Comparing Fractions Worksheets Grade 6 - Step-by-Step Solutions

Master advanced comparing fractions: complex fractions, mixed numbers, improper fractions, rational numbers, 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

Rational Number: Any number that can be expressed as a fraction \(\frac{p}{q}\) where p and q are integers and q ≠ 0

Like Fractions: Fractions with the same denominator

Unlike Fractions: Fractions with different denominators

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

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

Rule 7: Negative fractions: compare absolute values, then apply sign rules

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

Method 7 - Distance from 1: For fractions close to 1, compare how far each is from 1

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 \(2\frac{5}{6} > 2\frac{3}{4}\)

Advanced Grade 6: \(\frac{17}{24}\) vs \(\frac{13}{18}\) → LCD = 72: \(\frac{51}{72}\) vs \(\frac{52}{72}\), so \(\frac{13}{18} > \frac{17}{24}\)

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

Tip 5: Use prime factorization to efficiently find LCD for complex denominators

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

Pitfall 4: Confusing negative fraction comparisons

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

Advanced Shortcut: For fractions close to 1, compare their distances from 1

Solution: Exercises 1 to 3
1 Advanced Mixed Numbers with Large Denominators
Exercise 1
Compare: \(5\frac{11}{15}\) and \(5\frac{13}{18}\)
Rule:

Advanced Mixed Numbers: Compare whole numbers first. If equal, compare fractional parts using LCD or cross multiplication.

Visual Model:

vs

\(5\frac{11}{15}\) vs \(5\frac{13}{18}\)

Step-by-Step Method:
  1. Compare whole numbers: 5 vs 5 → Equal
  2. Compare fractional parts: \(\frac{11}{15}\) vs \(\frac{13}{18}\)
  3. Find LCD of 15 and 18: \(15 = 3 \times 5\), \(18 = 2 \times 3^2\), LCD = \(2 \times 3^2 \times 5 = 90\)
  4. Convert: \(\frac{11}{15} = \frac{66}{90}\), \(\frac{13}{18} = \frac{65}{90}\)
  5. Compare numerators: 66 vs 65 → 66 > 65
Step 1: Compare whole numbers

Both have whole number 5, so they are equal

Step 2: Compare fractional parts

Now compare \(\frac{11}{15}\) vs \(\frac{13}{18}\)

Step 3: Find LCD using prime factorization

\(15 = 3 \times 5\), \(18 = 2 \times 3^2\)

LCD = highest power of each prime = \(2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90\)

Step 4: Convert fractions

\(\frac{11}{15} = \frac{11 \times 6}{15 \times 6} = \frac{66}{90}\)

\(\frac{13}{18} = \frac{13 \times 5}{18 \times 5} = \frac{65}{90}\)

Step 5: Compare numerators

66 vs 65 → 66 is greater than 65

Step 6: Apply conclusion

Since \(\frac{66}{90} > \frac{65}{90}\), then \(\frac{11}{15} > \frac{13}{18}\)

Therefore, \(5\frac{11}{15} > 5\frac{13}{18}\)

\(5\frac{11}{15} > 5\frac{13}{18}\)
Final answer:

\(5\frac{11}{15}\) is greater than \(5\frac{13}{18}\)

Applied rules:

• Mixed Number Comparison: Whole numbers first, then fractions

• Prime Factorization: Efficient method to find LCD for complex denominators

• Alternative Method: Cross multiplication: \(11 \times 18 = 198\), \(15 \times 13 = 195\), confirming \(198 > 195\)

2 Complex Improper Fractions with Very Large Numbers
Exercise 2
Compare: \(\frac{127}{84}\) and \(\frac{95}{63}\)
Rule:

Very Large Improper Fractions: Use cross multiplication or convert to mixed numbers for easier comparison. Cross multiplication is often most efficient for large numbers.

