Question

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Answer 1

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