If a/b=c/d, then a/b=(a+c)/(b+d) mathematical proof

In this post I’ll be proving to you that if:

a/b=c/d

Then:

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

To prove this, the first thing you need to know is that:

$\frac { a }{ b } =\frac { c }{ d } \\ \\ \therefore \quad ad=bc$

The rest is just mathematical / algebraic trickery… Let me show you…

Proof 1:

$LHS\\ \\ =\frac { a }{ b } \\ \\ =\frac { a }{ b } \cdot 1\\ \\ =\frac { a }{ b } \cdot \frac { \left( b+d \right) }{ \left( b+d \right) } \\ \\ =\frac { ab+ad }{ b\left( b+d \right) } \\ \\ =\frac { ab }{ b\left( b+d \right) } +\frac { ad }{ b\left( b+d \right) } \\ \\ =\frac { a }{ b+d } +\frac { bc }{ b\left( b+d \right) } \\ \\ =\frac { a }{ b+d } +\frac { c }{ b+d } \\ \\ =\frac { a+c }{ b+d } \\ \\ =RHS$

Proof 2:

$LHS\\ \\ =\frac { a }{ b } \\ \\ =\frac { a }{ b } \cdot 1\\ \\ =\frac { a }{ b } \cdot \frac { \left( b+d \right) }{ \left( b+d \right) } \\ \\ =\frac { ab+ad }{ b\left( b+d \right) } \\ \\ =\frac { ab+bc }{ b\left( b+d \right) } \\ \\ =\frac { b\left( a+c \right) }{ b\left( b+d \right) } \\ \\ =\frac { a+c }{ b+d } \\ \\ =RHS$

Proof 1 Video:

Proof 2 Video:

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