Runner Distance Calculation Setup

Let the total distance of the run be represented by D kilometers.

Medical Aid Station Position Calculation

The first water station is reached after covering $\frac{1}{7}$th of the total distance. The medical aid is found after covering an additional $\frac{1}{6}$th of the total distance.

• Distance to water station = $\frac{1}{7}D$
• Additional distance to medical aid = $\frac{1}{6}D$
• Total distance covered to reach the medical aid station = $(\frac{1}{7} + \frac{1}{6})D$

To add the fractions:

$ \frac{1}{7} + \frac{1}{6} = \frac{6}{42} + \frac{7}{42} = \frac{13}{42} $

So, the medical aid station is located at $\frac{13}{42}D$ from the start.

Second Runner's Journey Analysis

A second runner joins the race 4 km after the medical-aid station. This means the second runner starts at a position of $(\frac{13}{42}D + 4)$ km.

The second runner stops 4 km before the completion of the run. This means the second runner's stopping point is at $(D - 4)$ km.

The distance covered by this second runner is given as $\frac{1}{2}$ of the total distance, which is $\frac{1}{2}D$.

Formulating the Distance Equation

The distance covered by the second runner can also be expressed as the difference between their stopping point and starting point:

$ (\text{Stopping Point}) - (\text{Starting Point}) = \text{Distance Covered} $ $ (D - 4) - (\frac{13}{42}D + 4) = \frac{1}{2}D $

Solving for Total Distance (D)

Simplify the equation:

$ D - 4 - \frac{13}{42}D - 4 = \frac{1}{2}D $ $ D - \frac{13}{42}D - 8 = \frac{1}{2}D $

Combine the terms involving D on one side:

$ D - \frac{13}{42}D - \frac{1}{2}D = 8 $

Find a common denominator (42) for the fractions:

$ \frac{42}{42}D - \frac{13}{42}D - \frac{21}{42}D = 8 $ $ (\frac{42 - 13 - 21}{42})D = 8 $ $ (\frac{8}{42})D = 8 $ $ \frac{4}{21}D = 8 $

Solve for D:

$ D = 8 \times \frac{21}{4} $ $ D = 2 \times 21 $ $ D = 42 $

Final Distance Result

The total distance of the run is 42 km.