Question 1157274
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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.<br>
(1) formal algebra....<br>
let x be the number of liters of 12% acid
then (60-x) is the number of liters of 16% acid<br>
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:<br>
{{{.12(x)+.16(60-x) = .13(60)}}}
{{{.12x+9.6-.16x = 7.8}}}
{{{1.8 = .04x}}}
{{{x = 1.8/.04 = 45}}}<br>
ANSWER: 45 liters of 12% acid and 15 liters of 16% acid<br>
(2) alternate method<br>
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.<br>
In this problem, with answer choices being given, you can guess the answer without doing any calculations at all.<br>
Clearly answer choice B doesn't make sense; the whole 60 liters can't be 16% acid.<br>
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%.<br>
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.<br>
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.<br>
(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.<br>
(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.<br>