Using Common Numerators to Compare Fractions in Grade 3 - Mathematics - Exercises with solutions

Master fraction comparison using common numerators: visual representations, fraction models, cross multiplication, and grade 3 methods through these 10 detailed exercises with step-by-step solutions.

Solutions: Exercises 1 to 10
1 Basic common numerators
Exercise 1
Compare 2/3 and 2/5 using the common numerator method. Which fraction is larger?
Difficulty: Beginner Time: ~4 minutes Skills: Numerator Comparison
Definition:

Common numerator comparison: When fractions have the same numerator, the fraction with the smaller denominator is larger because it represents larger pieces of the whole.

Note: This method works because when the numerators are the same, we're comparing how many equal parts we take from wholes divided into different numbers of parts.

Step-by-step common numerator method:
  1. Identify that both fractions have the same numerator (2 in both cases)
  2. Compare the denominators (3 and 5)
  3. Remember: Smaller denominator means larger pieces
  4. State which fraction is larger based on denominator comparison
Fractions to Compare
2/3 and 2/5
Common Numerator
2
Result
2/3 > 2/5
Step 1: Identify common numerator

Both fractions have numerator 2

Step 2: Compare denominators

3 < 5, so 2/3 represents larger pieces than 2/5

Step 3: Apply the rule

When numerators are equal, smaller denominator means larger fraction

Step 4: Conclusion

2/3 is larger than 2/5

2/3 > 2/5
Final answer:

2/3 is larger than 2/5 because when the numerators are the same, the fraction with the smaller denominator is larger.

Applied rules:

• Common numerator rule: When numerators are equal, smaller denominator = larger fraction

• Denominator effect: More pieces (larger denominator) means smaller individual pieces

• Visual representation: Thirds are larger pieces than fifths

• Practice Tip: When numerators match, compare denominators directly

Related Examples:
  • 3/4 vs 3/7: Since 4 < 7, then 3/4 > 3/7
  • 1/2 vs 1/3: Since 2 < 3, then 1/2 > 1/3
  • 5/6 vs 5/8: Since 6 < 8, then 5/6 > 5/8
Quick Tips:
  • When numerators are the same, just compare denominators
  • Smaller denominator means larger fraction
  • This method is faster than finding common denominators
Frequently Asked Questions:

Q: What if denominators are the same?
A: When denominators are the same, compare numerators - larger numerator means larger fraction.

Q: What if neither numerator nor denominator is common?
A: Use cross multiplication or find common denominators.

2 Finding common numerators
Exercise 2
Find equivalent fractions for 3/4 and 2/5 that have the same numerator, then compare them.
Difficulty: Intermediate Time: ~5 minutes Skills: Equivalent Fractions, Comparison
Definition:

Finding common numerators: Creating equivalent fractions by multiplying both numerator and denominator of each fraction by a number that results in the same numerator.

Note: This method involves finding the least common multiple (LCM) of the numerators to create equivalent fractions with matching numerators.

Step-by-step common numerator creation method:
  1. Find the LCM of the numerators (3 and 2)
  2. Determine what to multiply each numerator by to reach the LCM
  3. Multiply both numerator and denominator of each fraction by the same number
  4. Compare the resulting fractions using the common numerator rule
Original Fractions
3/4 and 2/5
Common Numerator
6
Equivalent Forms
6/8 and 6/15
Result
6/8 > 6/15
Step 1: Find LCM of numerators

LCM of 3 and 2 is 6

Step 2: Convert 3/4 to numerator 6

3/4 × 2/2 = 6/8

Step 3: Convert 2/5 to numerator 6

2/5 × 3/3 = 6/15

Step 4: Compare using common numerator

Since 6/8 and 6/15 have the same numerator, compare denominators: 8 < 15, so 6/8 > 6/15

3/4 > 2/5
Final answer:

3/4 is larger than 2/5 because when converted to have the same numerator (6), 3/4 becomes 6/8 and 2/5 becomes 6/15, and 6/8 > 6/15 since 8 < 15.

