Yes, \(\frac{4}{6}\) and \(\frac{2}{3}\) are equivalent fractions.
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 the first fraction and compare
For \(\frac{a}{b}\) and \(\frac{c}{d}\), check if \(ad = bc\)
\(\frac{4}{6}\) vs \(\frac{2}{3}\): \(4 × 3 = 12\) and \(6 × 2 = 12\)
Since \(12 = 12\), the fractions are equivalent
\(\frac{4}{6} = \frac{4÷2}{6÷2} = \frac{2}{3}\) ✓
• 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
\(\frac{3}{8} > \frac{2}{5}\)
Common denominator: When fractions have the same denominator, the one with the larger numerator is greater
Denominators are 8 and 5. LCM(8,5) = 40
\(\frac{3}{8} = \frac{3×5}{8×5} = \frac{15}{40}\)
\(\frac{2}{5} = \frac{2×8}{5×8} = \frac{16}{40}\)
Since 15 < 16, we have \(\frac{15}{40} < \frac{16}{40}\)
• Common denominator rule: When denominators are equal, compare numerators
• LCM method: Find smallest common multiple of denominators
\(\frac{7}{9} > \frac{5}{7}\)
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 \(b×c\)
\(\frac{7}{9}\) vs \(\frac{5}{7}\): \(7×7\) vs \(9×5\)
\(7×7 = 49\) and \(9×5 = 45\)
Since \(49 > 45\), we have \(\frac{7}{9} > \frac{5}{7}\)
LCD of 9 and 7 is 63: \(\frac{49}{63}\) vs \(\frac{45}{63}\) → \(\frac{49}{63} > \frac{45}{63}\) ✓
• Cross multiplication: If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\)
• Efficiency: Cross multiplication is faster than finding LCD for large denominators
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 question: Carefully identify the fractions to compare
- Choose method: Select the most efficient comparison method
- Apply technique: Execute the chosen method step-by-step
- Provide answer: State the answer clearly with explanation
- Verify: Check the answer using an alternative method
• 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
\(1\frac{2}{3} > 1\frac{3}{5}\)
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 (1), so we compare the fractional parts.
\(\frac{2}{3}\) vs \(\frac{3}{5}\)
Using cross multiplication: \(2×5 = 10\) and \(3×3 = 9\)
Since \(10 > 9\), we have \(\frac{2}{3} > \frac{3}{5}\)
Since the whole numbers are equal and \(\frac{2}{3} > \frac{3}{5}\), we have \(1\frac{2}{3} > 1\frac{3}{5}\)
• Same whole numbers: Compare fractional parts when whole numbers are equal
• Mixed to improper: Convert to improper fractions for easier comparison
\(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{4}{5}\)
Ascending order: Arranging from smallest to largest value
\(\frac{1}{2} = 0.5\), \(\frac{2}{3} = 0.667\), \(\frac{3}{4} = 0.75\), \(\frac{4}{5} = 0.8\)
0.5 < 0.667 < 0.75 < 0.8
\(\frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{4}{5}\)
Check a few pairs: \(\frac{1}{2}\) vs \(\frac{2}{3}\): \(1×3 = 3 < 4 = 2×2\) → \(\frac{1}{2} < \frac{2}{3}\) ✓
• Pattern recognition: In \(\frac{n}{n+1}\), as n increases, the value approaches 1
• Decimal conversion: Useful for comparing multiple fractions simultaneously
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 question 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
- Provide clear answer: Explain the reasoning and show all steps
• 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)