RATIO & PROPORTION

• Compounded Ratio of two ratios a/b and c/d is ac/bd, i.e., ac : bd.
  Duplicate ratio of a : b is a² : b²
  Triplicate ratio of a : b is a³ : b³
  Sub-duplicate ratio of a : b is a : b
  Sub-triplicate ratio of a : b is³a :³b
  Reciprocal ratio of a : b is b : a
• Proportion. Four (non-zero) quantities of the same kind a, b, c, d are in proportion,
  written as a : b :: c : d iff a/b = c/d
• The non-zero quantities of the same kind a, b, c, d, ... are in continued proportion iff
    a/b = b/c = c/d = ...
  In particular, a, b, c are in continued proportion iff a/b = c/d. In this case b is called the mean proportion; b = ac; c is called third proportional. If a, b, c, d are in proportion, then d is called fourth proportional.
• Invertendo. If a : b :: c : d then b : a :: d : c
  Alternendo. If a : b :: c : d then a : c :: b : d
  Componendo. If a : b :: c : d then (a +b) : b :: (c +d) : d
  Dividendo. If a : b :: c : d then (a -b) : b :: (c -d) : d
  Componendo and dividendo.
     If a : b :: c : d then (a +b) : (a -b) :: (c +d) : (c -d)
      i.e., a/b = c/d => (a +b)/(a - b) = (c +d)/(c +d)
• If a/b = c/d = e/f = ..., then each ratio = (a +c +e +...)/(b +d +f +...)