Answers:
Gabe needs 30 liters of the 50% solution
Gabe needs 30 liters of the 70% solution
Both answers are 30
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Explanation:
We have two beakers, each of which we don't know how much is inside. Let's call this x and y
x = amount of liquid in the first beaker (water+acid)
y = amount of liquid in the second beaker (water+acid)
We're told that the beakers have a 50% solution and a 70% solution of acid. This means that the first beaker has 0.5*x liters of pure acid, and the second beaker has 0.7*y liters of pure acid. In total, there is 0.5x+0.7y liters of pure acid when we combine the two beakers. This is out of x+y = 60 liters total, which we can solve for y to get y = 60-x. We will use y = 60-x later on when it comes to the substitution step
We can divide the total amount of pure acid (0.5x+0.7y liters) over the total amount of solution (x+y = 60) to get the following
(0.5x+0.7y)/(x+y) = (0.5x+0.7y)/60
We want this to be equal to 0.6 because Gabe wants a 60% solution when everything is said and done, so
(0.5x+0.7y)/60 = 0.60
0.5x+0.7y = 0.60*60 .... multiply both sides by 60
0.5x+0.7y = 36
0.5x+0.7( y ) = 36
0.5x+0.7(60-x) = 36 ........ replace y with 60-x (substitution step)
0.5x+0.7(60)+0.7(-x) = 36 .... distribute
0.5x+42-0.7x = 36
-0.2x+42 = 36
-0.2x = 36-42 .... subtract 42 from both sides
-0.2x = -6
x = -6/(-0.2) .... divide both sides by -0.2
x = 30
He needs 30 liters of the 50% solution
Use this x value to find y
y = 60-x
y = 60-30
y = 30
So he needs 30 liters of the 70% solution
