A map of a highway has a scale of 2 inchesequals45 miles. The length of the highway on the map is 9 inches. There are 7 rest stops equally spaced on the​ highway, including one at each end. You are making a new map with a scale of 1-inch equals 30 miles. How far apart are the rest stops on the new​ map?

Question

The rest stops on the new map are__ ​inch(es) apart.

Answer 1

To determine the distance between the rest stops on the new map, we can follow these steps:

1. Convert the map distance to the actual distance on the highway: The length of the highway on the original map is 9 inches, and the scale of the map is 2 inches = 45 miles. To find the actual length of the highway:

   \[ \text{Actual distance} = \left(\frac{45 \text{ miles}}{2 \text{ inches}}\right) \times 9 \text{ inches} = \frac{45 \times 9}{2} \text{ miles} \]

   Calculating this:

   \[ = \frac{405}{2} = 202.5 \text{ miles} \]

2. Determine the distance between the rest stops: There are 7 rest stops along the highway, including one at each end. Since they are equally spaced, we have:

   \[ \text{Number of segments} = 7 - 1 = 6 \text{ segments} \]

   To find the distance between the rest stops:

   \[ \text{Distance between rest stops} = \frac{\text{Total distance}}{\text{Number of segments}} = \frac{202.5 \text{ miles}}{6} \]

   Calculating this:

   \[ = 33.75 \text{ miles} \]

3. Convert this distance to the new map scale: The new map has a scale of 1 inch = 30 miles. To find the distance between the rest stops on the new map:

   \[ \text{Distance on new map} = \frac{\text{Distance}}{\text{Scale}} = \frac{33.75 \text{ miles}}{30 \text{ miles/inch}} \]

   Calculating this:

   \[ = 1.125 \text{ inches} \]

Thus, the rest stops on the new map are 1.125 inches apart.