Solved Exercises on Comparing Fractions with Like Denominators in Grade 4

Master comparing fractions with like denominators: identifying like fractions, comparing numerators, and ordering strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic comparison of like fractions
Exercise 1
Compare: \( \frac{2}{5} \) and \( \frac{3}{5} \). 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 5)
  2. Compare the numerators directly
  3. The fraction with the larger numerator is the larger fraction
Like fractions
\( \frac{2}{5}, \frac{3}{5} \)
Compare numerators
\( 2 < 3 \)
Result
\( \frac{2}{5} < \frac{3}{5} \)
Step 1: Identify like fractions

Both fractions have the same denominator (5), so they are like fractions.

Step 2: Compare numerators

Since 2 < 3, the fraction with numerator 2 is smaller than the fraction with numerator 3.

Step 3: Write the comparison

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

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

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

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 Multiple like fractions comparison
Exercise 2
Compare: \( \frac{1}{6}, \frac{4}{6}, \frac{3}{6} \). Order from least to greatest.
Definition:

Multiple Like Fractions: Several fractions with the same denominator. Order by comparing all numerators.

Like fractions
\( \frac{1}{6}, \frac{4}{6}, \frac{3}{6} \)
Compare numerators
\( 1 < 3 < 4 \)
Order
\( \frac{1}{6} < \frac{3}{6} < \frac{4}{6} \)
Step 1: Identify like fractions

All fractions have the same denominator (6), so they are like fractions.

Step 2: Compare all numerators

The numerators are 1, 4, and 3. In order: 1 < 3 < 4.

Step 3: Write the ordered fractions

\( \frac{1}{6} < \frac{3}{6} < \frac{4}{6} \)

\( \frac{1}{6} < \frac{3}{6} < \frac{4}{6} \)
Final answer:

\( \frac{1}{6} < \frac{3}{6} < \frac{4}{6} \)

Applied rules:

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

• Ordering Strategy: Arrange numerators in order, then match with corresponding fractions

• Systematic Approach: Compare all numerators before writing the final order

3 Real-world application with like fractions
Exercise 3
Sarah ate \( \frac{2}{8} \) of a pizza and Mark ate \( \frac{5}{8} \) of the same pizza. Who ate more pizza?
Definition:

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

Like fractions
\( \frac{2}{8}, \frac{5}{8} \)
Compare numerators
\( 2 < 5 \)
Result
\( \frac{2}{8} < \frac{5}{8} \)
Step 1: Identify the fractions to compare

Sarah ate \( \frac{2}{8} \) and Mark ate \( \frac{5}{8} \)

Step 2: Check if denominators are the same

Both fractions have denominator 8, so they are like fractions.

Step 3: Compare numerators

Since 2 < 5, Sarah ate less pizza than Mark.

Step 4: Interpret the result

Mark ate more pizza since \( \frac{5}{8} > \frac{2}{8} \).

Mark ate more pizza (\( \frac{2}{8} < \frac{5}{8} \))
Final answer:

Mark ate more pizza because \( \frac{5}{8} > \frac{2}{8} \)

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

Detailed Summary: Comparing Fractions with Like Denominators
\( \frac{a}{b} \) 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
Size Comparison
Larger numerator = larger fraction (same denominator)
Rule 3
Ordering
Arrange numerators in order, then match fractions
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

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

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

Complete methodology:
  1. Identify like fractions: Check if all fractions have the same denominator
  2. Compare numerators: Look at the top numbers directly
  3. Determine relationship: Decide which fraction is larger, smaller, or if they are equal
  4. Write the answer: Use the correct inequality symbol (<, >, or =)
  5. Verify: Double-check that denominators are indeed the same
Tip 1: Always check that denominators are the same before comparing numerators!
Tip 2: When ordering multiple like fractions, arrange numerators first, then write the corresponding fractions.
Tip 3: Remember that when denominators are the same, the larger numerator creates the larger fraction.
Tip 4: Think of like fractions as sharing the same size pieces - whoever has more pieces wins!
Tip 5: Always verify your work by double-checking that denominators are identical.
Common errors: Comparing numerators without checking denominators, making calculation errors when ordering multiple fractions, forgetting to verify that fractions are actually like fractions.
Key notes to remember: Like fractions are the easiest to compare! Just focus on the numerators. The larger the numerator (with same denominator), the larger the fraction. Always double-check your work!
Fraction Comparison Laws:

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

• Equality Law: \( \frac{a}{b} = \frac{c}{b} \) if and only if \( a = c \)

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

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

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

Solution: Exercises 4 to 5
4 Complex comparison with simplification
Exercise 4
Compare: \( \frac{4}{10} \) and \( \frac{6}{10} \). Then simplify both fractions and compare again.
Definition:

Simplified Fractions: Fractions reduced to their lowest terms by dividing both numerator and denominator by their greatest common factor.

