Comparing Fractions Worksheets PDF - Step-by-Step Solutions

Complete guide to comparing fractions: comprehensive exercises with solutions, rules, and visual explanations for all grade levels.

Download Free PDF Worksheet: Comparing Fractions Practice Sheets
Complete Guide: Comparing Fractions
Key Definitions:

Fraction: A number representing a part of a whole, written as \(\frac{\text{numerator}}{\text{denominator}}\)

Numerator: The top number showing how many parts we have

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

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

Mixed Number: A number combining a whole number and a proper fraction

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

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

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

Rule 8: When comparing negative fractions, reverse the inequality: if \(\frac{a}{b} > \frac{c}{d}\), then \(-\frac{a}{b} < -\frac{c}{d}\)

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

Method 8 - Absolute Value: For negative fractions, compare absolute values first

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

Grade 7 Advanced: \(-\frac{5}{7}\) vs \(-\frac{3}{4}\) → Compare absolute values: \(\frac{5}{7}\) vs \(\frac{3}{4}\) → \(\frac{5}{7} < \frac{3}{4}\), so \(-\frac{5}{7} > -\frac{3}{4}\)

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

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

Tip 6: For negative fractions, compare absolute values first, then reverse the inequality

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

Pitfall 5: Forgetting to reverse inequality for negative fractions

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

Negative Rule: The larger the absolute value of a negative number, the smaller the number itself

Solution: Exercises 1 to 3
1 Basic Fraction Comparison
Exercise 1
Compare: \(\frac{3}{4}\) and \(\frac{5}{6}\)
Rule:

Cross Multiplication: For \(\frac{a}{b}\) vs \(\frac{c}{d}\), compare \(a \times d\) vs \(b \times c\). If \(a \times d > b \times c\), then \(\frac{a}{b} > \frac{c}{d}\).

Visual Model:

vs

\(\frac{3}{4}\) vs \(\frac{5}{6}\)

Step-by-Step Method:
  1. Cross multiply: \(3 \times 6 = 18\) vs \(4 \times 5 = 20\)
  2. Compare results: 18 vs 20
  3. Since 20 > 18, then \(\frac{5}{6} > \frac{3}{4}\)
Step 1: Cross multiply

\(3 \times 6 = 18\) and \(4 \times 5 = 20\)

Step 2: Compare products

18 vs 20 → 20 is greater than 18

Step 3: Apply the rule

Since \(4 \times 5 > 3 \times 6\), then \(\frac{5}{6} > \frac{3}{4}\)

Decimal Verification: \(\frac{3}{4} = 0.75\) and \(\frac{5}{6} ≈ 0.833\), confirming \(0.833 > 0.75\)

\(\frac{5}{6} > \frac{3}{4}\)
Final answer:

\(\frac{5}{6}\) is greater than \(\frac{3}{4}\)

Applied rules:

• Cross Multiplication: Multiply diagonally and compare products

• Comparison: Larger product indicates larger fraction

• Verification: Convert to decimals to confirm

2 Mixed Numbers Comparison
Exercise 2
Compare: \(2\frac{3}{5}\) and \(2\frac{4}{7}\)
Rule:

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

Visual Model:

vs

\(2\frac{3}{5}\) vs \(2\frac{4}{7}\)

Step-by-Step Method:
  1. Compare whole numbers: 2 vs 2 → Equal
  2. Compare fractional parts: \(\frac{3}{5}\) vs \(\frac{4}{7}\)
  3. Cross multiply: \(3 \times 7 = 21\) vs \(5 \times 4 = 20\)
  4. Since 21 > 20, then \(\frac{3}{5} > \frac{4}{7}\)
Step 1: Compare whole numbers

Both have whole number 2, so they are equal

Step 2: Compare fractional parts

Now compare \(\frac{3}{5}\) vs \(\frac{4}{7}\)

Step 3: Cross multiply

\(3 \times 7 = 21\) and \(5 \times 4 = 20\)

