Simplifying fractions: The process of reducing a fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).
- Find the factors of the numerator and denominator
- Identify the greatest common factor (GCF)
- Divide both numerator and denominator by the GCF
- Verify the result using cross multiplication
Factors of 15: 1, 3, 5, 15
Factors of 20: 1, 2, 4, 5, 10, 20
Common factors: 1, 5
Greatest common factor: 5
Numerator: \(15 \div 5 = 3\)
Denominator: \(20 \div 5 = 4\)
\(\frac{15}{20} = \frac{3}{4}\)
\(15 \times 4 = 60\) and \(20 \times 3 = 60\)
Since both products are equal, the fractions are equivalent
\(\frac{15}{20} = \frac{3}{4}\)
• Division Property: Divide both numerator and denominator by the same non-zero number
• Greatest Common Factor: The largest number that divides both parts evenly
• Simplified Form: When numerator and denominator have no common factors other than 1
Common factors: Numbers that divide both the numerator and denominator evenly, allowing us to create equivalent fractions through division.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
Greatest common factor: 12
Numerator: \(24 \div 12 = 2\)
Denominator: \(36 \div 12 = 3\)
\(\frac{24}{36} = \frac{2}{3}\)
Cross multiplication: \(24 \times 3 = 72\) and \(36 \times 2 = 72\)
Since both products are equal, the fractions are equivalent
\(\frac{24}{36} = \frac{2}{3}\)
• Common Factor Division: Both numerator and denominator must be divisible by the same number
• Simplification Process: Repeatedly divide by common factors until no more exist
• Verification: Use cross multiplication to confirm equivalence
Simplest form: A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. This is achieved through repeated division by common factors.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Common factors: 1, 2, 3, 4, 6, 8, 12, 24
Greatest common factor: 24
\(\frac{48}{72} = \frac{48 \div 2}{72 \div 2} = \frac{24}{36}\)
\(\frac{24}{36} = \frac{24 \div 2}{36 \div 2} = \frac{12}{18}\)
\(\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}\)
Factors of 2: 1, 2
Factors of 3: 1, 3
Only common factor is 1, so \(\frac{2}{3}\) is in simplest form
Cross multiplication: \(48 \times 3 = 144\) and \(72 \times 2 = 144\) ✓
\(\frac{48}{72} = \frac{2}{3}\) in simplest form
• Repeated Division: Continue dividing until no common factors remain
• Simplest Form Criterion: When GCD of numerator and denominator is 1
• Efficiency: Divide by GCF directly for fastest simplification
Equivalent fractions: Fractions that represent the same value or the same portion of a whole.
Greatest Common Factor (GCF): The largest number that divides two or more numbers evenly.
Simplest form: A fraction where the numerator and denominator have no common factors other than 1.
- Identify the original fraction
- Find common factors of numerator and denominator
- Select the GCF for most efficient simplification
- Divide both parts by the chosen factor
- Verify equivalence using cross multiplication
• Basic division: \(\frac{a}{b} = \frac{a \div n}{b \div n}\) where n ≠ 0
• Simplification: \(\frac{a}{b} = \frac{a \div \text{GCF}(a,b)}{b \div \text{GCF}(a,b)}\)
• Verification: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a \times d = b \times c\)
Divisibility rules: Quick methods to determine if one number divides another evenly. These help identify common factors quickly.
54: 5+4=9 → divisible by 9
81: 8+1=9 → divisible by 9
\(\frac{54}{81} = \frac{54 \div 9}{81 \div 9} = \frac{6}{9}\)
6: 6 is divisible by 3
9: 9 is divisible by 3
\(\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}\)
Factors of 2: 1, 2
Factors of 3: 1, 3
Only common factor is 1 → simplest form
Cross multiplication: \(54 \times 3 = 162\) and \(81 \times 2 = 162\) ✓
\(\frac{54}{81} = \frac{2}{3}\) in simplest form
• Divisibility Rules: Sum of digits divisible by 3 → divisible by 3; sum of digits divisible by 9 → divisible by 9
• Systematic Simplification: Apply divisibility rules to identify common factors quickly
• Efficiency: Divide by largest possible common factor at each step
Word problem translation: Convert real-world situations into mathematical fractions, then simplify using the division method.
