\(\frac{2}{3}\) by multiplying both numerator and denominator by 2.
Generating equivalent fractions: Creating new fractions that have the same value as the original by multiplying both numerator and denominator by the same non-zero number. For example, \(\frac{2}{3}\) and \(\frac{4}{6}\) both represent the same portion of a whole.
- Take the original fraction (numerator and denominator)
- Choose a non-zero number to multiply both parts by
- Multiply both numerator and denominator by this number
- The result is an equivalent fraction
We start with \(\frac{2}{3}\) where numerator = 2 and denominator = 3
We multiply both numerator and denominator by 2
New numerator: \(2 \times 2 = 4\)
New denominator: \(3 \times 2 = 6\)
\(\frac{2}{3} = \frac{4}{6}\)
\(\frac{2}{3} = \frac{4}{6}\)
• Multiplication Property: Multiply both numerator and denominator by the same number
• Identity Property: Multiplying by \(\frac{2}{2}\) is the same as multiplying by 1
• Equivalence: Both fractions represent the same part of a whole
\(\frac{3}{5} = \frac{?}{10}\)
Finding missing values in equivalent fractions: Determining the unknown numerator or denominator that makes two fractions equal. This involves finding the factor used to convert one fraction to its equivalent form.
We have 5 and 10 as denominators. We need to find what multiplied by 5 gives 10.
\(10 \div 5 = 2\), so we multiply denominator by 2
New numerator: \(3 \times 2 = 6\)
\(\frac{3}{5} = \frac{6}{10}\) - both represent the same value
The missing number is 6, so \(\frac{3}{5} = \frac{6}{10}\)
• Division Method: Divide new denominator by old denominator to find multiplier
• Consistency: Apply the same operation to both numerator and denominator
• Cross Multiplication Check: \(3 \times 10 = 5 \times 6 = 30\) ✓
\(\frac{2}{4}\)
Multiple equivalent fractions: A single fraction can have many equivalent forms by multiplying numerator and denominator by different numbers. \(\frac{2}{4}\) can be simplified to \(\frac{1}{2}\) and then expanded to other forms.
We have \(\frac{2}{4}\) which represents two fourths of a whole
\(\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}\)
\(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\)
\(\frac{1}{2} \times \frac{4}{4} = \frac{4}{8}\)
Three equivalent fractions for \(\frac{2}{4}\) are: \(\frac{1}{2}, \frac{3}{6}, \frac{4}{8}\)
• Infinite Equivalents: A fraction has infinitely many equivalent forms
• Multiplication by 1: Multiplying by \(\frac{n}{n}\) doesn't change the value
• Pattern Recognition: Numerator and denominator increase proportionally
Numerator: The top number in a fraction representing the number of parts taken
Denominator: The bottom number in a fraction representing the total number of equal parts
Equivalent fractions: Fractions that represent the same value but have different numerators and denominators
- Identify the original fraction: Recognize numerator and denominator
- Choose your multiplier: Decide by what number to multiply both parts
- Apply multiplication: Multiply both numerator and denominator
- Verify equivalence: Check using cross multiplication
• Generating equivalent fractions: \(\frac{a}{b} = \frac{a \times n}{b \times n}\)
• Cross multiplication test: If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\)
• Simplification: \(\frac{a}{b} = \frac{a \div d}{b \div d}\) where d is a common divisor
Word problems with equivalent fractions: Real-world situations that require comparing fractional amounts to determine if they are equal, using the concept of equivalent fractions.
Alex: \(\frac{3}{6}\) of pizza, Sam: \(\frac{6}{12}\) of pizza
Alex: \(\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}\)
Sam: \(\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}\)
Both equal \(\frac{1}{2}\), so they ate the same amount
\(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\) and \(\frac{1}{2} \times \frac{6}{6} = \frac{6}{12}\)
Yes, Alex and Sam ate the same amount of pizza. \(\frac{3}{6} = \frac{6}{12} = \frac{1}{2}\) of the pizza.
• Simplification: Divide both numerator and denominator by their GCD
• Comparison: Simplified equivalent fractions are identical
• Real-world application: Equivalent fractions represent equal parts of a whole
\(\frac{2}{5} = \frac{4}{?} = \frac{?}{20}\)
Equivalent fraction chains: Problems requiring multiple steps to find missing values in a sequence of equivalent fractions using different multipliers.
Find multiplier: \(4 \div 2 = 2\)
Missing denominator: \(5 \times 2 = 10\)
So \(\frac{2}{5} = \frac{4}{10}\)
Find multiplier: \(20 \div 5 = 4\)
Missing numerator: \(2 \times 4 = 8\)
So \(\frac{2}{5} = \frac{8}{20}\)
\(\frac{2}{5} = \frac{4}{10} = \frac{8}{20}\)
Cross check: \(2 \times 10 = 5 \times 4 = 20\) ✓
\(2 \times 20 = 5 \times 8 = 40\) ✓
\(\frac{2}{5} = \frac{4}{10} = \frac{8}{20}\)
The missing denominator is 10 and the missing numerator is 8.
