Step 1: Set up the idea of the gap.
Both runners cover the same straight 100.00 m track at their own constant speeds. The instant the faster runner (Noah) touches the tape, the slower runner (Kishane) is still on the track. The distance between them at that instant is simply how far Kishane still has to run, so let us track Kishane's progress.
Step 2: Use a single ratio instead of computing speed.
Since Kishane moves uniformly, the fraction of the track he has covered equals the fraction of his own race-time that has elapsed. When Noah finishes, the elapsed time is Noah's time $t_N = 9.784$ s, while Kishane needs $t_K = 9.789$ s for the full 100 m.
Step 3: Find the fraction of the track still left for Kishane.
The fraction of distance remaining equals the fraction of time remaining: \[ \frac{d_{\text{gap}}}{100} = \frac{t_K - t_N}{t_K} = \frac{9.789 - 9.784}{9.789} \]
Step 4: Plug in the small time difference.
The numerator is the head-start in time: \[ t_K - t_N = 0.005 \text{ s} \] so the fraction left is \[ \frac{0.005}{9.789} \approx 5.108 \times 10^{-4} \]
Step 5: Convert the fraction back into a distance.
Multiply this fraction by the full track length: \[ d_{\text{gap}} = 100 \times 5.108 \times 10^{-4} \text{ m} \approx 0.0511 \text{ m} \]
Step 6: Express in centimetres and round.
Converting, \[ d_{\text{gap}} \approx 0.0511 \times 100 = 5.11 \text{ cm} \] which is closest to 5 cm. Notice this ratio method never needed the actual speed value, yet it lands on exactly the same answer as the key.
\[ \boxed{d_{\text{gap}} \approx 5 \text{ cm}} \]