \)

Applied rules:

• Cross Multiplication: Most efficient for larger denominators

• Multiple Verification: Use different methods to confirm results

• Pattern Recognition: Look for common factors to simplify

Complete Guide: Comparing Fractions with Unlike Denominators
\( \frac{a}{b} \text{ vs } \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \text{ vs } \frac{c \times b}{d \times b} \)
Cross Multiplication Formula
Key definitions:

Unlike Denominators: Fractions with different bottom numbers (e.g., \( \frac{2}{3} \) and \( \frac{5}{7} \))

Like Denominators: Fractions with the same bottom number (e.g., \( \frac{2}{7} \) and \( \frac{5}{7} \))

Numerator: The top number in a fraction (how many parts)

Denominator: The bottom number in a fraction (how many total parts)

Comparison: Determining which fraction is greater, lesser, or equal to another

Equivalent Fractions: Different fractions that represent the same amount

Complete methodology:
  1. Identify the fractions: Recognize the numerators and denominators
  2. Choose comparison method: LCD, cross multiplication, or visual models
  3. Apply the method: Follow steps for chosen technique
  4. Compare: Determine which fraction is greater
  5. Verify: Check with alternative method if needed
Tip 1: Use LCD method when denominators are small numbers (≤ 12).
Tip 2: Use cross multiplication when denominators are large or don't share obvious factors.
Tip 3: Always multiply both numerator and denominator by the same number.
Tip 4: Use visual models to verify your answer.
Common errors: Forgetting to multiply both numerator and denominator, confusing numerators with denominators, making calculation mistakes.
Memory aid: "Bigger bottom, smaller piece" - larger denominators mean smaller parts.
Essential formulas and rules:

• LCD Rule: \( \frac{a}{b} > \frac{c}{d} \) if \( a \times d > b \times c \)

• Equality: \( \frac{a}{b} = \frac{c}{d} \) if \( a \times d = b \times c \)

• Less Than: \( \frac{a}{b} < \frac{c}{d} \) if \( a \times d < b \times c \)

• Ordering: To order fractions with unlike denominators, convert to like denominators

• Verification: Always check with an alternative method

Comparison Guide

📊
LCD Method

• Find common denominator

• Convert fractions

• Compare numerators

Cross Multiply

• Multiply diagonally

• Compare products

• Faster for large denominators

Visual Model

• Draw representations

• Compare visually

• Good for verification

Worksheet Questions & Answers

Question: How do I know which method to use when comparing fractions with unlike denominators?

Answer: Here's the decision tree:

  • Use LCD when: Denominators are small numbers (≤ 12) and easy to work with
  • Use cross multiplication when: Denominators are large or don't share obvious common factors
  • Use visual models when: You're learning the concept or want to verify your answer

Examples:

  • \( \frac{2}{3} \) vs \( \frac{3}{4} \): LCD (12) is easy - use LCD method
  • \( \frac{7}{13} \) vs \( \frac{5}{11} \): Large denominators - use cross multiplication

With practice, you'll recognize which method is faster for each problem!

Question: Why can't I just compare the numerators when denominators are different?

Answer: Because different denominators mean different-sized pieces:

  • \( \frac{2}{3} \) means 2 pieces of a pizza cut into 3 pieces
  • \( \frac{3}{4} \) means 3 pieces of a pizza cut into 4 pieces
  • The pieces in the first pizza are bigger than pieces in the second pizza
  • So 2 big pieces might be more than 3 small pieces!

Think of it like comparing 2 large cookies to 3 small cookies - the 2 large ones could be more than the 3 small ones.

That's why we need to make the pieces the same size (find common denominator) before comparing!

Question: How do I order multiple fractions with unlike denominators?

Answer: Follow these steps:

  1. Find the LCD: Find the least common denominator for all fractions
  2. Convert all fractions: Change each fraction to an equivalent fraction with the LCD
  3. Compare numerators: Order the fractions by comparing their numerators
  4. Write in original form: Present the answer using the original fractions

Example: Order \( \frac{2}{3}, \frac{3}{4}, \frac{1}{2} \)

  • LCD of 3, 4, and 2 is 12
  • \( \frac{2}{3} = \frac{8}{12}, \frac{3}{4} = \frac{9}{12}, \frac{1}{2} = \frac{6}{12} \)
  • Order numerators: 6 < 8 < 9
  • Therefore: \( \frac{1}{2} < \frac{2}{3} < \frac{3}{4} \)