A survey of the 300 employees of a Bengaluru office recorded which of three languages — Hindi, Tamil and Bengali — each employee can speak. The following was found:
- 60 employees speak none of the three languages.
- 135 employees speak Hindi, 90 speak Tamil and 105 speak Bengali.
- 160 employees speak exactly one of the three languages, and 10 employees speak all three.
- 25 employees speak both Tamil and Bengali (some of them may also speak Hindi).
- The number of employees who speak Hindi and Bengali but not Tamil is 5 more than the number who speak Hindi and Tamil but not Bengali.
- 60 employees speak none of the three languages.
- 135 employees speak Hindi, 90 speak Tamil and 105 speak Bengali.
- 160 employees speak exactly one of the three languages, and 10 employees speak all three.
- 25 employees speak both Tamil and Bengali (some of them may also speak Hindi).
- The number of employees who speak Hindi and Bengali but not Tamil is 5 more than the number who speak Hindi and Tamil but not Bengali.
Question 1
Numerical answer
How many employees speak only Hindi?
Show answer and solution
Answer: 70
Let the 'exactly two' regions be x (H&T only), y (H&B only), z (T&B only), with all three = 10.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
Hindi only = 70.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
Hindi only = 70.
Question 2
Single correct option
How many employees speak Hindi and Bengali but not Tamil?
- A. 25
- B. 30
- C. 35
- D. 40
Show answer and solution
Answer: (B) 30
Let the 'exactly two' regions be x (H&T only), y (H&B only), z (T&B only), with all three = 10.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
The answer is 30. (25 is the Hindi-and-Tamil-only count; 40 includes the 10 who speak all three.)
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
The answer is 30. (25 is the Hindi-and-Tamil-only count; 40 includes the 10 who speak all three.)
Question 3
Single correct option
How many employees speak at least two of the three languages?
- A. 70
- B. 75
- C. 80
- D. 85
Show answer and solution
Answer: (C) 80
Let the 'exactly two' regions be x (H&T only), y (H&B only), z (T&B only), with all three = 10.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
At least two = exactly two + all three = (25 + 30 + 15) + 10 = 80.
Alternatively: 300 − 60 (none) − 160 (exactly one) = 80.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
At least two = exactly two + all three = (25 + 30 + 15) + 10 = 80.
Alternatively: 300 − 60 (none) − 160 (exactly one) = 80.
Question 4
Single correct option
Suppose 20 of the employees who speak only Tamil learn Hindi, and nobody else learns a new language. How many employees will then speak exactly two of the three languages?
- A. 70
- B. 80
- C. 85
- D. 90
Show answer and solution
Answer: (D) 90
Let the 'exactly two' regions be x (H&T only), y (H&B only), z (T&B only), with all three = 10.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
The 20 Tamil-only speakers move into the 'Hindi & Tamil only' region, which becomes 45. Exactly two = 45 + 30 + 15 = 90 (previously 70).
Nobody enters 'all three', since these 20 did not know Bengali.
- Clue: Tamil and Bengali (with or without Hindi) = 25 ⇒ z = 15.
- Exactly two = 70 ⇒ x + y = 55, and y = x + 5 ⇒ x = 25, y = 30.
- Hindi only = 135 − 25 − 30 − 10 = 70; Tamil only = 90 − 25 − 15 − 10 = 40; Bengali only = 105 − 30 − 15 − 10 = 50.
- Check: 70 + 40 + 50 = 160 = exactly one ✓; 160 + 70 + 10 + 60 = 300 ✓.
| Region | Employees |
|---|---|
| Hindi only | 70 |
| Tamil only | 40 |
| Bengali only | 50 |
| Hindi & Tamil only | 25 |
| Hindi & Bengali only | 30 |
| Tamil & Bengali only | 15 |
| All three | 10 |
| None | 60 |
The 20 Tamil-only speakers move into the 'Hindi & Tamil only' region, which becomes 45. Exactly two = 45 + 30 + 15 = 90 (previously 70).
Nobody enters 'all three', since these 20 did not know Bengali.
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