Applied rules:

• LCM method: Find least common multiple of numerators

• Equivalent fraction creation: Multiply both parts by same number

• Common numerator comparison: Smaller denominator means larger fraction

• Practice Tip: This method is useful when denominators are difficult to work with

Related Examples:
  • 2/3 and 3/4: LCM of 2,3 is 6 → 6/9 and 6/8 → 6/8 > 6/9 → 3/4 > 2/3
  • 1/4 and 2/7: LCM of 1,2 is 2 → 2/8 and 2/7 → 2/7 > 2/8 → 2/7 > 1/4
  • 4/9 and 3/5: LCM of 4,3 is 12 → 12/27 and 12/20 → 12/20 > 12/27 → 3/5 > 4/9
Quick Tips:
  • Find LCM of numerators to create common numerator
  • Multiply both numerator and denominator by the same number
  • After creating common numerator, compare denominators
Frequently Asked Questions:

Q: What if the LCM is very large?
A: The method still works, but cross multiplication might be more efficient for large numbers.

Q: Can I use this method for more than two fractions?
A: Yes, find LCM of all numerators and convert each fraction.

3 Cross multiplication
Exercise 3
Use cross multiplication to verify that the comparison in Exercise 2 (3/4 vs 2/5) is correct.
Difficulty: Intermediate Time: ~3 minutes Skills: Cross Multiplication, Verification
Definition:

Cross multiplication: For fractions a/b and c/d, if a×d = b×c, then the fractions are equivalent; if a×d > b×c, then a/b > c/d.

Note: Cross multiplication provides a mathematical way to verify fraction comparisons without needing common denominators or numerators.

Step-by-step cross multiplication method:
  1. Identify the numerators and denominators of both fractions
  2. Multiply the numerator of the first fraction by the denominator of the second
  3. Multiply the denominator of the first fraction by the numerator of the second
  4. Compare the two products to determine which fraction is larger
Fractions to Compare
3/4 and 2/5
Cross Products
3×5 = 15 and 4×2 = 8
Result
3/4 > 2/5
Step 1: Set up cross multiplication

We have 3/4 and 2/5

Step 2: Cross multiply

3 × 5 = 15 and 4 × 2 = 8

Step 3: Compare products

Since 15 > 8, then 3/4 > 2/5

Step 4: Verify previous result

Our previous result using common numerators was 3/4 > 2/5, which matches

3/4 > 2/5
Final answer:

Cross multiplication confirms that 3/4 is larger than 2/5 because 3×5 (15) is greater than 4×2 (8).

Applied rules:

• Cross multiplication: Multiply diagonally across the fractions

• Product comparison: Larger product indicates larger original fraction

• Verification: Confirms results from other methods

• Practice Tip: Cross multiplication works for any two fractions

Related Examples:
  • 2/3 vs 3/5: 2×5 = 10 and 3×3 = 9 → 10 > 9 → 2/3 > 3/5
  • 1/2 vs 2/5: 1×5 = 5 and 2×2 = 4 → 5 > 4 → 1/2 > 2/5
  • 3/7 vs 2/5: 3×5 = 15 and 7×2 = 14 → 15 > 14 → 3/7 > 2/5
Quick Tips:
  • Always multiply numerator of first by denominator of second
  • Then multiply denominator of first by numerator of second
  • Larger cross product indicates the larger original fraction
Frequently Asked Questions:

Q: Does cross multiplication work for all fractions?
A: Yes, it works for any two fractions regardless of their values.

Q: Can I use cross multiplication to compare more than two fractions?
A: Compare pairs of fractions one at a time using cross multiplication.

Solutions: Exercises 4 to 5
4 Comparing multiple fractions
Exercise 4
Order these fractions from least to greatest: 3/7, 3/5, 3/8, 3/4 using the common numerator method.
Definition:

Ordering with common numerators: When all fractions have the same numerator, order them by comparing denominators in reverse order (since smaller denominator means larger fraction).

