SOLUTION: Mr. Rosoff works in the lab at a pharmaceutical company. He needs to make 60 liters of a 13% acid solution to test a new product. His supplier only ships a 16% and a 12% solution.
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Question 1157274: Mr. Rosoff works in the lab at a pharmaceutical company. He needs to make 60 liters of a 13% acid solution to test a new product. His supplier only ships a 16% and a 12% solution. Mr. Rosoff decides to make the 13% solution by mixing the 16% solution with the 12% solution. How much of the 16% solution will Mr. Rosoff need to use?
[A] 10 L [B] 60 L [C] 45 L [D] 15 L Found 3 solutions by josgarithmetic, Boreal, greenestamps:Answer by josgarithmetic(39617) (Show Source):
You can put this solution on YOUR website! x=16% sol in l
60-x= 12% sol
60*0.13=7.8 liter of pure acid
.16x+.12(60-x)=7.8
.16x+7.2-.12x=7.8
0.04x=0.6
=15 liters of 16%
D
Notice that 13% is 1/4 of the way from 12 to 16%
That means 3/4 of the acid is weighted for 12% of 45 l, leaving 15 l for 16%
First a traditional solution using a standard algebraic method, which you should understand and be able to use. Then I'll show a couple of ways you can solve this kind of problem much more easily with just a little mental arithmetic.
(1) formal algebra....
let x be the number of liters of 12% acid
then (60-x) is the number of liters of 16% acid
The total amount of acid in x liters of 12% acid and (60-x) liters of 16% acid is equal to 13% of the total 60 liters:
ANSWER: 45 liters of 12% acid and 15 liters of 16% acid
(2) alternate method
The ratio in which the two ingredients must be mixed is directly related to where the percentage of the mixture lies between the percentages of the two ingredients.
In this problem, with answer choices being given, you can guess the answer without doing any calculations at all.
Clearly answer choice B doesn't make sense; the whole 60 liters can't be 16% acid.
Answer choice C says that 45 of the 60 liters will be the 16% acid. That means more 16% acid than 12% acid is used; but logically that would mean the percentage of the mixture would be closer to 16% than to 12%.
That leaves A and D as the only possible correct answers. And we can easily determine which is right with a couple of simple calculations.
So with that discussion behind us, let's look at two different ways to quickly find the answer to the question. The two methods are very similar; they are both based on observing where the percentage of the mixture lies between the percentages of the two ingredients. We just use slightly different calculations to find the answer.
(2a) One easy way to find the answer is to see that the final percentage of 13% is "one-fourth of the way from 12% to 16%" -- meaning that 1/4 of the mixture needs to be the 16% acid. That gives us 1/4 of 60 liters, or 15 liters, of the 16% acid, leaving 45 liters of the 12% acid.
(2b) Another way of comparing the three percentages is to say that the 13% of the mixture is "three times as close to 12% as it is to 16%" -- meaning that the mixture must contain 3 times as much 12% acid as 16% acid. So in this way of thinking we need to divide the 60 liters in the ratio 3:1, leading us again to the answer of 45 liters of 12% acid and 15 liters of 16% acid.
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