Componendo and dividendo are mathematical techniques used to simplify and solve proportions involving fractions or rational functions. The componendo rule states that if a : b = c : d, then (a + b) : b = (c + d) : d. The dividendo rule states that if a : b = c : d, then (a - b) : b = (c - d) : d. The componendo and dividendo rule states that if a : b = c : d, then (a + b) : (a - b) = (c + d) : (c - d) . These rules are based on direct proportion and can be used to reduce the number of operations and the complexity of the ratios.

The componendo and dividendo rules are particularly useful when dealing with equations involving fractions or rational functions in mathematical Olympiads, algebra, and other areas of mathematics. They allow for a quick way to perform calculations and reduce the amount of expansions needed. The key properties of componendo and dividendo are that they can simplify and solve proportions, and that they involve adding or subtracting the numerator and denominator of the given ratios.