It is not true that \[ \frac{a+b}{a+c}=\frac{b}{c} \] If it were, then I could pull a rabbit out of nowhere. Let's call this the rabbit rule.
\[ \frac{0}{1}=0 \ rabbits \]
If the rabbit rule were true
\[ 0=\frac{0}{1}=\frac{1+0}{1+1}=\frac{1}{2} \] Okay, so that's only half a rabbit. Picky, picky! But let's keep going: \[ \frac{1}{2}=\frac{1}{2}=\frac{1+1}{1+2}=\frac{2}{3} \] I'm slowly building a whole rabbit. In the limit (ah ha! we knew that there'd be some calculus in here somewhere...), as we do this over and over again forever, we get a whole rabbit.
Lesson: Don't use the rabbit rule. It's a horrible rule for calculus (but perhaps this explains why rabbits are so prolific?).
Some people think that $(a+b)^2=a^2+b^2$. Ouch! I can use this rabbit rule to pull a rabbit out of my ... hat.
Let's start with 0 rabbits: \[ 0 = (-1+1)= (-1+1)^2 = (-1)^2+1^2 = 2\ rabbits \] Hey, a pair! A pair of rabbits! Now I'm really on to something, because they'll generate lots more rabbits....
This is more commonly abused as \[ (a+b)^\frac{1}{2}=a^\frac{1}{2}+b^\frac{1}{2} \] but is equally horrific.... Let's start with 0 rabbits: \[ 0 = (-1+1)= (-1+1)^\frac{1}{2} = (-1)^\frac{1}{2}+1^\frac{1}{2} = i+1\ rabbits \] We get complex rabbits. That's really strange!