A hiking trail is 24 miles from start to finish. There are two rest areas located along the trail. The first rest area is located such that the ratio of the distance from the start of the trail to the rest area and the distance from the rest area to the end of the trail is 2:9. To the nearest hundredth of a mile, how far is the first rest area from the starting point of the trail? 

To solve this problem, we need to find the distance from the starting point of the trail to the first rest area, given the ratio of distances and the total trail length.

### Step-by-Step Solution:

1. **Total trail length** = 24 miles.
2. The ratio of the distance from the start of the trail to the first rest area, compared to the distance from the first rest area to the end of the trail, is given as 2:9.

   Let:
   - The distance from the start of the trail to the first rest area be \( 2x \).
   - The distance from the first rest area to the end of the trail be \( 9x \).

3. The total length of the trail is the sum of these two distances:
   \[
   2x + 9x = 24
   \]
   Simplifying:
   \[
   11x = 24
   \]
   Solving for \( x \):
   \[
   x = \frac{24}{11} = 2.18 \, \text{miles}
   \]

4. Now, the distance from the start of the trail to the first rest area is \( 2x \):
   \[
   2x = 2 \times 2.18 = 4.36 \, \text{miles}
   \]

### Final Answer:
The first rest area is approximately **4.36 miles** from the starting point of the trail.

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Would you like further details or have any questions about this solution? 

Here are 5 related questions for practice:

1. If the second rest area is located at a 1:3 ratio of distances to the total remaining distance, how far is it from the start?
2. If the trail were 30 miles long with the same 2:9 ratio, how far would the first rest area be from the start?
3. What if the ratio was reversed (9:2)? How would this affect the location of the first rest area?
4. How far is the first rest area from the end of the trail in this problem?
5. How would the solution change if the total trail length were 50 miles?

**Tip:** Always double-check ratios by adding them and ensuring they match the total distance being measured!

A hiking trail is 24 miles from start to finish. There are two rest areas located along the trail. The first rest area is located such that the ratio of the distance from the start of the trail to the rest area and the distance from the rest area to the end of the trail is 2:9. To the nearest hundredth of a mile, how far is the first rest area from the starting point of the trail?

Discover how to solve a ratio problem involving a 24-mile hiking trail and determine the location of the first rest area based on a 2:9 ratio of distances. This step-by-step guide covers key concepts like ratios, proportions, and linear equations, suitable for students in Grades 7-9.

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