A) \(\frac{6}{8}\)
B) \(\frac{9}{12}\)
C) \(\frac{12}{16}\)
D) All of the above
Equivalent fractions: Fractions that represent the same part of a whole, e.g., \(\frac{a}{b} = \frac{na}{nb}\) where n is any non-zero number
- Use cross multiplication to verify equivalence
- Or simplify each option and compare
\(\frac{3}{4}\) vs \(\frac{6}{8}\)
Using cross multiplication: \(3 × 8 = 24\) and \(4 × 6 = 24\)
Since \(24 = 24\), these fractions are equivalent.
\(\frac{3}{4}\) vs \(\frac{9}{12}\)
Using cross multiplication: \(3 × 12 = 36\) and \(4 × 9 = 36\)
Since \(36 = 36\), these fractions are equivalent.
\(\frac{3}{4}\) vs \(\frac{12}{16}\)
Using cross multiplication: \(3 × 16 = 48\) and \(4 × 12 = 48\)
Since \(48 = 48\), these fractions are equivalent.
\(\frac{6}{8} = \frac{6÷2}{8÷2} = \frac{3}{4}\) ✓
\(\frac{9}{12} = \frac{9÷3}{12÷3} = \frac{3}{4}\) ✓
\(\frac{12}{16} = \frac{12÷4}{16÷4} = \frac{3}{4}\) ✓
All options are equivalent to \(\frac{3}{4}\) because they can all be simplified to \(\frac{3}{4}\) by dividing numerator and denominator by their greatest common divisor.
• Equivalence test: Use cross multiplication: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(ad = bc\)
• Simplification: Reduce fractions to lowest terms to compare easily
• Pattern recognition: Multiplying numerator and denominator by the same number creates equivalent fractions
A) \(\frac{5}{6}\)
B) \(\frac{7}{9}\)
C) They are equal
D) Cannot be determined
Common denominator: When fractions have the same denominator, the one with the larger numerator is greater
Denominators are 6 and 9. LCM(6,9) = 18
\(\frac{5}{6} = \frac{5×3}{6×3} = \frac{15}{18}\)
\(\frac{7}{9} = \frac{7×2}{9×2} = \frac{14}{18}\)
Since 15 > 14, we have \(\frac{15}{18} > \frac{14}{18}\)
A) \(\frac{5}{6}\) is larger because when converted to common denominators, \(\frac{5}{6} = \frac{15}{18}\) and \(\frac{7}{9} = \frac{14}{18}\), so \(\frac{15}{18} > \frac{14}{18}\).
• Common denominator rule: When denominators are equal, compare numerators
• LCM method: Find smallest common multiple of denominators
• Prime factorization: Use prime factorization to find LCM efficiently
A) \(\frac{8}{15} < \frac{5}{9}\)
B) \(\frac{8}{15} > \frac{5}{9}\)
C) \(\frac{8}{15} = \frac{5}{9}\)
D) Cannot be compared
Cross multiplication: For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a×d\) with \(b×c\)
For \(\frac{a}{b}\) vs \(\frac{c}{d}\): multiply \(a×d\) and \(b×c\)
\(\frac{8}{15}\) vs \(\frac{5}{9}\): \(8×9\) vs \(15×5\)
\(8×9 = 72\) and \(15×5 = 75\)
Since \(72 < 75\), we have \(\frac{8}{15} < \frac{5}{9}\)
LCD of 15 and 9 is 45: \(\frac{24}{45}\) vs \(\frac{25}{45}\) → \(\frac{24}{45} < \frac{25}{45}\) ✓
A) \(\frac{8}{15} < \frac{5}{9}\) because \(8×9 = 72\) and \(15×5 = 75\), and since \(72 < 75\), we have \(\frac{8}{15} < \frac{5}{9}\).
• Cross multiplication: If \(ad < bc\), then \(\frac{a}{b} < \frac{c}{d}\)
• Efficiency: Cross multiplication is faster than finding LCD for large denominators
• Verification: Always verify with an alternative method
Fraction: A number representing part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\)
Numerator: The top number indicating how many parts are taken
Denominator: The bottom number indicating total equal parts in the whole
Equivalent fractions: Different fractions that represent the same value
Proper fraction: A fraction where the numerator is smaller than the denominator
Improper fraction: A fraction where the numerator is greater than or equal to the denominator
Like fractions: Fractions with the same denominator
Unlike fractions: Fractions with different denominators
- Read the problem: Carefully identify the fractions to compare
- Choose method: Select the most efficient comparison method
- Apply technique: Execute the chosen method step-by-step
- Check answer: Verify using an alternative method
- Reflect: Consider which method worked best for this problem
• If denominators are equal: \(\frac{a}{c} > \frac{b}{c}\) when \(a > b\)
• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) when \(ad > bc\)
• If numerators are equal: \(\frac{a}{b} > \frac{a}{c}\) when \(b < c\)
• Equivalent fractions: \(\frac{a}{b} = \frac{ka}{kb}\) for any non-zero k
A) \(2\frac{3}{5}\)
B) \(2\frac{5}{8}\)
C) They are equal
D) Cannot be determined
Mixed number: A number combining a whole number and a proper fraction, e.g., \(a\frac{b}{c} = \frac{ac+b}{c}\)
Both mixed numbers have the same whole number part (2), so we compare the fractional parts.
