Problem 46 - A Graphical Approach to College Algebra

Question

Mixing Glue\(\quad\) A glue company needs to make some glue that it can sell for \(\$ 120\) per barrel. It wants to use 150 barrels of glue worth \(\$ 100\) per barrel, along with some glue worth \(\$ 150\) per barrel and glue worth \(\$ 190\) per barrel. It must use the same number of barrels of \(\$ 150\) and \(\$ 190\) glue. How much of the \(\$ 150\) and \(\$ 190\) glue will be needed? How many barrels of \(\$ 120\) glue will be produced

Step-by-Step Solution

Answer

30 barrels each of \$150 and \$190 glue are used, totaling 210 barrels.

Step 1: Define Variables

Let \( x \) be the number of barrels of glue worth \( \\(150 \) per barrel and \( \\)190 \) per barrel. The problem states that these are equal, so we use the same variable for both.

Step 2: Set Up the Equation

The company wants to mix 150 barrels of \( \\(100 \) glue with \( x \) barrels each of \( \\)150 \) and \( \\(190 \) glue to create a glue worth \( \\)120 \) per barrel. The total number of barrels produced will be \( 150 + 2x \). The equation representing the total value of the glue is:\[ 100(150) + 150x + 190x = 120(150 + 2x) \]

Step 3: Simplify and Solve the Equation

First, simplify both sides: \[ 15000 + 150x + 190x = 18000 + 240x \]Combine like terms: \[ 15000 + 340x = 18000 + 240x \]Subtract \(240x\) from both sides: \[ 15000 + 100x = 18000 \]Subtract 15000 from both sides: \[ 100x = 3000 \]Divide by 100: \[ x = 30 \]

Step 4: Calculate Total Barrels Produced

Now that we know \( x = 30 \), calculate the total number of barrels produced: \[ 150 + 2x = 150 + 2(30) = 210 \]

Key Concepts

Linear Equations

In algebra, a linear equation is an equation that forms a straight line when graphed, with a general format of \( ax + b = c \). Linear equations are crucial for solving word problems where relationships between quantities need to be represented mathematically.

For the given glue problem, the total cost and the number of barrels are related linearly. By aligning the cost per barrel to form the equation:

• Combine the cost of different types of glue: \( 150 \times 150 + 150x + 190x \)
• Set this equal to the value they need to achieve with mix: \( 120(150 + 2x) \)

This forms a linear equation that helps in determining the number of barrels needed. Real-life situations, such as this one, often require translating words into mathematical expressions, offering a practical understanding of linear relationships in day-to-day operations.

Variable Assignment

Assigning variables is a key step when tackling word problems in algebra. Variables are symbols used to represent unknown quantities, making them flexible and powerful tools for solving equations.

In this problem, we decided to use the variable \( x \) for the number of barrels of glue worth \( \\(150 \) and \( \\)190 \) each because their quantities are equal. This choice simplifies the problem, transforming it into a manageable mathematical equation.

• Using a single variable reduces complexity, as it seamlessly holds the value for both quantities, making the equation easier to solve.
• This way of assigning variables ensures that the underlying relationships between these elements are maintained throughout the problem-solving process.

Being strategic about variable assignment is essential, particularly in problems involving multiple unknowns.

Problem-Solving Steps

Solving word problems systematically involves several clear steps, each building on the previous one. This approach ensures that you understand the problem and organize your thoughts effectively.

For instance, the glue problem follows a well-defined process:

• Define Variables: Establish variables for unknown quantities, in this case, \( x \) for the two kinds of glue.
• Formulate an Equation: Translate the problem's conditions into an equation, linking the quantities and the total value of glue.
• Simplify the Equation: Remove unnecessary components by combining like terms and isolating the variable.
• Solve for the Variable: Perform operations systematically to find the value of \( x \), representing the number of barrels needed.
• Validate and Interpret the Solution: Calculate the total results and interpret what this means in terms of the problem's original context.

This methodical approach aids in breaking down complex real-world problems into simpler, solvable steps, enhancing problem-solving skills across various mathematical challenges.