Answer: 14 gallons of the 64% acid solution is needed
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Work Shown:
x = amount, in gallons, of the 64% acid solution
0.64x = amount of pure acid from the stuff we're adding in
0.38*12 = 4.56 = amount of pure acid from 38% solution
In total, we have 0.64x+4.56 gallons of pure acid out of x+12 gallons total
Dividing the two amounts will give us the decimal form of the percentage. Set this equal to 0.52 and solve for x
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(0.64x+4.56)/(x+12) = 0.52
0.64x+4.56 = 0.52(x+12)
0.64x+4.56 = 0.52x+6.24
0.64x-0.52x = 6.24-4.56
0.12x = 1.68
x = 1.68/0.12
x = 14 gallons is the amount of the 64% acid solution we need to add in