Original fractions
\( \frac{4}{10}, \frac{6}{10} \)
Simplify
\( \frac{2}{5}, \frac{3}{5} \)
Comparison
\( \frac{2}{5} < \frac{3}{5} \)
Step 1: Compare original like fractions

Since denominators are the same (10), compare numerators: 4 < 6, so \( \frac{4}{10} < \frac{6}{10} \).

Step 2: Simplify both fractions

\( \frac{4}{10} = \frac{4 ÷ 2}{10 ÷ 2} = \frac{2}{5} \)

\( \frac{6}{10} = \frac{6 ÷ 2}{10 ÷ 2} = \frac{3}{5} \)

Step 3: Compare simplified fractions

Both simplified fractions have the same denominator (5), so compare numerators: 2 < 3.

Step 4: Verify the result

The comparison remains the same: \( \frac{2}{5} < \frac{3}{5} \), confirming our original answer.

\( \frac{4}{10} < \frac{6}{10} \) and \( \frac{2}{5} < \frac{3}{5} \)
Final answer:

\( \frac{4}{10} < \frac{6}{10} \) (which simplifies to \( \frac{2}{5} < \frac{3}{5} \))

Applied rules:

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

• Simplification: Equivalent fractions maintain the same relationship

• Verification: Simplified fractions should preserve the original comparison

5 Ordering multiple like fractions with different contexts
Exercise 5
Four friends ran different distances: Anna ran \( \frac{3}{7} \) of a mile, Ben ran \( \frac{1}{7} \) of a mile, Carol ran \( \frac{6}{7} \) of a mile, and David ran \( \frac{4}{7} \) of a mile. Order them from least distance to greatest distance.
Definition:

Multiple Like Fractions in Context: Several like fractions representing different quantities in a real-world scenario.

Like fractions
\( \frac{3}{7}, \frac{1}{7}, \frac{6}{7}, \frac{4}{7} \)
Numerators in order
\( 1 < 3 < 4 < 6 \)
Ordered fractions
\( \frac{1}{7} < \frac{3}{7} < \frac{4}{7} < \frac{6}{7} \)
Step 1: Identify all like fractions

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

Step 2: List the numerators

The numerators are: 3 (Anna), 1 (Ben), 6 (Carol), 4 (David)

Step 3: Order the numerators from least to greatest

1 < 3 < 4 < 6

Step 4: Match numerators with corresponding names

Ben (1/7), Anna (3/7), David (4/7), Carol (6/7)

Step 5: Write the final order

Ben, Anna, David, Carol - from least to greatest distance

Ben < Anna < David < Carol (\( \frac{1}{7} < \frac{3}{7} < \frac{4}{7} < \frac{6}{7} \))
Final answer:

From least to greatest distance: Ben (\( \frac{1}{7} \)), Anna (\( \frac{3}{7} \)), David (\( \frac{4}{7} \)), Carol (\( \frac{6}{7} \))

Applied rules:

• Systematic Ordering: List numerators first, then match with corresponding fractions

• Context Preservation: Maintain connection between fractions and their real-world meanings

• Verification: Double-check that denominators are identical before ordering

Advanced Like Fraction Comparison Techniques
\( \frac{a}{b} \) vs \( \frac{c}{b} \Rightarrow \frac{a}{b} < \frac{c}{b} \) if \( a < c \)
Fundamental Like Fractions 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 \( \frac{3}{7} \))

Greatest Common Factor (GCF): The largest number that divides both numerator and denominator evenly

Advanced methodology:
  1. Identify like fractions: Confirm that all fractions have identical denominators
  2. Compare numerators: Use basic number comparison for the top numbers
  3. Apply inequality symbols: Use <, >, or = appropriately
  4. Verify results: Double-check that denominators are the same and comparisons are accurate
  5. Contextual interpretation: Apply results to the given situation if applicable
Tip 1: When denominators are the same, compare numerators directly. Larger numerator = larger fraction.
Tip 2: For ordering multiple like fractions, list all numerators in order first, then match with fractions.
Tip 3: Always double-check that denominators are identical before using the like fraction rule.
Tip 4: Simplify fractions when possible to make comparisons clearer.
Tip 5: Remember: like fractions are the simplest type of fraction comparison!
Common errors: Comparing numerators without verifying identical denominators, making arithmetic mistakes when ordering multiple fractions, confusing like and unlike fraction rules.
Memory aids: "Same bottom, compare top" - when denominators are equal, focus on numerators. "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 \( a < c \)

