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A man has a barrel which is full of vinegar.  He removes 4 gallons and refills the barrel with water. He then removes 8 gallons.  An original vinegar amount of 21 gallons remains in the barrel.  Calculate capacity of barrel.  Thanks.

"A man has a barrel which is full of vinegar."
The barrel contains v gallons of vinegar.

"He removes 4 gallons and refills the barrel with water."
The barrel contains 4 gallons of water and (v-4) gallons of vinegar.
(v-4)/v of the contents of the barrel is vinegar.

"He then removes 8 gallons."
(v-8) gallons of liquid remains.

"An original vinegar amount of 21 gallons remains in the barrel."
(v-4)/v of (v-8) gallons = 21 gallons
(v-4)/v × (v-8) = 21
(v² - 12v + 32)/v = 21
(v² - 12v + 32) = 21v
v² - 33v + 32 = 0

v = [33 ± √(33² – 4·1·32)] / [2·1]
= [33 ± √961] / 2
= [33 ± 31] /2
= 1, 32
1 is an extraneous solution
v = 32
The barrel holds 32 gallons.
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#### Janet Yang

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