Steve wants to make 12 gallons of a 46% acid solution by mixing together a 65% acid solution and a 5% acid solution. How many gallons of each solution should he use?

Answer 1

So, for this problem, we want 12 total gallons by the end. That means that whatever number of gallons of both 65% and 5% solution must add up to 12.

So, to start, we can say that x = number of gallons of the 5% solution, and y = number of gallons of the 65% solution.

It would then give the equation of

x * .05 + y * .65 = 12 * .46

But this gives us two unknowns, and we have only one equation. We need another way to write the unknowns so that we can get both of our answers only solving for one variable.

Think about it this way: If we know how many gallons we have of one of the acid solutions, all we have to do is subtract that from 12 to get the other amount of gallons for the solution. In other words, if we know how many gallons we have of the first solution, which is x, then we know how many gallons of the second solution, y, must be equal to 12-x. So, we can rewrite the equation now this way.

x * .05 + (12-x) * .65 = 12 * .46

This is a much nicer equation to work with, and now all we have to do is the usual solve for x

.05x + (12 * .65) -.65x = 12 * .46

Use a calculator to get 12 * .65 = 7.8

12 * .46 = 5.52

.05x + 7.8 - .65x = 5.52

Combine like terms

-.6x + 7.8 = 5.52

Subtract 7.8 from both sides

-.6x = -2.28

Divide both sides by -.6

x = 3.8 = number of gallons of the 5% solution

12 - 3.8 = 8.2 = number of gallons of the 65% solution

Now, double check just to confirm

(3.8 * .05) + (8.2 * .65) = 5.52

12 * .46 = 5.52

Hope this is clear and able to help!