Solved Exercises on Comparing Fractions with Same Numerator in Grade 3

Master comparing fractions with same numerator: visual models, comparison strategies, and core principles through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Pie Chart Comparison
Exercise 1
Compare 1/3 and 1/5 using pie charts. Which fraction is larger?
Definition:

Same numerator: Fractions with identical top numbers (numerators)

Denominator effect: Larger denominator means smaller individual pieces

Same numerator comparison method:
  1. Identify that numerators are the same (both are 1)
  2. Compare denominators: 3 vs 5
  3. Since 3 < 5, then 1/3 > 1/5
  4. Draw visual models to confirm
1/3
vs
1/5
Step 1: Identify same numerator

Both fractions have numerator 1, meaning we're taking 1 part from each whole

Step 2: Compare denominators

1/3 means 1 part of 3 equal parts

1/5 means 1 part of 5 equal parts

Step 3: Understand denominator effect

More parts (larger denominator) = smaller individual parts

So 1/3 (3 parts) > 1/5 (5 parts)

Step 4: Visual confirmation

The 1/3 pie has a larger shaded area than the 1/5 pie

1/3 > 1/5
Final answer:

1/3 is larger than 1/5 because when numerators are the same, the fraction with the smaller denominator is larger. This is because fewer total parts mean each part is bigger.

Applied rules:

• Same numerator rule: When numerators are equal, compare denominators inversely

• Denominator effect: Larger denominator means smaller individual pieces

• Visual verification: Pie charts confirm mathematical result

2 Bar Model Comparison
Exercise 2
Compare 2/4 and 2/6 using bar models. Which fraction is larger?
Definition:

Bar model: Rectangular representation showing proportional parts of a whole

Denominator impact: More divisions mean smaller individual segments

2/4
vs
2/6
Step 1: Recognize same numerator

Both fractions have numerator 2, meaning we take 2 parts from each whole

Step 2: Analyze denominators

2/4 means 2 parts of 4 equal parts total

2/6 means 2 parts of 6 equal parts total

Step 3: Compare part sizes

Parts of 4 are larger than parts of 6

So 2 larger parts > 2 smaller parts

Step 4: Conclusion

Since 4 < 6, then 2/4 > 2/6

2/4 > 2/6
Final answer:

2/4 is larger than 2/6 because when numerators are the same, the fraction with the smaller denominator is greater. This is because the individual parts are larger when the whole is divided into fewer pieces.

Applied rules:

• Same numerator principle: Compare denominators inversely when numerators match

• Part size reasoning: Fewer divisions mean larger individual parts

• Visual comparison: Bar models show proportional differences clearly

3 Ordering Multiple Fractions
Exercise 3
Order these fractions from greatest to least: 3/4, 3/7, 3/5, 3/8
Definition:

Ordering fractions: Arranging fractions in ascending or descending order

Descending order: From largest to smallest

Step 1: Identify all numerators

All fractions have numerator 3, so we compare denominators inversely

Step 2: List the denominators

Denominators: 4, 7, 5, 8

Step 3: Order denominators from least to greatest

4, 5, 7, 8

Step 4: Match to fractions (inverse relationship)

Since smaller denominators mean larger fractions: 3/4 > 3/5 > 3/7 > 3/8

Step 5: Verify with visual models

Each fraction represents 3 parts of different-sized wholes

3/4 > 3/5 > 3/7 > 3/8
Final answer:

The fractions ordered from greatest to least are: 3/4, 3/5, 3/7, 3/8. Since all have the same numerator, we order by comparing denominators in reverse: smallest denominator first.

Applied rules:

• Same numerator ordering: Order fractions by ordering denominators in reverse

• Inverse relationship: Larger denominator means smaller fraction when numerator is same

• Systematic approach: List denominators separately, order them, then map back to fractions

Summary: Comparing Fractions with Same Numerator
Detailed Summary

Core Concepts and Definitions

Comparing fractions with same numerator involves understanding the inverse relationship between denominators and fraction size. When numerators are identical, the fraction with the smaller denominator is larger because each piece represents a larger portion of the whole.

