SOLUTION: Misty wishes to obtain 85 ounces of a 40% acid solution by combining a 72% acid solution with a 25% acid solution. How much of each solution should misty use?

Algebra ->  Rational-functions -> SOLUTION: Misty wishes to obtain 85 ounces of a 40% acid solution by combining a 72% acid solution with a 25% acid solution. How much of each solution should misty use?      Log On


   

Question 1156383: Misty wishes to obtain 85 ounces of a 40% acid solution by combining a 72% acid solution with a 25% acid solution. How much of each solution should misty use?
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the number of ounces of the stronger (72%)
Let y = the number of ounces of the weaker (25%) 

We make this chart and fill in what is given:


Acid-H20 solution | Oz. of liquid | % Pure | Oz. of Pure acid | 
--------------------------------------------------------------- 
Stronger Strength |      x        |  72%   |                  |
Weaker Strength   |      y        |  25%   |                  |
------------------|--------------------------------------------
Medium Strength   |     85        |  40%   |                  |

Then we take the percentages to get the number of ounces of PURE
acid in each solution, of the stronger and weaker before mixing, 
and of the medium strength after mixing.


Acid-H20 solution | Oz. of liquid | % Pure | Oz. of Pure acid | 
--------------------------------------------------------------- 
Stronger Strength |      x        |  72%   |    0.72x         |
Weaker Strength   |      y        |  25%   |    0.25y         |
------------------|--------------------------------------------
Medium Strength   |     85        |  40%   |    0.40(85)      |

Our two equations come from the first and last columns:

system%28x+%2B+++++y+=+85%2C%0D%0A0.72x+%2B+0.25y+=+0.40%2885%29%29

system%28x+%2B+++++y+=+85%2C%0D%0A72x+%2B+25y+=+40%2885%29%29

system%28x+%2B+++++y+=+85%2C%0D%0A72x+%2B+25y+=+3400%29

Solve the system by substitution or elimination.

x=1275%2F47        y=2720%2F47
x+=27%266%2F47      y=57%2641%2F47

As a visual partial check, since the percentage of the final 
mixture (40%) is nearer to the percentage of the weaker solution
(25%) than it is to the stronger solution, we expect that the
answer should be so that it takes more of the weaker solution
than of the stronger solution. 

Edwin

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


You should understand how to solve a two-part mixture problem like this using formal algebra. But if a formal algebraic solution is not required, there is a MUCH faster way to solve the problem.

A typical algebraic solution would look something like this....

x = ounces of 72% acid
85-x = ounces of 25% acid

The combined acid in the two ingredients is equal to the acid in 40% of the total 85 ounces:

.72%28x%29%2B.25%2885-x%29+=+.40%2885%29
.72x%2B21.25-.25x+=+34
.47x+=+12.75
x+=+12.75%2F.47+=+1275%2F47

The amount of 72% acid is 1275/47 ounces; the amount of 25% acid is 85-(1275/47) = 2720/47 ounces. Those are ugly fractions; and we had to do a lot of ugly calculations to find them.

Convert those numbers to decimals or percents if required....

Here is a much faster path to the same ugly answers.

On a number line, 40% is 15/47 of the way from 25% to 72%.
That means 15/47 of the mixture needs to be the 72% acid.
15/47 of the total 85 ounces is 1275/47 ounces.