Problem 257 - Elementary Algebra

Question

Jotham needs 70 liters of a \(50 \%\) alcohol solution. He has a \(\begin{array}{lllll}30 \% & \text { and an } & 80 \% & \text { solution }\end{array}\) available. How many liters of the \(30 \%\) and how many liters of the \(80 \%\) solutions should he mix to make the \(50 \%\) solution?

Step-by-Step Solution

Answer

42 liters of 30% solution and 28 liters of 80% solution.

Step 1: Identify Variables

Let \(x\) be the number of liters of the 30% alcohol solution and \(y\) be the number of liters of the 80% alcohol solution. The goal is to find the values of \(x\) and \(y\) that make up the 70 liters of 50% alcohol solution.

Step 2: Set Up Equations

The first equation comes from the total volume of the solution: \(x + y = 70\). The second equation relates to the concentration of alcohol: \(0.30x + 0.80y = 0.50 \times 70\).

Step 3: Simplify the Concentration Equation

Simplify the concentration equation: \(0.30x + 0.80y = 35\).

Step 4: Solve the System of Equations

Solve the system of linear equations: 1) \( x + y = 70 \) 2) \(0.30x + 0.80y = 35\). Solve the first equation for \(y\): \(y = 70 - x\). Substitute \(y\) into the second equation to get: \(0.30x + 0.80(70 - x) = 35\).

Step 5: Substitute and Solve for x

Substitute \(y = 70 - x\) in the second equation: \(0.30x + 0.80(70 - x) = 35\), which simplifies to \(0.30x + 56 - 0.80x = 35\). Combine like terms: \(-0.50x + 56 = 35\). Solve for \(x\):\(-0.50x = -21\) which implies \(x = 42\).

Step 6: Solve for y

Using \(y = 70 - x\) and \(x = 42\), substitute \(x\) into the equation to find \(y\): \(y = 70 - 42 = 28\).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra, serving as a backbone for solving various real-world problems. In essence, a linear equation is an equation between two variables that gives a straight line when plotted on a graph. The general form of a linear equation in two variables, say x and y, is:
ax + by = c,
where 'a', 'b', and 'c' are constants. In our example, we set up two linear equations to represent the problem scenario. The first equation, x + y = 70, represents the total volume of liquid we need. The second equation, 0.30x + 0.80y = 35, represents the concentration condition.
Steps to solve linear equations:

• Identify the variables and constants.
• Set up equations based on the given conditions.
• Simplify the equations, if necessary.
• Solve the equations using substitution or elimination method.

By understanding and following these steps, solving linear equations becomes more manageable.

Applying Algebraic Solutions

Algebraic solutions involve manipulating mathematical expressions to find unknown values. In our problem, we are tasked with solving a system of linear equations to determine the required volumes of two different concentrations.
The system of equations we formed is:
1) x + y = 70
2) 0.30x + 0.80y = 35
To solve these equations, we used the substitution method. This involves solving one equation for one variable and then substituting that expression into the other equation. Here's the process in detail:

• Solve for y in the first equation: y = 70 - x
• Substitute y in the second equation: 0.30x + 0.80(70 - x) = 35
• Simplify the expression: 0.30x + 56 - 0.80x = 35
• Combine like terms: -0.50x + 56 = 35
• Solve the equation for x: -0.50x = -21, so x = 42

After finding x, use its value in the first equation to find y: y = 70 - 42 = 28.
This way, algebraic manipulations help us determine the exact values for the variables.

Mastering Percent Concentration

Percent concentration is a measure that describes the amount of a substance in a mixture, expressed as a percentage of the total volume. It's commonly used in chemistry to represent the concentration of solutions.
To solve mixture problems, understanding percent concentration is crucial. In our exercise, we deal with different alcohol solutions measured by their percent concentrations:

• 30% alcohol solution
• 80% alcohol solution
• Target: 50% alcohol solution

To create a solution with a specific percent concentration, we need to combine two solutions in precise amounts.
The formula for calculating the concentration when mixing solutions is:
concentration = (amount of substance in solution / total volume of solution) × 100
For our problem, we set up the concentration equation: 0.30x + 0.80y = 0.50 × 70 (the desired 50% alcohol in 70 liters). Simplifying this, we get: 0.30x + 0.80y = 35.
By solving this and the volume equation, we manage to find the right amounts of each solution needed, achieving the desired percent concentration effectively.