Question

a. a runner gets a new map of her favorite running trail.
her old map has a scale of 1 centimeter to 100 meters.
her new map has a scale of 1 centimeter to 40 meters.
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: Analyze scale relationship

The old - map scale is 1 cm : 100 m, and the new - map scale is 1 cm : 40 m. A smaller number of real - world meters per centimeter on the new map means a larger scale. For the same real - world area, a larger - scale map will be larger in size.

Step2: Calculate length on new map

First, find the real - world length of the trail. On the old map, with a scale of 1 cm : 100 m and a length of 20 cm on the old map, the real - world length of the trail is \(20\times100 = 2000\) m. Then, use the new - map scale of 1 cm : 40 m. Let the length on the new map be \(x\) cm. We set up the proportion \(\frac{1}{40}=\frac{x}{2000}\). Cross - multiply to get \(40x = 2000\), and then solve for \(x\): \(x=\frac{2000}{40}=50\) cm.

Answer:

a. A. Larger
b. 50 centimeters. The real - world length of the trail is \(20\times100 = 2000\) m. Using the new - map scale of 1 cm : 40 m, we set up the proportion \(\frac{1}{40}=\frac{x}{2000}\) and solve for \(x\) to get \(x = 50\) cm.