Comparing Fractions Worksheets Grade 4 - Step-by-Step Solutions

Master comparing fractions: equivalent fractions, mixed numbers, improper fractions, and comparison strategies through these 5 detailed exercises.

Detailed Summary: Comparing Fractions
Key Definitions:

Fraction: A number representing a part of a whole, written as numerator/denominator

Numerator: The top number showing how many parts we have

Denominator: The bottom number showing total equal parts in the whole

Equivalent Fractions: Fractions that represent the same value but use different numbers

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator

Mixed Number: A combination of a whole number and a proper fraction

Core Rules and Principles:

Rule 1: When denominators are the same, compare numerators (larger numerator = larger fraction)

Rule 2: When numerators are the same, compare denominators (smaller denominator = larger fraction)

Rule 3: When both are different, use cross multiplication or find common denominators

Rule 4: Any fraction greater than 1 is an improper fraction; less than 1 is proper

Rule 5: Equivalent fractions have equal cross products

Step-by-Step Methods:

Method 1 - Same Denominator: Compare numerators directly

Method 2 - Same Numerator: Compare denominators (inverse relationship)

Method 3 - Cross Multiplication: Multiply diagonally and compare products

Method 4 - Common Denominator: Convert to same denominator, then compare numerators

Method 5 - Decimal Conversion: Convert fractions to decimals and compare

Examples (Simple to Advanced):

Simple: \(\frac{3}{5}\) vs \(\frac{2}{5}\) → Same denominator, 3 > 2, so \(\frac{3}{5} > \frac{2}{5}\)

Moderate: \(\frac{2}{3}\) vs \(\frac{2}{5}\) → Same numerator, 3 < 5, so \(\frac{2}{3} > \frac{2}{5}\)

Advanced: \(\frac{3}{4}\) vs \(\frac{5}{6}\) → Cross multiply: \(3 \times 6 = 18\), \(4 \times 5 = 20\), so \(\frac{5}{6} > \frac{3}{4}\)

Complex: \(\frac{7}{12}\) vs \(\frac{5}{8}\) → LCD = 24: \(\frac{14}{24}\) vs \(\frac{15}{24}\), so \(\frac{5}{8} > \frac{7}{12}\)

Tips and Common Pitfalls:

Tip 1: Always identify whether denominators or numerators are the same first

Tip 2: Remember: larger denominator = smaller fraction (when numerators are equal)

Tip 3: Use benchmarks like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\) to estimate

Pitfall 1: Don't just compare numerators without considering denominators

Pitfall 2: Forgetting that \(\frac{4}{4} = 1\) and \(\frac{5}{4} > 1\)

Key Notes for Memorization:

Memory Aid: "More pieces, smaller pieces" - when denominator increases, each piece gets smaller

Formula: For \(\frac{a}{b}\) vs \(\frac{c}{d}\): if \(a \times d > b \times c\), then \(\frac{a}{b} > \frac{c}{d}\)

Pattern: Fractions increase as they approach 1 (from below) or decrease as they move away from 1

Verification: Always check your answer by converting to decimals or drawing models

Solution: Exercises 1 to 3
1 Mixed Numbers Comparison
Exercise 1
Compare: \(1\frac{2}{3}\) and \(1\frac{3}{4}\)
Rule:

Mixed Numbers: Compare whole numbers first. If equal, compare fractional parts using fraction comparison rules.

Visual Model:

vs

\(1\frac{2}{3}\) vs \(1\frac{3}{4}\)

Step-by-Step Method:
  1. Compare whole numbers: 1 vs 1 → Equal
  2. Compare fractional parts: \(\frac{2}{3}\) vs \(\frac{3}{4}\)
  3. Use cross multiplication: \(2 \times 4 = 8\) vs \(3 \times 3 = 9\)
  4. Since 9 > 8, then \(\frac{3}{4} > \frac{2}{3}\)
Step 1: Compare whole numbers

Both have whole number 1, so they are equal

Step 2: Compare fractional parts

Now compare \(\frac{2}{3}\) vs \(\frac{3}{4}\)

Step 3: Cross multiply

\(2 \times 4 = 8\) and \(3 \times 3 = 9\)

Step 4: Compare products

8 vs 9 → 9 is greater than 8

Step 5: Apply conclusion

Since \(\frac{3}{4} > \frac{2}{3}\), then \(1\frac{3}{4} > 1\frac{2}{3}\)

\(1\frac{3}{4} > 1\frac{2}{3}\)
Final answer:

\(1\frac{3}{4}\) is greater than \(1\frac{2}{3}\)

Applied rules:

• Mixed Number Comparison: Whole numbers first, then fractions

• Cross Multiplication: Compare fractional parts when whole numbers are equal

• Conversion: Can convert to improper fractions if needed

2 Improper Fractions
Exercise 2
Compare: \(\frac{7}{4}\) and \(\frac{9}{5}\)
Rule:

Improper Fractions: Use same comparison methods as proper fractions. Both are greater than 1.

