Instead of working with abstract inequality symbols, try assigning actual trial ages to the four people and see which assignment survives every statement. Start with a simple guess: let $R = 10$, $P = 20$, $S = 30$, satisfying $P < S$ and $R < P$ directly from the first two clues. Now place Q somewhere and test Q's own claim. If we set $Q = 5$, the ranking becomes $Q, R, P, S$ from youngest to oldest, which makes Q the youngest overall. But Q insists 'I am neither the youngest nor the oldest,' so this trial fails and must be discarded. Next try $Q = 40$, giving the ranking $R, P, S, Q$; now Q is the oldest, which again breaks Q's own statement, so this trial also fails.
The only trials that keep Q's statement true are ones where Q lands strictly inside the existing chain, such as $Q = 15$ (ranking $R, Q, P, S$) or $Q = 25$ (ranking $R, P, Q, S$). Checking every one of these valid trials, the person occupying the bottom (youngest) spot never changes: it is always R, because R was already established as younger than both P and S before Q was even placed, and Q is forbidden from going below R by its own statement.
Since every scenario consistent with all three statements agrees on the same youngest person, the answer must be R.
\[\boxed{\text{Youngest person} = R}\]