Question

Six students P, Q, R, S, T and U, with distinct heights, compare their heights and make the following observations.
Observation I: S is taller than R.
Observation II: Q is the shortest of all.
Observation III: U is taller than only one student.
Observation IV: T is taller than S but is not the tallest.
The number of students that are taller than R is the same as the number of students shorter than \(\underline{\hspace{2cm}.}\)

• A. T
• B. R
• C. S
• D. P

Hint:

In ordering problems, first fix extreme positions (shortest/tallest), then place "second shortest/second tallest" conditions, and finally count ranks to match the required numbers.

Answer 1

The Correct Option is C

To solve this problem, let's analyze the given observations and deduction step by step:

1. Observation I: S is taller than R.
2. Observation II: Q is the shortest of all.
3. Observation III: U is taller than only one student.
4. Observation IV: T is taller than S but is not the tallest.

First, let's write down the status of each student based on the observations:

• As per Observation II, Q is the shortest, so we can place them at the bottom.
• Observation III tells us that U is taller than only Q, making U the second shortest.
• Observation I implies S is taller than R, so S > R.
• With Observation IV, T is taller than S, but not the tallest, which means S is taller than R: T > S > R.

Summarizing based on these observations gives us two fixed points and the following order:

• Q (1^st - Shortest)
• U (2^nd)
• R, S, T, P - Remaining positions need arrangement

Now, from Observation IV, T is taller than S but is not the tallest:

• This implies another student, possibly P, is taller than T, meaning T is the second tallest. Therefore, P is the tallest.
• The order from shortest to tallest is: Q, U, R, S, T, P.

This results in:

1. Q (Shortest)
2. U (2^nd shortest)
3. R (3^rd)
4. S (4^th)
5. T (5^th)
6. P (Tallest)

The question asks us to find who underline should represent, such that the number of students taller and shorter than this person are equal. Here, in this order:

• For R, there are two students (S, T, P) taller and two students (Q, U) shorter, satisfying equality in number.

Thus, the correct answer is S as R should occupy the middle placement for it to have the same number of students taller and shorter than S, rather than him.