What must be subtracted from $9\dfrac{1}{6}$ to get $6\dfrac{1}{9}$?

Let the number that can be subtracted be x.

$\Rightarrow 9\dfrac{1}{6} - x = 6\dfrac{1}{9}\\[1em] \Rightarrow \dfrac{55}{6} - x = \dfrac{55}{9}\\[1em] \Rightarrow x = \dfrac{55}{6} - \dfrac{55}{9}\\[1em]$

LCM of 6 and 9 = 18

$\Rightarrow x = \dfrac{55 \times 3}{6 \times 3} - \dfrac{55 \times 2}{9 \times 2}\\[1em] \Rightarrow x = \dfrac{165}{18} - \dfrac{110}{18}\\[1em] \Rightarrow x = \dfrac{165 - 110}{18}\\[1em] \Rightarrow x = \dfrac{55}{18}\\[1em] \Rightarrow x = 3\dfrac{1}{18}\\[1em]$

Hence, the number = $3\dfrac{1}{18}$.