Problem 26 - Prealgebra

Question

Model Trains The size of a model train relative to an actual train is referred to as its scale. Each scale is associated with a ratio as shown in the table. For example, an HO model train has a ratio of 1 to 87, meaning it is \(\frac{1}{87}\) as large as an actual train. Map Reading The scale on a map is drawn so that 4.5 inches corresponds to an actual distance of 250 miles. If two cities are 7.25 inches apart on the map, how many miles apart are they? (Round to the nearest tenth.)

Step-by-Step Solution

Answer

The cities are approximately 402.8 miles apart.

Step 1: Understand the Map Scale

The map scale provided is 4.5 inches on the map corresponds to 250 miles in reality. This means every 1 inch on the map translates to a certain number of actual miles.

Step 2: Calculate the Scale Factor

To find out how many miles each inch on the map represents, divide the actual distance (250 miles) by the map distance (4.5 inches). This gives the scale factor \( \frac{250}{4.5} \).

Step 3: Determine Distance per Inch

Calculate \( \frac{250}{4.5} \) to find out how many miles 1 inch represents on the map. Perform this division: \( \frac{250}{4.5} \approx 55.5556 \). Therefore, 1 inch on the map equals approximately 55.6 miles.

Step 4: Calculate the Actual Distance Between Cities

Given that the two cities are 7.25 inches apart on the map, multiply 7.25 by the scale factor found in Step 3: \( 7.25 \times 55.5556 \approx 402.778 \) miles.

Step 5: Round the Result

Round the distance calculated in Step 4 to the nearest tenth. Therefore, 402.778 miles rounds to 402.8 miles.

Key Concepts

Scale Factor Calculation

When working with maps or model scales, understanding how to calculate the scale factor is crucial. The scale factor reveals how much smaller or larger a representation is compared to the real-world object it's depicting. To compute the scale factor, you should divide the real-world measurement by the map or model measurement.
For a given map problem, like the one described, if 4.5 inches on the map equates to 250 miles in reality, the scale factor can be found by performing:

• Real-world distance: 250 miles
• Map distance: 4.5 inches

The calculation will be \( \frac{250}{4.5} \) resulting in a scale factor of approximately 55.6. This means each inch on the map corresponds to 55.6 actual miles.

Distance Measurement

Once the scale factor is known, it becomes possible to determine real-world distances using map measurements. Suppose you need to find the actual distance between two cities that are 7.25 inches apart on a map. By multiplying the number of inches by the scale factor, you'll obtain the real-world distance.
In this example:

• Map distance between cities: 7.25 inches
• Scale factor per inch: 55.6 miles

To find the real-world distance, compute:\[ 7.25 \times 55.6 \approx 402.78 \]The cities are approximately 402.78 miles apart in reality. This calculation helps translate map measurements to meaningful, real-world distances.

Rounding Numbers

In many calculations, especially those involving real-world applications like distance, the result may need to be presented in a simpler form. This is where rounding comes into play. Rounding numbers makes them easier to interpret and communicate, especially when precision beyond a certain digit is unnecessary.
To round a number like 402.78 to the nearest tenth, observe the tenths and hundredths places:

• Tenths place: 7 (from 402.7)
• Hundredths place: 8

Since the hundredths place is 5 or greater, you increase the tenths place by 1, making it 402.8. Thus, the rounded distance is 402.8 miles, providing a clean and readable figure. This process ensures your numbers are both accurate and accessible.