# A nurse has a 30% alcohol solution and an 80% alcohol solution. He needs 10L of a 40% alcohol solution.

## .3 x + .8 (10 - x) = .4 * 10

## To determine how much of each solution you need to mix, we can use the concept of weighted averages.

Let's assume the nurse needs x liters of the 30% alcohol solution and (10 - x) liters of the 80% alcohol solution.

To find the amount of alcohol in the 30% solution, we multiply the amount of solution (x liters) by the concentration of alcohol (30% or 0.3). Similarly, for the 80% solution, we multiply the amount of solution ((10 - x) liters) by the concentration of alcohol (80% or 0.8).

The total amount of alcohol in the final mixture should be equivalent to 10L of a 40% alcohol solution, which means we can set up the equation:

(x * 0.3) + ((10 - x) * 0.8) = 10 * 0.4

Now, we can solve this equation to find the value of x:

0.3x + 8 - 0.8x = 4

0.3x - 0.8x = 4 - 8

-0.5x = -4

x = (-4) / (-0.5)

x = 8

So, the nurse needs to mix 8 liters of the 30% alcohol solution with (10 - 8) = 2 liters of the 80% alcohol solution to obtain 10 liters of a 40% alcohol solution.