Answers:
Use 60 gallons of the 30% solution
Mix with 40 gallons of the 80% solution
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Work Shown:
x = amount of the 30% solution (in gallons)
y = amount of the 80% solution (in gallons)
The two amounts must add to 100 as this is the total we want, so x+y = 100 which becomes y = 100-x after subtracting x from both sides
The expression 0.30*x represents the amount of pure alcohol from the first batch, while 0.80*y represents the amount of pure alcohol from the second batch. In total, we have 0.30*x+0.80*y gallons of pure alcohol. We want 50 gallons of pure alcohol (50% of 100 gallons is 50 gallons), therefore we end up with this equation: 0.30*x+0.80*y = 50
Let's use substitution to isolate the variables.
0.30*x+0.80*y = 50
0.30*x+0.80*(100-x) = 50 ... replace y with 100-x
0.30*x+0.80*(100)+0.80*(-x) = 50 ... distribute
0.30*x+80-0.80x = 50
-0.50*x+80 = 50
-0.50*x+80-80 = 50-80 ... subtract 80 from both sides
-0.50*x = -30
x = -30/(-0.50) .... divide both sides by -0.50
x = 60
If x = 60, then y is...
y = 100-x
y = 100-60
y = 40