Answer: Spike, Boris, Perceval


Scaling the number of stairs run
Boris:     5 steps                  100 - finished
Spike:    4 steps       $\times$20      80 - finished (21 + 80 $\gt$ 99)
Percival: 3 steps                   60 - not finished (38 + 60 $\lt$ 99) Last place

                              $\times$19      95 (4 steps to go)
                                           76   (21 + 76 = 97 $\Rightarrow$ 2 steps to go) winner


Working out how long each person takes
Boris:    $99$ steps
Spike:    $99 - 21 = 78$ steps
Percival: $99 - 38 =61$ steps

In each time unit, Boris runs $5$ steps, Spike runs $4$ steps, Percival runs $3$ steps.

Boris:    $99$ steps, $99 \div 5 = 19\frac{4}{5}$
Spike:    $78$ steps, $78 \div 4 = 19\frac{1}{2}$
Percival: $61$ steps, $61 \div 3 = 20\frac{1}{3}$

Therefore they finish in the order Spike, Boris, Percival.