Solved Exercises on Ordering Fractions in Grade 4

Master ordering fractions: methods, properties, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Ordering like denominators
Exercise 1
Order these fractions from least to greatest: 3/7, 1/7, 6/7, 4/7
Definition:

Ordering Like Denominators: Arranging fractions with the same denominator by comparing their numerators.

Method for ordering fractions with like denominators:
  1. Verify that all fractions have the same denominator
  2. Compare the numerators of all fractions
  3. Arrange the numerators in ascending order
  4. Place the corresponding fractions in the same order
Fractions
3/7, 1/7, 6/7, 4/7
Numerators
1, 3, 4, 6
Ordered
1/7, 3/7, 4/7, 6/7
1/7
→
3/7
→
4/7
→
6/7
Step 1: Verify all denominators are the same

3/7 has denominator 7

1/7 has denominator 7

6/7 has denominator 7

4/7 has denominator 7

All denominators are 7 (like denominators)

Step 2: List the numerators

Numerators: 3, 1, 6, 4

Step 3: Order the numerators from least to greatest

Order: 1, 3, 4, 6

Step 4: Match the numerators to their original fractions

1 → 1/7

3 → 3/7

4 → 4/7

6 → 6/7

Order: 1/7, 3/7, 4/7, 6/7
Final answer:

The fractions ordered from least to greatest are: 1/7, 3/7, 4/7, 6/7.

Applied rules:

• Like Denominator Rule: When denominators are equal, compare numerators only

• Ascending Order: Arrange from smallest to largest value

• Preservation: Keep the original denominator in each fraction

2 Ordering unlike denominators
Exercise 2
Order these fractions from greatest to least: 1/2, 3/4, 2/3
Definition:

Ordering Unlike Denominators: Arranging fractions with different denominators by converting them to equivalent fractions with a common denominator.

Fractions
1/2, 3/4, 2/3
LCM
12
Ordered
9/12, 8/12, 6/12
3/4
→
2/3
→
1/2
Step 1: Find the LCM of denominators (2, 4, 3)

Factors of 2: 2, 4, 6, 8, 10, 12, 14...

Factors of 4: 4, 8, 12, 16...

Factors of 3: 3, 6, 9, 12, 15...

LCM is 12

Step 2: Convert each fraction to equivalent with denominator 12

1/2 = (1 × 6)/(2 × 6) = 6/12

3/4 = (3 × 3)/(4 × 3) = 9/12

2/3 = (2 × 4)/(3 × 4) = 8/12

Step 3: Order the equivalent fractions from greatest to least

9/12, 8/12, 6/12

Because 9 > 8 > 6

Step 4: Convert back to original fractions

9/12 = 3/4

8/12 = 2/3

6/12 = 1/2

Order: 3/4, 2/3, 1/2
Final answer:

The fractions ordered from greatest to least are: 3/4, 2/3, 1/2.

Applied rules:

• Common Denominator: Convert all fractions to have the same denominator

• Descending Order: Arrange from largest to smallest value

• Conversion: Maintain equivalence during conversion process

3 Mixed ordering problem
Exercise 3
Order these fractions from least to greatest: 2/5, 1/3, 4/5, 1/6
Definition:

Mixed Ordering: Arranging fractions with various denominators, some like and some unlike, by finding a common denominator for all.

Fractions
2/5, 1/3, 4/5, 1/6
LCM
30
Ordered
5/30, 10/30, 12/30, 24/30
1/6
→
1/3
→
2/5
→
4/5
Step 1: Find the LCM of all denominators (5, 3, 5, 6)

Prime factors: 5, 3, 5, 2×3

LCM = 2 × 3 × 5 = 30

Step 2: Convert each fraction to equivalent with denominator 30

2/5 = (2 × 6)/(5 × 6) = 12/30

1/3 = (1 × 10)/(3 × 10) = 10/30

4/5 = (4 × 6)/(5 × 6) = 24/30

1/6 = (1 × 5)/(6 × 5) = 5/30

Step 3: Order the equivalent fractions from least to greatest

5/30, 10/30, 12/30, 24/30

Because 5 < 10 < 12 < 24

Step 4: Convert back to original fractions

5/30 = 1/6

10/30 = 1/3

12/30 = 2/5

24/30 = 4/5

Order: 1/6, 1/3, 2/5, 4/5
Final answer:

The fractions ordered from least to greatest are: 1/6, 1/3, 2/5, 4/5.

Applied rules:

• Universal Common Denominator: Find LCM for all denominators simultaneously

• Systematic Conversion: Convert each fraction individually

• Ascending Order: Arrange from smallest to largest value

Key Concepts: Ordering Fractions Properties and Methods
a/c < b/c when a < b | a/b vs c/d → convert to common denominator
Ordering Rules
Rule 1
Like Denominators: Compare numerators
When denominators are equal
Rule 2
Unlike Denominators: Find common denominator
Convert to like denominators first
Rule 3
Order Direction: Ascending vs Descending
Pay attention to problem requirements
Key definitions:

Ordering Fractions: Arranging fractions in a sequence from least to greatest (ascending) or greatest to least (descending)