Key Rules and Principles

  • Same Numerator Rule: When numerators are equal, compare denominators inversely
  • Denominator Effect: Larger denominator means smaller fraction
  • Inverse Relationship: Smaller denominator = larger fraction
  • Part Size Concept: More divisions create smaller individual parts

Step-by-Step Methods

  1. Identify Numerators: Confirm that all fractions have the same numerator
  2. Compare Denominators: Focus on the bottom numbers
  3. Apply Inverse Logic: Smaller denominator means larger fraction
  4. Visual Confirmation: Draw models to verify results

Examples from Simple to Advanced

  • Simple: 1/2 vs 1/3 (clear difference in denominators)
  • Intermediate: 2/5 vs 2/7 (closer denominators to compare)
  • Advanced: Ordering 3/4, 3/6, 3/5, 3/8 from greatest to least

Tips and Tricks

  • Remember: "Same top, smaller bottom = bigger fraction"
  • Think of it like sharing: 1 slice of a 3-piece pizza vs 1 slice of a 5-piece pizza
  • Use visual models to confirm your reasoning
  • Focus on denominator differences when numerators match

Common Mistakes to Avoid

  • Thinking that larger denominators mean larger fractions (when numerators are same)
  • Applying this rule when numerators are different
  • Confusing numerators and denominators

Key Notes for Memorization

  • "Same top, compare bottoms (inversely)"
  • "Fewer pieces, bigger pieces"
  • "Larger denominator = smaller fraction (when numerator is same)"
When a/b vs a/c: Compare b vs c (inversely)
Same Numerator Rule
Rule
a/b vs a/c → Compare b vs c inversely
Same numerator, inverse denominator comparison
Principle
Smaller denominator = Larger fraction
When numerators match
Method
Inverse comparison
Denominator affects fraction size inversely
Same Numerator Rule
When numerators are equal, compare denominators in reverse: smaller denominator means larger fraction.
Inverse Relationship
As denominators increase, fraction size decreases when numerator is constant.
Part Size Concept
More equal divisions of a whole create smaller individual parts.
Tip 1: Always check that numerators are truly the same before applying this rule.
Tip 2: Remember: "Smaller denominator = bigger fraction" when numerators match.
Tip 3: Use visual models to reinforce the concept of part size.
Common errors: Thinking larger denominators mean larger fractions, applying this rule to fractions with different numerators, confusing numerator and denominator roles.
Key insight: When you take the same number of parts, the parts from fewer total pieces are larger.
Essential formulas:

• Same numerator: a/b vs a/c → Compare b vs c inversely

• Ordering: If n₁ = n₂ = ... = nₘ, then order by denominators in reverse

• Inverse relationship: Larger denominator means smaller fraction when numerator is constant

Solution: Exercises 4 to 5
4 Problem Solving Application
Exercise 4
In a pizza party, Emma ate 2/3 of her pizza, Jake ate 2/5 of his pizza, and Lily ate 2/4 of her pizza. Who ate the most pizza? Who ate the least?
Definition:

Problem solving: Applying fraction comparison to real-world scenarios

Multi-fraction comparison:
  1. Identify that all fractions have the same numerator (2)
  2. Compare denominators: 3, 5, 4
  3. Order denominators from least to greatest: 3, 4, 5
  4. Match back to people (inverse relationship)
Step 1: Identify numerators

All fractions have numerator 2, so we compare denominators inversely

Step 2: List denominators

Emma: 3, Jake: 5, Lily: 4

Step 3: Order denominators

From least to greatest: 3, 4, 5

Step 4: Apply inverse relationship

Smaller denominator = larger fraction

Emma (2/3) > Lily (2/4) > Jake (2/5)

Step 5: Answer the question

Most pizza: Emma (2/3), Least pizza: Jake (2/5)

Emma: 2/3 > Lily: 2/4 > Jake: 2/5
Final answer:

Emma ate the most pizza (2/3), and Jake ate the least (2/5). This is determined by comparing denominators since all fractions have the same numerator: the smaller the denominator, the larger the fraction.

Applied rules:

• Same numerator comparison: Compare denominators inversely

• Real-world application: Apply mathematical concepts to practical problems

• Data interpretation: Translate fractions to meaningful conclusions

5 Equivalent Fractions Context
Exercise 5
Compare 1/2 and 2/4. Notice that 2/4 is equivalent to 1/2. How does this relate to comparing fractions with same numerator?
Definition:

Equivalent fractions: Different fractions that represent the same value

Step 1: Recognize equivalence

2/4 = (2÷2)/(4÷2) = 1/2

Step 2: Rewrite as same numerator

We could write 1/2 as 2/4, giving us 2/4 vs 2/4

Step 3: Apply same numerator rule

With same numerators (2) and same denominators (4), the fractions are equal

Step 4: Conclusion

1/2 = 2/4 because they are equivalent fractions

Step 5: Relationship to same numerator

This shows that sometimes we need to convert fractions to have the same numerator before comparing

1/2 = 2/4
Final answer:

1/2 and 2/4 are equivalent fractions (both equal 1/2 when simplified). This demonstrates that to apply the same numerator rule, we sometimes need to convert fractions to equivalent forms with matching numerators.

