SOLUTION: A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 9 % vinegar, and the second brand contains 14 % vinegar. The chef wants to

Question 1112543: A chef is going to use a mixture of two brands of Italian dressing. The first brand contains
9
%
vinegar, and the second brand contains
14
%
vinegar. The chef wants to make
350
milliliters of a dressing that is
13
%
vinegar. How much of each brand should she use?
Found 3 solutions by addingup, mananth, greenestamps:
Answer by addingup(3677)   (Show Source): You can put this solution on YOUR website!
0.09x + 0.14(350 - x) = 0.13(350)
0.09x + 49 - 0.14x = 45.5
-0.05 = -3.5
x = 70
she needs 70 milliliters of the 9% and 350-70 = 280 milliliters of the 14%
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!

component percent ---------------- quantity
Dressing I 0.09 ---------------- x ml
Dressing II 0.14 ---------------- 350 - x ml
Mixture 0.13 ---------------- 350

0.09 x + 0.14 ( 350 - x ) = 350.00 * 0.13

0.09 x + 49 - 0.14 x = 45.50
0.09 x - 0.14 x = 45.5 - 49
-0.05 x = -3.5
/ -0.05
x = 70 ml 9.00% Dressing I
280 ml 14.00% Dressing II

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

The two solutions you have received for your question use the same traditional algebraic method to solve the problem. Let me show you a very different method that I find much easier and faster.

By the way.... I just timed myself in solving your problem. With pencil and paper and the traditional algebraic method, I took 1 minute 35 seconds to solve the problem; most of the time was spent multiplying unpleasant decimal numbers. Then I solved the problem again with my method; it took just under 10 seconds because all the arithmetic was easily done mentally.

So here is this alternate method for solving mixture problems involving only two ingredients.

With pencil and paper, the following array can be used to do the first part of the calculations; however, with a little practice you can do this part mentally.

The numbers in the first column are the percentages of the two ingredients -- 14% and 9%. The number in the second column is the desired percentage of the mixture.

The numbers in the third column are the differences, calculated diagonally, between the numbers in the first and second columns.

When the calculations are done this way, the numbers in the third column show the ratio in which the two ingredients need to be mixed.

So the numbers in the third column say that the mixture needs to be 4 parts the 14% ingredient to 1 part the 9% ingredient.

Then the rest of the solution involves converting the 4:1 ratio into amounts of each ingredient, given that the mixture is 350ml; the answer is 280ml of the 14% ingredient and 70ml of the 9% ingredient.

The explanation with all the word takes a long time; but the actual calculations are quite simple. Here is all I need to do mentally to solve this problem in a few seconds:

(1) 14-13=1; 13-9=4.
(2) 4:1 ratio with total 350ml --> 280ml and 70ml
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