Problem 50 - Elementary Algebra

Question

Suppose that one solution is \(50 \%\) alcohol and another solution is \(80 \%\) alcohol. How many liters of each solution should be mixed to make \(10.5\) liters of a \(70 \%\) alcohol solution?

Step-by-Step Solution

Answer

3.5 liters of the 50% solution, 7 liters of the 80% solution.

Step 1: Understanding the Problem

We need to mix two alcohol solutions of different percentages, one with 50% alcohol and the other with 80% alcohol, to create a 10.5-liter solution that is 70% alcohol. We need to determine the volume of each of the two solutions to use.

Step 2: Setting Up Variables

Let \( x \) be the volume in liters of the 50% solution, and \( y \) be the volume in liters of the 80% solution. The total volume of the mixture is 10.5 liters, so we can set up the equation: \( x + y = 10.5 \).

Step 3: Setting Up the Alcohol Content Equation

The 50% solution contributes \( 0.50x \) liters of alcohol and the 80% solution contributes \( 0.80y \) liters of alcohol. Combined, these should equate to 70% of the 10.5 liters of the new solution: \( 0.50x + 0.80y = 0.70 \times 10.5 \).

Step 4: Solving the System of Equations

We have the two equations: 1. \( x + y = 10.5 \), and 2. \( 0.50x + 0.80y = 7.35 \) (since \( 0.70 \times 10.5 = 7.35 \)). We can solve these equations simultaneously to find \( x \) and \( y \).

Step 5: Solving for One Variable

From equation 1, solve for \( x \): \( x = 10.5 - y \).

Step 6: Substituting to Find the Other Variable

Substitute \( x = 10.5 - y \) into equation 2: \( 0.50(10.5 - y) + 0.80y = 7.35 \). Simplifying, we get \( 5.25 - 0.50y + 0.80y = 7.35 \) which simplifies to \( 0.30y = 2.10 \).

Step 7: Solving for \( y \)

Solving \( 0.30y = 2.10 \), we find \( y = \frac{2.10}{0.30} = 7 \).

Step 8: Finding \( x \)

Substitute \( y = 7 \) back into the equation \( x = 10.5 - y \). Therefore, \( x = 10.5 - 7 = 3.5 \).

Step 9: Conclusion

The mixture should consist of 3.5 liters of the 50% alcohol solution and 7 liters of the 80% alcohol solution.

Key Concepts

System of Equations

In mixture problems like the one we're tackling, the use of a system of equations is crucial. A system of equations consists of two or more equations that share variables. When these equations are solved together, they provide a solution that satisfies all involved relationships.

In our example problem, we define two variables:

• \( x \), representing the number of liters of the 50% alcohol solution.
• \( y \), representing the number of liters of the 80% alcohol solution.

The problem gives us conditions that can be expressed as two distinct equations:

• The total volume of the mixture is 10.5 liters: \( x + y = 10.5 \).
• The amount of pure alcohol in the mixture must equal 70% of 10.5 liters: \( 0.50x + 0.80y = 7.35 \).

These equations form a system because they are both connected by the same variables, \( x \) and \( y \). Solving this system will give the values of \( x \) and \( y \), ensuring both the total volume and the desired alcohol concentration are achieved.

Percent Concentration

Understanding percent concentration is key to solving mixture problems, as it tells us how much of a solution is pure substance. In the context of alcohol solutions, percent concentration indicates the proportion of alcohol in the solution. For instance, a 50% alcohol solution means that half of the solution is pure alcohol, while the rest is something else, usually water.

In this exercise, you're given solutions with concentrations of 50% and 80% alcohol. The objective is to mix them to form a new solution with an exact concentration of 70% alcohol. The problem requires you to consider how these different concentrations combine.

• The 50% solution contributes \(0.50x\) liters of alcohol, where \(x\) is its volume.
• The 80% solution contributes \(0.80y\) liters of alcohol, where \(y\) is its volume.
• The resulting mixture should have \(0.70 \times 10.5\) or 7.35 liters of pure alcohol.

The concentration dictates how much pure substance will be present, making it possible to calculate how various solutions mix.

Algebraic Solution Methods

Solving the problem using algebra involves expressing the conditions as algebraic equations and then manipulating these equations to find the desired values. Here's a step-by-step guide on how we handled this task.

First, we derived two equations from the problem conditions:

• \( x + y = 10.5 \) relates to the total volume.
• \( 0.50x + 0.80y = 7.35 \) relates to the alcohol content.

One practical method to solve these equations is substitution. From the first equation, solve for one variable, such as \( x = 10.5 - y \). Substitute this expression into the second equation, allowing you to solve for \( y \).

After simplifying, you find the value of \( y \):

• \( 0.30y = 2.10 \) leading to \( y = 7 \).

With \( y \) known, substitute back to find \( x = 3.5 \). Using these values ensures that both conditions are satisfied: the mixture is 10.5 liters in total, and it has a 70% alcohol concentration. Algebraic methods like substitution showcase the power of algebra in breaking down and solving complex real-world problems.