Componendo Dividendo

Componendo and dividendo is a theorem on proportions that allows for a quick way to perform calculations and reduce the number of expansions needed. It is particularly useful when dealing with equations involving fractions or rational functions in mathematical Olympiads, especially when you see fractions.

The theorem is represented by the following formula: a/(b+c) = (a/b) + (a/c). The theorem is named for its two Latin terms: “componendo,” which means “by combining,” and “dividendo,” which means “by dividing.” The theorem can be applied to any type of number, including fractions, decimals, and negative numbers.

Componendo Dividendo Rule Proof

So, it can be expressed as a : b = c : d and (a + b ) : (a – b) = (c + d) : (c – d). It is true that if the ratio of a to b = c to d, then the ratio of (a + b) to (a – b) is the same as (c + d) to (c – d). This is termed the componendo and dividendo rule.

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Question - If (a / b) = (16 / 3), then find the value of (a – b) / (a + b).

Answer:

If (a / b) = (c / d)

 then [(a + b) / (a – b)] = [(c + d) / (c – d)].

[(a + b) / (a – b)]

= (16 + 3) / (16 – 3)

= 19 / 13