Question

Expain componendo and invertendo with example.

Answer 1

Let a,b,c,d be four numbers such that a/b = c/d  (b=d= non zero)
Then:
Componendo states that:
a+b/b = c+d/d
Eg: a= 3, b =4 , c = 9 , d = 12
[a/b = c/d== 3/4 = 9/12 or 3/4]
a+b/b = 3+4/4 = 7/4
c+d/d = 9+12/12 = 21/12 = 7/4
Hence, a+b/b = c+d/d...Satisfied.

Invertendo states that:
If a/b = c/d, then b/a = d/c
Eg: a= 3, b =4 , c = 9 , d = 12
[a/b = c/d== 3/4 = 9/12 or 3/4]
b/a = 4/3
d/c = 12/9 = 4/3
Hence, b/a = d/c or b:a = d:c...Hence satisfied.
Hope this is what you were looking for(and believe it helps).

Answer 2

Solution

(1) Invertendo

$\sf \: if \: \dfrac{a}{b} = \dfrac{c}{d} \: \: then \: \: \dfrac{b}{a} = \dfrac{d}{c}$

$\sf \implies \: \dfrac{a}{b} = \dfrac{c}{d}$

a × d = b × c

b × c = a × d

$\: \sf \dfrac{b×c}{a×c} = \: \dfrac {a×d}{a×c}$

.... (Dividing both sides by a × c)

$\: \sf \dfrac{b}{a} = \dfrac {d}{c}$

$\sf Therefore, if \dfrac{a}{b} = \: \dfrac{c}{d} then \sf \: \dfrac{b}{a} = \dfrac{d}{c}$

This property is known as Invertendo

(2) Componendo

$\sf If \: \dfrac {a}{b} = \dfrac{c}{d} then \: \dfrac {a+b}{b} = \: \dfrac{c+d}{d}$

$\sf \: \dfrac{a}{b} = \dfrac{c}{d}$

$\sf \dfrac{a}{b} + 1= \:\dfrac{c}{d}+1$

(Adding 1 to both the side)

$\: \sf \dfrac {a+b}{b} = \: \dfrac{c+d}{d}$

$\: \sf \dfrac{a}{b} = \dfrac{c}{d}\: \: \: then \: \: \: \dfrac {a+b}{b} = \dfrac{c+d}{d}$