Visual Model:

vs

\(\frac{127}{84}\) vs \(\frac{95}{63}\)

Step-by-Step Method:
  1. Use cross multiplication: \(127 \times 63 = 8001\) vs \(84 \times 95 = 7980\)
  2. Compare results: 8001 vs 7980
  3. Since 8001 > 7980, then \(\frac{127}{84} > \frac{95}{63}\)
Step 1: Cross multiply

\(127 \times 63 = 8001\) and \(84 \times 95 = 7980\)

Step 2: Compare products

8001 vs 7980 → 8001 is greater than 7980

Step 3: Apply the rule

Since \(127 \times 63 > 84 \times 95\), then \(\frac{127}{84} > \frac{95}{63}\)

Step 4: Alternative verification

Convert to mixed numbers: \(\frac{127}{84} = 1\frac{43}{84}\) and \(\frac{95}{63} = 1\frac{32}{63}\)

Compare fractional parts: \(\frac{43}{84}\) vs \(\frac{32}{63}\) → Cross multiply: \(43 \times 63 = 2709\), \(84 \times 32 = 2688\)

Since \(2709 > 2688\), then \(\frac{43}{84} > \frac{32}{63}\), confirming \(\frac{127}{84} > \frac{95}{63}\)

Decimal Verification: \(\frac{127}{84} ≈ 1.512\) and \(\frac{95}{63} ≈ 1.508\), confirming \(1.512 > 1.508\)

\(\frac{127}{84} > \frac{95}{63}\)
Final answer:

\(\frac{127}{84}\) is greater than \(\frac{95}{63}\)

Applied rules:

• Cross Multiplication: Works for any fractions regardless of size

• Improper Fraction Rule: Same comparison methods apply

• Verification: Multiple methods confirm the result

3 Advanced Fraction Comparison Using Distance from 1
Exercise 3
Compare: \(\frac{47}{50}\) and \(\frac{59}{62}\)
Rule:

Distance from 1 Method: For fractions close to 1, find how far each is from 1 and compare those distances. The fraction with the smaller distance is larger.

Visual Model:

vs

\(\frac{47}{50}\) vs \(\frac{59}{62}\)

Step-by-Step Method:
  1. Find distance from 1: \(1 - \frac{47}{50} = \frac{3}{50}\) and \(1 - \frac{59}{62} = \frac{3}{62}\)
  2. Compare distances: \(\frac{3}{50}\) vs \(\frac{3}{62}\)
  3. Since numerators are equal, compare denominators: 50 vs 62
  4. Since 50 < 62, then \(\frac{3}{50} > \frac{3}{62}\)
  5. Therefore, \(\frac{59}{62} > \frac{47}{50}\) (since it's closer to 1)
Step 1: Find distance from 1

\(1 - \frac{47}{50} = \frac{50}{50} - \frac{47}{50} = \frac{3}{50}\)

\(1 - \frac{59}{62} = \frac{62}{62} - \frac{59}{62} = \frac{3}{62}\)

Step 2: Compare distances

Now compare \(\frac{3}{50}\) vs \(\frac{3}{62}\)

Step 3: Apply same numerator rule

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

Since 50 < 62, then \(\frac{3}{50} > \frac{3}{62}\)

Step 4: Apply the distance rule

The fraction that is farther from 1 is smaller

Since \(\frac{47}{50}\) is farther from 1 than \(\frac{59}{62}\), then \(\frac{59}{62} > \frac{47}{50}\)

Step 5: Verify with cross multiplication

\(47 \times 62 = 2914\) and \(50 \times 59 = 2950\)

Since \(2950 > 2914\), then \(\frac{59}{62} > \frac{47}{50}\), confirming our result

Decimal Verification: \(\frac{47}{50} = 0.94\) and \(\frac{59}{62} ≈ 0.9516\), confirming \(0.9516 > 0.94\)

\(\frac{59}{62} > \frac{47}{50}\)
Final answer:

\(\frac{59}{62}\) is greater than \(\frac{47}{50}\)

Applied rules:

• Distance from 1: Efficient method for fractions close to 1

• Same Numerator Rule: Compare denominators when numerators are equal

• Verification: Cross multiplication confirms the result

Solution: Exercises 4 to 5
4 Advanced Equivalent Fractions Check with Algebraic Expressions
Exercise 4
Are \(\frac{39}{65}\) and \(\frac{42}{70}\) equivalent? Show your work using prime factorization.
Rule:

Advanced Equivalent Fractions: Two fractions are equivalent if their cross products are equal or if they simplify to the same fraction using GCD found through prime factorization.

