Solved Exercises on Comparing Fractions with Unlike Denominators in Grade 4

Master comparing fractions with unlike denominators: finding common denominators, cross multiplication, equivalent fractions, and comparison strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic comparison of unlike denominators
Exercise 1
Compare: \( \frac{2}{3} \) and \( \frac{3}{4} \). Which fraction is larger?
Definition:

Unlike Denominators: Fractions that have different denominators. When comparing unlike denominators, we must find equivalent fractions with the same denominator.

Unlike denominator comparison method:
  1. Find the least common denominator (LCD) of both fractions
  2. Convert each fraction to an equivalent fraction with the LCD
  3. Compare the numerators of the equivalent fractions
  4. The fraction with the larger numerator is the larger fraction
Original fractions
\( \frac{2}{3}, \frac{3}{4} \)
LCD = 12
\( \frac{8}{12}, \frac{9}{12} \)
Comparison
\( \frac{8}{12} < \frac{9}{12} \)
Step 1: Find the LCD

The denominators are 3 and 4. Multiples of 3: 3, 6, 9, 12... Multiples of 4: 4, 8, 12... 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, we have \( \frac{8}{12} < \frac{9}{12} \)

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

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

Applied rules:

• Least Common Denominator: Find the smallest number that both denominators divide evenly into

• Equivalent Fractions: Multiply both numerator and denominator by the same number to maintain equality

• Numerator Comparison: When denominators are equal, compare numerators directly

2 Cross multiplication method
Exercise 2
Compare: \( \frac{3}{5} \) and \( \frac{2}{7} \). Which fraction is larger?
Definition:

Cross Multiplication: A shortcut method to compare two fractions by multiplying diagonally across the fractions.

Original fractions
\( \frac{3}{5}, \frac{2}{7} \)
Cross products
\( 3 \times 7 = 21, 2 \times 5 = 10 \)
Comparison
\( 21 > 10 \)
Step 1: Apply cross multiplication

Multiply the numerator of the first fraction by the denominator of the second: \( 3 \times 7 = 21 \)

Multiply the numerator of the second fraction by the denominator of the first: \( 2 \times 5 = 10 \)

Step 2: Compare the products

Since 21 > 10, the first fraction is larger than the second fraction

Step 3: Write the comparison

\( \frac{3}{5} > \frac{2}{7} \)

\( \frac{3}{5} > \frac{2}{7} \)
Final answer:

\( \frac{3}{5} > \frac{2}{7} \)

Applied rules:

• Cross Multiplication: \( \frac{a}{b} \) vs \( \frac{c}{d} \) → compare \( a \times d \) vs \( c \times b \)

• Product Comparison: Larger product corresponds to the larger fraction

• Efficiency: Cross multiplication is often faster than finding LCD

3 Multiple fractions comparison
Exercise 3
Order from least to greatest: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \)
Definition:

Ordering Unlike Denominators: To order multiple fractions with unlike denominators, convert them to equivalent fractions with a common denominator or use cross multiplication to compare pairs.

Original fractions
\( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \)
LCD = 12
\( \frac{6}{12}, \frac{8}{12}, \frac{9}{12} \)
Order
\( \frac{6}{12} < \frac{8}{12} < \frac{9}{12} \)
Step 1: Find the LCD for all fractions

Denominators: 2, 3, 4. LCM of 2, 3, 4 is 12

Step 2: Convert each fraction to equivalent with LCD

\( \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \)

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

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

Step 3: Order by comparing numerators

Since 6 < 8 < 9, the order is \( \frac{6}{12} < \frac{8}{12} < \frac{9}{12} \)

Step 4: Write the final order

\( \frac{1}{2} < \frac{2}{3} < \frac{3}{4} \)

\( \frac{1}{2} < \frac{2}{3} < \frac{3}{4} \)
Final answer:

\( \frac{1}{2} < \frac{2}{3} < \frac{3}{4} \)

Applied rules:

• Multiple Fraction Ordering: Find LCD for all fractions simultaneously

• Systematic Approach: Convert all fractions before comparing

• Verification: Double-check each conversion to ensure accuracy

Detailed Summary: Comparing Fractions with Unlike Denominators
\( \frac{a}{b} \) vs \( \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \) vs \( \frac{c \times b}{d \times b} \)
Cross Multiplication Method
Method 1
Cross Multiplication
\( \frac{a}{b} \) vs \( \frac{c}{d} \) → compare \( a \times d \) vs \( c \times b \)
Method 2
Common Denominator
Convert to same denominator, then compare numerators
Method 3
Decimal Conversion
Divide numerator by denominator to get decimals
Key definitions:

