SOLUTION: Thirty gallons of a 20% acid solution is needed for an experiment. Only 40% and 10% acid solutions are available. How much of the 10% acid solution is needed?

Question 1203656: Thirty gallons of a 20% acid solution is needed for an experiment. Only 40% and 10% acid solutions are available. How much of the 10% acid solution is needed?
Found 2 solutions by greenestamps, josgarithmetic:
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

First, here is a typical standard formal algebraic solution.

x gallons of 10% acid, plus (30-x) gallons of 40% acid, makes 30 gallons of 30% acid:

ANSWER: x=10 gallons of 10% acid and (30-x)=20 gallons of 40% acid

And now here are two variations of an informal solution method that is nearly always much faster than formal algebra.

First variation....

Picture the three percentages -- 10, 30, and 40 -- on a number line and observe/calculate that 30 is 2/3 of the way from 10 to 40.
That means 2/3 of the mixture is the higher percentage acid.
2/3 of 30 gallons is 20 gallons, so again we get the answer of 20 gallons of 40% acid and 10 gallons of 10% acid.
Second variation....

Again looking at the three percentages, find that the difference between 10 and 30 is 20 and the difference between 30 and 40 is 10.
The ratio of those differences, 20 and 10, is 20:10 or 2:1.
That means the two ingredients must be mixed in the ratio 2:1; and since 30 is closer to 40 than to 10, the large portion must be the 40% acid.
And 30 gallons in the ratio 2:1 means 20 gallons of the 40% acid and 10 gallons of the 10% acid.

Here is a crude diagram of how this "alligation" method is apparently taught in pharmacology:
10 10 \ / 30 / \ 40 20

The two numbers on the left are the percentages of the two ingredients.
The number in the middle is the percentage of the mixture.
The two numbers on the right show the ratio (10:20, or 2:1) in which the two ingredients must be mixed, showing that the mixture needs to be 2 parts 40% acid and 1 part 10% acid.
And 30 gallons with 2 parts 40% acid and 1 part 10% acid means 20 gallons of 40% acid and 10 gallons of 10% acid.

Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
Two Thirds the way from one to the other between the two available acid stock solutions

If you WANT the amount of the 10% to be unknown variable then x.

Amount of the 40% is 30-x.

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----------which is 20 gallons of the 10% acid.
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