• Equality Law: \( \frac{a}{b} = \frac{c}{b} \) if and only if \( a = c \)

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

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

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

• Consistency Property: Equivalent fractions maintain the same comparison relationships

Visual Like Fraction Comparison: Understanding Numerator Relationships
Visualizing Like Fractions
Compare these like fractions visually:
\( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \)
Notice how as the numerator increases, the fraction gets larger (same denominator).

Pattern Recognition: The chart shows how like fractions increase as the numerator increases.

  • \( \frac{1}{5} = 0.2 \) (20%)
  • \( \frac{2}{5} = 0.4 \) (40%)
  • \( \frac{3}{5} = 0.6 \) (60%)
  • \( \frac{4}{5} = 0.8 \) (80%)

Questions & Answers

Question: Why is comparing like fractions so much easier than comparing unlike fractions?

Answer: Great question! Here's why:

  • Like fractions: Have the same denominator, which means they represent parts of the same whole divided into equal pieces. You only need to compare how many pieces (numerators) you have.
  • Unlike fractions: Have different denominators, which means they represent parts of wholes divided differently. You need to convert them to like fractions first to make a fair comparison.

Think of it like comparing slices of pizza: if all pizzas are cut into 8 slices (like fractions), comparing 3 slices vs 5 slices is easy. But if one pizza is cut into 8 slices and another into 6 slices, you need to think about how to make the comparison fair.

For like fractions: just look at the top numbers! For unlike fractions: you need extra steps to make denominators the same.

Question: My child sometimes forgets to check if denominators are the same before comparing numerators. How can I help them remember?

Answer: Here are some helpful strategies:

  • Create a checklist: Before comparing, always ask: "Are the denominators the same?"
  • Visual reminders: Circle or highlight the denominators first to draw attention to them
  • Memory phrase: "Same bottom, compare top" or "Check the bottom first!"
  • Practice identification: Give exercises where children must identify whether fractions are like or unlike before solving

You could also use the acronym "SLID" (Same, Look, Identify, Denominator) to remind them to check denominators first.

Practice with worksheets that mix like and unlike fractions to strengthen discrimination skills!

Question: How can I help my child visualize why \( \frac{3}{5} \) is larger than \( \frac{2}{5} \)?

Answer: Visualization is key! Try these approaches:

  • Pie analogy: Draw a circle divided into 5 equal slices. Shade 2 slices for \( \frac{2}{5} \) and 3 slices for \( \frac{3}{5} \). Compare the shaded areas.
  • Rectangle model: Draw a rectangle divided into 5 equal sections. Shade 2 sections for \( \frac{2}{5} \) and 3 sections for \( \frac{3}{5} \).
  • Number line: Plot both fractions on a number line from 0 to 1. \( \frac{3}{5} \) will be further to the right than \( \frac{2}{5} \).

You can also use physical objects like chocolate bars divided into equal pieces, or paper folding activities where children physically see that having 3 out of 5 equal pieces is more than having 2 out of 5 equal pieces.

The key is that the pieces are the same size (same denominator), so more pieces (larger numerator) means more total amount.

Question: What if I have like fractions but one is simplified and the other isn't? Like comparing \( \frac{2}{4} \) and \( \frac{3}{4} \)?

Answer: Great question! Even if one fraction is simplified and the other isn't, if they have the same denominator, they are still like fractions and you can compare the numerators directly.

  • For \( \frac{2}{4} \) and \( \frac{3}{4} \): Both have denominator 4, so they are like fractions.
  • Compare numerators: 2 < 3, so \( \frac{2}{4} < \frac{3}{4} \).

You could also simplify \( \frac{2}{4} \) to \( \frac{1}{2} \), but then you'd have unlike fractions (\( \frac{1}{2} \) and \( \frac{3}{4} \)), which would require additional steps.

It's often easier to keep like fractions with the same denominator for direct comparison!

Question: Can I always use the same method for comparing any like fractions?

Answer: Yes! The method for comparing like fractions is always the same:

If you have like fractions \( \frac{a}{b} \) and \( \frac{c}{b} \), simply compare the numerators \( a \) and \( c \).

Rules:

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

This rule works for any like fractions, regardless of how large or small the numerators or denominators are!