What number should be added to the two terms of a fraction to obtain the inverse of that same fraction?
Let $\frac{a}{b}$ be the sought fraction and let $c$ be a number such that $\frac{a+c}{b+c}=\frac{b}{a}$. This equality is equivalent to $(a+c) a=(b+c) b$. Thus we have:

$$
(a+c) a=(b+c) b \Longrightarrow a^{2}+a c-b^{2}-b c=0 \Longrightarrow\left(a^{2}-b^{2}\right)+c(a-b)=0
$$

Hence

$$
0=\left(a^{2}-b^{2}\right)+c(a-b)=(a-b)(a+b)+c(a-b)=(a-b)(a+b+c)
$$

Therefore $(a-b)(a+b+c)=0$. We have two cases:

$\left.1^{o}\right) a-b=0 \Longrightarrow a=b$. In this case the fraction is equal to $1=\frac{a}{a}$ and we can add any number.

$\left.2^{o}\right) a+b+c=0 \Longrightarrow c=-(a+b)$. In this case we need to add $-a-b$.