Solved Exercises on Comparing Fractions with Same Numerator in Grade 3

Master comparing fractions with same numerator: identifying larger fractions, ordering fractions, and solving comparison word problems with visual representations through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing Fractions with Same Numerator
Exercise 1
Compare 1/4 and 1/6. Which fraction is larger? Use visual models to explain your answer.
Definition:

Comparing fractions with same numerator: When fractions have the same numerator, the fraction with the smaller denominator is larger.

Comparison method:
  1. Verify that numerators are the same
  2. Compare the denominators
  3. The fraction with the smaller denominator is larger
  4. Draw visual models to confirm
Fractions
1/4 vs 1/6
Numerator
1 = 1
Denominator
4 vs 6
1
2
3
4
vs
1
2
3
4
5
6
1/4
>
1/6
One fourth is larger than one sixth because fourths are bigger pieces than sixths.
Step 1: Verify numerators are the same

Both fractions have numerator 1, so we can compare denominators

Step 2: Compare denominators

4 < 6, so 1/4 > 1/6

Step 3: Verify with visual model

1 part out of 4 equal parts is larger than 1 part out of 6 equal parts

Step 4: Write the comparison

1/4 > 1/6 (1/4 is greater than 1/6)

Key Rule: When numerators are equal, smaller denominator means larger fraction!
1/4 > 1/6
Final answer:

1/4 is larger than 1/6 because both fractions have the same numerator (1), and 4 < 6. When the numerator is the same, the fraction with the smaller denominator is larger.

Applied rules:

• Same numerator rule: Compare denominators when numerators match

• Inverse relationship: Smaller denominator = larger fraction

• Part size principle: More equal parts mean smaller individual parts

2 Ordering Fractions with Same Numerator
Exercise 2
Order these fractions from smallest to largest: 1/8, 1/3, 1/5, 1/7
Explain your reasoning.
Definition:

Ordering fractions with same numerator: Arranging fractions from smallest to largest by comparing their denominators in reverse order.

1/8
1/3
1/5
1/7
Numerator
1
Denominators
8, 3, 5, 7
Order
8>7>5>3
1
2
3
4
5
6
7
8
1/8 (smallest)
1
2
3
1/3 (largest)
1
2
3
4
5
1/5
1
2
3
4
5
6
7
1/7
Step 1: Verify all numerators are the same

All fractions have numerator 1

Step 2: List the denominators

The denominators are: 8, 3, 5, 7

Step 3: Order denominators from largest to smallest

8 > 7 > 5 > 3

Step 4: Write fractions in reverse order

Since larger denominator = smaller fraction: 1/8 < 1/7 < 1/5 < 1/3

1/8
<
1/7
<
1/5
<
1/3
Final answer:

From smallest to largest: 1/8, 1/7, 1/5, 1/3

Applied rules:

• Same numerator ordering: Arrange denominators in reverse order

• Consistency: All fractions must have the same numerator

• Visual confirmation: Draw models to verify the ordering

3 Real-World Application
Exercise 3
Three friends are sharing candy bars of the same size. Emma gets 1/4 of a candy bar, Liam gets 1/6 of a candy bar, and Olivia gets 1/3 of a candy bar. Who gets the largest piece? Who gets the smallest piece?
Definition:

Real-world fraction comparison: Applying fraction comparison skills to solve practical problems involving quantities with the same numerator.

E
2
3
4
Emma: 1/4
L
2
3
4
5
6
Liam: 1/6
O
2
3
Olivia: 1/3
Numerators
All = 1
Denominators
4, 6, 3
Order
3<4<6
Step 1: Identify the fractions

Emma: 1/4, Liam: 1/6, Olivia: 1/3

Step 2: Verify numerators are the same

All fractions have numerator 1

Step 3: Compare denominators

3 < 4 < 6

Step 4: Apply the rule

Smaller denominator = larger fraction: 1/3 > 1/4 > 1/6

1/3
>
1/4
>
1/6
Final answer:

Olivia gets the largest piece (1/3), and Liam gets the smallest piece (1/6).

