Step 1: Try real numbers for the ages instead of just symbols.
Pick any set of four different ages that fits the clues, then read off who is youngest.
From P is younger than S, try $S = 40$ and $P = 30$.
From P is older than R (R's own statement), try $R = 20$.
Step 2: Place Q using Q's own clue.
Q says Q is neither the youngest nor the oldest of the four.
With $R = 20$, $P = 30$, $S = 40$ already fixed, Q must sit strictly between the smallest and largest of this whole group once Q is added, so Q cannot be given a value below 20 or above 40.
Pick $Q = 35$ as one valid choice, since $20 < 35 < 40$ and Q is not the extreme in either direction.
Step 3: List the ages side by side.
$R = 20$, $P = 30$, $Q = 35$, $S = 40$.
Sorted from youngest to oldest this reads R, P, Q, S.
Step 4: Check this against every statement again.
P younger than S, $30 < 40$, true.
R younger than P, $20 < 30$, true.
Q neither youngest nor oldest, Q is 35, which is not the smallest (20) or the largest (40), true.
All three statements hold with this set of numbers.
Step 5: Try a different valid set of numbers to be sure R always comes out youngest.
Try $R = 5$, $P = 6$, $S = 50$, $Q = 25$. Every statement still checks out, and sorting gives R, P, Q, S again, with R always at the bottom.
No matter which specific numbers are chosen, as long as they satisfy the three statements, R keeps landing as the smallest, because R must be less than P and Q can never be forced below R without breaking Q's own statement.
Final Answer:
Testing with actual numbers confirms R is always the youngest of the four people.
\[ \boxed{R} \]