Comparing Unlike Fractions Questions - Complete Solutions

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

Complete Guide: Comparing Unlike Fractions
Key Definitions:

Unlike Fractions: Fractions with different denominators (e.g., \(\frac{2}{3}\) and \(\frac{5}{7}\))

Like Fractions: Fractions with the same denominator (e.g., \(\frac{2}{5}\) and \(\frac{3}{5}\))

Cross Multiplication: A method for comparing unlike fractions by multiplying the numerator of one by the denominator of the other

Least Common Denominator (LCD): The smallest number that is a multiple of all denominators in a set of fractions

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

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

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: Cross multiplication: For \(\frac{a}{b}\) vs \(\frac{c}{d}\), compare \(a \times d\) vs \(b \times c\)

Rule 5: If \(a \times d > b \times c\), then \(\frac{a}{b} > \frac{c}{d}\)

Rule 6: If \(a \times d < b \times c\), then \(\frac{a}{b} < \frac{c}{d}\)

Rule 7: If \(a \times d = b \times c\), then \(\frac{a}{b} = \frac{c}{d}\) (equivalent fractions)

Step-by-Step Methods:

Method 1 - Cross Multiplication: Multiply diagonally and compare products

Method 2 - Common Denominator: Convert to same denominator, then compare numerators

Method 3 - Decimal Conversion: Convert fractions to decimals and compare

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

Method 5 - Prime Factorization: Find LCD using prime factors

Examples (Simple to Advanced):

Simple: \(\frac{2}{3}\) vs \(\frac{3}{5}\) → Cross multiply: \(2 \times 5 = 10\), \(3 \times 3 = 9\), so \(\frac{2}{3} > \frac{3}{5}\)

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

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

Complex: \(\frac{13}{18}\) vs \(\frac{11}{15}\) → Cross multiply: \(13 \times 15 = 195\), \(18 \times 11 = 198\), so \(\frac{11}{15} > \frac{13}{18}\)

Very Complex: \(\frac{17}{24}\) vs \(\frac{19}{28}\) → Cross multiply: \(17 \times 28 = 476\), \(24 \times 19 = 456\), so \(\frac{17}{24} > \frac{19}{28}\)

Tips and Common Pitfalls:

Tip 1: Cross multiplication is often the fastest method for comparing two unlike fractions

Tip 2: Use benchmarks like \(\frac{1}{2}\) to estimate before calculating

Tip 3: For multiple unlike fractions, find the LCD to convert all to like fractions

Tip 4: Prime factorization helps efficiently find LCD for complex denominators

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

Pitfall 2: Forgetting to multiply the correct numerator with the opposite denominator

Pitfall 3: Making arithmetic errors in cross multiplication

Pitfall 4: Not simplifying fractions before comparing

Pitfall 5: Confusing the direction of the inequality symbol

Key Notes for Memorization:

Memory Aid: "Cross multiply: diagonal numbers, compare results" - numerator of first with denominator of second

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

Pattern: Larger cross product indicates larger fraction

Verification: Always check your answer by converting to decimals or using an alternative method

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 Basic Unlike Fraction Comparison
Exercise 1
Compare: \(\frac{2}{3}\) and \(\frac{3}{5}\)
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

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

Step-by-Step Method:
  1. Cross multiply: \(2 \times 5 = 10\) vs \(3 \times 3 = 9\)
  2. Compare results: 10 vs 9
  3. Since 10 > 9, then \(\frac{2}{3} > \frac{3}{5}\)
Step 1: Cross multiply

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

Step 2: Compare products

10 vs 9 → 10 is greater than 9

Step 3: Apply the rule

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

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

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

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

Applied rules:

• Cross Multiplication: Multiply numerator of first with denominator of second

• Comparison: Larger cross product indicates larger fraction

• Verification: Convert to decimals to confirm

2 Advanced Unlike Fraction Comparison
Exercise 2
Compare: \(\frac{7}{12}\) and \(\frac{5}{8}\)
Rule:

Common Denominator Method: Find LCD and convert fractions to like fractions, then compare numerators.

