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Question 1154729: Gabe Amodeo, a nuclear physicist, needs 60liters of a 40% acid solution. He currently has a 30 % solution and a 60 % solution. How many liters of each does he need to make the needed 6060 liters of 40% acid solution?
Found 2 solutions by jim_thompson5910, greenestamps:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
I'm assuming you meant to write "60" instead of "6060".
Gabe needs 60 liters of the 40% solution. This solution is a mix of water and pure acid. Specifically 60*0.4 = 24 liters of pure acid are needed.
Let
x = amount of the 30% solution (in liters)
y = amount of the 60% solution (in liters)
Based on those definitions above, we can say
0.30x = amount of pure acid from the 30% mix
0.60y = amount of pure acid from the 60% mix
0.30x+0.60y = amount of pure acid in total
0.30x+0.60y = 24, since Gabe wants 24 liters of pure acid in the end
At the same time, we can also say x+y = 60 because the two individual solution amounts (x and y) will combine to the overall amount of solution of 60 liters. Solve for y to get y = 60-x
Plug this into 0.30x+0.60y = 24, and solve for x
0.30x+0.60y = 24
0.30x+0.60( y ) = 24
0.30x+0.60( 60-x ) = 24 ... replace y with 60-x
0.30x+0.60(60)+0.60(-x) = 24 ... distribute
0.30x+36-0.60x = 24
-0.30x+36 = 24
-0.30x+36-36 = 24-36 ... subtract 36 from both sides
-0.30x = -12
-0.30x/(-0.30) = -12/(-0.30) .... divide both sides by -0.30
x = 40
We need 40 liters of the 30% solution
y = 60-x
y = 60-40
y = 20
and 20 liters of the 60% solution
With those x and y values in mind, note that
30% of x = 30% of 40 = 0.30*40 = 12
60% of y = 60% of 20 = 0.60*20 = 12
Those values sum to 12+12 = 24, which was the amount of pure acid Gabe needs.
This helps confirm we have the right answer.
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Answers:
40 liters of the 30% solution
20 liters of the 60% solution
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
You should know how to solve problems like this using a formal algebraic method, such as the one shown by the other tutor.
If an algebraic solution is not required, here is a fast and easy way to solve two-part mixture problems like this:
(1) The target 40% is "twice as close" to 30% as it is to 60%.
(2) Therefore, the mixture must contain twice as much of the 30% ingredient as the 60% ingredient.
Twice as much of the 30% as the 60%, and a total of 60 liters, means 40 liters of the 30% acid solution and 20 liters of the 60% acid solution.
ANSWER: 40 liters of the 30% acid; 20 liters of the 60% acid.
Here is another way to look at the same solution method.
Think of starting with 30% acid and adding 60% acid until the mixture is 40% acid.
You started at 30% and moved towards 60%, stopping when you reached 40%.
40% is one-third of the way from 30% to 60%.
Therefore, 1/3 of the mixture is the 60% acid you are adding.
ANSWER: 1/3 of 60 liters, or 20 liters, of the 60% acid solution; the other 40 liters are the 30% solution.
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