  A trail runner gets a new map of her favorite  mountain. 

●  Her old map has a scale of 1 cm to 100 m. 

●  Her new map has a scale of 1 cm to 500 m. 

a)  She ran a trail that was  40 cm long on her old map. 

How long is this trail on her new map  ? 

Explain/show your thinking. 

b)  Rewrite the scale 1 cm to 500 m so it does not require the use of units. 

Explain/show your thinking.

### Part A: Converting the Length of the Trail on the New Map

#### Step 1: Understand the scale of both maps.

- **Old map:** 1 cm = 100 m
- **New map:** 1 cm = 500 m

This means that 1 cm on the old map represents 100 meters in real life, while 1 cm on the new map represents 500 meters in real life.

#### Step 2: Calculate the real distance of the trail based on the old map.

The trail is **40 cm** long on the old map. Using the old map's scale (1 cm = 100 m), the actual distance of the trail is:
\[
\text{Real distance} = 40 \, \text{cm} \times 100 \, \text{m/cm} = 4000 \, \text{m}
\]

So, the real trail distance is **4000 meters**.

#### Step 3: Convert this real distance to the new map scale.

Using the new map's scale (1 cm = 500 m), we convert the real distance of 4000 meters back to the new map's length:
\[
\text{Length on new map} = \frac{4000 \, \text{m}}{500 \, \text{m/cm}} = 8 \, \text{cm}
\]

Thus, the trail is **8 cm** long on the new map.

### Part B: Rewrite the scale 1 cm to 500 m without units.

To express the scale without units, we convert the distance in meters to centimeters, since both sides of the ratio should be in the same units.

\[
1 \, \text{m} = 100 \, \text{cm}
\]

So, 500 meters is equivalent to:
\[
500 \, \text{m} = 500 \times 100 = 50{,}000 \, \text{cm}
\]

Therefore, the scale of 1 cm to 500 m can be written as:
\[
1 \, \text{cm} : 50{,}000 \, \text{cm} \quad \text{or} \quad 1 : 50{,}000
\]

This ratio means that 1 unit on the map corresponds to 50,000 units in reality, regardless of the specific unit being used.

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### Do you want any more details or have any further questions?

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Here are 5 relative questions for further understanding:
1. What would the length of the trail be on the old map if it were 20 cm on the new map?
2. How do map scales affect planning when estimating trail distances for hikes?
3. If a map had a scale of 1:10,000, how would you interpret this in terms of real-life distances?
4. How would you calculate the real-life distance of a trail that is 15 cm long on a map with a 1:1000 scale?
5. How do different scales affect the accuracy of a map for detailed navigation?

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**Tip:** When converting between different scales, always ensure both sides of the ratio use the same unit for a clearer understanding.

A trail runner gets a new map of her favorite mountain. Her old map has a scale of 1 cm to 100 m. Her new map has a scale of 1 cm to 500 m. a) She ran a trail that was 40 cm long on her old map. How long is this trail on her new map? Explain/show your thinking. b) Rewrite the scale 1 cm to 500 m so it does not require the use of units. Explain/show your thinking.

Learn how to convert a trail length from one map scale to another. This math problem focuses on ratios and proportions with scales of 1 cm to 100 m and 1 cm to 500 m, offering a clear step-by-step explanation. Ideal for students in Grades 6-8.

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