Visual Model:

vs

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

Step-by-Step Method:
  1. Find LCD of 12 and 8: \(12 = 2^2 \times 3\), \(8 = 2^3\), LCD = \(2^3 \times 3 = 24\)
  2. Convert: \(\frac{7}{12} = \frac{14}{24}\), \(\frac{5}{8} = \frac{15}{24}\)
  3. Compare numerators: 14 vs 15
  4. Since 15 > 14, then \(\frac{5}{8} > \frac{7}{12}\)
Step 1: Find LCD using prime factorization

\(12 = 2^2 \times 3\), \(8 = 2^3\)

LCD = highest power of each prime = \(2^3 \times 3 = 8 \times 3 = 24\)

Step 2: Convert fractions

\(\frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24}\)

\(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\)

Step 3: Compare numerators

14 vs 15 → 15 is greater than 14

Step 4: Apply the rule

Since denominators are the same, larger numerator means larger fraction

Step 5: Alternative verification

Cross multiply: \(7 \times 8 = 56\) vs \(12 \times 5 = 60\), confirming \(60 > 56\)

Decimal Verification: \(\frac{7}{12} ≈ 0.583\) and \(\frac{5}{8} = 0.625\), confirming \(0.625 > 0.583\)

\(\frac{5}{8} > \frac{7}{12}\)
Final answer:

\(\frac{5}{8}\) is greater than \(\frac{7}{12}\)

Applied rules:

• Common Denominator: Convert to same denominator for direct comparison

• Prime Factorization: Efficient method to find LCD

• Alternative Method: Cross multiplication provides verification

3 Complex Unlike Fraction Comparison
Exercise 3
Compare: \(\frac{13}{18}\) and \(\frac{11}{15}\)
Rule:

Cross Multiplication for Complex Denominators: When denominators are large, cross multiplication is often more efficient than finding LCD.

Visual Model:

vs

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

Step-by-Step Method:
  1. Cross multiply: \(13 \times 15 = 195\) vs \(18 \times 11 = 198\)
  2. Compare results: 195 vs 198
  3. Since 198 > 195, then \(\frac{11}{15} > \frac{13}{18}\)
Step 1: Cross multiply

\(13 \times 15 = 195\) and \(18 \times 11 = 198\)

Step 2: Compare products

195 vs 198 → 198 is greater than 195

Step 3: Apply the rule

Since \(18 \times 11 > 13 \times 15\), then \(\frac{11}{15} > \frac{13}{18}\)

Step 4: Alternative method verification

Find LCD of 18 and 15: \(18 = 2 \times 3^2\), \(15 = 3 \times 5\), LCD = \(2 \times 3^2 \times 5 = 90\)

\(\frac{13}{18} = \frac{65}{90}\), \(\frac{11}{15} = \frac{66}{90}\), confirming \(\frac{66}{90} > \frac{65}{90}\)

Decimal Verification: \(\frac{13}{18} ≈ 0.722\) and \(\frac{11}{15} ≈ 0.733\), confirming \(0.733 > 0.722\)

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

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

Applied rules:

• Cross Multiplication: Efficient method for complex denominators

• Alternative Method: Common denominator provides verification

• Verification: Convert to decimals to confirm result

Solution: Exercises 4 to 5
4 Equivalent Unlike Fractions Check
Exercise 4
Are \(\frac{14}{21}\) and \(\frac{18}{27}\) equivalent? Show your work.
Rule:

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

Visual Model:

=

\(\frac{14}{21}\) vs \(\frac{18}{27}\) (both equal \(\frac{2}{3}\))

Step-by-Step Method:
  1. Method 1: Cross multiply: \(14 \times 27 = 378\) vs \(21 \times 18 = 378\)
  2. Method 2: Simplify both fractions to lowest terms
  3. \(\frac{14}{21} = \frac{2}{3}\) and \(\frac{18}{27} = \frac{2}{3}\)
Step 1: Cross multiply

