Suppose you have entered aa 55​-mile biathlon that consists of a run and a bicycle race. During your​ run, your average velocity is 7 miles per​ hour, and during your bicycle​ race, your average velocity is 27 miles per hour. You finish the race in 5 hours. What is the distance of the​ run? What is the distance of the bicycle​ race?

Let's denote the distance of the run as \( x \) miles, and the distance of the bicycle race as \( 55 - x \) miles (since the total distance of the biathlon is 55 miles).

### Step 1: Time equations
We know the total time is 5 hours, and time is calculated by dividing distance by speed. So:

- The time spent running \( t_{\text{run}} \) is \( \frac{x}{7} \) hours (since the running speed is 7 miles per hour).
- The time spent cycling \( t_{\text{bike}} \) is \( \frac{55 - x}{27} \) hours (since the cycling speed is 27 miles per hour).

### Step 2: Set up the equation
Since the total time is 5 hours, the sum of the running time and the cycling time equals 5:

\[
\frac{x}{7} + \frac{55 - x}{27} = 5
\]

### Step 3: Solve the equation
To eliminate the fractions, we can multiply through by the least common denominator of 7 and 27, which is 189:

\[
189 \times \left( \frac{x}{7} \right) + 189 \times \left( \frac{55 - x}{27} \right) = 189 \times 5
\]

Simplifying the terms:

\[
27x + 7(55 - x) = 945
\]

Now, expand and simplify:

\[
27x + 385 - 7x = 945
\]

\[
20x + 385 = 945
\]

Subtract 385 from both sides:

\[
20x = 560
\]

Solve for \( x \):

\[
x = \frac{560}{20} = 28
\]

### Step 4: Interpret the result
The distance of the run is 28 miles. Since the total distance is 55 miles, the distance of the bicycle race is:

\[
55 - 28 = 27 \text{ miles}
\]

### Conclusion:
- The distance of the run is **28 miles**.
- The distance of the bicycle race is **27 miles**.

Would you like further details on any of the steps? Here are some related questions for deeper exploration:

1. What would happen if the average running speed was slower, say 6 miles per hour?
2. How would the total time change if you increased the bicycle speed to 30 miles per hour?
3. Can we set up a similar equation for a triathlon with three different activities?
4. What is the relationship between speed and time in such mixed-distance races?
5. If you increased both running and cycling speeds, how would the distances change?

**Tip:** When solving problems involving distance, speed, and time, remember the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to break down complex multi-part journeys.

Suppose you have entered a 55​-mile biathlon that consists of a run and a bicycle race. During your​ run, your average velocity is 7 miles per​ hour, and during your bicycle​ race, your average velocity is 27 miles per hour. You finish the race in 5 hours. What is the distance of the​ run? What is the distance of the bicycle​ race?

Learn how to solve a biathlon problem where the total distance is 55 miles, the running speed is 7 mph, and the cycling speed is 27 mph. This step-by-step solution shows how to calculate the distances for the run and bicycle race, using algebra and the relationship between distance, speed, and time. Suitable for students in Grades 8-10.

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