SOLUTION: Pola a massage therapist, needs 5 ounces of a 14% lavender oil solution. She has only 5% and 30% lavender oil solution available. How many ounces of each should Pola mix to obtain

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Question 1096590: Pola a massage therapist, needs 5 ounces of a 14% lavender oil solution. She has only 5% and 30% lavender oil solution available. How many ounces of each should Pola mix to obtain the desired solution.
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!

First, a solution by a traditional algebraic method....

You want to get 5 ounces of the mixture, so let x be the number of ounces of the 5% solution; then (5-x) will be the number of ounces of the 30% solution.

Then write and solve an equation that says the number of ounces of actual lavender oil in the mixture (14% of 5 ounces) is equal to the sum of the numbers of ounces in the two ingredients (5% of the x ounces, plus 30% of the (5-x) ounces):

.14%285%29+=+.05%28x%29+%2B+.30%285-x%29
.7+=+.05x%2B1.5-.30x
.25x+=+.8
x+=+.8%2F.25+=+3.2

You need 3.2 ounces of the 5% lavender solution and 5-3.2=1.8 ounces of the 30% lavender solution.

Now here is a method that, if you understand it, will always get you to an answer faster than the tradition algebraic method....

The key to this method is that the ratio in which the two ingredients need to be mixed is exactly determined by how far the mixture percentage is from the percentages of the two ingredients. Let's see how that works with your example.

We are mixing 5% and 30% to get 14%. The differences are 30-14 = 16 and 14-5 = 9. Those differences of 16 and 9 mean that the two ingredients must be mixed in the ratio 16:9. So one of the ingredients must be 16/25 of the total, and the other must be 9/25 of the total.

%2816%2F25%29%2A5+=+16%2F5+=+3.2 ; %289%2F25%29%2A5+=+9%2F5+=+1.8

We need 3.2 ounces of one ingredient and 1.8 ounces of the other.

But which is which? The answer to that comes from noting that 14% is closer to 5% than it is to 30%; that means the larger portion must be the 5% ingredient.

So again we have the answer that we need 3.2 ounces of the 5% lavender solution and 1.8 ounces of the 30% lavender solution.

All the words of explanation make this look like a lengthy process. But here is all that is really involved in finding the answer by this method, without all the words:

30-14+=+16 ; 14-5+=+9 --> ratio is 16:9 --> fractions are 16/25 and 9/25.
%2816%2F25%29%2A5+=+3.2 ; %289%2F25%29%2A5+=+1.8
3.2 ounces of the 5%; 1.8 ounces of the 30%.