Boris has to run up $99$ steps, and runs up $5$ in each unit of time, so it takes him $99 \div 5 = 19\frac{4}{5}$ units of time to reach the top.

Spike has to run up $78$ steps, and runs up $4$ in each unit of time, so it takes him $78 \div 4 = 19\frac{1}{2}$ units of time.

Percival has to run up $61$ steps, and runs up $3$ in each unit of time, so it takes him $61 \div 3 = 20\frac{1}{3}$ units of time.

Therefore they finish in the order Spike, Boris Percival.

*This problem is taken from the UKMT Mathematical Challenges.*

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