To solve this, we need to understand the relationship between the two map scales and the real-world distances they represent.

Given:

• The scale of the old map is $1 \, \text{cm} = 100 \, \text{m}$.
• The scale of the new map is $1 \, \text{cm} = 500 \, \text{m}$.
• She ran a trail that was 40 cm long on her old map.

Step 1: Calculate the real-world length of the trail based on the old map.

$\text{Real-world length} = 40 \, \text{cm} \times 100 \, \text{m/cm} = 4000 \, \text{m}$

So, the real-world trail is 4000 meters long.

Step 2: Convert the real-world length into the new map's scale.

Now, using the new map scale where $1 \, \text{cm} = 500 \, \text{m}$:

$\text{Length on the new map} = \frac{4000 \, \text{m}}{500 \, \text{m/cm}} = 8 \, \text{cm}$

Conclusion:

The trail will be 8 cm long on the new map.

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Would you like more details or have any questions? Here are 5 related questions you might want to explore:

1. What if the scale on the new map was $1 \, \text{cm} = 1 \, \text{km}$?
2. How do we calculate the area if we have two maps with different scales?
3. What is the general formula for converting lengths between maps with different scales?
4. How would the distance appear on a map with a scale of $1 \, \text{cm} = 200 \, \text{m}$?
5. How would you calculate the trail length on the ground if the map scale was different?

Tip: When converting between scales, always convert to real-world units first before switching between maps.