Question

1. a runner gets a new map of her favorite running trail. her old map has a scale of 1 centimeter to 400 meters. her new map has a scale of 1 centimeter to 100 meters. a. if the maps represent the same area, will the new map be larger, smaller, or the same size as the old map? a. larger b. smaller c. the same size explain your thinking. b. her favorite running trail was 20 centimeters long on her old map. how long is this trail on her new map? explain your thinking.

Explanation:

Step1: Understand map - scale concept

The scale of a map is a ratio that represents the relationship between a distance on the map and the corresponding distance on the ground. A smaller scale (e.g., 1 cm to 400 m) means that a larger area on the ground is represented by a smaller area on the map, while a larger scale (e.g., 1 cm to 100 m) means that a smaller area on the ground is represented by a larger area on the map.

Step2: Answer part a

Since the new map has a larger scale (1 cm to 100 m) compared to the old map (1 cm to 400 m), for the same area on the ground, the new map will be larger. So the answer to part a is A. Larger.

Step3: Find the actual - length of the trail

On the old map, the scale is 1 cm to 400 m. If the trail is 20 cm long on the old map, the actual length of the trail is $20\times400 = 8000$ m.

Step4: Calculate the length on the new map

On the new map, the scale is 1 cm to 100 m. Let the length of the trail on the new map be $x$ cm. We know that the actual length of the trail is 8000 m. Using the scale - ratio $\frac{1}{100}=\frac{x}{8000}$. Cross - multiplying gives $x=\frac{8000}{100}=80$ cm.

Answer:

a. A. Larger
b. 80 cm