Mixing Alcohol Solutions of Different Concentrations to Achieve a Desired Mixture

Mixing different concentrations of alcohol solutions to achieve a specific concentration can often seem like a complex task. However, with a systematic approach, it can be easily managed. Let's explore how to mix 10% and 50% alcohol solutions to get 15 liters of a 26% solution.

Understanding the Problem

The goal is to determine the amount of 10% alcohol solution (let’s denote it as (x)) and 50% alcohol solution (denoted as (y)) needed to produce 15 liters of a 26% alcohol solution.

Step-by-Step Calculation

To solve this problem, we start by setting up a system of equations based on the volumes and concentrations of alcohol in the mixtures.

The total volume of the mixture is 15 liters, which is the sum of the volumes of the 10% and 50% solutions:

[x y 15]

Setting Up the Equations

Next, we set up the equations based on the alcohol content in each solution. The amount of alcohol in the 10% solution is 0.1x, and in the 50% solution, it is 0.5y. The total amount of alcohol in the 26% solution is 15 liters times 26%, or 15 * 0.26.

The equation for the alcohol content is as follows:

[0.1x 0.5y 15 times 0.26]

Solving the Equations

First, solve for one of the variables using the volume equation:

[x 15 - y]

Substitute (x 15 - y) into the alcohol content equation:

[0.1(15 - y) 0.5y 15 times 0.26]

Simplify and solve for (y):

[1.5 - 0.1y 0.5y 3.9]

[0.4y 2.4]

[y 6]

Now, substitute (y 6) back into the volume equation:

[x 6 15]

[x 9]

Conclusion

From this, we find that 9 liters of 10% alcohol solution and 6 liters of 50% alcohol solution need to be mixed to yield 15 liters of a 26% solution.

Discussion

This problem can be generalized to other scenarios involving different concentrations and volumes. Understanding the principles behind this calculation is crucial for applications in pharmacy, chemistry, and many other fields where precise mixtures are required.

Frequently Asked Questions (FAQs)

Q: How do I calculate the amount of alcohol in a certain volume of solution with a given concentration?

A: Multiply the volume by the concentration expressed as a decimal to find the volume of alcohol. For example, 10 liters of 50% alcohol solution would contain 5 liters of pure alcohol (10 * 0.5).

Q: What if I want to mix three or more solutions to achieve a specific concentration?

A: Create a system of equations based on the total volume and the concentrations. Solve these equations simultaneously to find the required volumes.

Q: How can I check if my calculations are correct?

A: Double-check by calculating the total volume and total alcohol content. Ensure that the volume sums to the desired amount and the alcohol content matches the target concentration.