Using the Fundamental Theorem of Fractions

The Fundamental Theorem of Fractions:

a/b = ac/bc, where a and c are not equal to zero. The Fundamental Theorem of Fractions is used in
  • Reducing Fractions
  • Building Fractions

Reducing Fractions

Note if the numerator and denominator of a fraction have a common factor we can divide both numerator and denominator of the fraction by that common factor and reduce the fraction. Examples: Reduce each of the following:
  • 10/14 = [2(5)]/[2(7)] = 5/7 Divide both numerator and denominator by 2.
  • [3x]/[7x] = 3/7 Divide both numerator and denominator by x.
  • [12ab³ ]/[18a² b² ] = [(2² )(3)(a)(b³ )]/[(2)(3² )(a² )(b² )] = [2b]/[3a] Divide numerator and denominator by 2(3)(a)(b² ). If numerator and denominator of a fraction are polynomials we must factor numerator and denominator and then reduce.
  • [x² + 5x + 6]/[x² - 4] = [(x + 3)(x + 2)]/[(x + 2)(x - 2)] = (x + 3)/(x - 2) Divide numerator and denominator by x + 2.

Building Fractions

To build a fraction we multiply both numerator and denominator of the fraction by the same nonzero number to obtain an equivalent fraction. To write the fraction 3/4 as 6/8 we we multiply the numerator and demonimator by the building factor of 2. To find the building factor divide the desired denominator 8 by the original denominator 4 to obtain the building factor 2. Examples: Write each of the following fractions as an equivalent fraction with specified denominator.
  • 3/5 = ?/35 Building Factor: 35/5 = 7 3/5 = [(3)(7)]/[(5)(7)] = 21/35
  • [7a]/[5b] = ?/[10b² ] Building Factor: [10b² ]/[5b] = 2b [7a]/[5b] = [7a(2b)]/[5b(2b)] = [14ab]/[10b² ]
  • [x - 1]/[x + 5] = ?/[(x + 5)(x - 2)] Building Factor: [(x + 5)(x - 2)]/[x + 5] = (x - 2) [x - 1]/[x + 5] = [(x - 1)(x - 2)]/[(x + 5)(x - 2)]
  • 2/[x - 4] = ?/[x² + x - 20] To find the building factor first factor denominator. 2/[x - 4] = ?/[(x - 4)(x + 5)] Building Factor: [(x - 4)(x + 5)]/[x - 4] = (x + 5) 2/[x - 4] = [2(x + 5)]/[(x - 4)(x + 5)] = [2x + 10]/[x² + x - 20]

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