Comparing Fractions Worksheets Grade 5 - Step-by-Step Solutions

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

Detailed Summary: Advanced Comparing Fractions
Key Definitions:

Proper Fraction: A fraction where the numerator is less than the denominator (\(\frac{a}{b}\) where \(a < b\))

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (\(\frac{a}{b}\) where \(a ≥ b\))

Mixed Number: A number combining a whole number and a proper fraction (e.g., \(2\frac{3}{4}\))

Equivalent Fractions: Fractions that represent the same value but use different numerators and denominators

Unit Fraction: A fraction with numerator 1 (e.g., \(\frac{1}{5}\))

Like Fractions: Fractions with the same denominator

Unlike Fractions: Fractions with different denominators

Core Rules and Principles:

Rule 1: When denominators are the same, compare numerators directly

Rule 2: When numerators are the same, compare denominators inversely

Rule 3: For unlike fractions, find common denominator or use cross multiplication

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

Rule 5: Equivalent fractions have equal cross products

Rule 6: To compare mixed numbers, compare whole numbers first, then fractional parts

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

Method 6 - Benchmark Comparison: Compare to known values like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\)

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 \(\frac{5}{8} > \frac{7}{12}\)

Very Complex: \(2\frac{3}{4}\) vs \(2\frac{5}{6}\) → Compare fractional parts: \(\frac{3}{4} = \frac{9}{12}\), \(\frac{5}{6} = \frac{10}{12}\), so \(2\frac{5}{6} > 2\frac{3}{4}\)

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

Tip 4: For mixed numbers, convert to improper fractions if easier to compare

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

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

Pitfall 3: Not simplifying fractions before comparing

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

Shortcut: If one fraction is clearly greater than 1 and the other is less than 1, the comparison is obvious

Solution: Exercises 1 to 3
1 Complex Mixed Numbers
Exercise 1
Compare: \(3\frac{5}{6}\) and \(3\frac{7}{8}\)
Rule:

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

Visual Model:

vs

\(3\frac{5}{6}\) vs \(3\frac{7}{8}\)

Step-by-Step Method:
  1. Compare whole numbers: 3 vs 3 → Equal
  2. Compare fractional parts: \(\frac{5}{6}\) vs \(\frac{7}{8}\)
  3. Use cross multiplication: \(5 \times 8 = 40\) vs \(6 \times 7 = 42\)
  4. Since 42 > 40, then \(\frac{7}{8} > \frac{5}{6}\)
Step 1: Compare whole numbers

Both have whole number 3, so they are equal

Step 2: Compare fractional parts

Now compare \(\frac{5}{6}\) vs \(\frac{7}{8}\)

Step 3: Cross multiply

\(5 \times 8 = 40\) and \(6 \times 7 = 42\)

Step 4: Compare products

40 vs 42 → 42 is greater than 40

Step 5: Apply conclusion

Since \(\frac{7}{8} > \frac{5}{6}\), then \(3\frac{7}{8} > 3\frac{5}{6}\)

\(3\frac{7}{8} > 3\frac{5}{6}\)
Final answer:

\(3\frac{7}{8}\) is greater than \(3\frac{5}{6}\)

Applied rules:

• Mixed Number Comparison: Whole numbers first, then fractions

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

• Alternative Method: Convert to improper fractions: \(\frac{23}{6}\) vs \(\frac{31}{8}\)

2 Improper Fractions with Large Numbers
Exercise 2
Compare: \(\frac{23}{12}\) and \(\frac{19}{10}\)
Rule:

Large Improper Fractions: Use cross multiplication or convert to mixed numbers for easier comparison.

Visual Model:

vs

\(\frac{23}{12}\) vs \(\frac{19}{10}\)

Step-by-Step Method:
  1. Use cross multiplication: \(23 \times 10 = 230\) vs \(12 \times 19 = 228\)
  2. Compare results: 230 vs 228
  3. Since 230 > 228, then \(\frac{23}{12} > \frac{19}{10}\)
Step 1: Cross multiply

\(23 \times 10 = 230\) and \(12 \times 19 = 228\)

Step 2: Compare products

230 vs 228 → 230 is greater than 228

Step 3: Apply the rule

Since \(23 \times 10 > 12 \times 19\), then \(\frac{23}{12} > \frac{19}{10}\)

Step 4: Alternative verification

Convert to mixed numbers: \(\frac{23}{12} = 1\frac{11}{12}\) and \(\frac{19}{10} = 1\frac{9}{10}\)

Compare fractional parts: \(\frac{11}{12}\) vs \(\frac{9}{10}\) → \(11 \times 10 = 110\), \(12 \times 9 = 108\)

Since \(110 > 108\), then \(\frac{11}{12} > \frac{9}{10}\), confirming \(\frac{23}{12} > \frac{19}{10}\)

