$\dfrac{x}{30}+\dfrac{x}{25}+\dfrac{1}{3}=5\dfrac{5}{6}$
$⇔\dfrac{x}{30}+\dfrac{x}{25}+\dfrac{1}{3}=\dfrac{35}{6}$
$⇔\dfrac{25x}{750}+\dfrac{30x}{750}+\dfrac{250}{750}=\dfrac{4375}{750}$
$⇔25x+30x+250=4375$
$⇔55x=4375-250$
$⇔55x=4125$
$⇔x=\dfrac{4125}{55}$
$⇔x=75$
Vậy $S=\{75\}$