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To solve this problem, we need to find the length of the biking leg in a 23-mile race that includes running, paddling, and biking. The given lengths are:

- **Running leg**: \(6 \frac{1}{2}\) miles
- **Paddling leg**: \(11 \frac{2}{5}\) miles

### Step-by-Step Solution:

1. **Convert the mixed numbers to improper fractions:**
   - Running leg: \(6 \frac{1}{2} = \frac{13}{2}\) miles
   - Paddling leg: \(11 \frac{2}{5} = \frac{57}{5}\) miles

2. **Convert the improper fractions to have a common denominator:**
   - The least common denominator (LCD) for 2 and 5 is 10.
   - Convert each fraction to have a denominator of 10:
     - Running leg: \(\frac{13}{2} = \frac{13 \times 5}{2 \times 5} = \frac{65}{10}\)
     - Paddling leg: \(\frac{57}{5} = \frac{57 \times 2}{5 \times 2} = \frac{114}{10}\)

3. **Add the running and paddling distances together:**
   \[
   \text{Total of running and paddling legs} = \frac{65}{10} + \frac{114}{10} = \frac{179}{10} = 17.9 \text{ miles}
   \]

4. **Calculate the biking leg length:**
   The total race length is 23 miles, so we subtract the combined length of the running and paddling legs from the total:
   \[
   \text{Biking leg} = 23 - 17.9 = 5.1 \text{ miles}
   \]

### Final Answer:
The biking leg is **5.1 miles** long.

Would you like further details, or do you have any questions? Here are some related questions for further exploration:

1. How would the answer change if the paddling leg was 12 miles instead?
2. What is the fraction representation of 5.1 miles?
3. How can you convert fractions to decimals and vice versa?
4. How would you solve the problem if the total race length changed to 30 miles?
5. Can you calculate the total time taken for the race if each leg has a different speed?

**Tip:** When working with mixed numbers, always convert them to improper fractions first to make arithmetic operations easier.

You compete in a 23-mile-long race that includes running, paddling, and biking. The running leg is 6 1/2 miles long, and the paddling leg is 11 2/5 miles long. How long is the biking leg?

Learn how to solve a real-life word problem involving fractions, addition, and subtraction. This problem teaches you how to calculate the biking leg distance in a 23-mile race, where the running and paddling legs are expressed as mixed fractions. Suitable for Grades 6-8.

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