Applied rules:

• Real-world application: Connect mathematical concepts to practical scenarios

• Same numerator comparison: Compare denominators in reverse order

• Logical reasoning: Apply mathematical results to answer the question

Fractions with Same Numerator Rules
1/a vs 1/b → Compare a vs b (reverse)
Comparison Rule
Same Numerator
1/3 vs 1/5
Compare denominators
Ordering
1/7, 1/4, 1/2
Reverse denominator order
Visual Model
Larger parts
Smaller denominator
Key Rule: When numerators are the same, the fraction with the smaller denominator is larger.
Comparison Logic: More equal parts mean smaller individual parts.
Tip 1: Always verify that numerators are equal before comparing denominators.
Tip 2: Draw visual models to confirm your comparisons.
Tip 3: Remember: More pieces = smaller individual pieces when numerators match.
Key Rules to Remember:

• Same numerator: Compare denominators only

• Inverse relationship: Smaller denominator means larger fraction

• Ordering: Arrange denominators in reverse order for fractions

• Visual verification: Draw models to confirm comparisons

If a < b, then 1/a > 1/b (when numerators are equal)
Comparison Formula
Fractions Numerator Denominators Comparison 1/4 vs 1/7 1 4 vs 7 1/4 > 1/7 1/2 vs 1/5 1 2 vs 5 1/2 > 1/5 1/8 vs 1/3 1 8 vs 3 1/8 < 1/3 1/6 vs 1/6 1 6 vs 6 1/6 = 1/6
Additional Practice
4 Mixed Practice
Exercise 4
Compare the following pairs of fractions:
a) 1/9 and 1/4
b) 1/7 and 1/10
c) 1/2 and 1/2
Definition:

Mixed practice: Combining different fraction comparisons to strengthen understanding of same numerator rules.

Part a: 1/9 vs 1/4

Same numerator (1), compare denominators: 9 > 4, so 1/9 < 1/4

Part b: 1/7 vs 1/10

Same numerator (1), compare denominators: 7 < 10, so 1/7 > 1/10

Part c: 1/2 vs 1/2

Same numerator and denominator, so 1/2 = 1/2

Final answers:

a) 1/9 < 1/4

b) 1/7 > 1/10

c) 1/2 = 1/2

Applied rules:

• Consistency: Apply the same method to all problems

• Same numerator: Compare denominators in reverse order

• Equality: When both numerator and denominator are identical

5 Ordering Challenge
Exercise 5
Order these fractions from largest to smallest: 1/5, 1/9, 1/3, 1/7, 1/2
Then write them in ascending order.
Definition:

Descending and ascending order: Arranging fractions from greatest to least or least to greatest by comparing denominators when numerators are equal.

Numerators
All = 1
Denominators
5, 9, 3, 7, 2
Order
2<3<5<7<9
Step 1: List all denominators

5, 9, 3, 7, 2

Step 2: Order denominators from smallest to largest

2 < 3 < 5 < 7 < 9

Step 3: Write fractions in descending order

Smallest denominator first: 1/2, 1/3, 1/5, 1/7, 1/9

Step 4: Write fractions in ascending order

Largest denominator first: 1/9, 1/7, 1/5, 1/3, 1/2

1/2
>
1/3
>
1/5
>
1/7
>
1/9
Final answer:

Descending order: 1/2, 1/3, 1/5, 1/7, 1/9

Ascending order: 1/9, 1/7, 1/5, 1/3, 1/2

Applied rules:

• Ordering: Arrange denominators in reverse sequence for unit fractions

• Consistency: All fractions must have same numerator

• Reversal: Ascending is opposite of descending

Comprehensive Summary: Comparing Fractions with Same Numerator
1/a vs 1/b → If a < b then 1/a > 1/b
Comparison Rule
Key Definitions:

Unit fractions: Fractions with numerator 1 (like 1/2, 1/3, 1/4).

