Dale bought a map of the city. It uses a scale of 1 inch = 8 miles. Dale’s house and school are 1 1/2 inches apart on the map. How far apart would his house and school be on the map if the scale were 1 INCH = 6 MILES???
The correct answer and explanation is :
To solve this problem, we need to use a proportional relationship between the distances on the map and the actual distances in the real world. The problem asks for the new distance between Dale’s house and school on the map if the scale changes from 1 inch = 8 miles to 1 inch = 6 miles.
Step 1: Understand the relationship of the original scale
The original scale is 1 inch = 8 miles, and the distance between Dale’s house and school on the map is 1 1/2 inches. To calculate the real-world distance between the two places, we multiply the map distance by the scale factor.
Real-world distance:
$$
1.5 \text{ inches} \times 8 \text{ miles per inch} = 12 \text{ miles}
$$
So, the real-world distance between Dale’s house and school is 12 miles.
Step 2: Apply the new scale
Now, we need to find how far apart Dale’s house and school would be on the map if the scale were changed to 1 inch = 6 miles. The real-world distance between Dale’s house and school remains 12 miles, but we need to find the map distance that corresponds to this distance at the new scale.
Using the formula:
$$
\text{Map distance} = \frac{\text{Real-world distance}}{\text{Scale factor}}
$$
We substitute the values:
$$
\text{Map distance} = \frac{12 \text{ miles}}{6 \text{ miles per inch}} = 2 \text{ inches}
$$
Step 3: Conclusion
The new distance between Dale’s house and school on the map, with the scale of 1 inch = 6 miles, would be 2 inches.
Explanation
The key to solving this problem is recognizing that the map distance changes in proportion to the scale. When the scale changes from 1 inch = 8 miles to 1 inch = 6 miles, the map distance increases. This is because the scale represents a smaller real-world distance per inch on the map. Since the real-world distance (12 miles) remains constant, the map distance must increase when the scale is reduced (from 8 miles per inch to 6 miles per inch). Thus, the distance between Dale’s house and school on the map would be 2 inches instead of 1 1/2 inches.