Four people P, Q, R, and S, of different ages, make the following observations.
P: I am younger than S.
Q: I am neither the youngest nor the oldest.
R: P is older than me.
Based on these observations, the youngest person is ______.
Show Hint
Chain the two direct clues to get R < P < S, then use Q's statement to show Q cannot be below R or above S. R ends up below everyone.
Step 1: Understanding the Question:
Four people, P, Q, R, and S, describe their relative ages, and we need to find who is youngest. Instead of chaining inequalities, this method tests each person as a candidate for "youngest" and checks whether that choice breaks any of the three statements.
Step 2: Key Formula or Approach:
First convert the clues to plain relations: P's statement gives $P < S$ (P younger than S). R's statement gives $R < P$ (R younger than P). Q's statement rules Q out from being either the youngest or the oldest. Now test each person as "the youngest" one at a time and see if it is possible.
Step 3: Detailed Explanation:
Test P as youngest: but $R < P$ tells us R is younger than P, so P cannot be the youngest. This fails.
Test Q as youngest: Q's own statement says Q is not the youngest, a direct contradiction. This fails immediately.
Test S as youngest: but $P < S$ tells us P is younger than S, so S cannot be the youngest either. This fails.
Test R as youngest: check this against every statement. $R < P$ is satisfied, since R being the smallest is consistent with R being younger than P. $P < S$ does not involve R directly and stays valid. Q's statement only requires Q to avoid the two extremes; since R sits at the bottom of the order and S sits at least as high as P, Q can sit anywhere between R and S without becoming an extreme. No contradiction arises. R is the only candidate that survives every test.
Step 4: Final Answer:
Testing each candidate against the three statements eliminates P, Q, and S, leaving R as the only person who can consistently be the youngest.