Step 4: Compare products

21 vs 20 → 21 is greater than 20

Step 5: Apply conclusion

Since \(\frac{3}{5} > \frac{4}{7}\), then \(2\frac{3}{5} > 2\frac{4}{7}\)

Decimal Verification: \(2\frac{3}{5} = 2.6\) and \(2\frac{4}{7} ≈ 2.571\), confirming \(2.6 > 2.571\)

\(2\frac{3}{5} > 2\frac{4}{7}\)
Final answer:

\(2\frac{3}{5}\) is greater than \(2\frac{4}{7}\)

Applied rules:

• Mixed Number Comparison: Whole numbers first, then fractions

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

• Verification: Convert to decimals to confirm result

3 Equivalent Fractions Check
Exercise 3
Are \(\frac{12}{18}\) and \(\frac{8}{12}\) equivalent? Show your work.
Rule:

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

Visual Model:

=

\(\frac{12}{18}\) vs \(\frac{8}{12}\) (both equal \(\frac{2}{3}\))

Step-by-Step Method:
  1. Cross multiply: \(12 \times 12 = 144\) vs \(18 \times 8 = 144\)
  2. Compare results: 144 vs 144
  3. Since 144 = 144, then \(\frac{12}{18} = \frac{8}{12}\)
Step 1: Cross multiply

\(12 \times 12 = 144\) and \(18 \times 8 = 144\)

Step 2: Compare products

144 vs 144 → Products are equal

Step 3: Simplify both fractions

\(\frac{12}{18} = \frac{12 ÷ 6}{18 ÷ 6} = \frac{2}{3}\)

\(\frac{8}{12} = \frac{8 ÷ 4}{12 ÷ 4} = \frac{2}{3}\)

Step 4: Apply the rule

Since both fractions simplify to \(\frac{2}{3}\), they are equivalent

Decimal Verification: \(\frac{12}{18} = 0.667\) and \(\frac{8}{12} = 0.667\), confirming they are equivalent

\(\frac{12}{18} = \frac{8}{12}\)
Final answer:

Yes, \(\frac{12}{18}\) and \(\frac{8}{12}\) are equivalent fractions

Applied rules:

• Equivalence Test: Cross products must be equal

• Simplification: Reduce to lowest terms using GCD

• Verification: Both methods confirm equivalence

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

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

Number Line:

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

LCD of 2, 3, 4, 6 is 12

Step 2: Convert all fractions

\(\frac{1}{2} = \frac{6}{12}\), \(\frac{2}{3} = \frac{8}{12}\), \(\frac{3}{4} = \frac{9}{12}\), \(\frac{5}{6} = \frac{10}{12}\)

Step 3: Compare numerators

6, 8, 9, 10 → Order from least to greatest: 6, 8, 9, 10

Step 4: Write in order

\(\frac{6}{12} < \frac{8}{12} < \frac{9}{12} < \frac{10}{12}\)

Step 5: Convert back to original fractions

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

Decimal Verification: 0.5, 0.667, 0.75, 0.833, confirming the order

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

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

Applied rules:

• Common Denominator: Convert all fractions to same denominator

• Ordering: Compare numerators once denominators are equal

• Verification: Convert to decimals to confirm order

5 Negative Fraction Comparison
Exercise 5
Compare: \(-\frac{3}{5}\) and \(-\frac{2}{3}\)
Rule:

Negative Fractions: Compare absolute values first, then reverse the inequality. The larger absolute value corresponds to the smaller negative number.

