UPSC Prelims 2006 Q96: Science & Technology – Logical Reasoning & Puzzles
Each of the persons A, B, C, D and E possesses unequal number (<10) of similar items. A, B
and C possess 21 items in all, while C, D and E possess 7 items in all. How many items do A and B
possess in all?
- (a) 15
- (b) 17
- (c) 18
- (d) Data is insufficient
Answer: (b) 17
Explanation
Unequal numbers of similar items
Each of A, B, C, D and E has a different positive number below 10.
| Step | Reasoning |
|---|---|
| C + D + E = 7 | D and E must be unequal and positive, so D + E is at least 1 + 2 = 3, giving C at most 4 |
| A + B + C = 21 | Three distinct digits below 10 summing to 21 are {9, 8, 4}, {9, 7, 5}, {8, 7, 6}; C at most 4 forces {9, 8, 4} with C = 4 |
| Check | C = 4 leaves D + E = 3 = 1 + 2, all five numbers distinct |
| A + B | 9 + 8 = 17 |
Answer: (b) 17.
Why each option is right or wrong
- (a) Incorrect. A + B = 15 would need C = 6, which leaves D + E = 1 and is impossible for distinct positive numbers.
- (b) CORRECT. C must be 4, so A and B are 9 and 8 and together have 17.
- (c) Incorrect. A + B = 18 would need C = 3, which no valid triple of distinct digits gives.
- (d) Incorrect. The conditions fix C = 4, so the data are sufficient.
Key point
With C + D + E = 7 and all numbers distinct positive digits, C = 4 and {A, B} = {9, 8}, so A + B = 17.
Remember
C = 4, A and B = 9 and 8, so A + B = 17.
Common mistakes
- Allowing a person to have zero items.
- Letting two persons have an equal number of items.
- Missing that C belongs to both groups.
Exam tip
Use the smaller total (7) to bound the shared person first, then the larger total (21).
Sources
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Puzzle Test)
- Arun Sharma — How to Prepare for Logical Reasoning for the CAT (number puzzles with constraints)