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In a gathering on 100 people, 70 of them can speak Hindi, 60 can speak English and 30 can speak French. Further, 30 of them can speak both Hindi and English, 20 can speak both Hindi and French. If x is the number of people who can speak both English and French, then which one of the following is correct? (Assume that everyone can speak at least one of the three languages.)
- a)9 < × ≤ 30
- b)0 ≤ × ≤ 8
- c)x = 9
- d)x = 8
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
In a gathering on 100 people, 70 of them can speak Hindi, 60 can speak...
Let the number of people who speak all three languages be y
Total number of people = ∑Number of people speaking English, French and Hindi – ∑Number of people speaking two languages + Number of people speaking three languages
⇒ 100 = 70 + 60 + 30 – 30 – 20 – x + y
⇒ 100 = 110 – x + y
⇒ x = 10 + y
⇒ Least possible value of y = 0,
⇒ x ≥ 10
⇒ 100 = 110 – x + y
⇒ x = 10 + y
⇒ Least possible value of y = 0,
⇒ x ≥ 10
y cannot exceed 20, because for knowing all three languages, they should know Hindi and French too, only 20 people know Hindi and French
If y = 20
then x ≤ 30
∴ 10 ≤ × ≤ 30 or 9 < × ≤ 30
then x ≤ 30
∴ 10 ≤ × ≤ 30 or 9 < × ≤ 30
Most Upvoted Answer
In a gathering on 100 people, 70 of them can speak Hindi, 60 can speak...
We can use the principle of inclusion-exclusion to solve this problem.
The number of people who can speak at least one language is the sum of those who can speak Hindi, English, or French, minus the sum of those who can speak two languages, plus the number who can speak all three languages:
100 = 70 + 60 + 30 - 30 - 20 + x + y
where y is the number of people who can speak all three languages.
Simplifying this equation, we get:
y = 10 - x
Now, we need to find x. We can use the fact that the total number of people who can speak English or French is 60 + 30 - x (i.e. the sum of those who can speak English only, those who can speak French only, and those who can speak both).
So:
60 + 30 - x = 100 - 70
Solving for x, we get:
x = 20
Therefore, y = 10 - x = 10 - 20 = -10, which doesn't make sense in the context of the problem. This means that there is an error in the information given, and none of the answer choices are correct.
The number of people who can speak at least one language is the sum of those who can speak Hindi, English, or French, minus the sum of those who can speak two languages, plus the number who can speak all three languages:
100 = 70 + 60 + 30 - 30 - 20 + x + y
where y is the number of people who can speak all three languages.
Simplifying this equation, we get:
y = 10 - x
Now, we need to find x. We can use the fact that the total number of people who can speak English or French is 60 + 30 - x (i.e. the sum of those who can speak English only, those who can speak French only, and those who can speak both).
So:
60 + 30 - x = 100 - 70
Solving for x, we get:
x = 20
Therefore, y = 10 - x = 10 - 20 = -10, which doesn't make sense in the context of the problem. This means that there is an error in the information given, and none of the answer choices are correct.
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In a gathering on 100 people, 70 of them can speak Hindi, 60 can speak English and 30 can speak French. Further, 30 of them can speak both Hindi and English, 20 can speak both Hindi and French. If x is the number of people who can speak both English and French, then which one of the following is correct? (Assume that everyone can speak at least one of the three languages.)a)9 < × ≤ 30b)0 ≤ × ≤ 8c)x = 9d)x = 8Correct answer is option 'A'. Can you explain this answer?
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