Let's call one pack of gum X.

The ratio of # of cherry to # of grape is .\(\\(\dfrac{20}{30} = \dfrac{2}{3})\) Using this, we have the following equality.

\((\dfrac{20-X}{30} = \dfrac{20}{30 + 5X})\). The ratio to the left is the ratio of gum if chewman loses 1 pack of cherry, and the ratio to the right is the ratio of gum if chewman finds 5 packs of grape.

Cross multiplying, we have \((20-X)(30+5X) = (20)(30) \Rightarrow (20-X)(30+5X) = 600\). Factoring, we have \(-5x^2 + 70x + 600 = 600 \Rightarrow -5x^2 + 70x = 0 \Rightarrow 5x^2 = 70x \Rightarrow \boxed{x = 14}\).

There are 14 pieces of gum in one pack of gum.