# How does addition in the denominator work?

This is part of a series on common misconceptions.

Is this true or false?

\[ \dfrac{a}{b+c} = \dfrac{a}{b} + \dfrac{a}{c} \]

**Why some people say it's true:** It works when we add numerators like \(\dfrac b a + \dfrac c a = \dfrac{b+c} a \), so it's the same for denominators.

**Why some people say it's false:** Division is complicated, and you can't just add the things you're dividing by and still get a correct result.

The statement is \( \color{red}{\textbf{false}}\).

Proof:Let us take the left hand side of the equation: \(\dfrac{a}{b} + \dfrac{a}{c}\).

Taking common denominator, we get\[\dfrac{a}{b} + \dfrac{a}{c} = \dfrac{ac + ab}{bc} =\dfrac{a(b + c)}{bc} \neq \frac{a}{b+c}.\]

Rebuttal: When \(a=0,\) \(\dfrac 0b+\dfrac 0c = \dfrac 0{b+c}.\)

Reply: We have just proved that it is true for this case only, which doesn't mean that it is true in general, i.e. we haven't proved it for all real values.

**See Also**

**Cite as:**How does addition in the denominator work?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/how-does-addition-in-the-denominator-work/