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i.e.,b:a d:c
(2) If a:b c:dthena:c b:d [alternendo]

Proof : Given that,
d

c

b

a

?ad bc [multiplying both the sides by bd]

or,

cd

bc

cd

ad

[dividing both the sides by cdwhere cz 0 ,dz 0 ]

or,

d

b

c

a

i.e.,a:c b:d

(3) If a:b c:d then
d

c d

b

a b 



[componendo]

Proof : Given that,
d

c

b

a

?  1  1

d

c

b

a [Adding 1 to both the sides]

i.e.,

d

c d

b

a b 



(4) If a:b c:d then
d

c d

b

a b 



[ dividendo]

Proof : a:b c:d

?  1  1

d

c

b

a

[subtracting 1 from both the sides]

i.e.,

d

c d

b

a b 



(5) If a:b c:d then
c d

c d

a b

a b







 [componendo – dividendo]

Proof : Given that,
d

c

b

a

By componendo, ..............(i)

d

c d

b

a b 



Again by dividendo,

d

c d

b

a b 



or,

c d

d

a b

b





[by invertendo] ............. ( ii)