Problem 30 - Prealgebra

Question

Google Earth The Google Earth image shows Disney World in Florida. A scale indicates that one inch is 200 meters. If the distance between Splash Mountain and the Jungle Cruise is 190 meters, what is the distance on the map in inches? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The distance on the map is 0.95 inches.

Step 1: Understand the Scale

The scale given in the problem is that 1 inch on the map equals 200 meters in real life.

Step 2: Set Up the Proportion

Since we want to find out how many inches correspond to 190 meters on the map, we set up the proportion \( \frac{1 \text{ inch}}{200 \text{ meters}} = \frac{x \text{ inches}}{190 \text{ meters}} \).

Step 3: Solve for x

Cross-multiply in the proportion to get the equation: \( 200x = 190 \).

Step 4: Simplify your Calculation

Divide both sides of the equation by 200 to isolate \( x \): \( x = \frac{190}{200} \).

Step 5: Simplify the Fraction

Calculate \( x = \frac{190}{200} = 0.95 \) inches.

Key Concepts

Map Scale

A map scale is a crucial tool that helps us understand the relationship between distances on a map and their corresponding real-world distances. Let's say you have a scale where 1 inch represents 200 meters, as given in our exercise. This means that for every inch you measure on the map, it equates to 200 meters out there in the real world.

Using a map scale is like having a special ruler that lets you translate map measurements into reality. This concept is especially useful when you're planning trips or understanding the layout of an area without actually being there. Imagine a tiny utopia where you can simply measure it out on a piece of paper!

• Helps in representing large areas on smaller maps.
• Maintains proportionality between the map and real-world dimensions.
• Essential for navigation and spatial planning.

Understanding map scale allows for accurate interpretations and practical applications, whether plotting a course on land or dreaming about your next adventure.

Cross-Multiplication

Cross-multiplication is a powerful mathematical technique used to solve proportions, making it especially useful in scaling problems like those involving map scales. When you have a proportion such as \( \frac{1 \, \text{inch}}{200 \, \text{meters}} = \frac{x \, \text{inches}}{190 \, \text{meters}} \), you can use cross-multiplication to find the unknown value \( x \).

To use cross-multiplication, you multiply across the equals sign diagonally: one numerator by the opposite denominator. So, in our example, you multiply 1 inch by 190 meters and set it equal to 200 meters multiplied by \( x \). This results in the equation \( 200x = 190 \).

• Streamlines solving proportions.
• Creates a straightforward algebraic equation.
• Makes it easy to find unknown values in proportional relationships.

Cross-multiplication simplifies the problem-solving process, and it is a technique you can use in many other situations involving proportions.

Unit Conversion

When solving problems involving scales and measurements, understanding unit conversion is essential. Let's break it down: unit conversion is the process of converting one unit of measure to another. This is frequently necessary when dealing with measurements like lengths, weights, or distances that are given in different units.

In our given problem, we initially have a distance of 190 meters that needs to be converted into an equivalent distance on the map, measured in inches. By employing the concept of unit conversion, we convert a real-world measurement (meters) into a scale on a map (inches).

• Ensures consistent and comparable measurements.
• Enables us to work between different units effectively.
• Crucial for interpreting maps and performing calculations accurately.

Mastering unit conversion is an indispensable skill, especially in fields like science, engineering, and even map reading, where accuracy and consistency are vital.