Problem 8 - 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 A map is drawn so that every 2.5 inches on the map corresponds to an actual distance of 100 miles. If the actual distance between two cities is 350 miles, how far apart are they on the map?

Step-by-Step Solution

Answer

The cities are 8.75 inches apart on the map.

Step 1: Understand the Problem Statement

You are given that 2.5 inches on a map corresponds to 100 miles in real distance. We need to find how many inches correspond to an actual distance of 350 miles.

Step 2: Set Up the Proportion

The problem involves a direct proportion between map distance and actual distance. Set up a proportion: \[ \frac{2.5}{100} = \frac{x}{350} \]Here, \(x\) represents the map distance in inches for the actual distance of 350 miles.

Step 3: Solve the Proportion

To solve the proportion, cross-multiply and solve for \(x\):\[ 2.5 \times 350 = 100 \times x \] \[ 875 = 100x \] \[ x = \frac{875}{100} \] \[ x = 8.75 \] So, the two cities are 8.75 inches apart on the map.

Key Concepts

Understanding Word Problems

Word problems can sometimes feel intimidating because they blend words with mathematical elements. However, they are a fantastic way to apply math to real-world situations. The key is to carefully read and understand what the problem is asking.

• Identify what you are solving for: In our example, we need to find the distance between two cities on a map.
• Extract the given information: Know that 2.5 inches equals 100 miles.
• Understand the relationship between elements: We're dealing with a proportional relationship in this scenario.

By breaking down the problem into these steps, word problems can be made simpler and more approachable.

Exploring Map Reading

Map reading involves understanding the scale of a map, which is crucial for translating between map measurements and actual distances. Every map comes with a scale that shows the ratio of map units to real-world units.

• Scale Interpretation: A scale of 2.5 inches equals 100 miles helps us determine distances by using a proportion.
• Practical Application: Use proportions to find unknown distances, as seen with the 350 miles conversion in our problem.
• Map Scaling: The concept of scaling helps convert large real-world distances into manageable map distances.

Maps are not just navigational tools, but also a great way to practice mathematical reasoning and proportional thinking.

Strategies for Problem Solving

Effective problem-solving requires a clear approach. Here, we applied a strategic process to tackle the word problem:

• Identify Relationships: Recognizing the direct proportion relationship between map and real-world distances.
• Set Up Equations: Translating the given information into a mathematical proportion is crucial. Example: \( \frac{2.5}{100} = \frac{x}{350} \)
• Apply Mathematical Operations: Use cross multiplication to solve the equation.
• Check Results: Once a solution is found, ensure it logically fits the real-world scenario.

By following these steps, any complex problem can turn into a structured and manageable task.

Applying Mathematical Reasoning

Mathematical reasoning goes beyond mere number crunching; it's about making logical connections and inferences.

• Understand Proportions: Recognize that if two ratios are equivalent, they form a proportion.
• Cross Multiplication: A powerful technique to solve proportions, ensuring the equality holds true across different scales.
• Logical Deductions: Apply logic to verify that the resulting measurement makes sense, like ensuring that the 8.75 inches on the map aligns with the expected real-world distance.
• Real-world Connections: Bridging abstract math to tangible scenarios enhances comprehension and retention of concepts.

This systematic approach helps in effectively applying mathematics to real-world problems, enhancing both understanding and problem-solving skills.