\(14 \times 27 = 378\) and \(21 \times 18 = 378\)

Step 2: Compare products

378 vs 378 → Products are equal

Step 3: Simplify \(\frac{14}{21}\)

GCD of 14 and 21: Factors of 14: 1, 2, 7, 14; Factors of 21: 1, 3, 7, 21

GCD = 7: \(\frac{14 ÷ 7}{21 ÷ 7} = \frac{2}{3}\)

Step 4: Simplify \(\frac{18}{27}\)

GCD of 18 and 27: Factors of 18: 1, 2, 3, 6, 9, 18; Factors of 27: 1, 3, 9, 27

GCD = 9: \(\frac{18 ÷ 9}{27 ÷ 9} = \frac{2}{3}\)

Step 5: Conclusion

Both fractions simplify to \(\frac{2}{3}\), so they are equivalent

Decimal Verification: \(\frac{14}{21} ≈ 0.667\) and \(\frac{18}{27} ≈ 0.667\), confirming they are equivalent

\(\frac{14}{21} = \frac{18}{27}\)
Final answer:

Yes, \(\frac{14}{21}\) and \(\frac{18}{27}\) 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 Unlike Fractions
Exercise 5
Order from least to greatest: \(\frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{2}{3}\)
Rule:

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

Number Line:

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

3=3, 4=\(2^2\), 6=\(2 \times 3\), 8=\(2^3\)

LCD = highest power of each prime = \(2^3 \times 3 = 8 \times 3 = 24\)

Step 2: Convert all fractions

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

Step 3: Compare numerators

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

Step 4: Write in order

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

Step 5: Convert back to original fractions

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

Decimal Verification: 0.667, 0.75, 0.833, 0.875, confirming the order

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

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

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 \(\frac{5}{7}\) and \(\frac{4}{5}\). They're both close to 1, but I don't know which is bigger. Cross multiplication gives me 25 and 28, so which one is larger?

Answer: Great question! When you cross multiply \(\frac{5}{7}\) and \(\frac{4}{5}\):

  • \(5 \times 5 = 25\) (numerator of first × denominator of second)
  • \(7 \times 4 = 28\) (denominator of first × numerator of second)
  • Since \(25 < 28\), then \(\frac{5}{7} < \frac{4}{5}\)

The key is: the cross product involving the numerator of the first fraction tells you about the first fraction. Since 25 < 28, the first fraction \(\frac{5}{7}\) is smaller than the second fraction \(\frac{4}{5}\).

You can verify: \(\frac{5}{7} ≈ 0.714\) and \(\frac{4}{5} = 0.8\), confirming \(0.714 < 0.8\).

Question: My child is struggling with comparing unlike fractions like \(\frac{3}{8}\) and \(\frac{2}{5}\). How can I help them understand this better?

Answer: For unlike fractions, use the cross multiplication method:

  • Cross multiply: \(3 \times 5 = 15\) and \(8 \times 2 = 16\)
  • Compare results: 15 vs 16
  • Since \(15 < 16\), then \(\frac{3}{8} < \frac{2}{5}\)

Visual aid: Draw two rectangles divided into 8 and 5 equal parts respectively. Shade 3 parts of the first and 2 parts of the second to see which is larger.

Memory trick: "Cross multiply, compare the answers - the bigger answer belongs to the bigger fraction!"

You can also find a common denominator: LCD of 8 and 5 is 40, so \(\frac{3}{8} = \frac{15}{40}\) and \(\frac{2}{5} = \frac{16}{40}\), confirming \(\frac{15}{40} < \frac{16}{40}\).

Question: When ordering unlike fractions like \(\frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{2}{3}\), should I always find the LCD of all denominators or are there better strategies?

Answer: Finding the LCD is often the most systematic approach for ordering multiple unlike fractions:

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

Alternative strategies include: pairwise comparison using cross multiplication, or converting to decimals. However, the LCD method ensures accuracy and works well when denominators aren't too large after factorization. For this specific set, the LCD method is quite efficient since the LCD (24) is manageable.