Problem 88 - 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. Find the number of bricks per row in the fireplace.

Step-by-Step Solution

Answer

The number of bricks that can fit in a row of the fireplace is 10.

Step 1: Calculate the space occupied by one brick including the adjoining mortar

The total space occupied by a brick including the mortar is 8 inches (brick) + \(\frac{1}{2}\) inch (mortar) = \(8.5\) inches.

Step 2: Find the number of bricks based on the total fireplace width

To find the number of bricks that will fit, divide the total space available (93 inches) by the space used by each brick (8.5 inches). Hence, \(n = \frac{93}{8.5}\) = 10.94.

Step 3: Round down the number of bricks

The value obtained, 10.94, must be rounded down to the nearest whole number because you cannot use a fraction of a brick. So, \(\lfloor 10.94 \rfloor = 10\). Therefore, ten bricks will fit in the fireplace row.

Key Concepts

Division

Division is a crucial mathematical operation in elementary algebra that helps us split a total into equal parts. In this context, division is used to determine how many bricks can fit in the width of the fireplace, considering the size of each brick along with the mortar.

• Understanding Division in Context: We begin by dividing the total width of the fireplace, 93 inches, by the width of a single brick plus the mortar, 8.5 inches. This helps us figure out how many complete sections of 8.5 inches can fit across the total space.
• Practical Application of Division: Once we divide, we get a quotient of 10.94. This number represents how many complete bricks, with mortar, could theoretically fit across the fireplace if partial bricks were possible.

Division lays the groundwork for further adjustments based on real-world constraints, such as the impossibility of using parts of a brick, leading us to the next concept—rounding numbers.

Rounding Numbers

Rounding numbers is an essential concept when dealing with numbers that result from division and cannot be used as fractions in practical scenarios. In our fireplace example, after calculating the number of bricks as 10.94, we need to make this number whole.

• Why Round Numbers: Since you cannot use a partial brick, you must round down to ensure all bricks are fully used and fit within the fireplace. This is where we round 10.94 down to 10, allowing complete bricks to fit with the desired width of mortar.
• Methods of Rounding: In this situation, rounding down is essential. Most mathematical problems require rounding up or down to the nearest whole number, depending on the needs of the situation.

Rounding simplifies the number to match it to real-world applications, making it possible to use the derived numbers practically in construction or design.

Problem Solving

Problem solving in algebra often requires applying multiple mathematical concepts to reach a practical solution. In this case, the challenge was to determine how many bricks fit per row within a fixed width.

• Identifying the Problem: The primary problem was to figure out how many bricks, with added mortar, will fit across 93 inches.
• Applying Mathematical Operations: By using division to calculate the raw number of bricks, and rounding numbers to create practical solutions, mathematics helps solve real-life challenges accurately.
• Logical and Sequential Thinking: Each step—figuring out the total width usage per brick, dividing, and then rounding—is part of a logical sequence that ensures a comprehensive solution.

Successful problem solving involves breaking down the task into smaller, manageable parts, understanding each part thoroughly, and logically working through the math until you reach a solution that fits the real-world needs.