Note: When numerators are equal, the fraction with the largest denominator is smallest, and the fraction with the smallest denominator is largest.

Step-by-step ordering method:
  1. Confirm all fractions have the same numerator (3 in all cases)
  2. List the denominators in order from largest to smallest
  3. Order the fractions in the same order as the reversed denominators
  4. State the final ordered sequence from least to greatest
Fractions to Order
3/7, 3/5, 3/8, 3/4
Denominators
8, 7, 5, 4
Ordered Fractions
3/8, 3/7, 3/5, 3/4
Step 1: Identify common numerator

All fractions have numerator 3

Step 2: List denominators

Denominators are 7, 5, 8, 4

Step 3: Order denominators from largest to smallest

8, 7, 5, 4

Step 4: Match fractions to ordered denominators

3/8, 3/7, 3/5, 3/4

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

From least to greatest: 3/8, 3/7, 3/5, 3/4

Applied rules:

• Common numerator ordering: When numerators match, order by denominator in reverse

• Denominator effect: Larger denominator means smaller fraction

• Reverse ordering: Largest denominator first when ordering from least to greatest

• Practice Tip: Remember: smaller denominator = larger fraction when numerators are equal

Related Examples:
  • 2/9, 2/7, 2/5, 2/3 → Ordered: 2/9 < 2/7 < 2/5 < 2/3
  • 5/12, 5/10, 5/8, 5/6 → Ordered: 5/12 < 5/10 < 5/8 < 5/6
  • 1/6, 1/5, 1/4, 1/3 → Ordered: 1/6 < 1/5 < 1/4 < 1/3
Quick Tips:
  • When numerators are equal, list denominators in reverse order for ascending sequence
  • Smaller denominator always means larger fraction with equal numerators
  • This method is much faster than finding common denominators
Frequently Asked Questions:

Q: What if not all fractions have the same numerator?
A: Find equivalent fractions with common numerators first, then order.

Q: Can I use this method for descending order?
A: Yes, list denominators from smallest to largest when ordering from greatest to least.

5 Visual models
Exercise 5
Draw fraction circles to compare 2/3 and 2/5 using the common numerator method.
Difficulty: Beginner Time: ~5 minutes Skills: Visual Representation
Definition:

Visual comparison with common numerators: Using circular or rectangular models to represent fractions with the same numerator, allowing students to see that smaller denominators result in larger pieces.

Note: Visual models provide concrete evidence of the common numerator rule.

Step-by-step visual comparison method:
  1. Draw two identical circles representing the same whole
  2. Divide the first circle into 3 equal parts and shade 2 parts (2/3)
  3. Divide the second circle into 5 equal parts and shade 2 parts (2/5)
  4. Compare the shaded areas to verify that 2/3 > 2/5
Circle 1
2/3
Circle 2
2/5
Result
2/3 > 2/5
Step 1: Draw first circle

Divide into 3 equal parts, shade 2 parts: 2/3

Step 2: Draw second circle

Divide into 5 equal parts, shade 2 parts: 2/5

Step 3: Compare shaded areas

2/3 covers more area than 2/5

Step 4: Conclusion

Since both fractions take 2 parts, but 2/3 has larger parts, 2/3 > 2/5

2/3 > 2/5
Final answer:

2/3 is larger than 2/5 because when the numerators are the same, the fraction with the smaller denominator represents larger pieces of the whole.

Applied rules:

• Visual representation: Shows fractions as portions of a whole

• Equal division: Parts must be equal in size to compare accurately

• Shading: Shaded portions represent the numerator

• Practice Tip: Use manipulatives or drawings to visualize fraction relationships

Related Examples:
  • 3/4 vs 3/5: Visual shows 3/4 covers more area
  • 1/2 vs 1/3: Visual shows 1/2 covers more area
  • 4/6 vs 4/7: Visual shows 4/6 covers more area
Quick Tips:
  • Draw circles with precise divisions using a compass or template
  • Ensure each part is exactly equal in size
  • Shade the same number of parts to demonstrate common numerators
Frequently Asked Questions:

Q: Can I use rectangles instead of circles?
A: Yes, rectangles, squares, or any shape can be used as long as the whole is divided equally.

