Problem 55 - Basic College Mathematics

Question

Map Scaling. On a map, 1 in. represents \(240 \mathrm{mi} .\) What distance does \(\frac{2}{3}\) in. represent?

Step-by-Step Solution

Answer

160 miles

Step 1: Understand the Problem

Determine what distance \(\frac{2}{3}\) inches represents based on the given map scale where 1 inch corresponds to 240 miles.

Step 2: Set Up the Proportion

Establish the proportion between the known scale and the unknown distance. Since \(1 \text{ in}\) equals \(240 \text{ mi}\), we can use the proportion: \(\frac{1 \text{ in}}{240 \text{ mi}} = \frac{2/3 \text{ in}}{x \text{ mi}}\).

Step 3: Solve for the Unknown Distance

Solve for the unknown distance \(x\) by cross-multiplying and dividing: \[ \frac{1 \text{ in}}{240 \text{ mi}} = \frac{2/3 \text{ in}}{x \text{ mi}} \Rightarrow x = \frac{240 \text{ mi} \times 2/3 \text{ in}}{1 \text{ in}} \Rightarrow x = 160 \text{ mi}. \]

Key Concepts

headline of the respective core concept

Proportion is a relationship that shows how two quantities are related to each other. In this problem, we are comparing inches to miles. If we have a map where 1 inch represents 240 miles, we can set up a proportion to find out what \( \frac{2}{3} \) inch represents. This proportion is written as \( \frac{1 \text{ in}}{240 \text{ mi}} = \frac{2/3 \text{ in}}{x \text{ mi}} \). By setting up this equation, we can solve for any missing value when given a certain scale. Whether you are increasing or decreasing the size, the proportional relationship remains constant.

headline of the respective core concept

Distance measurement on a map involves understanding the relationship between the map's scale and the actual distance. In our exercise, the map tells us that 1 inch equals 240 miles. When we measure \( \frac{2}{3} \) inch on the map, we need to calculate the corresponding real-world distance. This requires understanding the scale and using it to convert the map distance to actual miles. The known scale gives us a proportion, which is a crucial step in determining the actual distance.

headline of the respective core concept

Cross-multiplication is a technique used to solve proportions. It involves multiplying across the equals sign in the proportion equation. In our problem, the proportion \( \frac{1 \text{ in}}{240 \text{ mi}} = \frac{2/3 \text{ in}}{x \text{ mi}} \) can be solved by cross-multiplying: \[ 1 \text{ in} \times x = 240 \text{ mi} \times \frac{2}{3} \text{ in} \] This simplifies to \[ x = \frac{240 \text{ mi} \times 2/3 \text{ in}}{1 \text{ in}} \], giving us \[ x = 160 \text{ mi} \]. This method ensures that we correctly find the unknown distance, making cross-multiplication an essential process in solving scale and map-related problems.