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

Denominator: The bottom number in a fraction (how many equal parts the whole is divided into)

Unlike Denominators: Fractions where the bottom numbers are different

Like Denominators: Fractions where the bottom numbers are the same

Equivalent Fractions: Fractions that represent the same amount but have different numerators and denominators

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

Proper Fraction: A fraction where the numerator is smaller than the denominator

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator

Complete methodology:
  1. Identify unlike denominators: Recognize that the denominators are different
  2. Choose the comparison method: Cross multiplication, common denominator, or decimal conversion
  3. Apply the method: Follow the steps of the chosen technique systematically
  4. Compare results: Determine which fraction is larger, smaller, or if they are equal
  5. Write the answer: Use the correct inequality symbol (<, >, or =)
Tip 1: Cross multiplication is often the fastest method for comparing two fractions with unlike denominators.
Tip 2: When ordering multiple fractions, finding a common denominator is usually more efficient than pairwise comparisons.
Tip 3: Remember that the larger the numerator (with same denominator), the larger the fraction.
Tip 4: When denominators are the same, focus on numerators. When numerators are the same, focus on denominators (smaller denominator = larger fraction).
Tip 5: Always verify your work by converting back to the original fractions or checking with an alternative method.
Common errors: Confusing numerators with denominators, making calculation errors when finding LCD, forgetting to multiply both numerator and denominator when creating equivalent fractions, misapplying cross multiplication.
Key notes to remember: Equivalent fractions have the same value even though they look different. The larger the denominator, the smaller each part becomes. Always double-check your work!
Fraction Comparison Laws:

• Transitive Property: If \( \frac{a}{b} > \frac{c}{d} \) and \( \frac{c}{d} > \frac{e}{f} \), then \( \frac{a}{b} > \frac{e}{f} \)

• Reflexive Property: Any fraction is equal to itself: \( \frac{a}{b} = \frac{a}{b} \)

• Antisymmetric Property: If \( \frac{a}{b} > \frac{c}{d} \), then \( \frac{c}{d} < \frac{a}{b} \)

• Equivalence Property: \( \frac{a}{b} = \frac{c}{d} \) if and only if \( a \times d = b \times c \)

Solution: Exercises 4 to 5
4 Real-world application with unlike denominators
Exercise 4
Emma read \( \frac{2}{5} \) of her book and John read \( \frac{3}{7} \) of his book. Who read more of their book?
Definition:

Real-World Application: Comparing fractions with unlike denominators helps us determine which quantity is greater in practical situations.

Fractions to compare
\( \frac{2}{5}, \frac{3}{7} \)
LCD = 35
\( \frac{14}{35}, \frac{15}{35} \)
Comparison
\( \frac{14}{35} < \frac{15}{35} \)
Step 1: Identify the fractions to compare

Emma read \( \frac{2}{5} \) and John read \( \frac{3}{7} \)

Step 2: Find the LCD

Denominators are 5 and 7. LCM of 5 and 7 is 35

Step 3: Convert to equivalent fractions

\( \frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \)

\( \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \)

Step 4: Compare the numerators

Since 14 < 15, we have \( \frac{14}{35} < \frac{15}{35} \)

Step 5: Interpret the result

John read more of his book since \( \frac{3}{7} > \frac{2}{5} \)

John read more (\( \frac{3}{7} > \frac{2}{5} \))
Final answer:

John read more of his book because \( \frac{3}{7} > \frac{2}{5} \)

Applied rules:

• Problem Solving: Translate word problems into mathematical expressions

• Real-World Context: Apply fraction comparison to practical situations

• Interpretation: Connect mathematical results back to the original context

5 Complex comparison with mixed numbers
Exercise 5
Compare: \( 1\frac{1}{4} \) and \( 1\frac{2}{5} \). Which mixed number is larger?
Definition:

Mixed Numbers: Numbers that contain both a whole number and a proper fraction. To compare mixed numbers with unlike fractional parts, focus on comparing the fractional parts since the whole numbers are the same.