\(\frac{23}{12} > \frac{19}{10}\)
Final answer:

\(\frac{23}{12}\) is greater than \(\frac{19}{10}\)

Applied rules:

• Cross Multiplication: Works for any fractions regardless of size

• Improper Fraction Rule: Same comparison methods apply

• Alternative Method: Converting to mixed numbers can provide verification

3 Complex Fraction Comparison
Exercise 3
Compare: \(\frac{17}{24}\) and \(\frac{13}{18}\)
Rule:

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

Visual Model:

vs

\(\frac{17}{24}\) vs \(\frac{13}{18}\)

Step-by-Step Method:
  1. Find LCD of 24 and 18 = 72
  2. Convert: \(\frac{17}{24} = \frac{51}{72}\) and \(\frac{13}{18} = \frac{52}{72}\)
  3. Compare numerators: 51 vs 52
  4. Since 52 > 51, then \(\frac{52}{72} > \frac{51}{72}\)
Step 1: Find LCD

LCD of 24 and 18: Prime factorization: \(24 = 2^3 \times 3\), \(18 = 2 \times 3^2\)

LCD = \(2^3 \times 3^2 = 8 \times 9 = 72\)

Step 2: Convert fractions

\(\frac{17}{24} = \frac{17 \times 3}{24 \times 3} = \frac{51}{72}\)

\(\frac{13}{18} = \frac{13 \times 4}{18 \times 4} = \frac{52}{72}\)

Step 3: Compare numerators

51 vs 52 → 52 is greater than 51

Step 4: Apply the rule

Since denominators are the same, larger numerator means larger fraction

Step 5: Alternative verification

Cross multiply: \(17 \times 18 = 306\) vs \(24 \times 13 = 312\), confirming \(312 > 306\)

\(\frac{13}{18} > \frac{17}{24}\)
Final answer:

\(\frac{13}{18}\) is greater than \(\frac{17}{24}\)

Applied rules:

• Common Denominator: Convert to same denominator for direct comparison

• Prime Factorization: Efficient method to find LCD

• Alternative Method: Cross multiplication provides verification

Solution: Exercises 4 to 5
4 Advanced Equivalent Fractions Check
Exercise 4
Are \(\frac{35}{42}\) and \(\frac{45}{54}\) equivalent? Show your work.
Rule:

Advanced Equivalent Fractions: Two fractions are equivalent if their cross products are equal or if they simplify to the same fraction.

Visual Model:

=

\(\frac{35}{42}\) vs \(\frac{45}{54}\) (both equal \(\frac{5}{6}\))

Step-by-Step Method:
  1. Method 1: Cross multiply: \(35 \times 54 = 1890\) vs \(42 \times 45 = 1890\)
  2. Method 2: Simplify both fractions to lowest terms
  3. \(\frac{35}{42} = \frac{5}{6}\) and \(\frac{45}{54} = \frac{5}{6}\)
Step 1: Cross multiply

\(35 \times 54 = 1890\) and \(42 \times 45 = 1890\)

Step 2: Compare products

1890 vs 1890 → Products are equal

Step 3: Simplify \(\frac{35}{42}\)

GCD of 35 and 42: Factors of 35: 1, 5, 7, 35; Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

GCD = 7: \(\frac{35 \div 7}{42 \div 7} = \frac{5}{6}\)

Step 4: Simplify \(\frac{45}{54}\)

GCD of 45 and 54: Factors of 45: 1, 3, 5, 9, 15, 45; Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54

GCD = 9: \(\frac{45 \div 9}{54 \div 9} = \frac{5}{6}\)

Step 5: Conclusion

Both fractions simplify to \(\frac{5}{6}\), so they are equivalent

\(\frac{35}{42} = \frac{45}{54}\)
Final answer:

Yes, \(\frac{35}{42}\) and \(\frac{45}{54}\) 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 Advanced Ordering Multiple Fractions
Exercise 5
Order from least to greatest: \(\frac{7}{9}, \frac{5}{6}, \frac{11}{12}, \frac{3}{4}, \frac{8}{9}\)
Rule:

Advanced Ordering: Find a common denominator and compare numerators, or convert to decimals. For large denominators, consider using prime factorization to find LCD efficiently.