Number Line Model:

0
\(-\frac{3}{5}\)
\(-\frac{2}{3}\)
Step-by-Step Method:
  1. Compare absolute values: \(\frac{3}{5}\) vs \(\frac{2}{3}\)
  2. Cross multiply: \(3 \times 3 = 9\) vs \(5 \times 2 = 10\)
  3. Since \(9 < 10\), then \(\frac{3}{5} < \frac{2}{3}\)
  4. Reverse the inequality for negatives: \(-\frac{3}{5} > -\frac{2}{3}\)
Step 1: Compare absolute values

Compare \(\left|-\frac{3}{5}\right| = \frac{3}{5}\) vs \(\left|-\frac{2}{3}\right| = \frac{2}{3}\)

Step 2: Cross multiply

\(3 \times 3 = 9\) and \(5 \times 2 = 10\)

Step 3: Compare products

9 vs 10 → 9 is less than 10

Step 4: Apply the absolute value comparison

Since \(9 < 10\), then \(\frac{3}{5} < \frac{2}{3}\)

Step 5: Apply the negative fraction rule

For negative numbers, if \(\frac{3}{5} < \frac{2}{3}\), then \(-\frac{3}{5} > -\frac{2}{3}\)

Decimal Verification: \(-\frac{3}{5} = -0.6\) and \(-\frac{2}{3} ≈ -0.667\), confirming \(-0.6 > -0.667\)

\(-\frac{3}{5} > -\frac{2}{3}\)
Final answer:

\(-\frac{3}{5}\) is greater than \(-\frac{2}{3}\)

Applied rules:

• Negative Fraction Rule: Compare absolute values first, then reverse inequality

• Cross Multiplication: Compare absolute values using standard method

• Verification: Convert to decimals to confirm result

Questions & Answers

Question: I'm confused about comparing \(\frac{3}{4}\) and \(\frac{5}{6}\). They're both close to 1, but I don't know which is bigger.

Answer: Great question! Here are two effective methods:

Method 1 - Cross Multiplication:

  • Multiply diagonally: \(3 \times 6 = 18\) and \(4 \times 5 = 20\)
  • Compare results: 18 vs 20
  • Since 20 > 18, then \(\frac{5}{6} > \frac{3}{4}\)

Method 2 - Distance from 1:

  • \(\frac{3}{4}\) is \(\frac{1}{4}\) away from 1
  • \(\frac{5}{6}\) is \(\frac{1}{6}\) away from 1
  • Since \(\frac{1}{6} < \frac{1}{4}\), then \(\frac{5}{6}\) is closer to 1
  • Therefore: \(\frac{5}{6} > \frac{3}{4}\)

The cross multiplication method always works, while the distance method is efficient for fractions close to 1.

Question: My child is struggling with comparing mixed numbers like \(3\frac{2}{5}\) and \(3\frac{3}{7}\). 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 3, compare \(\frac{2}{5}\) vs \(\frac{3}{7}\)
  • Use cross multiplication: \(2 \times 7 = 14\) vs \(5 \times 3 = 15\)
  • Since \(15 > 14\), then \(\frac{3}{7} > \frac{2}{5}\)
  • Therefore, \(3\frac{3}{7} > 3\frac{2}{5}\)

Visual aid: Draw two rectangles divided into 5 and 7 equal parts respectively. Shade 2 parts of the first and 3 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: When ordering fractions like \(\frac{2}{3}, \frac{3}{4}, \frac{5}{6}, \frac{1}{2}\), should I always find the LCD of all denominators or are there better strategies?

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

  1. Prime factorization approach: 2=2, 3=3, 4=\(2^2\), 6=\(2 \times 3\), LCD = \(2^2 \times 3 = 12\)
  2. Benchmark comparison: Compare each fraction to \(\frac{1}{2}\), \(\frac{3}{4}\), etc., to get rough estimates
  3. Pairwise comparison: Compare pairs using cross multiplication and build the order gradually

For this specific set, the LCD method is quite efficient since the LCD (12) is relatively small. Convert: \(\frac{1}{2} = \frac{6}{12}\), \(\frac{2}{3} = \frac{8}{12}\), \(\frac{3}{4} = \frac{9}{12}\), \(\frac{5}{6} = \frac{10}{12}\). Then order by numerators: \(\frac{6}{12} < \frac{8}{12} < \frac{9}{12} < \frac{10}{12}\).