\(\frac{3}{5}\) vs \(\frac{5}{8}\)
Using cross multiplication: \(3×8 = 24\) and \(5×5 = 25\)
Since \(24 < 25\), we have \(\frac{3}{5} < \frac{5}{8}\)
Since the whole numbers are equal and \(\frac{3}{5} < \frac{5}{8}\), we have \(2\frac{3}{5} < 2\frac{5}{8}\)
B) \(2\frac{5}{8}\) is larger because \(\frac{3}{5} < \frac{5}{8}\) (since \(3×8 = 24 < 25 = 5×5\)), so \(2\frac{3}{5} < 2\frac{5}{8}\).
• Same whole numbers: Compare fractional parts when whole numbers are equal
• Mixed to improper: Convert to improper fractions for easier comparison
• Pattern recognition: Larger fractional parts make the mixed number larger
A) \(\frac{9}{10} > \frac{7}{8} > \frac{5}{6} > \frac{4}{5}\)
B) \(\frac{4}{5} > \frac{5}{6} > \frac{7}{8} > \frac{9}{10}\)
C) \(\frac{7}{8} > \frac{9}{10} > \frac{5}{6} > \frac{4}{5}\)
D) \(\frac{5}{6} > \frac{4}{5} > \frac{9}{10} > \frac{7}{8}\)
Descending order: Arranging from largest to smallest value
\(\frac{4}{5} = 0.8\), \(\frac{7}{8} = 0.875\), \(\frac{5}{6} = 0.833\), \(\frac{9}{10} = 0.9\)
0.9 > 0.875 > 0.833 > 0.8
\(\frac{9}{10} > \frac{7}{8} > \frac{5}{6} > \frac{4}{5}\)
Check a few pairs: \(\frac{9}{10}\) vs \(\frac{7}{8}\): \(9×8 = 72 > 70 = 10×7\) → \(\frac{9}{10} > \frac{7}{8}\) ✓
A) The correct descending order is \(\frac{9}{10} > \frac{7}{8} > \frac{5}{6} > \frac{4}{5}\) because as the numerator and denominator approach equality, the value approaches 1.
• Pattern recognition: In \(\frac{n}{n+k}\), as n increases (with k constant), the value approaches 1
• Decimal conversion: Useful for comparing multiple fractions simultaneously
• Verification: Always check with an alternative method
Proper fraction: Numerator is smaller than denominator (\(\frac{a}{b}\) where \(a < b\))
Improper fraction: Numerator is greater than or equal to denominator (\(\frac{a}{b}\) where \(a ≥ b\))
Mixed number: Combination of whole number and proper fraction (\(a\frac{b}{c}\))
Like fractions: Fractions with the same denominator
Unlike fractions: Fractions with different denominators
Equivalent fractions: Fractions that represent the same value despite having different numerators and denominators
- Read the problem carefully: Identify what is being asked
- Identify fraction types: Determine if proper, improper, or mixed numbers
- Choose appropriate method: Common denominator, cross multiplication, or decimal conversion
- Apply the method: Follow specific steps for the chosen technique
- Verify the result: Use an alternative method to confirm accuracy
- Record the answer: Note the result and reflect on the process
• Common denominator: \(\frac{a}{b} = \frac{an}{bn}\) for any non-zero n
• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) if and only if \(ad > bc\)
• Mixed to improper: \(a\frac{b}{c} = \frac{ac+b}{c}\)
• Same numerator: \(\frac{a}{b} > \frac{a}{c}\) if \(b < c\)
• Same denominator: \(\frac{a}{c} > \frac{b}{c}\) if \(a > b\)
\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\)
Analysis: As the numerator and denominator increase proportionally while maintaining the relationship where numerator is one less than denominator, the value approaches 1.
- \(\frac{1}{2} = 0.5\) (50% of whole)
- \(\frac{2}{3} ≈ 0.667\) (66.7% of whole)
- \(\frac{3}{4} = 0.75\) (75% of whole)
- \(\frac{4}{5} = 0.8\) (80% of whole)
- \(\frac{5}{6} ≈ 0.833\) (83.3% of whole)