Same numerator comparison: When fractions have the same numerator, compare denominators to determine which is larger.

Numerator: The top number in a fraction that represents how many parts are selected.

Denominator: The bottom number in a fraction that represents how many equal parts the whole is divided into.

Ordering: Arranging fractions from smallest to largest or largest to smallest.

Visual model: A drawing or diagram that represents the fraction.

Complete Methodology:
  1. Verify numerators: Ensure all fractions have the same numerator
  2. Compare denominators: Look at the bottom numbers
  3. Determine relationship: Smaller denominator means larger fraction
  4. Visual verification: Draw models to confirm comparisons
  5. Apply to problems: Use the rule in various contexts
Tip 1: When numerators are the same, focus only on the denominators.
Tip 2: Draw pictures to visualize which fraction is larger.
Tip 3: Remember: More equal parts = smaller individual parts.
Tip 4: The fraction with the smaller denominator is always greater when numerators match.
Common errors: Comparing denominators directly instead of using inverse relationship, forgetting to verify same numerator.
Key insights: This rule provides a foundation for comparing all fractions.
Core Rules and Principles:

• Inverse relationship: When numerators are equal, smaller denominator = larger fraction

• Greater than: Smaller denominator means larger fraction

• Ordering principle: Arrange denominators in reverse order for ascending sequence

• Visual confirmation: Draw models to verify comparisons

• Consistency: All fractions must have the same numerator

Examples: 1/2 > 1/3, 1/5 < 1/4, 1/8 > 1/10
Comparison Examples
Memory aid: "When tops are the same, smaller bottom means bigger fraction!"
Success check: If 1/3 > 1/4, then 3 < 4. ✓

Questions & Answers

Question: Why is 1/4 bigger than 1/8? Doesn't 8 seem like a bigger number than 4?

Answer: This is a common confusion! Think of it like sharing pizza:

With 1/4:

  • The pizza is cut into 4 equal pieces
  • You get 1 piece out of 4
  • Each piece is quite large

With 1/8:

  • The same pizza is cut into 8 equal pieces
  • You get 1 piece out of 8
  • Each piece is smaller

The rule: When the numerator is the same (you get 1 piece), the fraction with fewer total pieces (smaller denominator) gives you a larger piece!

So 1/4 > 1/8 because 4 pieces are bigger than 8 pieces!

Question: My child is struggling with ordering fractions like 1/2, 1/3, 1/4. How can I help them remember the correct order?

Answer: Here are effective strategies to help remember the order:

Memory trick:

  • Remember: "The smaller the bottom number, the bigger the piece"
  • Think of it like cake: 1/2 of a cake is more than 1/4 of a cake

Visual strategy:

  • Draw circles divided into 2, 3, and 4 parts
  • Shade 1 part in each circle
  • Compare the sizes of the shaded parts

Ordering method:

  • List denominators: 2, 3, 4
  • Order them from smallest to largest: 2 < 3 < 4
  • The fractions go in the same order: 1/2 > 1/3 > 1/4

Practice with real objects like pizza slices or chocolate bars!

Question: What happens if I compare fractions with different numerators? Can I still use the same rule?

Answer: No, you cannot use the same rule when numerators are different!

Example with 2/3 and 1/4:

  • 2/3 means 2 parts out of 3 equal parts
  • 1/4 means 1 part out of 4 equal parts
  • The numerators are different (2 vs 1) and denominators are different (3 vs 4)

What to do instead:

  • Find equivalent fractions with the same denominator
  • Or convert to decimals or percentages
  • Or draw visual models to compare

The rule "smaller denominator = larger fraction" only works when numerators are the same!

For 2/3 vs 1/4: 2/3 is actually larger because 2 parts of 3 equal parts is more than 1 part of 4 equal parts.