UPSC Prelims 2006 Q103: Science & Technology – Logical Reasoning & Puzzles
In an office, the number of persons who take tea is twice the number of persons who take
only coffee. The number of persons who take coffee is twice the number of persons who take
only tea.
Consider the following statements:
1. The sum of the number of persons who take either tea or coffee or both is four times the
number of persons who take both coffee and tea.
2. The sum of the number of persons who take only coffee and those who take only tea is twice
the number of persons who take both coffee and tea.
Which of the statements given above is/are correct?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Answer: (b) 2 only
Explanation
Tea and coffee drinkers (Venn diagram)
Let only-tea = T, only-coffee = C and both = x.
| Condition | Equation |
|---|---|
| Tea-takers are twice only-coffee | T + x = 2C |
| Coffee-takers are twice only-tea | C + x = 2T |
| Solving | Subtracting gives T = C, then x = T, so T = C = x |
Checking the statements:
| Statement | Verdict | Reason |
|---|---|---|
| 1. Either or both = 4 times both | False | Total = T + C + x = 3x, not 4x |
| 2. Only-coffee + only-tea = twice both | True | T + C = 2x |
Answer: (b) 2 only.
Why each option is right or wrong
- (a) Incorrect. The total of tea-or-coffee takers is 3x, not 4x.
- (b) CORRECT. Only-tea plus only-coffee equals 2x, twice the both-group.
- (c) Incorrect. Statement 1 is false because the total equals three times the both-group.
- (d) Incorrect. Statement 2 is true, so it is not 'neither'.
Key point
From the two conditions, only-tea = only-coffee = both = x, so the total is 3x and only-tea plus only-coffee is 2x.
Remember
Only tea = only coffee = both; total = 3 x both.
Common mistakes
- Treating 'persons who take tea' as only tea.
- Not solving the two equations together.
- Using 4x as the total instead of 3x.
Exam tip
Write total-tea = only-tea + both and total-coffee = only-coffee + both before comparing.
Sources
- NCERT Class XI Mathematics — Sets
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Logical Venn Diagrams)