Mixed numbers
\( 1\frac{1}{4}, 1\frac{2}{5} \)
Compare fractional parts
\( \frac{1}{4}, \frac{2}{5} \)
LCD = 20
\( \frac{5}{20}, \frac{8}{20} \)
Step 1: Observe the mixed numbers

Both have the same whole number part (1), so we only need to compare the fractional parts: \( \frac{1}{4} \) and \( \frac{2}{5} \)

Step 2: Find the LCD for the fractional parts

Denominators are 4 and 5. LCM of 4 and 5 is 20

Step 3: Convert fractional parts to equivalent fractions

\( \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \)

\( \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \)

Step 4: Compare the fractional parts

Since 5 < 8, we have \( \frac{5}{20} < \frac{8}{20} \), which means \( \frac{1}{4} < \frac{2}{5} \)

Step 5: Combine with whole number parts

Since the whole number parts are equal and \( \frac{1}{4} < \frac{2}{5} \), we have \( 1\frac{1}{4} < 1\frac{2}{5} \)

\( 1\frac{1}{4} < 1\frac{2}{5} \)
Final answer:

\( 1\frac{1}{4} < 1\frac{2}{5} \)

Applied rules:

• Mixed Number Comparison: When whole numbers are equal, compare only the fractional parts

• Comparison Strategy: Focus on the different components first (fractional parts)

• Systematic Approach: Handle the whole number and fractional parts separately when appropriate

Advanced Unlike Denominator Comparison Techniques
\( \frac{a}{b} \) vs \( \frac{c}{d} \Rightarrow \frac{a \times d}{b \times d} \) vs \( \frac{c \times b}{d \times b} \)
Cross Multiplication Rule
Advanced definitions:

Like Fractions: Fractions with the same denominator (e.g., \( \frac{2}{5} \) and \( \frac{3}{5} \))

Unlike Fractions: Fractions with different denominators (e.g., \( \frac{2}{3} \) and \( \frac{4}{5} \))

Proper Fraction: A fraction where the numerator is smaller than the denominator (e.g., \( \frac{3}{4} \))

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \( \frac{5}{3} \))

Mixed Number: A combination of a whole number and a proper fraction (e.g., \( 2\frac{1}{3} \))

Unit Fraction: A fraction with numerator 1 (e.g., \( \frac{1}{4} \))

Like Numerators: Fractions with the same numerator (e.g., \( \frac{3}{4} \) and \( \frac{3}{7} \))

Unlike Numerators: Fractions with different numerators (e.g., \( \frac{2}{5} \) and \( \frac{3}{5} \))

Advanced methodology:
  1. Analyze the fractions: Look at numerators and denominators to determine the best approach
  2. Choose the optimal method: Cross multiplication for two fractions, common denominator for multiple fractions, or decimal conversion for complex numbers
  3. Apply the chosen method: Follow the steps systematically, paying attention to detail
  4. Verify the result: Check calculations and make sure the answer makes sense in context
  5. Express the answer: Use the correct inequality symbol (<, >, or =) and provide context if needed
Tip 1: When denominators are the same, compare numerators directly. Larger numerator = larger fraction.
Tip 2: When numerators are the same, compare denominators. Smaller denominator = larger fraction.
Tip 3: For quick mental comparisons, try to estimate if the fraction is more or less than 1/2, 1/4, or 3/4.
Tip 4: Always double-check your LCD calculations by ensuring each original denominator divides evenly into the LCD.
Tip 5: When comparing to 1, determine if the fraction is proper (less than 1) or improper (greater than or equal to 1).
Common errors: Forgetting to find LCD, making arithmetic mistakes, confusing numerators and denominators, not simplifying fractions before comparison, applying cross multiplication incorrectly.
Memory aids: "Bigger bottom, smaller fraction" - when numerators are equal, a larger denominator creates a smaller fraction. "More parts taken" - when denominators are equal, more parts (larger numerator) means a larger fraction.
Advanced Fraction Laws:

• Transitive Property: If \( \frac{a}{b} > \frac{c}{d} \) and \( \frac{c}{d} > \frac{e}{f} \), then \( \frac{a}{b} > \frac{e}{f} \)

• Reflexive Property: Any fraction is equal to itself: \( \frac{a}{b} = \frac{a}{b} \)

• Antisymmetric Property: If \( \frac{a}{b} > \frac{c}{d} \), then \( \frac{c}{d} < \frac{a}{b} \)

• Equivalence Property: \( \frac{a}{b} = \frac{c}{d} \) if and only if \( a \times d = b \times c \)

• Consistency Property: All comparison methods will yield the same result when applied correctly

Visual Fraction Comparison: Understanding Unlike Denominator Relationships
Visualizing Fraction Sizes
Compare these fractions visually:
\( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \)
Notice how as the numerator and denominator increase proportionally, the fraction gets closer to 1.

Pattern Recognition: The chart shows how fractions with consecutive numerators and denominators approach 1 as the numbers get larger.

  • \( \frac{1}{2} = 0.5 \) (50%)
  • \( \frac{2}{3} ≈ 0.667 \) (66.7%)
  • \( \frac{3}{4} = 0.75 \) (75%)
  • \( \frac{4}{5} = 0.8 \) (80%)
  • \( \frac{5}{6} ≈ 0.833 \) (83.3%)

Questions & Answers

Question: I'm confused about when to use cross multiplication versus finding a common denominator. How do I know which method to use?

Answer: Great question! Here's how to decide:

  • Cross multiplication: Best when comparing just two fractions quickly. It's faster and doesn't require finding LCD.
  • Common denominator: Better when comparing multiple fractions or when you need to order several fractions.

For example, comparing \( \frac{3}{4} \) and \( \frac{5}{7} \): Cross multiplication (3×7 vs 5×4) is faster.

But for ordering \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \): Finding a common denominator (12) makes it easier to order all three at once.

Both methods will give the same result - choose the one that feels more comfortable for the specific problem!

Question: My child is struggling with finding the least common denominator. Are there any tricks to make this easier?

Answer: Yes! Here are some helpful strategies:

  • Start with the larger denominator: List its multiples and see if the smaller denominator divides evenly into any of them.
  • Look for patterns: If denominators are consecutive numbers (like 3 and 4), their LCD is often their product (12).
  • Use prime factorization: Break down each denominator into prime factors and take the highest power of each prime.

Example: For denominators 6 and 8:

Multiples of 8: 8, 16, 24... Does 6 divide 8? No. Does 6 divide 16? No. Does 6 divide 24? Yes! So LCD = 24.

Practice with smaller numbers first to build confidence!

Question: How can I help my child visualize why \( \frac{3}{4} \) is larger than \( \frac{2}{3} \) without doing calculations?

Answer: Visualization is key! Try these approaches:

  • Pizza analogy: \( \frac{3}{4} \) means 3 slices out of 4 total. \( \frac{2}{3} \) means 2 slices out of 3 total. Draw both scenarios.
  • Number line: Plot both fractions on a number line from 0 to 1. \( \frac{3}{4} \) will be further to the right.
  • Think in terms of halves: \( \frac{3}{4} = 0.75 \) (more than half), \( \frac{2}{3} ≈ 0.67 \) (less than 0.75).

For \( \frac{3}{4} \) vs \( \frac{2}{3} \): Both are close to 1, but \( \frac{3}{4} \) needs only 1 piece to make a whole, while \( \frac{2}{3} \) needs 1 piece too. However, the pieces in \( \frac{3}{4} \) are smaller (fourths vs thirds), so having 3 of the smaller pieces vs 2 of the larger pieces needs calculation, which is why we use the methods taught!

Question: What happens when I have to compare a fraction to a whole number? Like comparing \( \frac{5}{3} \) to 2?

Answer: Great question! You can handle this in two ways:

  • Method 1: Convert the whole number to a fraction with the same denominator. \( 2 = \frac{6}{3} \). Now compare \( \frac{5}{3} \) and \( \frac{6}{3} \). Since 5 < 6, \( \frac{5}{3} < 2 \).
  • Method 2: Convert the fraction to a mixed number. \( \frac{5}{3} = 1\frac{2}{3} \). Since \( 1\frac{2}{3} < 2 \), we have \( \frac{5}{3} < 2 \).

The key is to have the same form for both numbers before comparing!

For \( \frac{5}{3} \) vs 2: Since \( \frac{5}{3} = 1\frac{2}{3} \) and 1 is less than 2, we know \( \frac{5}{3} < 2 \).

Question: Can I always use cross multiplication for comparing any two fractions?

Answer: Yes, cross multiplication works for comparing any two fractions! Here's why:

If you have \( \frac{a}{b} \) and \( \frac{c}{d} \), cross multiplication gives you \( a \times d \) and \( c \times b \).

This works because you're essentially comparing \( \frac{a \times d}{b \times d} \) and \( \frac{c \times b}{b \times d} \) without actually calculating the denominators.

Rules:

  • If \( a \times d > c \times b \), then \( \frac{a}{b} > \frac{c}{d} \)
  • If \( a \times d < c \times b \), then \( \frac{a}{b} < \frac{c}{d} \)
  • If \( a \times d = c \times b \), then \( \frac{a}{b} = \frac{c}{d} \)

Cross multiplication is especially useful when the denominators are large or when you want a quick comparison!