Answer:
There needs to be 6 gallons of 35 % alcohol solution
and 3 gallons of 50 % solution.
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Step-by-step explanation:
x = number of gallons of the 35% solution
y = number of gallons of the 50% solution
The two amounts must add to 9 gallons, so x+y = 9
This solves to y = 9-x
Since we want 9 gallons of 40% alcohol solution, this means we want 0.40*9 = 3.6 gallons of pure alcohol.
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0.35x = amount of pure alcohol from the first batch
0.50y = amount of pure alcohol from the second batch
0.35x+0.50y = total amount of pure alcohol
Set that expression equal to the 3.6 figure calculated earlier (the total amount of pure alcohol we're after) and we get this equation
0.35x+0.50y = 3.6
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Next, we replace y with 9-x and then solve for x.
0.35x+0.50y = 3.6
0.35x+0.50(9-x) = 3.6
0.35x+4.5-0.50x = 3.6
-0.15x+4.5 = 3.6
-0.15x = 3.6-4.5
-0.15x = -0.9
x = -0.9/(-0.15)
x = 6
We'll need 6 gallons of the 35% solution
Use this x value to find y
y = 9-x
y = 9-6
y = 3
We'll also need 3 gallons of the 50% solution