21 students passed out of 28 total students
This is \(\frac{21}{28}\)
Factors of 21: 1, 3, 7, 21
Factors of 28: 1, 2, 4, 7, 14, 28
Common factors: 1, 7
Greatest common factor: 7
Numerator: \(21 \div 7 = 3\)
Denominator: \(28 \div 7 = 4\)
\(\frac{21}{28} = \frac{3}{4}\)
\(\frac{3}{4}\) of the students passed the test, which is the same as 21 out of 28 students
\(\frac{3}{4}\) of the students passed the test in simplest form
• Problem Translation: Convert real-world ratios to mathematical fractions
• Application of Division: Use division method to simplify word problem results
• Contextual Interpretation: Understand the meaning of the simplified fraction
Equivalent fractions: Different fractions that represent the same value or the same portion of a whole. They occupy the same position on a number line.
Division method: A technique for generating equivalent fractions by dividing both the numerator and denominator by the same non-zero number.
Greatest Common Factor (GCF): The largest number that divides two or more numbers evenly without remainder.
Simplest form: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.
- Identify the original fraction you want to simplify
- Find the factors of both the numerator and denominator
- Determine the GCF of the numerator and denominator
- Divide both parts of the fraction by the GCF
- Verify equivalence using cross multiplication
• Division Property: \(\frac{a}{b} = \frac{a \div n}{b \div n}\) where n ≠ 0
• Simplification Formula: \(\frac{a}{b} = \frac{a \div \text{GCF}(a,b)}{b \div \text{GCF}(a,b)}\)
• Equivalence Test: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a \times d = b \times c\)
• Simplest Form Test: A fraction is in simplest form when \(\text{GCD}(\text{numerator}, \text{denominator}) = 1\)
Simplify \(\frac{8}{12}\):
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
GCF = 4
\(\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\)
Simplify \(\frac{45}{75}\):
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 75: 1, 3, 5, 15, 25, 75
GCF = 15
\(\frac{45}{75} = \frac{45 \div 15}{75 \div 15} = \frac{3}{5}\)
- Division maintains fraction value while changing appearance
- Always divide both numerator and denominator by the same number
- Use GCF for most efficient simplification
- Verify results with cross multiplication
- Simplification makes fractions easier to work with
Simplify the following fractions using the division method:
- \(\frac{12}{18}\)
- \(\frac{20}{30}\)
- \(\frac{16}{24}\)
- \(\frac{28}{42}\)
- \(\frac{35}{50}\)
Solve these word problems:
- A pizza was cut into 16 slices. 12 slices were eaten. What fraction of the pizza was eaten? Simplify your answer.
- In a bag of 24 marbles, 18 are blue. What fraction of the marbles are blue? Simplify your answer.
- A recipe calls for 8 cups of flour out of 12 total cups of ingredients. What fraction of the ingredients is flour? Simplify your answer.
Practice Problem 1:
- 1. \(\frac{12}{18} = \frac{2}{3}\)
- 2. \(\frac{20}{30} = \frac{2}{3}\)
- 3. \(\frac{16}{24} = \frac{2}{3}\)
- 4. \(\frac{28}{42} = \frac{2}{3}\)
- 5. \(\frac{35}{50} = \frac{7}{10}\)
Practice Problem 2:
- 1. \(\frac{12}{16} = \frac{3}{4}\) of the pizza was eaten
- 2. \(\frac{18}{24} = \frac{3}{4}\) of the marbles are blue
- 3. \(\frac{8}{12} = \frac{2}{3}\) of the ingredients is flour