Jean runs 10 mi and then rides 21 mi on her bicycle in a biathlon. She rides 6 mph faster than she runs. If the total time for her to complete the race is 2.75 hr, determine her average speed running and her average speed riding her bicycle.

We can solve this problem using a system of equations. Let’s define:

- \( r \) as Jean's running speed in miles per hour (mph).
- \( r + 6 \) as Jean's biking speed, since she rides 6 mph faster than she runs.

### Step 1: Express the time for each part of the race

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

#### Running:
The time spent running is \( \frac{10}{r} \).

#### Biking:
The time spent biking is \( \frac{21}{r + 6} \).

### Step 2: Total time equation

The total time for the race is 2.75 hours. So, we can set up the equation:

\[
\frac{10}{r} + \frac{21}{r + 6} = 2.75
\]

### Step 3: Solve the equation

To solve this, we need to find \( r \). Let's first eliminate the fractions by multiplying the entire equation by \( r(r + 6) \), the common denominator.

\[
10(r + 6) + 21r = 2.75r(r + 6)
\]

Expanding both sides:

\[
10r + 60 + 21r = 2.75r^2 + 16.5r
\]

Simplify:

\[
31r + 60 = 2.75r^2 + 16.5r
\]

Now, move all terms to one side:

\[
2.75r^2 + 16.5r - 31r - 60 = 0
\]

Simplify further:

\[
2.75r^2 - 14.5r - 60 = 0
\]

### Step 4: Solve the quadratic equation

We have the quadratic equation:

\[
2.75r^2 - 14.5r - 60 = 0
\]

We can solve this using the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 2.75 \), \( b = -14.5 \), and \( c = -60 \). Let's calculate the discriminant and find \( r \).The two solutions for \( r \) are:

\[
r_1 = 8 \quad \text{and} \quad r_2 = -2.73
\]

Since speed cannot be negative, Jean's running speed is \( r = 8 \) mph.

### Step 5: Calculate her biking speed

Since her biking speed is 6 mph faster than her running speed:

\[
\text{Biking speed} = r + 6 = 8 + 6 = 14 \, \text{mph}
\]

### Final Answer:
- Jean's average running speed is **8 mph**.
- Jean's average biking speed is **14 mph**.

Would you like further details, or have any other questions?

Here are 5 related questions:
1. How would the total time change if Jean ran slower but biked faster?
2. What if Jean's total time was extended to 3 hours? How would that affect her speeds?
3. Can we calculate the exact time spent on each activity?
4. How would the result change if Jean rode 25 miles instead of 21?
5. What would happen to her average speeds if the biking distance was halved?

**Tip:** When solving a quadratic equation, always check for meaningful solutions based on the context (e.g., speed must be positive).

This detailed solution explains how to calculate Jean's running and biking speeds in a biathlon where she rides 6 mph faster than she runs. Learn key algebraic concepts, including the time-distance-speed relationship and solving quadratic equations. Suitable for high school students.

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