Problem 10 - Prealgebra

Question

On a set of architectural drawings for a new school, the scale is \(\frac{1}{2}\) inch \(=\) 9 feet. Find the actual length of each room. $$\begin{array}{|l|c|}\hline \quad \quad \quad \text { Room } & \text { Drawing Distance } \\ \hline \text { Computer Room } & 3.8 \text { inches } \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The actual length of the Computer Room is 68.4 feet.

Step 1: Understand the Scale

The scale given is \( \frac{1}{2} \text{ inch} = 9 \text{ feet} \). This means every \( \frac{1}{2} \) inch on the drawing corresponds to 9 feet in reality.

Step 2: Set Up the Proportion

To find the actual length of the Computer Room, set up a proportion. Let \( x \) represent the actual length in feet. The proportion is \( \frac{1}{2} : 9 = 3.8 : x \).

Step 3: Solve the Proportion

Multiply both sides of the proportion by 9 to eliminate the fraction: \( 3.8 \times 9 = \frac{x}{2} \times 9 \). This simplifies to \( x = 9 \times \frac{3.8}{\frac{1}{2}} \), which is \( x = 9 \times 7.6 \).

Step 4: Calculate the Actual Length

Calculate \( x = 68.4 \). This represents the actual length of the Computer Room in feet.

Key Concepts

Proportion in Scale Drawings

Understanding proportions is crucial when working with scale drawings. A scale drawing is a representation of an object or space, like a building or a room, where all the dimensions are reduced or enlarged by a certain factor. This factor is the scale. In the example exercise, the scale is given as \( \frac{1}{2} \) inch = 9 feet.

Proportions help us maintain accurate relationships between dimensions:

• The scale \( \frac{1}{2} : 9 \) tells us that for every \( \frac{1}{2} \) inch on the drawing, it equates to 9 feet in reality.
• To find a real-world measurement, you set up a proportion to compare the drawing measurement to its real-world counterpart.

In our example, the drawing measurement for the Computer Room is 3.8 inches. Using the proportion \( \frac{1}{2} : 9 = 3.8 : x \), where \( x \) is the length of the room in feet, helps bridge the gap between the scale drawing and the actual space.

Solving the proportion requires basic algebraic manipulation where the cross multiplication method is applied. Understanding and setting up proportions correctly ensures that translation from drawings to the real world is accurate and effective, ensuring the integrity of the entire process.

Conversion of Units

When working with scale drawings and proportions, converting units correctly is essential for accuracy. In the original exercise, we deal with inches in the drawing and feet in reality. Let's break this down.

The scale \( \frac{1}{2} \) inch = 9 feet directly tells us how to convert measurements from the drawing to real-world dimensions:

• To convert from drawing inches to real feet, understand that \( 1 \) inch on the drawing represents twice what \( \frac{1}{2} \) inch does. Therefore, it equals 18 feet in reality (9 feet per \( \frac{1}{2} \) inch).
• For our specific example, the drawing dimension is 3.8 inches. This needs to be scaled according to our understanding of the conversion rate.

Using these conversions guarantees that your calculations maintain the accurate size relationships defined by the scale, preventing potential errors. Always double-check the scales for correct unit conversions, particularly when translating drawings into real-world sized models or constructions.

Mathematical Reasoning

Mathematical reasoning allows us to make sense of numbers and the relationships within equations, especially when applied to real-world problems like scale drawings. Here’s how it comes into play in our exercise.

In the solution:

• We identified the connection between the drawer measurement (3.8 inches) and its real-world equivalent through a proportion.
• To further solve for \( x \) (the real length), mathematical reasoning guided us through multiplying both sides of the proportion equation by 9 to eliminate any fractions.

This step-by-step reasoning:

• Helps to eliminate confusion by guiding us through the logical sequence of mathematical operations.
• Ensures that each step follows logically from the last, resulting in a clear and correct solution.

By honing mathematical reasoning skills, students can confidently tackle similar exercises with assurance, applying logical methods to complex problems involving scales, conversions, and more.