Which rational number should be added to $\dfrac{-7}{8}$ to get $\dfrac{5}{9}$?

Let $x$ be added to $\dfrac{-7}{8}$.

$\dfrac{-7}{8} + x = \dfrac{5}{9}\\[1em] ⇒ x = \dfrac{5}{9} - \dfrac{-7}{8}$

LCM of 9 and 8 is 2 x 2 x 2 x 3 x 3 = 72

$\Rightarrow x = \dfrac{5 \times 8}{9 \times 8} - \dfrac{-7 \times 9}{8 \times 9}\\[1em] \Rightarrow x = \dfrac{40}{72} - \dfrac{-63}{72}\\[1em] \Rightarrow x = \dfrac{40 - (-63)}{72}\\[1em] \Rightarrow x = \dfrac{40 + 63}{72}\\[1em] \Rightarrow x = \dfrac{103}{72}\\[1em] \Rightarrow x = 1 \dfrac{31}{72}$

The number added to $\dfrac{-7}{8}$ to get $\dfrac{5}{9}$ is $1\dfrac{31}{72}$.