4.1 The Rule and Its Derivation

If a/b = c/d, then:

Componendo: (a + b)/b = (c + d)/d Dividendo: (a − b)/b = (c − d)/d Componendo-Dividendo: (a + b)/(a − b) = (c + d)/(c − d)

Derivation: Start with a/b = c/d. Add 1 to both sides: a/b + 1 = c/d + 1 → (a+b)/b = (c+d)/d … (Componendo) Subtract 1 from both sides: a/b − 1 = c/d − 1 → (a−b)/b = (c−d)/d … (Dividendo) Now divide the componendo result by the dividendo result: [(a+b)/b] ÷ [(a−b)/b] = [(c+d)/d] ÷ [(c−d)/d] The b’s and d’s cancel out, giving (a+b)/(a−b) = (c+d)/(c−d).

This rule is extremely useful for solving equations where a variable is trapped inside sums and differences of fractions — it avoids messy cross-multiplication and quadratic expansion.