A chemist has one solution that has 50% acid. She has another solution that is 25% acid. How many liters of the 50% acid solution should she combine to get 10 liters of a 40% acid solution?

Step-by-step explanation:

Let's begin by assigning letters to represent our two unknowns:

   x (liters of 25% solution)

   y (liters of 50% solution)

Our system of equations will consist of two equations:

   Equation #1 (total volume of solution)

   Equation #2 (total concentration of acid)

Our total volume of solution is 10 liters, which can be expressed as the sum of our unknowns:

   Equation #1:  x + y = 10

Our total concentration of acid can be expressed as the sum of the individual acid concentrations to make up the concentration of the final solution:

   Equation #2:  (0.25)(x) + (0.50)(y) =

                            (0.40)(10)

We can use Equation #1 to express one unknown in terms of the other and then plug that expression into Equation #2 to solve for one of the unknowns:

   x + y = 10

         y = 10 - x

Now we'll plug our expression for y in terms of x into Equation #2 and solve for x:

   0.25(x) + 0.50(10 - x) = 0.40(10)

   0.25x + 5 - 0.50x = 4

   -0.25x = 4 - 5

   -0.25x = -1

          x = (-1)/(-0.25)

          x = 4 (liters of 25% solution)

Now we'll plug our value for x into Equation #1 and solve for y:

   4 + y = 10

         y  = 10 - 4

         y = 6 (liters of 50% solution)

Finally, we will verify the correctness of our answers by plugging these values into Equation #2 to see if the sum of the component acid concentrations equals the final solution concentration:

   0.25(4) + (0.50)(6) = 0.40(10)

   1 + 3 = 4

         4 = 4 (our answers are correct)