How much water must be added to 60 liters of 60 percent alcohol mixture to reduce it to a mixture of 50 percent alcohol concentration? What if 30 percent alcohol mixture is added instead of water?

(60 * .6) + (x * 0) = (60 + x) * .5

(60 * .6) + (x * .3) = (60 + x) * .5

To solve the first part of the question, we need to calculate the amount of water that needs to be added to the 60 liters of 60% alcohol mixture to achieve a 50% alcohol concentration.

Step 1: Determine the amount of alcohol in the initial mixture.
In 60 liters of a 60% alcohol mixture, the amount of alcohol is 60% of 60 liters, which is (60/100) * 60 = 36 liters.

Step 2: Calculate the total amount of the mixture after adding water.
The total amount of mixture will be 60 liters (initial mixture) + x liters (water added).

Step 3: Calculate the amount of alcohol in the final mixture.
Since we want the final mixture to have a 50% alcohol concentration, the amount of alcohol in the final mixture will be 50% of the total mixture, which is (50/100) * (60 + x) liters.

Step 4: Set up the equation.
Now we can set up the equation by considering the conservation of alcohol:
36 liters (initial alcohol) = (50/100) * (60 + x) liters (final alcohol)

Step 5: Solve the equation.
36 = (50/100) * (60 + x)

To solve for x, we can multiply both sides by 100/50:
36 * (100/50) = 60 + x

Rearranging the equation:
x = (36 * 100/50) - 60 = 72 - 60 = 12 liters

Therefore, 12 liters of water must be added to the 60 liters of 60% alcohol mixture to reduce it to a mixture of 50% alcohol concentration.

Now, let's consider the second part of the question where 30% alcohol mixture is added instead of water.

Step 1: Determine the amount of alcohol in the 30% alcohol mixture.
In a 30% alcohol mixture, 30% of the mixture is alcohol. So, if we add x liters of this mixture, the amount of alcohol added will be (30/100) * x liters.

Step 2: Set up the equation.
The equation can be set up by considering the conservation of alcohol in the final mixture:
36 liters (initial alcohol) = [(50/100) * 60] liters (initial alcohol) + (30/100) * x liters (added alcohol)

Step 3: Solve the equation.
36 = (50/100) * 60 + (30/100) * x

Multiplying and simplifying:
36 = 30 + (3/10) * x

Rearranging the equation:
(3/10) * x = 36 - 30 = 6

Multiplying by the reciprocal of (3/10):
x = 6 * (10/3) = 20

So, if a 30% alcohol mixture is added instead of water, 20 liters of the mixture should be added to the 60 liters of 60% alcohol mixture to achieve a final mixture with a 50% alcohol concentration.