omponendo and dividendo is a fundamental concept in mathematics. It helps in solving complex problems.

if a / b = c / d , then by applying componendo and dividendo , we get

(a + b) / ( a - b ) = (c + d ) / ( c - d ), where a is not equal to b.

This proof is invalid for a = b and c= d .

because if a/b = c/d and a = b and c= d / then ( a+ b ) / ( a - b ) = ( c + d ) / ( c- d )

(a+b ) / 0 = ( c+d) / 0

which is not defined.

componendo and dividendo is a combination of componendo and dividendo.

## Proof of Componendo and Dividendo Rule

**componendo rule proof**

If a / b = c / d,

then by using componendo , we get ,

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

Proof:

adding 1 both side , we get

a / b + 1 = c / d + 1

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

hence proved.

**Proof of dividendo rule**

If a / b = c / d , then ( a - b ) / b = ( c - d ) / d

proof:

subtracting 1 from both side , we get

a / b - 1 = c / d - 1

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

hence proved.

## Proof of Componendo and Dividendo

if a / b = c / d , then ( a + b ) / ( a - b ) = ( c + d ) / ( c - d )

proof :

Taking L. H . S, we get ( a + b ) / ( a - b ) = ( a / b + 1 ) / ( a / b - 1 ) = ( c / d + 1 ) / ( c / d - 1 )

= ( c + d ) / ( c - d )

Hence proved.

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