• Sequential solving: Solve one part at a time
• Multiplication factor: Determine what factor transforms one denominator to another
• Cross verification: Always verify using cross multiplication
Generating equivalent fractions: The process of creating new fractions that have the same value as the original by multiplying both numerator and denominator by the same non-zero number.
Numerator: The top number in a fraction indicating how many parts are taken.
Denominator: The bottom number in a fraction indicating how many equal parts the whole is divided into.
Multiplier: The number used to multiply both numerator and denominator to create an equivalent fraction.
Equivalent fractions: Different fractions that represent the same portion of a whole.
- Basic Generation: Multiply both numerator and denominator by the same number
- Missing Value: Divide known denominators to find the multiplier, then apply to the other part
- Verification: Use cross multiplication to confirm equivalence
- Simplification: Divide both parts by their greatest common divisor to get simplest form
- Comparison: Convert fractions to equivalent forms with common denominators to compare
Basic:
- \(\frac{1}{2} = \frac{2}{4}\) (multiplied by 2)
- \(\frac{1}{3} = \frac{3}{9}\) (multiplied by 3)
- \(\frac{2}{5} = \frac{4}{10}\) (multiplied by 2)
Intermediate:
- \(\frac{3}{4} = \frac{6}{8} = \frac{9}{12}\) (multiplied by 2 and 3)
- \(\frac{2}{3} = \frac{4}{6} = \frac{6}{9}\) (multiplied by 2 and 3)
Advanced:
- \(\frac{4}{6} = \frac{2}{3}\) (simplified by dividing by 2)
- \(\frac{8}{12} = \frac{2}{3}\) (simplified by dividing by 4)
- \(\frac{1}{2} = \frac{3}{6} = \frac{5}{10} = \frac{7}{14}\) (multiplied by 3, 5, and 7)
• Multiplication Property: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0
• Cross Multiplication: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a \times d = b \times c\)
• Identity Principle: Multiplying by \(\frac{n}{n}\) is equivalent to multiplying by 1
• Consistency: Operations must be applied to both numerator and denominator equally
• Simplification: Dividing by the GCD gives the simplest equivalent fraction
• "What you do to the top, you must do to the bottom"
• "A fraction over itself equals 1: \(\frac{2}{2} = 1\), \(\frac{3}{3} = 1\), etc."
• "Cross multiply to check: diagonal products should be equal"
• "Fractions are equivalent if they represent the same part of a whole"
• "Always multiply both numerator and denominator by the same number"
Questions & Answers
Question: I don't understand why \(\frac{2}{3}\) and \(\frac{4}{6}\) are the same amount. They look completely different!
Answer: Great question! Let me explain with a visual analogy:
- Imagine you have a pizza cut into 3 equal slices, and you eat 2 of them. That's \(\frac{2}{3}\) of the pizza.
- Now imagine the same pizza cut into 6 equal slices, and you eat 4 of them. That's \(\frac{4}{6}\) of the pizza.
- In both cases, you've eaten the same amount of pizza - just in different-sized pieces!
Mathematically: \(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\). You're essentially multiplying by \(\frac{2}{2}\), which equals 1, so the value doesn't change.
Cross check: \(2 \times 6 = 12\) and \(3 \times 4 = 12\). Since both products are equal, the fractions are equivalent!
Question: When generating equivalent fractions, what numbers can I use to multiply by?
Answer: You can multiply by any non-zero number:
- Small numbers (2, 3, 4): These are easiest for beginners
- Larger numbers (5, 6, 7, etc.): These create more complex equivalents
- Never use 0: This would make the denominator zero, which is undefined
Examples:
- \(\frac{1}{2} \times \frac{2}{2} = \frac{2}{4}\) (simple)
- \(\frac{1}{2} \times \frac{5}{5} = \frac{5}{10}\) (more complex)
- \(\frac{1}{2} \times \frac{10}{10} = \frac{10}{20}\) (even more complex)
Start with small numbers to build confidence, then try larger ones!
Question: How can I quickly check if two fractions are equivalent without doing complex calculations?
Answer: The fastest method is the cross multiplication shortcut:
- To check if \(\frac{a}{b} = \frac{c}{d}\), multiply diagonally: \(a \times d\) and \(b \times c\)
- If the products are equal, the fractions are equivalent
- If the products are different, the fractions are not equivalent
Example: Check if \(\frac{2}{3} = \frac{4}{6}\)
Cross multiply: \(2 \times 6 = 12\) and \(3 \times 4 = 12\)
Since both products equal 12, the fractions are equivalent!
Alternative quick check: Try to reduce both fractions to simplest form. If they become identical, they're equivalent.