Number Line:

\(\frac{3}{4}\)
\(\frac{7}{9}\)
\(\frac{5}{6}\)
\(\frac{8}{9}\)
\(\frac{11}{12}\)
Step-by-Step Method:
  1. Find LCD of denominators 4, 6, 9, 12 = 36 (using prime factorization)
  2. Convert each fraction to have denominator 36
  3. Compare numerators of converted fractions
  4. Order by increasing numerator values
Step 1: Find LCD using prime factorization

4 = \(2^2\), 6 = \(2 \times 3\), 9 = \(3^2\), 12 = \(2^2 \times 3\)

LCD = \(2^2 \times 3^2 = 4 \times 9 = 36\)

Step 2: Convert all fractions

\(\frac{3}{4} = \frac{27}{36}\), \(\frac{5}{6} = \frac{30}{36}\), \(\frac{7}{9} = \frac{28}{36}\), \(\frac{8}{9} = \frac{32}{36}\), \(\frac{11}{12} = \frac{33}{36}\)

Step 3: Compare numerators

27, 28, 30, 32, 33 → Order from least to greatest: 27, 28, 30, 32, 33

Step 4: Write in order

\(\frac{27}{36} < \frac{28}{36} < \frac{30}{36} < \frac{32}{36} < \frac{33}{36}\)

Step 5: Convert back to original fractions

\(\frac{3}{4} < \frac{7}{9} < \frac{5}{6} < \frac{8}{9} < \frac{11}{12}\)

\(\frac{3}{4} < \frac{7}{9} < \frac{5}{6} < \frac{8}{9} < \frac{11}{12}\)
Final answer:

From least to greatest: \(\frac{3}{4}, \frac{7}{9}, \frac{5}{6}, \frac{8}{9}, \frac{11}{12}\)

Applied rules:

• Common Denominator: Convert all fractions to same denominator

• Prime Factorization: Efficient method to find LCD for multiple fractions

• Verification: Convert to decimals to confirm order: 0.75, 0.778, 0.833, 0.889, 0.917

Questions & Answers

Question: I'm trying to compare \(\frac{19}{24}\) and \(\frac{17}{20}\). They're both close to 1, but finding the LCD seems hard. Is there an easier way?

Answer: Yes, there are several efficient approaches for comparing fractions close to 1:

  • Distance from 1 method: \(\frac{19}{24}\) is \(\frac{5}{24}\) away from 1, \(\frac{17}{20}\) is \(\frac{3}{20}\) away from 1
  • Compare \(\frac{5}{24}\) vs \(\frac{3}{20}\): Cross multiply: \(5 \times 20 = 100\) vs \(24 \times 3 = 72\)
  • Since \(100 > 72\), then \(\frac{5}{24} > \frac{3}{20}\), meaning \(\frac{19}{24}\) is farther from 1 than \(\frac{17}{20}\)
  • Therefore, \(\frac{17}{20} > \frac{19}{24}\)

Alternatively, use cross multiplication: \(19 \times 20 = 380\) vs \(24 \times 17 = 408\). Since \(408 > 380\), then \(\frac{17}{20} > \frac{19}{24}\).

The distance method is especially helpful when fractions are close to 1, as it reduces the comparison to smaller fractions.

Question: My child is struggling with comparing mixed numbers like \(4\frac{3}{7}\) and \(4\frac{5}{9}\). How can I help them understand this better?

Answer: For mixed numbers with the same whole number, focus on the fractional parts:

  • Since both have whole number 4, compare \(\frac{3}{7}\) vs \(\frac{5}{9}\)
  • Use cross multiplication: \(3 \times 9 = 27\) vs \(7 \times 5 = 35\)
  • Since \(35 > 27\), then \(\frac{5}{9} > \frac{3}{7}\)
  • Therefore, \(4\frac{5}{9} > 4\frac{3}{7}\)

Visual aid: Draw two rectangles divided into 7 and 9 equal parts respectively. Shade 3 parts of the first and 5 parts of the second to see which is larger.

Memory trick: "Same whole number? Just compare the fractions!" The whole number doesn't affect the comparison when they're equal.

Question: What's the most efficient method for ordering 5 or more fractions like \(\frac{2}{3}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12}\)? Should I find the LCD of all denominators?

Answer: For multiple fractions, finding the LCD of all denominators can be efficient if done systematically:

  1. Prime factorization approach: 3=3, 4=\(2^2\), 6=\(2 \times 3\), 8=\(2^3\), 12=\(2^2 \times 3\)
  2. LCD = \(2^3 \times 3 = 24\)
  3. Convert: \(\frac{2}{3} = \frac{16}{24}\), \(\frac{3}{4} = \frac{18}{24}\), \(\frac{5}{6} = \frac{20}{24}\), \(\frac{7}{8} = \frac{21}{24}\), \(\frac{11}{12} = \frac{22}{24}\)
  4. Order by numerators: 16, 18, 20, 21, 22 → \(\frac{2}{3} < \frac{3}{4} < \frac{5}{6} < \frac{7}{8} < \frac{11}{12}\)

Alternative strategies: For fractions close to benchmarks, compare to \(\frac{1}{2}\), \(\frac{3}{4}\), etc. Or use pairwise comparisons with cross multiplication.

For this specific set, the LCD method is quite efficient since the LCD (24) is relatively small compared to the product of all denominators (6912).