Problem 7 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Map Reading The scale on a map indicates that 1 inch corresponds to an actual distance of 95 miles. Two cities are 4.5 inches apart on the map. What is the actual distance between the two cities?

Step-by-Step Solution

Answer

The actual distance is 427.5 miles.

Step 1: Understand the problem

We need to find the actual distance between two cities using the scale given on a map. The map indicates that 1 inch corresponds to 95 miles.

Step 2: Set up the known proportion

We have a scale of 1 inch to 95 miles. This can be expressed as the proportion \( \frac{1 \text{ inch}}{95 \text{ miles}} \).

Step 3: Set up the unknown proportion

We know the distance between the cities on the map is 4.5 inches, but we need to find the actual distance in miles. This is expressed as the proportion \( \frac{4.5 \text{ inches}}{x \text{ miles}} \), where \( x \) is the unknown actual distance.

Step 4: Write the proportion equation

Using the known proportion and the unknown proportion, set up the equation: \( \frac{1}{95} = \frac{4.5}{x} \).

Step 5: Solve the proportion equation

Cross-multiply to solve for \( x \). This gives us \( 1 \cdot x = 95 \cdot 4.5 \). Simplify to get \( x = 427.5 \text{ miles} \).

Step 6: Conclude the solution

We've found that the actual distance between the two cities is 427.5 miles based on the map scale.

Key Concepts

Map Reading

When you look at a map, you are actually looking at a scaled-down representation of a much larger area. Maps are incredibly useful tools for understanding geography, planning routes, and solving problems like the one in the exercise. One important aspect of map reading is understanding and correctly interpreting the scale.

• Scale Information: The scale on a map tells you about the relationship between a certain distance on the map and the actual distance in real life. For example, a scale of 1 inch to 95 miles indicates that for every inch you measure on the map, the real-world distance is 95 miles.
• Interpreting Distances: To find out how far two locations are apart in reality, you measure the distance between them on the map and then use the scale to convert that measurement into an actual distance.

Reading and interpreting maps can seem challenging, but with practice, it becomes easier to understand various scales and accurately determine distances.

Distance Calculation

Calculating distances using map scales involves some straightforward math. Once you understand the scale, you can easily convert map measurements into real-world distances by applying that scale.

• Measuring Distance on the Map: Use a ruler to measure the distance between the two points you're interested in. Record this measurement carefully.
• Applying the Scale: With the scale known, multiply the map distance by the scale factor to find the real-world distance. In our exercise, 4.5 inches on the map equals 4.5 times 95 miles in reality.

This method allows you to quickly and effectively find out how far places are from each other without physically measuring that distance.

Cross-Multiplication

Cross-multiplication is a key technique in solving proportion problems, like finding actual distances on maps. It helps you find an unknown value in a proportion by multiplying across the equals sign.

• Proportion Setup: A proportion is an equation that states two ratios are equal. For the map problem, the ratios are the map scale and the measured distance.
• Cross-Multiplying: To solve for the unknown, cross-multiply the terms. For instance, with the equation \( \frac{1}{95} = \frac{4.5}{x} \), multiply 1 by \( x \) and 95 by 4.5 to get \( 1 \cdot x = 95 \cdot 4.5 \).
• Solving for the Variable: After cross-multiplying, solve for the variable by performing any necessary division or simplification. In this case, \( x = 427.5 \).

This method is quick, efficient, and makes solving similar problems much easier. It's especially helpful in translating real-world scenarios into mathematical equations for easier understanding and solution.