Question

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 500 meters.

if the maps represent the same area, are the lengths on the new map longer, shorter, or the same size as the old map?
longer shorter the same size

Explanation:

Step1: Understand scale concept

Scale represents ratio of map - distance to real - world distance.

Step2: Compare old and new scales

Old scale: 1 cm represents 100 m. New scale: 1 cm represents 500 m.

Step3: Analyze for same real - world length

For a given real - world length, on old map: if real - world length is $L$ meters, map length $l_{old}=\frac{L}{100}$ cm. On new map, map length $l_{new}=\frac{L}{500}$ cm.
Since $\frac{L}{500}<\frac{L}{100}$ (when $L > 0$), lengths on new map are shorter.

Answer:

Shorter