Visual Model:

=

\(\frac{39}{65}\) vs \(\frac{42}{70}\) (both equal \(\frac{3}{5}\))

Step-by-Step Method:
  1. Method 1: Cross multiply: \(39 \times 70 = 2730\) vs \(65 \times 42 = 2730\)
  2. Method 2: Simplify using prime factorization of numerator and denominator
  3. \(\frac{39}{65} = \frac{3 \times 13}{5 \times 13} = \frac{3}{5}\) and \(\frac{42}{70} = \frac{2 \times 3 \times 7}{2 \times 5 \times 7} = \frac{3}{5}\)
Step 1: Cross multiply

\(39 \times 70 = 2730\) and \(65 \times 42 = 2730\)

Step 2: Compare products

2730 vs 2730 → Products are equal

Step 3: Prime factorization of \(\frac{39}{65}\)

39 = \(3 \times 13\), 65 = \(5 \times 13\)

\(\frac{39}{65} = \frac{3 \times 13}{5 \times 13} = \frac{3}{5}\) (cancel common factor 13)

Step 4: Prime factorization of \(\frac{42}{70}\)

42 = \(2 \times 3 \times 7\), 70 = \(2 \times 5 \times 7\)

\(\frac{42}{70} = \frac{2 \times 3 \times 7}{2 \times 5 \times 7} = \frac{3}{5}\) (cancel common factors 2 and 7)

Step 5: Conclusion

Both fractions simplify to \(\frac{3}{5}\), so they are equivalent

Decimal Verification: \(\frac{39}{65} = 0.6\) and \(\frac{42}{70} = 0.6\), confirming they are equivalent

\(\frac{39}{65} = \frac{42}{70}\)
Final answer:

Yes, \(\frac{39}{65}\) and \(\frac{42}{70}\) are equivalent fractions

Applied rules:

• Equivalence Test: Cross products must be equal

• Prime Factorization: Efficient method to find and cancel common factors

• Verification: Both methods confirm equivalence

5 Advanced Ordering Multiple Complex Fractions
Exercise 5
Order from least to greatest: \(\frac{13}{15}, \frac{17}{20}, \frac{19}{24}, \frac{11}{12}, \frac{23}{30}\)
Rule:

Advanced Ordering: Find a common denominator using prime factorization and compare numerators, or convert to decimals. For large sets of fractions, prime factorization is most efficient.

Number Line:

\(\frac{23}{30}\)
\(\frac{19}{24}\)
\(\frac{17}{20}\)
\(\frac{13}{15}\)
\(\frac{11}{12}\)
Step-by-Step Method:
  1. Find LCD of denominators 12, 15, 20, 24, 30 using prime factorization
  2. Convert each fraction to have denominator 120
  3. Compare numerators of converted fractions
  4. Order by increasing numerator values
Step 1: Find LCD using prime factorization

12 = \(2^2 \times 3\), 15 = \(3 \times 5\), 20 = \(2^2 \times 5\), 24 = \(2^3 \times 3\), 30 = \(2 \times 3 \times 5\)

LCD = highest power of each prime = \(2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120\)

Step 2: Convert all fractions to denominator 120

\(\frac{11}{12} = \frac{11 \times 10}{12 \times 10} = \frac{110}{120}\)

\(\frac{13}{15} = \frac{13 \times 8}{15 \times 8} = \frac{104}{120}\)

\(\frac{17}{20} = \frac{17 \times 6}{20 \times 6} = \frac{102}{120}\)

\(\frac{19}{24} = \frac{19 \times 5}{24 \times 5} = \frac{95}{120}\)

\(\frac{23}{30} = \frac{23 \times 4}{30 \times 4} = \frac{92}{120}\)

Step 3: Compare numerators

92, 95, 102, 104, 110 → Order from least to greatest: 92, 95, 102, 104, 110

Step 4: Write in order

\(\frac{92}{120} < \frac{95}{120} < \frac{102}{120} < \frac{104}{120} < \frac{110}{120}\)

