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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.

 Jan 6, 2019
 #1
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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.

 

Hope this helps,

 

- PM cool

 #2
avatar+679 
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If you want to check, plugging in 14 for X, we have \(\dfrac{20-14}{30} = \dfrac{20}{30 + 5(14)} \Rightarrow \dfrac{6}{30} = \dfrac{20}{100} = \dfrac{1}{5}\)

This is true, so the answer is 14.

 

- PM

 #3
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Let the number of pieces in each pack =p
The total number of packs =50/p

2/3 - p/30 = 5p / 50, solve for p
p =5 - number of pieces in each pack
20/5 =4 packs of Cherry gum
30/5 =6 packs of Grapes gum
50/5 = 10 packs in total.

Check:
(20 - 5) / 30 =(5 x 5) /50
15/30 =25/50
1/2 = 1/2

 Jan 6, 2019
 #4
avatar+679 
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15/30 does not equal 25/30.

 

1/2 does not equal 5/6.


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