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Intermediate Algebra: Simplifying Fractions

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이 집합의 용어 (20)

  • What does it mean to simplify a fraction?
    To rewrite the fraction in its simplest form by dividing the numerator and denominator by their greatest common divisor.
  • How do you find the greatest common divisor (GCD) of two numbers?
    List the factors of each number and choose the largest factor they have in common.
  • Simplify the fraction \(\frac{12}{16}\).
    Divide numerator and denominator by 4 (GCD): \(\frac{3}{4}\).
  • What is the first step in simplifying a fraction?
    Find the greatest common divisor (GCD) of the numerator and denominator.
  • Why is it important to simplify fractions?
    Simplified fractions are easier to understand, compare, and use in calculations.
  • Simplify \(\frac{45}{60}\).
    GCD is 15, so \(\frac{45 \div 15}{60 \div 15} = \frac{3}{4}\).
  • Can a fraction be simplified if the numerator and denominator are relatively prime?
    No, if the GCD is 1, the fraction is already in simplest form.
  • Simplify \(\frac{7}{13}\).
    Since 7 and 13 have no common factors other than 1, the fraction is already simplified.
  • What is the role of prime factorization in simplifying fractions?
    Prime factorization helps identify common factors to find the GCD.
  • Simplify \(\frac{24}{36}\) using prime factorization.
    24 = 2^3 * 3, 36 = 2^2 * 3^2; common factors 2^2 * 3 = 12; so \(\frac{24 \div 12}{36 \div 12} = \frac{2}{3}\).
  • What happens if you multiply numerator and denominator by the same number?
    The value of the fraction does not change, but it is not simplified.
  • How do you simplify a fraction with negative signs?
    Move the negative sign to the numerator or in front of the fraction and simplify the absolute values.
  • Simplify \(\frac{-18}{24}\).
    GCD is 6, so \(\frac{-18 \div 6}{24 \div 6} = \frac{-3}{4}\).
  • What is an improper fraction?
    A fraction where the numerator is greater than or equal to the denominator.
  • Can improper fractions be simplified?
    Yes, they can be simplified just like proper fractions by dividing numerator and denominator by their GCD.
  • Simplify \(\frac{50}{20}\).
    GCD is 10, so \(\frac{50 \div 10}{20 \div 10} = \frac{5}{2}\).
  • What is a mixed number?
    A number consisting of a whole number and a proper fraction.
  • How do you convert an improper fraction to a mixed number?
    Divide numerator by denominator; the quotient is the whole number, and the remainder over denominator is the fraction.
  • Convert \(\frac{11}{4}\) to a mixed number.
    11 ÷ 4 = 2 remainder 3, so mixed number is \(2 \frac{3}{4}\).
  • Why should fractions be simplified before performing operations?
    Simplifying reduces complexity and helps avoid errors in calculations.