Like Denominators: Fractions that have the same denominator

Unlike Denominators: Fractions that have different denominators

Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers

Common Denominator: A denominator that is the same for two or more fractions

Ascending Order: Arranging from smallest to largest

Descending Order: Arranging from largest to smallest

Complete methodology:
  1. Identify the fractions: Note all numerators and denominators
  2. Determine strategy: Check if denominators are alike or different
  3. Apply appropriate method: Like denominators vs unlike denominators
  4. Perform conversions: Find common denominators if needed
  5. Compare and order: Arrange fractions according to requirements
Tip 1: Always find the LCM for the most efficient common denominator.
Tip 2: Pay careful attention to whether the problem asks for ascending or descending order.
Tip 3: When in doubt, convert all fractions to decimals for easier comparison.
Tip 4: Use visual models (bars, circles) to verify your ordering.
Common errors: Mixing up ascending/descending, forgetting to convert to common denominator, misordering numerators.
Memory aids: "ABC order" = Ascending (smallest to largest), "Z to A" = Descending (largest to smallest).
Essential ordering facts:

• Like denominators require only numerator comparison

• Unlike denominators require conversion to common denominator

• The LCM provides the most efficient common denominator

• Equivalent fractions maintain the same value during conversion

• Always verify the direction of order (ascending vs descending)

Ordering Visualization: Fraction Sequences
Exercise 6: Fraction Ordering
Visualize the ordering of fractions from least to greatest:
1/4, 1/3, 1/2, 2/3, 3/4
Convert to common denominator and see the sequence

Analysis: The chart shows how fractions form a sequence when properly ordered.

  • 1/4 = 6/24
  • 1/3 = 8/24
  • 1/2 = 12/24
  • 2/3 = 16/24
  • 3/4 = 18/24
  • Each fraction increases in value as the sequence progresses

Questions & Answers

Question: Why do I need to find a common denominator when ordering fractions with different denominators?

Answer: Finding a common denominator is necessary because:

Reasons for common denominator:

  • Denominators represent the size of the pieces
  • Different denominators mean different-sized pieces
  • You can't directly compare different-sized pieces
  • Converting to common denominator makes all pieces the same size
  • Then you can simply count how many pieces you have (compare numerators)

Example: Comparing 2/3 and 3/4

  • 2/3 means 2 pieces of size 1/3
  • 3/4 means 3 pieces of size 1/4
  • 1/3 pieces are larger than 1/4 pieces
  • Converting to common denominator: 2/3 = 8/12, 3/4 = 9/12
  • Now comparing 8 pieces vs 9 pieces of the same size (1/12)

Common denominators standardize the piece size, making comparison possible!

Question: How do I know if I should order from least to greatest or greatest to least?

Answer: Pay close attention to the specific wording in the problem:

Phrases indicating LEAST to GREATEST:

  • "from least to greatest"
  • "from smallest to largest"
  • "ascending order"
  • "increasing order"
  • "order from smallest"

Phrases indicating GREATEST to LEAST:

  • "from greatest to least"
  • "from largest to smallest"
  • "descending order"
  • "decreasing order"
  • "order from largest"

Memory aids:

  • Think of alphabet: A to Z = Ascending (smallest to largest)
  • Think of alphabet: Z to A = Descending (largest to smallest)
  • Read the problem carefully and underline the key words

Always read the problem twice to make sure you're ordering in the correct direction!

Question: Is there a quicker way to order fractions without finding common denominators?

Answer: Yes, there are alternative methods for ordering fractions:

Alternative methods:

  1. Decimal Conversion: Convert fractions to decimals and compare
  2. Reference Points: Compare to benchmarks like 1/2, 1/4, or 1
  3. Cross Multiplication: Compare pairs of fractions using cross multiplication
  4. Visual Estimation: Use mental pictures of fraction sizes

Examples:

  • For 2/3, 3/4, 1/2: Convert to 0.667, 0.75, 0.5 → Order: 1/2, 2/3, 3/4
  • For 1/3, 3/5, 1/4: 1/4 < 1/2, 1/3 ≈ 1/2, 3/5 > 1/2 → Order: 1/4, 1/3, 3/5

However, finding common denominators is the most reliable method, especially for exact comparisons!

Question: What if I have mixed numbers? How do I order those?

Answer: For mixed numbers, you have several approaches:

Method 1: Compare whole numbers first

  • If whole numbers are different, order by whole numbers
  • Example: 2 1/3, 1 3/4, 3 1/2 → Order by whole numbers: 1 3/4, 2 1/3, 3 1/2

Method 2: Convert to improper fractions

  • Convert each mixed number to an improper fraction
  • Order the improper fractions using common denominators
  • Example: 1 1/2 = 3/2, 2 1/4 = 9/4, 1 3/4 = 7/4

Method 3: Convert to decimals

  • Convert mixed numbers to decimal form
  • Compare decimal values directly
  • Example: 1 1/2 = 1.5, 2 1/4 = 2.25, 1 3/4 = 1.75 → Order: 1.5, 1.75, 2.25

The whole number comparison method is often the fastest when whole numbers differ!

Question: How does ordering fractions help in real life?

Answer: Ordering fractions has many practical applications:

  • Shopping: Comparing discount percentages (25% vs 1/3 off)
  • Cooking: Ordering ingredient amounts (1/2 cup, 2/3 cup, 3/4 cup)
  • Sports: Ranking player statistics with fractional measures
  • Finance: Comparing interest rates or investment returns
  • Academics: Ordering test scores with different total points

Real-world examples:

  • Ranking students by test scores: 18/20, 23/25, 15/18 → Convert and order
  • Comparing running times: 1/4 mile in 1:30 vs 1/3 mile in 2:00 → Calculate rates
  • Recipe scaling: Ordering ingredient amounts for different serving sizes

This skill helps make informed decisions when comparing different fractional quantities in real-world situations!