The boys mixed 10 gallons of the 20% pure lemon juice and 10 gallons of the 70% pure lemon juice mix, because they wanted 20 gallons of lemonade at a 40% lemon juice mixture. how much should they have used to get a final mixture of 20 gallons at 40% lemon juice?

The boys should use 140% of pure lemon juice mi to get a final mixture of 20 gallons at 40% lemon juice:

if 20:70 the 40:70 - it is more like ration and proportion. 

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First subtract 10 from both side
y^2+6y-16=0
(y+8)(y-2)=0
-8 and 2