Chewbacca has 20 pieces of cherry gum and 30 pieces of grape gum.
Some of the pieces are in complete packs, while others are loose. Each complete pack has
exactly x pieces of gum. If Chewbacca loses one pack of cherry gum, then the ratio of
the number of pieces of cherry gum he has to the number of pieces of grape gum will be
exactly the same as if he instead finds 5 packs of grape gum. Find x.
+126 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.
