Problem 24 - Elementary and Intermediate Algebra

Question

Higher quality paint typically contains more solids. Grant has available paint that contains \(45 \%\) solids and paint that contains \(25 \%\) solids. How much of each should he use to create 20 gal of paint that contains \(39 \%\) solids?

Step-by-Step Solution

Answer

Use 14 gallons of 45% solids paint and 6 gallons of 25% solids paint.

Step 1: Define Variables

Let \( x \) be the gallons of 45% solids paint and \( y \) be the gallons of 25% solids paint. The total amount of paint required is 20 gallons, so we have the equation \( x + y = 20 \).

Step 2: Set Up the Solids Equation

The paint mixture needs to contain 39% solids. Therefore, the total amount of solids in the mixture is \(0.39 \times 20 \) gallons. This leads to the equation \( 0.45x + 0.25y = 7.8 \).

Step 3: Solve the System of Equations

Solve the system of linear equations: \( x + y = 20 \) and \( 0.45x + 0.25y = 7.8 \).

Step 4: Substitute and Solve for One Variable

From the first equation, solve for \( y \): \( y = 20 - x \). Substitute \( y \) in the second equation: \( 0.45x + 0.25(20 - x) = 7.8 \).

Step 5: Simplify and Solve for \( x \)

Expand and simplify the equation: \( 0.45x + 5 - 0.25x = 7.8 \), which simplifies to \( 0.2x + 5 = 7.8 \). Solving for \( x \) gives \( x = \frac{2.8}{0.2} = 14 \).

Step 6: Find \( y \)

Using \( y = 20 - x \), calculate \( y = 20 - 14 = 6 \).

Key Concepts

systems of equations

When we talk about systems of equations, we refer to multiple equations that share the same variables. In our example, we have two equations. One represents the total amount of paint: \( x + y = 20 \), and the other represents the total amount of solids in the paint: \( 0.45x + 0.25y = 7.8 \). These equations help us find the values for the two unknowns, \( x \) and \( y \). Systems of equations are common in real-world problems like mixing solutions or planning budgets. Solving them typically involves combining the equations to isolate one of the variables.

percentage calculations

Percentage calculations involve expressing a number as a fraction of 100. They are useful in many contexts, including financial statements, statistical data, and concentration of solutions. In this problem, we are dealing with percentages of solids in paint. The 45% and 25% represent the concentration of solids in different paints. To find out the required amount of each type of paint, you need to set equations based on these percentages. For instance, 45% of \( x \) plus 25% of \( y \) must equal 39% of 20 gallons. This translates into the equation: \( 0.45x + 0.25y = 7.8 \).

substitution method

The substitution method is one way to solve a system of linear equations. This approach involves solving one of the equations for one variable and then substituting that expression into the other equation. In our case, after establishing the equations \( x + y = 20 \) and \( 0.45x + 0.25y = 7.8 \), we solve the first equation for \( y \): \( y = 20 - x \). We then substitute this expression for \( y \) into the second equation: \( 0.45x + 0.25(20 - x) = 7.8 \). This substitution helps us to isolate and solve for \( x \). Once \( x \) is known, substituting its value back into the equation for \( y \) gives us the value of \( y \).

linear algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear equations. The system of equations in our problem, \( x + y = 20 \) and \( 0.45x + 0.25y = 7.8 \), is an example of linear equations where the variables appear in a linear form. In simpler words, none of the variables are raised to a power other than one. Solving these equations involves techniques from linear algebra, such as substitution or elimination. These methods help us find the exact values that satisfy both equations simultaneously. Mastering these techniques is crucial for solving more complex problems in various fields such as engineering, physics, and economics.