1.5 Visualize Fractions

Fractions are a key part of math, showing how parts relate to wholes. They help us split things up, compare amounts, and do calculations. Understanding fractions is crucial for more advanced math and real-world problem-solving.

This section covers the basics of fractions, including how to simplify them and do math with them. We'll learn about equivalent fractions, multiplying and dividing fractions, and how to convert words into fraction form. These skills are essential for working with fractions confidently.

Fractions

Equivalent fractions

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• Represent the same value with different numerators and denominators
  • Multiplying or dividing both numerator and denominator by the same non-zero number creates an equivalent fraction ($\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$)
• Find an equivalent fraction by multiplying or dividing numerator and denominator by the same non-zero number
  • To find an equivalent fraction for $\frac{3}{5}$, multiply both numerator and denominator by 2: $\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$
• Equivalent fractions have the same value when simplified ($\frac{2}{4} = \frac{1}{2}$)
• Useful for comparing fractions with different denominators ($\frac{1}{2}$ and $\frac{2}{4}$ are equivalent, so they are equal)

Simplifying fractions

• Fraction is in simplest form (lowest terms) when numerator and denominator have no common factors other than 1
  • $\frac{6}{8}$ is not in simplest form because 6 and 8 have a common factor of 2
• Reduce fraction to simplest form by dividing numerator and denominator by their greatest common factor (GCF)
  • To reduce $\frac{6}{8}$, divide both 6 and 8 by their GCF of 2: $\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$
• Simplifying fractions makes them easier to compare and work with ($\frac{6}{8}$ and $\frac{3}{4}$ represent the same value, but $\frac{3}{4}$ is simpler)
• Always simplify the result of fraction multiplication and division

Multiplication and division of fractions

• Multiply fractions by multiplying numerators and multiplying denominators
  • $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$
• Divide fractions by multiplying the first fraction by the reciprocal of the second fraction
  • Reciprocal of a fraction is obtained by flipping numerator and denominator ($\frac{4}{5}$ has reciprocal $\frac{5}{4}$)
  • $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6}$
• Simplify the result of multiplication and division by reducing to lowest terms
• Remember to multiply by the reciprocal when dividing fractions

Fraction bars in expressions

• Act as grouping symbols, indicating expression above bar should be divided by expression below bar
  • $\frac{a+b}{c}$ means $(a+b) \div c$
• Simplify expressions with fraction bars by performing operations in numerator and denominator separately, then dividing results
  • $\frac{2+4}{3+1} = \frac{6}{4} = \frac{3}{2}$
• Fraction bars have higher precedence than addition and subtraction, but lower precedence than exponents
• Treat numerator and denominator as separate expressions to simplify

Verbal to fractional conversions

• Identify numerator and denominator based on verbal description
  • Words like "of," "out of," and "per" often indicate division and separate numerator and denominator ("Three out of five" can be written as $\frac{3}{5}$)
• Write fraction using identified numerator and denominator (fraction notation)
  • "The ratio of a to b" can be written as $\frac{a}{b}$
• Pay attention to the order of the words to determine numerator and denominator
• Look for key phrases that suggest a fraction, such as "out of," "divided by," and "ratio of"

Fractions and their representations

• Fractions express a part-whole relationship, where the numerator represents a part of the whole (denominator)
• Rational numbers can be expressed as fractions, where the numerator is divided by the denominator
• Fractions can be represented in different forms:
  • Decimal representation: dividing the numerator by the denominator (e.g., $\frac{1}{2} = 0.5$)
  • Percentage: multiplying the decimal form by 100 (e.g., $\frac{1}{2} = 0.5 = 50\%$)