Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $5$ km behind, and Foley is $6$ km behind. When Edith finishes the race, how far behind is Foley, in km?

Answer 1

Let the speeds of Dirk, Edith, and Foley be \(d\), \(e\), and \(f\) km/h, respectively. Since they all run at constant speeds and start at the same time, we can express the times taken to finish the race in terms of their speeds.

The time taken by Dirk to finish the \(24\) km race is given by:

\[ t_D = \frac{24}{d} \]

At this point, Edith has run:

\[ e \cdot t_D = e \cdot \frac{24}{d} \]

According to the problem, when Dirk finishes the race, Edith is \(5\) km behind, so:

\[ e \cdot \frac{24}{d} = 24 - 5 = 19 \] This leads to the equation:

\[ e \cdot \frac{24}{d} = 19 \]

Multiplying both sides by \(d\):

\[ e \cdot 24 = 19d \]

Thus, we have:

\[ e = \frac{19d}{24} \]

Next, let’s analyze Foley's situation. When Dirk finishes the race, Foley has run:

\[ f \cdot t_D = f \cdot \frac{24}{d} \]

Since Foley is \(6\) km behind Dirk, this means:

\[ f \cdot \frac{24}{d} = 24 - 6 = 18 \]

From this, we derive another equation:

\[ f \cdot \frac{24}{d} = 18 \]

Multiplying both sides by \(d\) gives:

\[ f \cdot 24 = 18d \]

Thus:

\[ f = \frac{18d}{24} = \frac{3d}{4} \]

Now that we have expressions for \(e\) and \(f\) in terms of \(d\), we can determine the relationship of the speeds when Edith finishes her race. The time taken by Edith to finish is:

\[ t_E = \frac{24}{e} \]

Substituting \(e = \frac{19d}{24}\):

\[ t_E = \frac{24}{\frac{19d}{24}} = \frac{24 \cdot 24}{19d} = \frac{576}{19d} \]

During this time, let’s find out how far Foley is behind when Edith finishes. The distance Foley runs in time \(t_E\) is:

\[ \text{Distance Foley runs} = f \cdot t_E \]

Substituting \(f = \frac{3d}{4}\) and \(t_E = \frac{576}{19d}\) gives:

\[ \text{Distance Foley runs} = \frac{3d}{4} \cdot \frac{576}{19d} \]

Simplifying this, we have:

\[ = \frac{3 \cdot 576}{4 \cdot 19} = \frac{1728}{76} = \frac{432}{19} \]

Now we can determine how far behind Foley is when Edith finishes. Since she runs \(24\) km, we need to calculate the difference in distance:

\[ \text{Distance behind} = 24 - \frac{432}{19} \]

To simplify this, we first convert \(24\) into a fraction with a denominator of \(19\):

\[ 24 = \frac{456}{19} \]

Now we find the distance behind:

\[ \text{Distance behind} = \frac{456}{19} - \frac{432}{19} = \frac{456 - 432}{19} = \frac{24}{19} \]

Thus, when Edith finishes the race, Foley is

\[ \boxed{\frac{24}{19}} \text{ km behind her}. \]