Logical Reasoning

Step 1: Analyze Statement (I) alone.

- There are 8 positions (1=bottom, 8=top).
- "Only three boxes are kept above D" D is at position 5. (Boxes at 6,7,8 are above).
- "one box is kept between D and Q" and "Q is kept lower than D" D is at 5, so Q must be at position 3.
- "Q is immediately below P" P is at position 4.
- From (I), we have the partial stack: __ __ __ P(4) Q(3) __ __. D is at 5. So: __ __ __ D(5) P(4) Q(3) __ __.
- We know the positions of D, P, Q. We don't know the position of R or the box above it. Statement I is not sufficient.

Step 2: Analyze Statement (II) alone.

- "Only one box is kept between A and C" A __ C or C __ A.
- "C is three boxes above Q" C is at position x, Q is at x-4. (e.g. C=5, Q=1 or C=8, Q=4).
- "As many boxes are kept above B as are kept below R" If B is at position n, R is at 9-n. (e.g. B=8, R=1; B=7, R=2 etc.). They are symmetrical.
- This statement gives relative positions but no absolute positions. We cannot locate R. Statement II is not sufficient.

Step 3: Analyze both statements together.

- From (I): D=5, P=4, Q=3.
- Now use (II) with this information: "C is three boxes above Q". Since Q=3, C must be at position 3+4=7.
- Now use "Only one box is kept between A and C". Since C=7, A must be at position 5. But D is at position 5. This creates a contradiction.

Let me re-read "C is three boxes above Q". This means C is at position x, Q is at x-3, or C __ __ Q. No, it means 3 boxes are between them. C __ __ __ Q. So if Q=3, C=7. This seems right.
Let's check the distance. Pos 7, Pos 3. Boxes at 4, 5, 6 are between them. Yes, three boxes. C=7.
"Only one box is kept between A and C". C=7, so A must be at 5 or 9. 9 is not possible. So A=5.
This means A and D are in the same spot (position 5). The statements are contradictory.

Let's re-read "C is kept three boxes above Q". This might mean C is at Q's position + 3 = 3+3=6.
If C=6, then "one box between A and C" means A=4 or A=8. A=4 is not possible as P is there. So A=8.
- So far: A=8, C=6, D=5, P=4, Q=3.
- Positions left: 1, 2, 7. Boxes left: B, R, S.
- Now use "As many boxes above B as below R". Let's test the remaining spots.
- If B=7 (1 above), then R=2 (1 below). This works. S would be at position 1.
- This gives a complete valid arrangement: A(8), B(7), C(6), D(5), P(4), Q(3), R(2), S(1).
- The question is "Which box is kept immediately above R?". In this arrangement, R is at position 2. The box at position 3 is Q.
- Since we have found a unique arrangement and can answer the question, both statements together are sufficient.

Step 1: Analyze Statement (I) alone.

- "A is taller than B" gives us .
- "B is shorter than A and E only" is the key. This means B is shorter than exactly two people: A and E. And B is taller than everyone else.
- The other people are C and D. So, and .
- Combining this, we have the order for four people: A, E \textgreater B \textgreater C, D. The relative order of A and E, and C and D is not known.
- However, we know that the only people taller than B are A and E. This means A and E are the two tallest people in the group.
- Therefore, exactly 3 people (B, C, D) are shorter than E. The question is "how many persons are taller than E?". Since E is one of the two tallest, either A is taller than E or E is taller than A. But no one else can be. So, either 0 or 1 person is taller than E. This is not sufficient to give a unique answer.

Let's re-read "B is shorter than A and E only". This means that for any person X in the group, if , then X must be either A or E. This implies that A and E are the only two people taller than B. Thus A and E are the two tallest people. The question is "how many are taller than E?". We don't know if or . So we can't answer.

Let's try another interpretation of "B is shorter than A and E only". Maybe it implies the complete order? . If this is the case, then 0 people are taller than E. This gives a unique answer. Or . Then 1 person is taller than E. The wording is ambiguous. Let's assume the first interpretation: A and E are the top two. Let's re-evaluate. Statement I is not sufficient.

Step 2: Analyze Statement (II) alone.

