Ratio and Proportion — Core Techniques

Entry	Year	Description	Impact
Invertendo	Classical	If a/b = c/d, then the inverse ratios are also equal, i.e., b/a = d/c. This property allows for the flipping of ratios while maintaining proportionality.	Useful when dealing with reciprocal relationships or when the unknown term is in the denominator. Simplifies algebraic manipulation in certain problem types.
Alternendo	Classical	If a/b = c/d, then a/c = b/d. This property allows for the interchange of the means (or extremes) while maintaining proportionality.	Highly effective for rearranging terms to group similar variables or constants, making it easier to solve for unknowns or simplify complex expressions in multi-ratio problems.
Componendo	Classical	If a/b = c/d, then (a+b)/b = (c+d)/d. This property adds the consequent to the antecedent in each ratio while keeping the consequent in the denominator.	Simplifies expressions where sums of terms are involved. Often used in conjunction with Dividendo to quickly solve equations with (x+y)/(x-y) type structures.
Dividendo	Classical	If a/b = c/d, then (a-b)/b = (c-d)/d. This property subtracts the consequent from the antecedent in each ratio.	Useful for problems involving differences of terms. When combined with Componendo, it forms a powerful tool (Componendo and Dividendo) for algebraic simplification.