Invertendo
If a:b=c:d then b:a = d:c
a:b=c:d



Alternendo
If a:b=c:d, then a:c=b:d.

a:c=b:d
Componendo
If a:b=c:d, then a+b:b=c+d:d
Adding 1 to both sides, we get


Dividendo
If a:b=c:d then a-b:b=c-d:d
Subtracting 1 from both sides, we get


Componendo and Dividendo
If a:b=c:d then a+b:a-b::c+d:c-d.
a:b=c:d
a+b:b=c+d:d by componendo
Dividing,


Illustrative examples
1) If a:b::c:d, show that 2a+3b:2a-3b::2c+3d:2c-3d.
Suggested answer:
a:b::c:d
Multiplying both sides by 2/3, we get
Applying componendo and dividendo,
2a+3b:2a-3b=2c+3d:2c-3d

Suggested answer:




we get
Dividing both sides by 2x, we get






4) If a, b, c and d are in continued proportion then prove that

Suggested answer:
a, b, c, d are in continued proportion.
then a=bk, b=ck, c=dk
Consider,




Suggested answer:

Consider,




Suggested answer:


Squaring both sides,
By componendo and dividendo, we get



Suggested answer:
(a2+b2) (m2+n2)=(am+bn)2 given
a2m2+b2m2+a2n2+b2n2=a2m2+b2n2+2ambnb2m2+a2n2-2ambn=0
(bm-an)2=0
bm=an

