SOLUTION: A mixture of 3% disinfectant solution is to be made from 6% and 2% disinfectant solutions. How much of each solution should be used if 24 gallons of 3% solution are needed?

Question 719507: A mixture of 3% disinfectant solution is to be made from 6% and 2% disinfectant solutions. How much of each solution should be used if 24 gallons of 3% solution are needed?
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 disinfectant.

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 disinfectant in the three solutions.

For the 6% solution, we do not know how much of it will be used. So the amount of disinfectant in this solution would be:
0.06 * x or 0.06x
For the 2% solution we do not know how much of that we will use, either. But we know that the total, after we combine the 6% and 2% solutions will be 24 liters. So we can use (24-x) for the amount of 2% solution. So the disinfectant in this solution will be:
0.02 * (24 - x) or 0.48 - 0.02x
For the 3% solution, we know there will be 24 liters. This makes the amount of disinfectant in this solution:
0.03 * 24 or 0.72

Now that we have the expressions we were looking for we are ready to solve the problem. Not only do the amounts of the 6% and 2% solutions add up the amount of 3% solution, but the amounts of disinfectant in these solutions add up, too:
0.06x + (0.48 - 0.02x) = 0.72
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:
6x + (48 - 2x) = 72
Combining the like terms on the left side:
4x + 48 = 72
Subtracting 48:
4x = 24
Dividing by 4:
x = 6
So we use 6 liters of the 6% solution. And since the amount of 2% solution is (24 - x), we use 24 - 6 or 18 liters of the 2% solution.

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