- "C is shorter than A" gives .
- "A is shorter than E" gives .
- Combining these gives . This gives the relative order of three people. It tells us nothing about B and D. We cannot determine how many people are taller than E. Statement II is not sufficient.

Step 3: Analyze both statements together.

- From (I), we know A and E are the two tallest people.
- From (II), we know .
- Combining these, E must be the single tallest person in the group.
- Therefore, exactly zero people are taller than E.
- This gives a definite answer. So both statements together are required. This contradicts the provided answer key.

Let's reconsider the wording of Statement I. "B is shorter than A and E only". This is a very strong statement. It means there is no one else taller than B. It defines the set of people taller than B as \{A, E\}. It does not say anything about people shorter than B.
The question is "how many are taller than E?". We still cannot tell if or . Statement I alone is NOT sufficient.

There seems to be a fundamental error in the question or the provided answer key. My analysis suggests both statements are needed. Let's assume there is a common interpretation I am missing. Perhaps "B is shorter than A and E only" implies a rank order, with A and E being just above B. Even so, their internal rank isn't specified.

Step 1: Determine R and S.

- The average of R and S is 63. So .
- The difference between R and S is 2. So (assuming ).
- Solving these two equations: . Then . So, two of the numbers are 62 and 64.

Step 2: Determine the set of five consecutive integers.

- Since 62 and 64 are part of a set of five consecutive integers, the set must be one of these: {60,61,62,63,64}, {61,62,63,64,65}, {62,63,64,65,66}.
- The problem states the smallest is . This eliminates the first set.
- The greatest is . This condition is met by both remaining sets.

Step 3: Use the remaining clues to identify the correct set and assign numbers.

- (iii) "Both the digits of P are same". In the range 61–66, the only number with identical digits is 66. So . This means the set must be {62, 63, 64, 65, 66}.
- (ii) "T is a multiple of 9". In this set, the only multiple of 9 is 63. So .
- We have .
- The remaining number in the set is 65, so .
- (i) "A and B both are prime numbers". This seems to be a typo and should refer to two of the numbers P,Q,R,S,T. Let's check the primality of our numbers. None of 62, 63, 64, 65, 66 are prime. This indicates a contradiction in the problem statement.

Let's reconsider the set {61,62,63,64,65}.
- is not in this set. This set is not possible.
Let's re-read clue (i). "A and B both are prime numbers." This is likely a typo for two of P,Q,R,S,T. If no numbers in the set are prime, the question is flawed.
Let's check the primes between 60 and 70. They are 61 and 67.
If the set contains two primes, it must contain 61 and 67. The only set of 5 consecutive integers that could contain one of them is {61,62,63,64,65}. It doesn't contain 67. A set of 5 cannot contain both.
This is a major contradiction. Let's ignore clue (i) and see if we can proceed.

Let's assume the set is {62, 63, 64, 65, 66}.
.

Step 4: Find the sum of the digits of Q.

Final Answer:

Step 1: Determine the set of seven consecutive integers.

- The integers are between 60 and 70 (exclusive). The possible range is from 61 to 69.
- Clue (I) states that A and B are both prime numbers. The only prime numbers between 61 and 69 are 61 and 67.
- For a set of seven consecutive integers to contain both 61 and 67, the set must be \{61, 62, 63, 64, 65, 66, 67\}.
- From Clue (I) and the condition "A is smaller than B", we can definitively say A = 61 and B = 67.

Step 2: Use the remaining clues to identify the other numbers.

- Clue (IV): The average of R and S is 63, so . Their difference is 2, so . Solving these equations gives R = 64 and S = 62. Both are in our set.
- Clue (III): Both digits of P are the same. In our set, the only number with identical digits is 66. So, P = 66.
- Clue (II): T is a multiple of 9. In our set, the only multiple of 9 is 63. So, T = 63.

Step 3: Identify the remaining number, Q.

- The numbers we have identified are A=61, B=67, R=64, S=62, P=66, and T=63.
- The only integer left in the set \{61, 62, 63, 64, 65, 66, 67\} is 65.
- Therefore, Q = 65.

Step 4: Calculate the required sum.

- The question asks for the sum of A and Q.

Final Answer: