Solved Exercises on Comparing Like and Unlike Fractions in Grade 4

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

Solution: Exercises 1 to 3
1 Comparing like fractions
Exercise 1
Compare: \( \frac{3}{7} \) and \( \frac{5}{7} \). Which fraction is larger?
Definition:

Like Fractions: Fractions that have the same denominator. When comparing like fractions, we only need to compare the numerators.

Like fraction comparison method:
  1. Check if denominators are the same (they are both 7)
  2. Compare the numerators directly
  3. The fraction with the larger numerator is the larger fraction
Like fractions
\( \frac{3}{7}, \frac{5}{7} \)
Compare numerators
\( 3 < 5 \)
Result
\( \frac{3}{7} < \frac{5}{7} \)
Step 1: Identify like fractions

Both fractions have the same denominator (7), so they are like fractions.

Step 2: Compare numerators

Since 3 < 5, the fraction with numerator 3 is smaller than the fraction with numerator 5.

Step 3: Write the comparison

\( \frac{3}{7} < \frac{5}{7} \)

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

\( \frac{3}{7} < \frac{5}{7} \)

Applied rules:

• Like Fractions Rule: When denominators are the same, compare numerators directly

• Size Comparison: Larger numerator means larger fraction (same denominator)

• Simplicity: Like fractions are easier to compare than unlike fractions

2 Comparing unlike fractions
Exercise 2
Compare: \( \frac{2}{3} \) and \( \frac{3}{5} \). Which fraction is larger?
Definition:

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

Unlike fractions
\( \frac{2}{3}, \frac{3}{5} \)
LCD = 15
\( \frac{10}{15}, \frac{9}{15} \)
Comparison
\( 10 > 9 \)
Step 1: Identify unlike fractions

The denominators are different (3 and 5), so these are unlike fractions.

Step 2: Find the LCD

Denominators are 3 and 5. LCM of 3 and 5 is 15.

Step 3: Convert to equivalent fractions

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

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

Step 4: Compare numerators

Since 10 > 9, we have \( \frac{10}{15} > \frac{9}{15} \).

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

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

Applied rules:

• Unlike Fractions Rule: Must convert to like fractions before comparing

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

• Systematic Approach: Find LCD, convert, then compare

3 Mixed comparison of like and unlike fractions
Exercise 3
Order from least to greatest: \( \frac{1}{4}, \frac{3}{8}, \frac{5}{8} \)
Definition:

Mixed Fractions: A set containing both like and unlike fractions. We identify like fractions first, then handle unlike fractions.

Original fractions
\( \frac{1}{4}, \frac{3}{8}, \frac{5}{8} \)
Like fractions: \( \frac{3}{8}, \frac{5}{8} \)
\( \frac{3}{8} < \frac{5}{8} \)
Compare \( \frac{1}{4} \) to others
\( \frac{2}{8} < \frac{3}{8} \)
Step 1: Identify like and unlike fractions

\( \frac{3}{8} \) and \( \frac{5}{8} \) are like fractions (same denominator). \( \frac{1}{4} \) is unlike with the others.

Step 2: Compare like fractions first

Between \( \frac{3}{8} \) and \( \frac{5}{8} \), since 3 < 5, we have \( \frac{3}{8} < \frac{5}{8} \).

Step 3: Compare unlike fraction to the group

Convert \( \frac{1}{4} \) to eighths: \( \frac{1}{4} = \frac{2}{8} \).

Since \( \frac{2}{8} < \frac{3}{8} < \frac{5}{8} \), we have \( \frac{1}{4} < \frac{3}{8} < \frac{5}{8} \).

Step 4: Write the final order

\( \frac{1}{4} < \frac{3}{8} < \frac{5}{8} \)

\( \frac{1}{4} < \frac{3}{8} < \frac{5}{8} \)
Final answer:

\( \frac{1}{4} < \frac{3}{8} < \frac{5}{8} \)

Applied rules:

• Mixed Comparison: Handle like fractions first, then convert unlike fractions

• Systematic Ordering: Organize by similarities before differences

• Verification: Double-check conversions and comparisons

Detailed Summary: Comparing Like and Unlike Fractions
\( \frac{a}{b} \) vs \( \frac{c}{b} \Rightarrow \frac{a}{b} < \frac{c}{b} \) if \( a < c \)
Like Fractions Rule
Rule 1
Like Fractions
Same denominator: compare numerators directly
Rule 2
Unlike Fractions
Different denominators: find common denominator first
Rule 3
Cross Multiplication
\( \frac{a}{b} \) vs \( \frac{c}{d} \) → compare \( a \times d \) vs \( c \times b \)
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)

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 \( \frac{4}{5} \))

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 fraction types: Determine if fractions are like or unlike
  2. Choose the comparison method: Direct comparison for like fractions, common denominator for unlike fractions
  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: Always check if fractions are like first - it's much easier to compare them!
Tip 2: When ordering mixed like/unlike fractions, handle like fractions first, then convert the rest.
Tip 3: Remember that when denominators are the same, the larger numerator creates the larger fraction.
Tip 4: When numerators are the same, the smaller denominator creates the larger fraction.
Tip 5: Always verify your work by converting back to the original fractions or checking with an alternative method.
Common errors: Comparing numerators without checking denominators, making calculation errors when finding LCD, forgetting to multiply both numerator and denominator when creating equivalent fractions, misapplying comparison rules.
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:

• Like Fractions Law: \( \frac{a}{b} < \frac{c}{b} \) if and only if \( a < c \)

• Unlike Fractions Law: Convert to like fractions before comparison

• 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 \( \frac{c}{d} < \frac{a}{b} \)

Solution: Exercises 4 to 5
4 Real-world application with like and unlike fractions
Exercise 4
Emma ate \( \frac{2}{6} \) of a cake and Liam ate \( \frac{1}{3} \) of the same cake. Who ate more cake?
Definition:

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

Fractions to compare
\( \frac{2}{6}, \frac{1}{3} \)
Simplify first fraction
\( \frac{2}{6} = \frac{1}{3} \)
Comparison
\( \frac{1}{3} = \frac{1}{3} \)
Step 1: Identify the fractions to compare

Emma ate \( \frac{2}{6} \) and Liam ate \( \frac{1}{3} \)

Step 2: Simplify if possible

\( \frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3} \)

Step 3: Compare the simplified fractions

Both fractions are now \( \frac{1}{3} \), so they are equal.

Step 4: Interpret the result

Both Emma and Liam ate the same amount of cake.

Both ate the same amount (\( \frac{2}{6} = \frac{1}{3} \))
Final answer:

Emma and Liam ate the same amount of cake because \( \frac{2}{6} = \frac{1}{3} \)

Applied rules:

• Problem Solving: Translate word problems into mathematical expressions

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

• Equivalent Fractions: Simplify fractions before comparison when possible

5 Complex comparison with multiple like and unlike fractions
Exercise 5
Order from greatest to least: \( \frac{3}{4}, \frac{2}{8}, \frac{6}{8}, \frac{1}{2} \)
Definition:

Complex Ordering: A set containing multiple like and unlike fractions requiring systematic conversion and comparison.

Original fractions
\( \frac{3}{4}, \frac{2}{8}, \frac{6}{8}, \frac{1}{2} \)
Convert to LCD = 8
\( \frac{6}{8}, \frac{2}{8}, \frac{6}{8}, \frac{4}{8} \)
Order by numerators
\( \frac{6}{8} = \frac{6}{8} > \frac{4}{8} > \frac{2}{8} \)
Step 1: Identify like and unlike fractions

\( \frac{2}{8} \) and \( \frac{6}{8} \) are like fractions. \( \frac{3}{4} \) and \( \frac{1}{2} \) are unlike with others.

Step 2: Find LCD for all fractions

Denominators are 4, 8, 8, 2. LCM of 4, 8, 2 is 8.

Step 3: Convert all fractions to equivalent with LCD

\( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \)

\( \frac{2}{8} = \frac{2}{8} \) (already has denominator 8)

\( \frac{6}{8} = \frac{6}{8} \) (already has denominator 8)

\( \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \)

Step 4: Order by comparing numerators

Since 6 = 6 > 4 > 2, the order is \( \frac{6}{8} = \frac{6}{8} > \frac{4}{8} > \frac{2}{8} \)

Step 5: Convert back to original forms

\( \frac{3}{4} = \frac{6}{8} > \frac{1}{2} = \frac{4}{8} > \frac{2}{8} \)

\( \frac{3}{4} = \frac{6}{8} > \frac{1}{2} > \frac{2}{8} \)
Final answer:

\( \frac{3}{4} = \frac{6}{8} > \frac{1}{2} > \frac{2}{8} \)

Applied rules:

• Systematic Conversion: Convert all fractions to like fractions before ordering

• Comparison Strategy: Handle like fractions first, then convert unlike fractions

• Verification: Double-check all conversions and ensure accuracy

Advanced Fraction 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 if they are like or unlike
  2. Choose the optimal method: Direct comparison for like fractions, common denominator for unlike fractions, or cross multiplication for two fractions
  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: Simplify fractions before comparing when possible to make calculations easier.
Common errors: Forgetting to find LCD for unlike fractions, making arithmetic mistakes, confusing numerators and denominators, not simplifying fractions before comparison, applying comparison rules 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:

• Like Fractions Law: \( \frac{a}{b} < \frac{c}{b} \) if and only if \( a < c \)

• Unlike Fractions Law: Convert to like fractions before comparison

• 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 Like vs Unlike Fractions
Visualizing Fraction Relationships
Compare these fractions visually:
Like: \( \frac{1}{4}, \frac{2}{4}, \frac{3}{4} \)
Unlike: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \)
Notice how like fractions have the same sized pieces, making comparison straightforward.

Pattern Recognition: The chart shows how like fractions are easier to compare than unlike fractions.

  • Like fractions: \( \frac{1}{4} = 0.25, \frac{2}{4} = 0.5, \frac{3}{4} = 0.75 \)
  • Unlike fractions: \( \frac{1}{2} = 0.5, \frac{2}{3} ≈ 0.667, \frac{3}{4} = 0.75 \)

Questions & Answers

Question: How do I know if fractions are like or unlike? Sometimes I get confused.

Answer: Great question! It's simple:

  • Like fractions: Have the SAME denominator (bottom number). Examples: \( \frac{2}{5} \) and \( \frac{3}{5} \), or \( \frac{1}{8} \) and \( \frac{7}{8} \).
  • Unlike fractions: Have DIFFERENT denominators. Examples: \( \frac{2}{3} \) and \( \frac{4}{5} \), or \( \frac{1}{4} \) and \( \frac{3}{7} \).

Just look at the bottom numbers! If they're the same, they're like fractions. If they're different, they're unlike fractions.

For like fractions: compare numerators directly. For unlike fractions: find common denominators first!

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).
  • Check if one divides the other: If 4 divides 8 evenly, then 8 is the LCD for fractions with denominators 4 and 8.

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:

  • Pie analogy: Draw two circles of the same size. Divide one into 4 equal parts and shade 3. Divide the other into 3 equal parts and shade 2. Compare the shaded areas.
  • Number line: Plot both fractions on a number line from 0 to 1. \( \frac{3}{4} \) will be further to the right than \( \frac{2}{3} \).
  • Think in terms of halves: \( \frac{3}{4} = 0.75 \) (more than half), \( \frac{2}{3} ≈ 0.67 \) (less than 0.75).

\( \frac{3}{4} \) takes 1 piece to make a whole, while \( \frac{2}{3} \) also needs 1 piece. But fourths are smaller pieces than thirds, so having 3 small pieces vs 2 larger pieces requires 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!