Why does the butterfly method work in math?

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What the butterfly method does is to make the fractions equivalent fractions with a new denominator (bottom number) the product of the original denominators (bottom numbers). Then to compare/add/subtract the fractions, the numerators (top numerators) can be compared/added/subtracted.

Let the two fractions be a/b and c/d, then the new denominator is bd (b times d). Then the equivalent fractions are:
  • a/b ⇒ (a x d)/(b x d) ⇒ ad/bd
  • c/d ⇒ (b x c)/(b x d) ⇒ bc/bd
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