Applied rules:

• Equivalence principle: 2/4 = 1/2 when numerator and denominator are halved

• Conversion necessity: Sometimes convert to common numerator before comparing

• Identity rule: Any fraction equals itself

Comprehensive Guide: Comparing Fractions with Same Numerator
a/b vs a/c → Compare b vs c (inversely)
Same Numerator Rule
Key definitions:

Comparing fractions with same numerator: Determining which fraction is greater when both have identical numerators

Same numerator rule: When numerators are equal, compare denominators inversely

Inverse relationship: Understanding that larger denominators create smaller individual parts

Complete methodology:
  1. Identify numerators: Confirm that fractions have the same numerator
  2. Focus on denominators: Compare the bottom numbers inversely
  3. Apply inverse logic: Smaller denominator means larger fraction
  4. Visual verification: Draw models to confirm results
  5. State conclusion: Write final comparison with appropriate symbol
Tip 1: Always double-check that numerators are identical before applying this rule.
Tip 2: Remember that smaller denominators mean larger fractions when numerators are the same.
Tip 3: Use visual models to reinforce the concept of part size.
Tip 4: This is the inverse of the same denominator rule.
Common errors: Thinking that larger denominators mean larger fractions, applying this rule to fractions with different numerators, confusing numerator and denominator roles.
Exam preparation: Master this rule as it's essential for more complex comparisons, practice with various denominators while keeping numerators constant.
Formulas to know by heart:

• Same numerator: a/b vs a/c → Compare b vs c inversely

• Ordering: If numerators match, order by denominators in reverse

• Equivalence: Sometimes convert to common numerator first

Fraction Comparison Visualization

Same Numerator Principle: When numerators are equal, compare denominators inversely.

  • Visual confirmation: Pie charts and bar models show the inverse relationship clearly
  • Numerical comparison: Smaller denominators create larger fractions
  • Part size reasoning: More divisions mean smaller individual parts
  • Real-world applications: Sharing, measurements, and comparisons

Questions & Answers

Question: I don't understand why 1/3 is bigger than 1/5 when 3 is smaller than 5. It seems backwards!

Answer: This is a very common confusion! Think of it this way:

  • If you have 1 slice of a pizza cut into 3 pieces, that slice is pretty big
  • If you have 1 slice of a pizza cut into 5 pieces, that slice is much smaller
  • The denominator tells you how many equal pieces the whole is divided into
  • More pieces (larger denominator) means each piece is smaller

So 1/3 means 1 of 3 big pieces, while 1/5 means 1 of 5 small pieces. That's why 1/3 is larger than 1/5!

It's like sharing cookies: 1 cookie shared among 3 people gives bigger pieces than 1 cookie shared among 5 people!

Question: How can I remember which rule to use - same numerator or same denominator?

Answer: Here are memory tricks for each rule:

  1. Same Denominator: "Same Bottom, Compare Tops" - Just look at numerators
  2. Same Numerator: "Same Top, Invert Bottom" - Compare denominators in reverse
  3. Visual reminder: Draw a picture of what each fraction looks like

For same numerator: "Fewer pieces, bigger pieces" - so smaller denominator means larger fraction

For same denominator: "More parts, bigger fraction" - so larger numerator means larger fraction

Always check first: Are the tops the same? Or are the bottoms the same?

Question: My child understands the concept but still gets confused when ordering multiple fractions. What strategies can help?

Answer: Here are systematic strategies for ordering multiple fractions:

  1. Separate denominators: Write down just the denominators in a list
  2. Order denominators: Put them in order from least to greatest
  3. Apply inverse logic: Since smaller denominator = larger fraction, reverse the order
  4. Map back to fractions: Match each ordered denominator to its original fraction

Example: For 2/3, 2/7, 2/5:

  • Step 1: Denominators are 3, 7, 5
  • Step 2: Ordered denominators: 3, 5, 7
  • Step 3: Inverse order (for fractions): 2/3, 2/5, 2/7

Encourage your child to write out each step clearly. This systematic approach prevents confusion and builds good mathematical habits!