Visual Model:

vs

\(\frac{7}{4}\) vs \(\frac{9}{5}\)

Step-by-Step Method:
  1. Use cross multiplication: \(7 \times 5 = 35\) vs \(4 \times 9 = 36\)
  2. Compare results: 35 vs 36
  3. Since 36 > 35, then \(\frac{9}{5} > \frac{7}{4}\)
Step 1: Cross multiply

\(7 \times 5 = 35\) and \(4 \times 9 = 36\)

Step 2: Compare products

35 vs 36 → 36 is greater than 35

Step 3: Apply the rule

Since \(4 \times 9 > 7 \times 5\), then \(\frac{9}{5} > \frac{7}{4}\)

Step 4: Verify with conversion

\(\frac{7}{4} = 1.75\) and \(\frac{9}{5} = 1.8\), confirming \(1.8 > 1.75\)

\(\frac{9}{5} > \frac{7}{4}\)
Final answer:

\(\frac{9}{5}\) is greater than \(\frac{7}{4}\)

Applied rules:

• Improper Fraction Rule: Same comparison methods apply

• Cross Multiplication: Works for any fractions

• Decimal Verification: Convert to decimals to confirm

3 Complex Fractions
Exercise 3
Compare: \(\frac{11}{12}\) and \(\frac{7}{8}\)
Rule:

Complex Fractions: Use LCD method or cross multiplication for accurate comparison.

Visual Model:

vs

\(\frac{11}{12}\) vs \(\frac{7}{8}\)

Step-by-Step Method:
  1. Find LCD of 12 and 8 = 24
  2. Convert: \(\frac{11}{12} = \frac{22}{24}\) and \(\frac{7}{8} = \frac{21}{24}\)
  3. Compare numerators: 22 vs 21
  4. Since 22 > 21, then \(\frac{22}{24} > \frac{21}{24}\)
Step 1: Find LCD

LCD of 12 and 8 is 24 (smallest number divisible by both)

Step 2: Convert fractions

\(\frac{11}{12} = \frac{11 \times 2}{12 \times 2} = \frac{22}{24}\)

\(\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}\)

Step 3: Compare numerators

22 vs 21 → 22 is greater than 21

Step 4: Apply the rule

Since denominators are the same, larger numerator means larger fraction

Step 5: Alternative verification

Cross multiply: \(11 \times 8 = 88\) vs \(12 \times 7 = 84\), confirming \(88 > 84\)

\(\frac{11}{12} > \frac{7}{8}\)
Final answer:

\(\frac{11}{12}\) is greater than \(\frac{7}{8}\)

Applied rules:

• Common Denominator: Convert to same denominator for direct comparison

• Equivalent Fractions: Maintain value while changing representation

• Alternative Method: Cross multiplication provides verification

Solution: Exercises 4 to 5
4 Equivalent Fractions Check
Exercise 4
Are \(\frac{15}{20}\) and \(\frac{9}{12}\) equivalent? Show your work.
Rule:

Equivalent Fractions: Two fractions are equivalent if their cross products are equal.

Visual Model:

=

\(\frac{15}{20}\) vs \(\frac{9}{12}\) (both equal \(\frac{3}{4}\))

Step-by-Step Method:
  1. Method 1: Cross multiply: \(15 \times 12 = 180\) vs \(20 \times 9 = 180\)
  2. Method 2: Simplify both fractions to lowest terms
  3. \(\frac{15}{20} = \frac{3}{4}\) and \(\frac{9}{12} = \frac{3}{4}\)
Step 1: Cross multiply

\(15 \times 12 = 180\) and \(20 \times 9 = 180\)

Step 2: Compare products

180 vs 180 → Products are equal

Step 3: Simplify \(\frac{15}{20}\)

GCD of 15 and 20 is 5: \(\frac{15 \div 5}{20 \div 5} = \frac{3}{4}\)

Step 4: Simplify \(\frac{9}{12}\)

GCD of 9 and 12 is 3: \(\frac{9 \div 3}{12 \div 3} = \frac{3}{4}\)

Step 5: Conclusion

Both fractions simplify to \(\frac{3}{4}\), so they are equivalent

\(\frac{15}{20} = \frac{9}{12}\)
Final answer:

Yes, \(\frac{15}{20}\) and \(\frac{9}{12}\) are equivalent fractions

Applied rules:

• Equivalence Test: Cross products must be equal

• Simplification: Reduce to lowest terms using GCD

• Verification: Both methods confirm equivalence

5 Ordering Multiple Fractions
Exercise 5
Order from least to greatest: \(\frac{2}{3}, \frac{5}{6}, \frac{3}{4}, \frac{7}{8}, \frac{1}{2}\)
Rule:

Ordering Fractions: Find a common denominator and compare numerators, or convert to decimals.

