Problem 61 - Elementary Algebra

Question

A blueprint has a scale in which 1 inch represents 6 foet. Find the dimensions of a rectangular room that measures \(2 \frac{1}{2}\) inches by \(3 \frac{1}{4}\) inches on the blueprint.

Step-by-Step Solution

Answer

The room's dimensions are 15 feet by 19.5 feet.

Step 1: Understand the Scale

The blueprint uses a scale where 1 inch equals 6 feet in real life. This means every inch on the drawing corresponds to 6 feet of actual length.

Step 2: Convert Blueprint Length

For the first dimension, convert the blueprint length of \(2 \frac{1}{2}\) inches to feet. Write \(2 \frac{1}{2}\) as a fraction: \(\frac{5}{2}\). Multiply \(\frac{5}{2}\) by 6 to find the real-life length:\[ \frac{5}{2} \times 6 = \frac{30}{2} = 15 \text{ feet} \]

Step 3: Convert Blueprint Width

For the second dimension, convert the blueprint width of \(3 \frac{1}{4}\) inches to feet. Write \(3 \frac{1}{4}\) as a fraction: \(\frac{13}{4}\). Multiply \(\frac{13}{4}\) by 6 to find the real-life width:\[ \frac{13}{4} \times 6 = \frac{78}{4} = 19.5 \text{ feet} \]

Step 4: State the Real Dimensions

Summarize the converted dimensions. The actual dimensions of the room are 15 feet by 19.5 feet.

Key Concepts

Fraction Multiplication

Fraction multiplication is a fundamental skill, especially when converting measurements in a blueprint. To multiply fractions, you simply multiply the numerators together and the denominators together. This means

• Numerator: The top number of a fraction, it shows how many parts you have.
• Denominator: The bottom number, it shows how many equal parts the whole is divided into.

For instance, if you have a fraction \( \frac{5}{2} \) (which is the mixed number \( 2 \frac{1}{2} \) converted to an improper fraction), and you need to multiply it by another number, such as 6 in our blueprint scale conversion, the process is straightforward. Multiply the numerator of the fraction by the integer: \( 5 \times 6 = 30 \). Then divide the result by the denominator: \( \frac{30}{2} = 15 \). This skill allows you to handle various problems where you scale measurements or areas directly.

Scale Drawings

Scale drawings are used to represent large objects accurately on smaller pieces of paper. Each measurement on a scale drawing corresponds to a real-world measurement. It's important to understand that:

• The scale is a ratio that dictates the proportionate difference between the drawing and its actual counterpart.
• A scale of 1 inch = 6 feet means that each inch on the blueprint equals 6 feet in real life.

When working with a scale drawing, you need to know how to convert dimensions from the blueprint to real-life dimensions. This often involves multiplying the measurement on the drawing by the scale conversion factor, demonstrating how proportionate scaling works efficiently for architects and engineers.

Metric Conversion

While the problem uses feet instead of metric units, understanding metric conversion is crucial. Metric conversions involve translating between different units of measurement within the metric system, such as meters and centimeters. It is important to:

• Know the basic units and their conversions: 1 meter = 100 centimeters, 1 kilometer = 1000 meters.
• Use multiplication or division to convert between units, depending on whether you are converting to a larger or smaller unit.

While this exercise primarily deals with scale in feet, similar principles apply when converting metric units in blueprints or any technical drawings, ensuring accurate and understandable measurements.