A) \(\frac{4}{10}\)
B) \(\frac{6}{15}\)
C) \(\frac{8}{20}\)
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 to original
\(\frac{2}{5}\) vs \(\frac{4}{10}\)
Using cross multiplication: \(2 × 10 = 20\) and \(5 × 4 = 20\)
Since \(20 = 20\), these fractions are equivalent.
\(\frac{2}{5}\) vs \(\frac{6}{15}\)
Using cross multiplication: \(2 × 15 = 30\) and \(5 × 6 = 30\)
Since \(30 = 30\), these fractions are equivalent.
\(\frac{2}{5}\) vs \(\frac{8}{20}\)
Using cross multiplication: \(2 × 20 = 40\) and \(5 × 8 = 40\)
Since \(40 = 40\), these fractions are equivalent.
\(\frac{4}{10} = \frac{4÷2}{10÷2} = \frac{2}{5}\) ✓
\(\frac{6}{15} = \frac{6÷3}{15÷3} = \frac{2}{5}\) ✓
\(\frac{8}{20} = \frac{8÷4}{20÷4} = \frac{2}{5}\) ✓
All options are equivalent to \(\frac{2}{5}\) because they can all be reduced to \(\frac{2}{5}\) by dividing numerator and denominator by their greatest common factor.
• 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) \(\frac{3}{7}\)
B) \(\frac{4}{9}\)
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 7 and 9. LCM(7,9) = 63
\(\frac{3}{7} = \frac{3×9}{7×9} = \frac{27}{63}\)
\(\frac{4}{9} = \frac{4×7}{9×7} = \frac{28}{63}\)
Since 27 < 28, we have \(\frac{27}{63} < \frac{28}{63}\)
B) \(\frac{4}{9}\) is larger because when converted to common denominators, \(\frac{3}{7} = \frac{27}{63}\) and \(\frac{4}{9} = \frac{28}{63}\), so \(\frac{27}{63} < \frac{28}{63}\).
• Common denominator rule: When denominators are equal, compare numerators
• LCM method: Find smallest common multiple of denominators
A) \(\frac{5}{8} < \frac{7}{12}\)
B) \(\frac{5}{8} > \frac{7}{12}\)
C) \(\frac{5}{8} = \frac{7}{12}\)
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 \(b×c\)
\(\frac{5}{8}\) vs \(\frac{7}{12}\): \(5×12\) vs \(8×7\)
\(5×12 = 60\) and \(8×7 = 56\)
Since \(60 > 56\), we have \(\frac{5}{8} > \frac{7}{12}\)
LCD of 8 and 12 is 24: \(\frac{15}{24}\) vs \(\frac{14}{24}\) → \(\frac{15}{24} > \frac{14}{24}\) ✓
B) \(\frac{5}{8} > \frac{7}{12}\) because \(5×12 = 60\) and \(8×7 = 56\), and since \(60 > 56\), we have \(\frac{5}{8} > \frac{7}{12}\).
• 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 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
• 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) \(1\frac{3}{4}\)
B) \(1\frac{5}{6}\)
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 (1), so we compare the fractional parts.
\(\frac{3}{4}\) vs \(\frac{5}{6}\)
Using cross multiplication: \(3×6 = 18\) and \(4×5 = 20\)
Since \(18 < 20\), we have \(\frac{3}{4} < \frac{5}{6}\)
Since the whole numbers are equal and \(\frac{3}{4} < \frac{5}{6}\), we have \(1\frac{3}{4} < 1\frac{5}{6}\)
B) \(1\frac{5}{6}\) is larger because \(\frac{3}{4} < \frac{5}{6}\) (since \(3×6 = 18 < 20 = 4×5\)), so \(1\frac{3}{4} < 1\frac{5}{6}\).
• Same whole numbers: Compare fractional parts when whole numbers are equal
• Mixed to improper: Convert to improper fractions for easier comparison
A) \(\frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6}\)
B) \(\frac{5}{6} < \frac{4}{5} < \frac{3}{4} < \frac{2}{3}\)
C) \(\frac{2}{3} < \frac{4}{5} < \frac{3}{4} < \frac{5}{6}\)
D) \(\frac{3}{4} < \frac{2}{3} < \frac{5}{6} < \frac{4}{5}\)
Ascending order: Arranging from smallest to largest value
\(\frac{2}{3} = 0.667\), \(\frac{3}{4} = 0.75\), \(\frac{4}{5} = 0.8\), \(\frac{5}{6} = 0.833\)
0.667 < 0.75 < 0.8 < 0.833
\(\frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6}\)
Check a few pairs: \(\frac{2}{3}\) vs \(\frac{3}{4}\): \(2×4 = 8 < 9 = 3×3\) → \(\frac{2}{3} < \frac{3}{4}\) ✓
A) The correct ascending order is \(\frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6}\) because as the numerator and denominator increase proportionally while maintaining the relationship where numerator is one less than denominator, the value approaches 1.
• 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 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)