Problem 87 - Elementary Algebra Within Reach

Question

Use the following information. A fireplace is 93 inches wide. Each brick in the fireplace has a length of 8 inches, and there is \(\frac{1}{2}\) inch of mortar between adjoining bricks (see figure). Let \(n\) be the number of bricks per row. Explain why the number of bricks per row is the solution of the equation \(8 n+\frac{1}{2}(n-1)=93\).

Step-by-Step Solution

Answer

The equation \(8n + \frac{1}{2}(n-1) = 93\) represents the total width of the fireplace, with \(8n\) accounting for the total length of the bricks lined up along the row and \(\frac{1}{2}(n - 1)\) accounting for the total width of the mortar between the bricks. Since there's always one fewer mortar segment than the number of bricks, we subtract 1 from \(n\) in the latter part of the equation.

Step 1: Context understanding and Identifying Variables

In the exercise, we learn that the fireplace's total width is 93 inches, and each brick has a length of 8 inches. Now, in-between these bricks, there is a layer of mortar that measures \(\frac{1}{2}\) inch between each adjoining pair of bricks. Since the number of bricks per row is represented by \(n\), the total length of the bricks, when added together, would be \(8n\) inches.

Step 2: Incorporating the Mortar Width

At this point, we've covered the length that the bricks contribute to the total width of the fireplace. Now, we need to take into account the width of the mortar. Between each pair of bricks, there is a mortar width of \(\frac{1}{2}\) inch, so if we have \(n\) bricks, we will only have \(n-1\) segments of the mortar because there is no mortar after the last brick. Therefore, total mortar width would be \(\frac{1}{2}(n-1)\) inches.

Step 3: Formulating the Equation

The total width of the fireplace consists of the total length of the bricks and the total width of the mortar. So, when we add up the total length of the bricks (\(8n\)) and the total width of the mortar (\(\frac{1}{2}(n-1)\)), it should equal the total width of the fireplace, which is 93 inches. This gives us the equation \(8n + \frac{1}{2}(n-1) = 93\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. In our exercise, the expression \(8n + \frac{1}{2}(n-1)\) represents the total width of the brick fireplace. Here, \(8n\) accounts for the total length contributed by the bricks, and \(\frac{1}{2}(n-1)\) adds up the mortars' contribution to the overall width.

Algebraic expressions allow us to succinctly describe real-world situations in math language. Each component in the expression is crucial:

• \(8n\) - the term "8" represents the length of each brick in inches, and the variable "\(n\)" indicates the number of bricks per row.
• \(\frac{1}{2}(n-1)\) - given that there is \(\frac{1}{2}\) inch of mortar between every two bricks, "\(n-1\)" accounts for the gaps when "\(n\)" bricks are aligned in a row.

Understanding these components and how they build the expression is vital for simplifying and solving equations. Recognizing the real-world application of these expressions in calculating lengths, areas, and more enhances your capability in solving practical problems.

Problem-Solving

Problem-solving in math often involves translating a real-world scenario into a mathematical problem. Here, the task was to figure out how many bricks fit in a 93-inch wide fireplace given specific measurements.

Effective problem-solving follows these steps:

• Identify the known values: The fireplace width (93 inches), brick length (8 inches), and gap size (\(\frac{1}{2}\) inch).
• Determine the unknown: The number of bricks, \(n\), is the missing value in our setup.
• Set up an equation by expressing the total width as a sum of the brick lengths and the mortars' width.
• Solve the equation to find the unknown value. In this case, solving \(8n + \frac{1}{2}(n-1) = 93\) gives the solution for \(n\).

Such structured approaches make complex problems easier by breaking them into manageable parts. Practice helps in recognizing patterns that lead to quicker formulation of equations.

Equations with Variables

Equations with variables are statements of equality between two expressions. They use letters (often, variables like \(n\) in our example) to represent unknown quantities.

Solving equations with variables involves finding the value(s) of the variable that make the equation true. In the scenario presented, the equation \(8n + \frac{1}{2}(n-1) = 93\) needs \(n\) to be determined in such a way that the fireplace's dimensions are accurately represented. These equations are foundational in algebra, with several key features:

• Variables like \(n\) stand for unknown numbers we need to discover.
• Equality signifies a balance; both sides of the equation must be equal for the value of \(n\) to be correct.

By isolating the variable on one side of the equation, we can determine its value step by step. In practical terms, this entails using inverse operations to manipulate the equation, so it becomes easier to solve. Successfully handling such equations builds skills not just in basic mathematics but in logical, methodical thinking across disciplines.