### Componendo et Dividendo

The following theorems on proportion are very useful. The names of the theorems in Latin are included for interest only.

 Name in Latin Theorem Alterando Invertendo Componendo Dividendo Componendo et Dividendo Proofs of Componendo et Dividendo

(1)        k-method

Let Then   a = bk,  c = dk   (2)        Combine the proofs of Componendo and Dividendo theorems  Divide (1) by (2) and cancel the “b” and “d” in the denominators of (1) and (2), (3)        Divide and multiply

Given:  Converse of Componendo et Dividendo

The point is : Can we go backward?

Can we prove: Proofs of the converse

(1)        Direct expansion (a + b)(c – d)  = (a – b)(c + d)

ac – ad + bc – bd = ac + ad – bc – bd

\ 2bc    = 2ad

ad    = bc (2)        Apply Componendo et Dividendo itself

This is quite interesting. You use the theorem to prove the converse of the theorem!    Exercise

Apply Componendo et Dividendo to the following proportional and solve the equation: Highlight the box below for answer:

 (Ans)  x = -2               (Note:  x = 2  is not an answer!)