Jordan and Amari run a​ 200-meter race, and Jordan wins by 13 meters. Let x be the distance behind the starting line that Jordan needs to start in order for the race to be a tie. Divide the total distance Amari needs to run by​ Amari's speed. Set this equal to the total​ distance, including​ x, that Jordan needs to run and divide by​ Jordan's speed. Then solve for x.

Let's solve the problem step by step, following a similar process as before.

### Given:
- Jordan wins the 200-meter race by 13 meters.
- This means when Jordan has completed 200 meters, Amari has only completed \(200 - 13 = 187\) meters.
- Let \( s_J \) be Jordan's speed and \( s_A \) be Amari's speed.

### Goal:
- Find the distance \( x \) behind the starting line that Jordan needs to start so that the race ends in a tie.

### Steps to Solve:

1. **Express Amari's time to complete the race:**

   Amari's time to run 200 meters is given by:
   \[
   t_A = \frac{200}{s_A}
   \]

2. **Express Jordan's time to complete the race including the distance \( x \):**

   If Jordan starts \( x \) meters behind the starting line, he needs to run \( 200 + x \) meters.
   So, Jordan's time is:
   \[
   t_J = \frac{200 + x}{s_J}
   \]

3. **Set the times equal (since the race ends in a tie):**
   \[
   t_A = t_J
   \]
   Substituting the expressions for \( t_A \) and \( t_J \):
   \[
   \frac{200}{s_A} = \frac{200 + x}{s_J}
   \]

4. **Express the relationship between Jordan's and Amari's speeds:**

   Since when Jordan runs 200 meters, Amari runs only 187 meters:
   \[
   \frac{s_A}{s_J} = \frac{187}{200}
   \]
   Therefore, Jordan's speed \( s_J \) in terms of Amari's speed \( s_A \) is:
   \[
   s_J = \frac{200}{187} s_A
   \]

5. **Substitute \( s_J \) into the time equation:**
   \[
   \frac{200}{s_A} = \frac{200 + x}{\frac{200}{187} s_A}
   \]

6. **Simplify and solve for \( x \):**
   \[
   \frac{200}{s_A} = \frac{187(200 + x)}{200 s_A}
   \]
   Multiply both sides by \( 200 s_A \) to eliminate the denominators:
   \[
   200 \times 200 = 187 \times (200 + x)
   \]
   Expand and simplify:
   \[
   40000 = 37400 + 187x
   \]
   Subtract 37400 from both sides:
   \[
   2600 = 187x
   \]
   Finally, solve for \( x \):
   \[
   x = \frac{2600}{187} \approx 13.9 \text{ meters}
   \]

### Conclusion:
Jordan needs to start approximately \( \boxed{13.9} \) meters behind the starting line for the race to end in a tie.

---

Do you want further details or have any questions about the solution?

### 5 Related Questions:
1. What would happen if Jordan had to start 10 meters behind the line instead of 13.9?
2. How does changing the distance of the race (e.g., 100 meters) affect the value of \( x \)?
3. If Amari's speed increased slightly, how would that impact \( x \)?
4. How does the problem change if the race is 400 meters instead of 200 meters?
5. Can this method be applied to races of different lengths, and how does it generalize?

### Tip:
When solving problems involving relative speeds, always ensure you correctly express the relationships between the competitors' speeds before proceeding to solve for the unknowns.

Explore how to solve a speed, distance, and time problem involving Jordan and Amari's race. Understand the concept of relative speeds and how to calculate the starting distance for Jordan to tie the race. Suitable for high school students learning about speed and motion in physics and mathematics.

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