Each of the 200 attendees at a certain conference speaks either

Question

Each of the 200 attendees at a certain conference speaks either exactly one or all four of the following languages: English, French, German and Italian. If 80 attendees speak English, 60 speak French, 70 speak German and 50 speak Italian, how many attendees speak all four languages?

A. 12
B. 15
C. 20
D. 30
E. 60

A. 12
B. 15
C. 20
D. 30
E. 60

gmatophobia · Answer

80 + 60 + 70 + 50 = 260

The number of attendees were only 200, hence the extra 60 represents those who have been counted multiple times.

Each person who speaks all four languages got counted four times, once in each language group, while they should have got counted only once.

Hence, the number of people who can speak all four languages have been counted thrice in error, and the extra 60 is because of that.

Let's assume the number of people who can speak all four languages = x

3x = 60

x = 20

Option C

kevincan · Answer

Suppose that x attendees speak all four languages. Then 200 - x is the number that speak exactly one of the four languages.

We can write
200- x = (80 - x) + (70 - x) + (50 - x) + (60 - x)

Solving for x , we see that x = 20

MartyMurray · Answer

Each of the 200 attendees at a certain conference speaks either exactly one or all four of the following languages: English, French, German and Italian. If 80 attendees speak English, 60 speak French, 70 speak German and 50 speak Italian, how many attendees speak all four languages?

A. 12
B. 15
C. 20
D. 30
E. 60

To answer this question, we can use an extension of the following commonly used triple overlapping sets formula:

Total = A + B + C - (Exactly Two Sets Overlaps) - 2(Three Sets Overlaps)

The extension is the following:

Total = A + B + C + D - (Exactly Two Sets Overlaps) - 2(Three Sets Overlaps) - 3(Four Sets Overlap)

The reason why that extension works is that the elements included in all four sets are counted four times, once in the total for each set, when, in reality, they exist only once. So, they are overcounted three times. Thus, for arriving at the correct total, they must be subtracted three times.

Thus, we have the following:

200 = 80 + 60 + 70 + 50 - (Exactly Two Sets Overlaps) - 2(Three Sets Overlaps) - 3(Four Sets Overlap)

We are told the following:

Each of the 200 attendees at a certain conference speaks either  exactly one or all four  of the following languages

So, we have to be concerned with only the four sets overlap.

Thus, we have the following:

200 = 260 - 3(Four Sets Overlap)

Four Sets Overlap = 20

A. 12
B. 15
C. 20
D. 30
E. 60

Correct answer:   C

HarshavardhanR · Answer

Each of the 200 attendees at a certain conference speaks either exactly one or all four of the following languages  
  
This means that ->

- If the number of attendees who speak all four languages is x, then, everyone else speaks exactly one language (200-x).
- 80 attendees speak English. Out of these 80, x attendees speak all languages. So, 80-x attendees speak only English.
 - 60 attendees speak French. Out of these 60, x attendees speak all languages. So, 60-x attendees speak only French. 
 - 70 attendees speak German. Out of these 70, x attendees speak all languages. So, 70-x attendees speak only Spanish. 
 - 50 attendees speak Italian. Out of these 50, x attendees speak all languages. So, 50-x attendees speak only Italian.

Overall

(80-x) + (70-x) + (60-x) + (50-x) = 200 - x = total number of attendees who speak only one language.

i.e.,

260-4x = 200-x

3x = 60

=> x = 20. Choice C.

---
Harsha

Each of the 200 attendees at a certain conference speaks either

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