Question

what is the amount (a) of alcohol contributed by the 70% alcohol solution in the following problem? a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution.

Answer 1

To achieve a 50% alcohol solution, 25 liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution.

Step-by-step explanation:

To solve the problem, we are going to use the concept of mixture solutions. The amount of alcohol in the final solution is the weighted average of the amount of alcohol in the two initial solutions. First, we calculate the total amount of alcohol in the 40% solution, then we use algebra to compute the amount of the 70% solution needed.

Step-by-Step Solution:

1. Calculate the amount of alcohol in 50 liters of the 40% solution:
   40% of 50 liters = 0.40 × 50
2. = 20 liters of alcohol.
3. Let a be the amount of the 70% alcohol solution that needs to be added. In this solution, the amount of pure alcohol will be 70% of a, which is 0.70 × a.
4. The total volume of the solution after adding a liters of the 70% solution will be 50 + a liters.
5. To achieve a 50% alcohol solution, the amount of pure alcohol in the combined solution should be 50% of the total volume:
   50% of (50 + a) liters = 0.50 × (50 + a).
6. Set up the equation:
   20 + (0.70 × a) = 0.50 × (50 + a).
7. Solve for a:
   20 + 0.70a = 25 + 0.50a
   0.70a - 0.50a = 25 - 20
   0.20a = 5
   a = 5 / 0.20
   a = 25 liters.

Therefore, 25 liters of the 70% alcohol solution must be added to the 50 liters of a 40% alcohol solution to get a 50% alcohol solution.