Q: What if my divisions aren't equal?
A: The fractions won't be represented accurately. Equal divisions are essential for accurate comparison.

Solutions: Exercises 6 to 10
6 Number line representation
Exercise 6
Show on a number line that 2/3 is greater than 2/5 using the common numerator method.
Definition:

Number line comparison: Placing fractions on a number line to show their relative positions, where fractions with the same numerator can be compared by their positions.

Note: On a number line, fractions with the same numerator but smaller denominator appear farther from 0.

Final answer:

2/3 is positioned farther to the right on the number line than 2/5, confirming that 2/3 > 2/5.

7 Area models
Exercise 7
Draw area models to show that 3/4 is greater than 3/7 using the common numerator method.
Definition:

Area model comparison: Using rectangular models divided into equal parts to represent fractions with the same numerator, showing that smaller denominators result in larger portions.

Note: Area models provide a rectangular approach to understanding the common numerator rule.

Final answer:

3/4 covers more area than 3/7 when both rectangles represent the same whole, confirming that 3/4 > 3/7.

8 Real-world problem
Exercise 8
Sarah ate 2/5 of her pizza and Tom ate 2/3 of his pizza. Who ate more pizza? Use the common numerator method to solve.
Definition:

Real-world fraction comparison: Word problems that involve comparing fractions in practical situations using the common numerator method.

Note: When both people take the same number of parts (numerator), the person with fewer total parts (smaller denominator) takes a larger portion.

Final answer:

Tom ate more pizza because 2/3 > 2/5 (same numerator, smaller denominator means larger fraction).

9 Ordering fractions
Exercise 9
Use the common numerator method to order these fractions from greatest to least: 4/5, 4/9, 4/7, 4/3
Definition:

Descending order with common numerators: When ordering fractions with the same numerator from greatest to least, arrange them by their denominators from smallest to largest.

Note: The fraction with the smallest denominator is the largest fraction when numerators are equal.

Final answer:

From greatest to least: 4/3, 4/5, 4/7, 4/9

10 Mixed operations
Exercise 10
Complete the pattern: 5/6, 5/7, 5/8, ___, ___ using the common numerator method.
Definition:

Patterns with common numerators: Sequences where fractions have the same numerator, allowing comparison based on increasing denominators.

Note: As denominators increase with the same numerator, the fractions decrease in value.

Final answer:

The pattern continues: 5/6, 5/7, 5/8, 5/9, 5/10 (fractions decrease as denominators increase)

Detailed Summary of Using Common Numerators to Compare Fractions
Key Definitions:

Common numerator comparison: A method for comparing fractions that have the same numerator by comparing their denominators. When numerators are equal, the fraction with the smaller denominator is larger.

Equivalent fractions: Fractions that represent the same value or occupy the same position on a number line, even though they have different numerators and denominators.

Numerator: The top number in a fraction that indicates how many parts are being considered.

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

Cross multiplication: A method to compare fractions by multiplying diagonally across the fractions to determine equivalence or relative size.

Core Rules and Principles:
  1. Common Numerator Rule: When fractions have the same numerator, the fraction with the smaller denominator is larger
  2. Denominator Effect: Larger denominators mean smaller individual parts
  3. Ordering Principle: With common numerators, order fractions by denominator in reverse order
  4. Verification Method: Cross multiplication can verify results obtained using common numerators
Basic Example: 2/3 > 2/5 because 3 < 5 (same numerator, smaller denominator wins)
Advanced Example: 3/4 > 3/7 because 4 < 7 (same numerator rule applies)
Real-World Application: If two people take the same number of slices from pizzas cut differently, the person with fewer total slices gets larger pieces
Memory Tip: "When tops are the same, bottoms tell the tale - smaller bottom means bigger fraction!"
Common Mistakes: Confusing the common numerator rule with the common denominator rule, forgetting that smaller denominator means larger fraction when numerators are equal, mixing up which comparison method to use.
Memorization Tips: Remember "numerator = same, denominator = opposite" - smaller denominator means larger fraction. Use visual models to reinforce the concept.
Key Procedures:

