Problem 15 - Prealgebra

Question

What is the scale factor if the scale is 8 inches \(=1\) foot?

Step-by-Step Solution

Answer

The scale factor is \( \frac{2}{3} \).

Step 1: Understand the Units and Context

Before finding the scale factor, identify the units involved. The exercise states that 8 inches is equivalent to 1 foot according to the given scale.

Step 2: Convert Units to a Common Measure

Convert the larger unit (foot) to inches to make comparison easier. Since 1 foot equals 12 inches, we can express the 1 foot as 12 inches.

Step 3: Set Up a Ratio

Set up the ratio of the scale as the proportion of the given inches to inches in a foot. This can be expressed as a fraction: \( \frac{8\text{ inches}}{12\text{ inches}} \).

Step 4: Simplify the Ratio

Simplify the ratio \( \frac{8}{12} \) by dividing both the numerator and denominator by their greatest common divisor, which is 4. This simplifies to \( \frac{2}{3} \).

Step 5: Interpret the Scale Factor

The ratio \( \frac{2}{3} \) represents the scale factor. This means that every 8 inches in the model equals 12 inches (1 foot) in reality.

Key Concepts

Understanding Ratios

Ratios are a fundamental concept used to compare two quantities. They are often represented as fractions or with a colon separating the two numbers, such as 8:12. In the context of scale factors, ratios help us understand how one measurement relates to another. For example, if something is 8 inches in a model and is equivalent to 1 foot (or 12 inches) in real life, the ratio is used to express this relationship. To set up a ratio:

• Identify the two numbers to compare.
• Write them as a fraction or with a colon, ensuring both units are the same.

Understanding ratios makes it easier to interpret scale factors and conversion in measurement-related problems.

The Art of Unit Conversion

Unit conversion is crucial when comparing measurements that use different units. In this exercise, the comparison involves inches and feet. Since 1 foot equals 12 inches, converting the foot to inches allows for a direct comparison. Always make sure you're using equivalent units to ensure your comparisons are accurate. Steps for unit conversion:

• Know the conversion factor (e.g., 1 foot = 12 inches).
• Apply the conversion factor to change units (e.g., multiply feet by 12 to get inches).
• Use the converted measurement in your ratio or equation.

This consistency in units is the key to making meaningful comparisons and calculations.

Simplifying Fractions

Simplifying fractions is an essential skill in this problem, as it reveals the scale factor. The goal is to reduce a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example, we simplified the fraction \( \frac{8}{12} \) by dividing by 4, the GCD, resulting in \( \frac{2}{3} \). Steps to simplify fractions:

• Find the GCD of the numerator and denominator.
• Divide both parts of the fraction by the GCD.
• Write the simplified fraction.

Simplification makes the fraction easier to work with and clearly indicates the relationship between the quantities.

Measurement Units and Their Importance

Understanding measurement units is vital for interpreting real-world quantities. This exercise involves inches and feet, common units for length in the Imperial system. Knowing how these units relate helps in correctly setting up ratios, converting units, and determining scale factors. Familiarize yourself with common measurement conversions:

• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 mile = 5280 feet

Being comfortable with these measurements allows for seamless conversions and accurate calculations, making it easier to solve scale-related problems.