Gina's running app tracked her average rate in minutes per kilometre. After starting the app, Gina stood still for 15 seconds and then ran at a constant rate of 7 minutes per kilometre for the rest of the run. How many kilometres did Gina run between when her app showed her average rate as 7 minutes 30 seconds per kilometre and when it showed 7 minutes 5 seconds per kilometre?
Let $d_{1}$ be the distance in kilometres that Gina had ran when her average rate was 7 minutes 30 seconds per kilometre. Let $d_{2}$ be the distance in kilometres that Gina had ran when her average rate was 7 minutes 5 seconds per kilometre.

The distance we seek is $d_{2}-d_{1}$.

Since 7 minutes 30 seconds was the average rate when the distance was $d_{1}$ kilometres, $\frac{15}{2} \times d_{1}$ minutes passed from the time Gina started the app to when she had ran $d_{1}$ kilometres. We are using that 7 minutes 30 seconds is equal to $\frac{15}{2}$ minutes.

On the other hand, Gina ran at a constant rate of 7 minutes per kilometre, so it took $7 d_{1}$ minutes for her to run $d_{1}$ kilometres once she started moving.

Since 15 seconds equals $\frac{1}{4}$ minute, this means that $\frac{1}{4}+7 d_{1}$ minutes passed from the time Gina started the app to the time she had ran $d_{1}$ kilometres.

This gives the equation $\frac{15}{2} d_{1}=\frac{1}{4}+7 d_{1}$ which can be solved for $d_{1}$ to get $d_{1}=\frac{1}{2}$ kilometre.

Since 5 seconds equals $\frac{1}{12}$ minute, 7 minutes 5 seconds equals $\frac{85}{12}$ minutes.

Using calculations similar to those involving $d_{1}$, we can obtain the equation $\frac{85}{12} d_{2}=\frac{1}{4}+7 d_{2}$, which can be solved for $d_{2}$ to get $d_{2}=3$ kilometres.

Therefore, the answer is $d_{2}-d_{1}=3-\frac{1}{2}=\frac{5}{2}$ kilometres or 2.5 kilometres.

ANSWER: $2.5 \mathrm{~km}$