Problem 42 - Prealgebra

Question

MAPS On a map of a state park, the scale is 0.5 inch \(=1.5\) miles. Find the actual distance from the ranger's station to the beach if the distance on the map is 1.75 inches. (lesson \(6-4\) )

Step-by-Step Solution

Answer

The actual distance is 5.25 miles.

Step 1: Understanding the Map Scale

The map has a scale of 0.5 inches representing 1.5 miles.

Step 2: Set Up the Proportion

We know that 0.5 inches on the map equals 1.5 miles in reality. If the map shows a distance of 1.75 inches, we can set up a proportion: \ \frac{0.5 \, \text{inches}}{1.5 \, \text{miles}} = \frac{1.75 \, \text{inches}}{x \, \text{miles}} \ where \(x\) is the actual distance in miles.

Step 3: Solve the Proportion

To solve for \(x\), cross-multiply and solve the equation: \(0.5 \times x = 1.75 \times 1.5\)\\(0.5x = 2.625\) miles.

Step 4: Isolate \(x\)

Divide both sides by 0.5 to solve for \(x\): \\(x = \frac{2.625}{0.5} = 5.25\).

Step 5: Interpret the Result

The actual distance from the ranger's station to the beach is 5.25 miles.

Key Concepts

Map Scales

Understanding map scales is key to navigating maps accurately. A map scale is a tool that helps us convert distances shown on a map to real-world distances.
A scale might be written as "1 inch represents 5 miles," which means for every inch you measure on the map, it corresponds to 5 miles in reality. Some typical scales include:

• A fractional scale, like 1:24,000, which shows that one unit on the map is equal to 24,000 of the same units in the real world.
• A graphical scale, shown as a bar or line on the map divided into labeled segments to indicate actual distances.

In our example, with a scale of 0.5 inch equaling 1.5 miles, understanding how to apply this scale means that measuring 0.5 inch translates directly to covering 1.5 miles on the ground.
Comprehending these scales is crucial for determining distances accurately, particularly when planning routes or estimating travel times.

Distance Conversion

Distance conversion is vital when working with maps and real-world navigation. It involves converting measurements taken in one unit to another unit, usually from a smaller unit to a larger one or vice versa. In our exercise, we needed to convert 1.75 inches on the map into miles using the scale provided. The conversion process involved setting up a proportion: since 0.5 inch equals 1.5 miles, 1.75 inches equals an unknown number of miles, which is what we worked out by using the proportion:\[\frac{0.5 \, \text{inches}}{1.5 \, \text{miles}} = \frac{1.75 \, \text{inches}}{x \, \text{miles}}\]Cross-multiplying helps solve the equation, translating the inch measurement into the desired mile measure. Mastering distance conversion is not just required for map-based problems but is also applicable in various real-world scenarios, like converting measurements when traveling to places using different measurement systems.

Mathematical Problem Solving

Mathematical problem solving involves logical reasoning to find solutions to problems. In our exercise, we dealt with proportions, requiring specific steps to determine the actual distance between two locations. Breaking down the problem:

• First, understand the map scale provided.
• Set up a proportion based on that scale to formulate the relationship between map distances and real-world distances.
• Apply basic algebra to solve for the unknown, which in our case was the actual distance.

In our example, using proportions allowed for determining the actual miles represented by a given map measurement. Solving this problem required cross-multiplication to develop and simplify the equation, isolating the variable:\[0.5 \times x = 1.75 \times 1.5\]Solving it by performing division, we found the distance in real terms was 5.25 miles. Learning these steps enhances problem-solving abilities, empowering one to tackle similar real-world problems with confidence and clarity.