Step 5: Convert back to original fractions

\(\frac{23}{30} < \frac{19}{24} < \frac{17}{20} < \frac{13}{15} < \frac{11}{12}\)

Decimal Verification: 0.767, 0.792, 0.85, 0.867, 0.917, confirming the order

\(\frac{23}{30} < \frac{19}{24} < \frac{17}{20} < \frac{13}{15} < \frac{11}{12}\)
Final answer:

From least to greatest: \(\frac{23}{30}, \frac{19}{24}, \frac{17}{20}, \frac{13}{15}, \frac{11}{12}\)

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

Questions & Answers

Question: I'm trying to compare \(\frac{123}{140}\) and \(\frac{119}{132}\). They're both really close to 1, and cross multiplication gives huge numbers. Is there a better method?

Answer: Yes, the distance from 1 method is perfect for fractions close to 1:

  • Distance from 1: \(\frac{123}{140}\) is \(\frac{17}{140}\) away from 1, \(\frac{119}{132}\) is \(\frac{13}{132}\) away from 1
  • Compare \(\frac{17}{140}\) vs \(\frac{13}{132}\): Cross multiply: \(17 \times 132 = 2244\) vs \(140 \times 13 = 1820\)
  • Since \(2244 > 1820\), then \(\frac{17}{140} > \frac{13}{132}\), meaning \(\frac{123}{140}\) is farther from 1 than \(\frac{119}{132}\)
  • Therefore, \(\frac{119}{132} > \frac{123}{140}\)

Alternatively, you could use the fact that both fractions are close to 1 and compare their differences: since \(\frac{17}{140} ≈ 0.121\) and \(\frac{13}{132} ≈ 0.098\), and \(0.121 > 0.098\), then \(\frac{119}{132} > \frac{123}{140}\).

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

Question: My child is working with fractions like \(\frac{47}{50}\) and \(\frac{59}{62}\). The denominators are getting quite large. How can they find the LCD efficiently?

Answer: For large denominators, use prime factorization to find the LCD efficiently:

  • Factor each denominator completely: \(50 = 2 \times 5^2\) and \(62 = 2 \times 31\)
  • Take the highest power of each prime factor: \(2^1\), \(5^2\), \(31^1\)
  • LCD = \(2 \times 25 \times 31 = 1550\)
  • Convert: \(\frac{47}{50} = \frac{47 \times 31}{50 \times 31} = \frac{1457}{1550}\)
  • Convert: \(\frac{59}{62} = \frac{59 \times 25}{62 \times 25} = \frac{1475}{1550}\)
  • Compare: Since \(1475 > 1457\), then \(\frac{59}{62} > \frac{47}{50}\)

Alternative method: For fractions close to 1, use the distance from 1 method (as shown in Exercise 3), which is often faster for this type of problem.

Memory trick: "Find all the prime building blocks and take the biggest ones!"

Question: When ordering multiple fractions like \(\frac{13}{15}, \frac{17}{20}, \frac{19}{24}, \frac{11}{12}, \frac{23}{30}\), should I always find the LCD of all denominators or are there better strategies?

Answer: Finding the LCD of all denominators is often the most systematic approach, but consider these strategies:

  1. Prime factorization approach: As shown in Exercise 5, this is efficient for multiple fractions
  2. Grouping method: Group fractions that share common denominators or are obviously close together
  3. Benchmark comparison: Compare each fraction to \(\frac{3}{4}\), \(\frac{4}{5}\), etc., to get rough estimates
  4. Pairwise comparison: Compare pairs using cross multiplication and build the order gradually

For the given example: \(\frac{13}{15}, \frac{17}{20}, \frac{19}{24}, \frac{11}{12}, \frac{23}{30}\)

Using prime factorization: 12=\(2^2 \times 3\), 15=\(3 \times 5\), 20=\(2^2 \times 5\), 24=\(2^3 \times 3\), 30=\(2 \times 3 \times 5\)

LCD = \(2^3 \times 3 \times 5 = 120\), which is manageable. The LCD method ensures accuracy and works well when denominators aren't too large after factorization.