SOLUTION: A chemist has three different acid solutions. The first acid solution contains 15% acid, the second contains 30% and the third contains 50%. They want to use all three solutions to

Algebra ->  Linear-equations -> SOLUTION: A chemist has three different acid solutions. The first acid solution contains 15% acid, the second contains 30% and the third contains 50%. They want to use all three solutions to      Log On


   

Question 1175368: A chemist has three different acid solutions. The first acid solution contains 15% acid, the second contains 30% and the third contains 50%. They want to use all three solutions to obtain a mixture of 96 liters containing 35% acid, using 3 times as much of the 50% solution as the 30% solution. How many liters of each solution should be used?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
SOLUTION        VOLUME         PURE
  15%            96-4v       0.15(96-4v)
  30%             v          0.3v
  50%            3v          0.5*3v
TOTAL            96          0.35*96

Simplify and solve for v:
0.15%2896-4v%29%2B0.3v%2B0.5%2A3v=0.35%2A96

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


On the chance that you will understand it, I will solve this problem quickly and easily with logical reasoning and simple mental arithmetic -- without the need to set up and solve an algebraic equation.

If you understand what I am doing in this solution, you will have a quick and easy way to solve a wide variety of 2-part mixture problems.

I'll mix the 30% and 50% solutions first.

Using 3 times as much 50% solution as 30% solution means that, when I mix these two, 3/4 of the mixture will be the 50% acid.
That means the percentage of the mixture will be 3/4 of the way from 30% to 50%.
3/4 of the way from 30% to 50% is 45%.

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NOTE: That's the key to using this method of solving 2-part mixture problems. 3/4 of this mixture being 50% acid and 1/4 being 30% acid means the percentage of the mixture will be 3/4 of the way from 30% to 50%.

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So mixing the 30% and 50% acid results in a mixture that is 45% acid.

Now the problem is mixing that 45% acid solution with the given 15% acid solution to obtain a mixture that is 35% acid.
35% is 2/3 of the way from 15% to 45%
So 2/3 of the final mixture is the 45% acid and 1/3 is the 15% acid.

Now I'm ready to find the amounts of each.

1/3 of the final mixture is the 15% acid. That's 1/3 of 96 liters, or 32 liters.

The other 2/3 of the mixture, or 64 liters, is the 45% acid, of which 3/4 is the original 50% acid and 1/4 is the original 30% acid. That makes 48 liters of the 50% acid and 16 liters of the 30% acid.

ANSWERS: 48 liters of 50% acid; 16 liters of 30% acid; 32 liters of 15% acid.

CHECK:
.50(48)+.30(16)+.15(32) = 24+4.8+.48 = 33.6
.35(96) = 33.6