SOLUTION: Pure acid is to be added to a 10% acid solution to obtain 126 L of a 20% acid solution. What amount of each should be used?

Question 719296: Pure acid is to be added to a 10% acid solution to obtain 126 L of a 20% acid solution. What amount of each should be used?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Often the key to these mixture problems is to find expressions for the actual amount of the dissolved/mixed-in element in each of the mixtures/solutions. In this case, we are going to look to express the amount of acid.

When "A" is some percent of some solution/mixture, then the actual amount of A in the solution/mixture will be:
(percent as a decimal or fraction) * (the total amount of that mixture/solution)
(Note: Don't use a percent as a percent in an equation. Use the decimal or fraction equivalent instead.) Let's use this expression to express the amount of acid in the three solutions.

For the pure (pure means 100%) acid solution, we do not know how much of it will be used. Let's call this "x". So the amount of acid in this solution would be:
1 * x or just x
For the 10% solution, we do not know the amount of that, either. But we know that we will end up with 126 liters after we combine this solution and the pure acid. So we can use (126 - x) for this amount. The acid in this solution will be:
0.10 * (126 - x) or 12.6 - 0.10x
For the 20% solution, we know that the amount will be 126 liters. So the amount of 20% solution will be x + 20. This makes the amount of alcohol in this solution:
0.20 * 126 or 25.2

Now that we have the expressions we were looking for we are ready to solve the problem. Not only do the amounts of the pure acid and 10% solutions add up the amount of 20% solution, but the amounts of acid in these solutions add up, too:
x + (12.6 - 0.10x) = 25.2
With this equation we can solve for x.

We can solve for x with the decimals in there. But I like to get rid of them. Multiplying by 100 will shift all the decimals over by two places:
100x + 1260 - 10x = 2520
Combining like terms on the left side:
90x + 1260 = 2520
Subtracting 1260:
90x = 1260
Dividing by 90:
x = 14
So we use 14 liters of the pure acid. And since the amount of 10% solution was (126 - x), we use 126 - 14 or 112 liters of 10% acid solution.

P.S. You get the exact same answer if you leave the decimals in while solving.
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