• Identifying Common Numerators: Check if fractions have the same numerator

• Comparison Method: When numerators match, compare denominators in reverse

• Ordering Fractions: With common numerators, order by denominator from largest to smallest for ascending sequence

• Verification: Use cross multiplication to confirm results

Rules and Methods for Using Common Numerators
\(\text{If } \frac{a}{b} \text{ and } \frac{a}{c}, \text{ then } \frac{a}{b} > \frac{a}{c} \text{ if } b < c\)
Common Numerator Comparison
Method 1
Direct Comparison
When numerators match
Method 2
Finding Common Numerator
When numerators differ
Method 3
Cross Multiplication
Verification method

Key Takeaways

  • When numerators are equal, smaller denominator means larger fraction
  • This method is faster than finding common denominators
  • Always verify results with cross multiplication
  • Visual models help understand why smaller denominators create larger pieces
  • Ordering fractions with common numerators requires reverse denominator ordering
  • Works best when fractions naturally have the same numerator
  • Common Core Standard 3.NF.A.3d focuses on comparing fractions with same numerator

Questions & Answers

Question: Why is 2/3 larger than 2/5 when both have the same numerator?

Answer: When the numerators are the same, the fraction with the smaller denominator is larger:

  • 2/3 means 2 parts out of 3 total parts
  • 2/5 means 2 parts out of 5 total parts
  • Thirds are larger pieces than fifths because the whole is divided into fewer parts
  • So 2 larger pieces (thirds) is more than 2 smaller pieces (fifths)

Think of it like pizza: 2 slices of a 3-slice pizza is more than 2 slices of a 5-slice pizza.

The key is that smaller denominators create larger individual pieces.

Question: How can I help my child understand the common numerator method?

Answer: Here are effective strategies for the common numerator method:

  • Use manipulatives: Fraction bars or circles that show the same numerator with different denominators
  • Draw pictures: Have your child draw rectangles divided into different parts with the same number shaded
  • Real-world examples: Use food, money, or measuring cups to show fraction concepts
  • Memory aids: "When tops match, bottoms matter - smaller bottom means bigger piece"

Start with simple fractions like 1/2 vs 1/3, then gradually introduce more complex comparisons.

Consistent practice with visual models builds strong conceptual understanding.

Question: What if the fractions don't have the same numerator?

Answer: You can create equivalent fractions with common numerators:

  • Find the least common multiple (LCM) of the numerators
  • Multiply both numerator and denominator of each fraction by the same number to get the common numerator
  • Then compare using the common numerator rule
  • Example: For 2/3 and 3/4, LCM of 2 and 3 is 6, so convert to 6/9 and 6/8, then 6/8 > 6/9

This method works well when denominators are easier to work with than finding common denominators.

Sometimes cross multiplication is faster for two fractions.

Fraction Comparison Glossary

Common Numerator Method
A comparison technique where fractions with the same numerator are compared by their denominators - smaller denominator means larger fraction.
Numerator
The top number in a fraction that indicates how many parts are being considered.
Denominator
The bottom number in a fraction that indicates how many equal parts the whole is divided into.
Cross Multiplication
A method to compare fractions by multiplying diagonally: a/b vs c/d compares a×d to b×c.
Equivalent Fractions
Fractions that represent the same value or occupy the same position on a number line.

Common Numerator Educational Team

Elementary Mathematics Specialists & Curriculum Developers

Our team of experienced elementary math teachers and curriculum specialists creates research-based, student-friendly fraction comparison resources. All content is aligned with Common Core State Standards (3.NF.A.3d) and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness for 3rd grade learners using the common numerator method for fraction comparison.