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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

By the quadratic formula,

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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