Number Line:

\(\frac{1}{2}\)
\(\frac{2}{3}\)
\(\frac{3}{4}\)
\(\frac{5}{6}\)
\(\frac{7}{8}\)
Step-by-Step Method:
  1. Find LCD of denominators 2, 3, 4, 6, 8 = 24
  2. Convert each fraction to have denominator 24
  3. Compare numerators of converted fractions
  4. Order by increasing numerator values
Step 1: Find LCD

LCD of 2, 3, 4, 6, 8 is 24

Step 2: Convert all fractions

\(\frac{1}{2} = \frac{12}{24}\), \(\frac{2}{3} = \frac{16}{24}\), \(\frac{3}{4} = \frac{18}{24}\), \(\frac{5}{6} = \frac{20}{24}\), \(\frac{7}{8} = \frac{21}{24}\)

Step 3: Compare numerators

12, 16, 18, 20, 21 → Order from least to greatest: 12, 16, 18, 20, 21

Step 4: Write in order

\(\frac{12}{24} < \frac{16}{24} < \frac{18}{24} < \frac{20}{24} < \frac{21}{24}\)

Step 5: Convert back to original fractions

\(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6} < \frac{7}{8}\)

\(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6} < \frac{7}{8}\)
Final answer:

From least to greatest: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}\)

Applied rules:

• Common Denominator: Convert all fractions to same denominator

• Ordering: Compare numerators once denominators are equal

• Verification: Convert to decimals to confirm order

Questions & Answers

Question: I'm confused about comparing \(\frac{5}{6}\) and \(\frac{7}{8}\). They both seem close to 1, but how do I know which is larger?

Answer: Great observation! When fractions are close to 1, finding the exact difference helps:

  • \(\frac{5}{6}\) is \(\frac{1}{6}\) away from 1 (since \(1 - \frac{5}{6} = \frac{1}{6}\))
  • \(\frac{7}{8}\) is \(\frac{1}{8}\) away from 1 (since \(1 - \frac{7}{8} = \frac{1}{8}\))
  • Since \(\frac{1}{8} < \frac{1}{6}\), then \(\frac{7}{8}\) is closer to 1

Alternatively, use cross multiplication: \(5 \times 8 = 40\) and \(6 \times 7 = 42\). Since \(42 > 40\), then \(\frac{7}{8} > \frac{5}{6}\).

Another approach is to find a common denominator: LCD of 6 and 8 is 24. \(\frac{5}{6} = \frac{20}{24}\) and \(\frac{7}{8} = \frac{21}{24}\). Since \(21 > 20\), then \(\frac{7}{8} > \frac{5}{6}\).

Question: My child is having trouble understanding that \(\frac{8}{8} = 1\) and \(\frac{9}{8} > 1\). How can I explain this better?

Answer: Use visual models to make this concrete:

  • Draw a circle divided into 8 equal parts. Shade all 8 parts - this is \(\frac{8}{8} = 1\) whole circle
  • Draw another circle divided into 8 equal parts. Shade all 8 parts plus one more part from another circle - this is \(\frac{9}{8}\)
  • Show that \(\frac{9}{8}\) is more than one full circle

Real-world analogy: If you have 8 slices of pizza and eat 8 slices, you've eaten exactly 1 pizza (\(\frac{8}{8} = 1\)). If you eat 9 slices, you've eaten more than 1 pizza (\(\frac{9}{8} > 1\)).

Memory trick: "Top number bigger than bottom number = more than one whole!" When numerator ≥ denominator, the fraction is ≥ 1.

Question: What's the most efficient method for comparing multiple fractions like \(\frac{2}{5}, \frac{3}{7}, \frac{4}{9}\)? Should I use common denominators or decimal conversion?

Answer: For multiple fractions, here are the most efficient approaches:

  1. Estimation Method: Compare each fraction to benchmarks like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\) \(\frac{2}{5} = 0.4 < \frac{1}{2}\), \(\frac{3}{7} ≈ 0.43 < \frac{1}{2}\), \(\frac{4}{9} ≈ 0.44 < \frac{1}{2}\)
  2. Decimal Conversion: Convert to decimals: \(\frac{2}{5} = 0.4\), \(\frac{3}{7} ≈ 0.429\), \(\frac{4}{9} ≈ 0.444\)
  3. Pairwise Cross Multiplication: Compare pairs systematically

Efficiency comparison: For 3 fractions, decimal conversion is fastest. For many fractions, finding LCD becomes cumbersome. Estimation is often sufficient for ordering.

For your example: \(0.4 < 0.429 < 0.444\), so \(\frac{2}{5} < \frac{3}{7} < \frac{4}{9}\). The LCD method would require finding LCD of 5, 7, 9 = 315, which is more work for this case.