At a conference, there are 250 attendees: 90 speak French, 8

Question

At a conference attended by ministers and interpreters, there are 250 attendees: 90 speak French, 80 speak Spanish, 100 speak German, and 100 speak Arabic. Each attendee speaks either one, three, or all four languages. Two-thirds of the attendees who speak more than one of these languages speak all four languages. How many attendees speak only one of these languages?

A. 130
B. 160
C. 190
D. 205
E. 220

A. 130
B. 160
C. 190
D. 205
E. 220

olalada · Answer

Know 4 languages - x people 
Know 3 languages - y people 
Know 1 language - z people

1) x + y + z = 250 
 
2) 90+80+100+100 = 370 (repeating those who know 3 languages 3 times and those who know 4 languages 4 times) 
=> 4x + 3y + z = 370

3) People who know more than 1 language = x+y
x = 2/3 (x+y) - people who know 4 languages 
=> x = 2/3x + 2/3y 
=> x - 2/3x = 2/3y 
=> 1/3x = 2/3y 
=> x = 2y

4) 2y + y + z = 250 => 3y + z = 250 
4*2y + 3y + z = 370 => 11y + z = 370

5) Subtract one equation from another: 
8y = 120 => y = 15

6) 2y + y + z = 250 
=> 45 + z = 250 
=> z = 205

SwethaReddyL · Answer

hi  kevincan , can you please provide the official solution? this felt complicated, can such questions appear in real test?

SwethaReddyL · Answer

the trouble here was to convert this stmt to an eqn - Two-thirds of the attendees who speak more than one of these languages speak all four languages.

x = people who spoke 1 language
y = people who spoke 3 languages
z = people who spoke 4 languages

given, 
x + y + z = 250 ---- eqn 1
x + 3y + 4z = 370 ---- eqn 2

Now, Two-thirds of the attendees who speak more than one of these languages speak all four languages.
translate, people who speak all 4 languages is 2/3 of the people who speak more than 1 langauage
z = 2/3 (y+z)
3z = 2y + 2z
z = 2y --- eqn 3
now substitute eqn 3 in 2, and eqn 1
x + 11y = 370 
x + 3y = 250

solving, we get y = 15, sub in eqn 3, you will get z = 30
y+z=45
250-45=205 people who speak only one language

thanks,
Swetha

kevincan · Answer

Here’s a nice method : try not to use three variables

Let y be the number of attendees who speak exactly three languages.

Since two-thirds of those who speak more than one language speak all four languages, the number who speak all four languages must be 2y.

Therefore:

* Exactly three languages: y
* All four languages: 2y
* Only one language: 250 − 3y

Now count total language incidences.

90 + 80 + 100 + 100 = 370

On the other hand, the total number of language incidences is

(250 − 3y)(1) + y(3) + (2y)(4) = 370

250 − 3y + 3y + 8y = 370

250 + 8y = 370

8y = 120

y = 15

Therefore, the number who speak only one language is

250 − 3(15) = 205

Answer: D. 205.

kevincan · Answer

On the actual GMAT, questions often look complicated on the surface but are designed so that one good observation collapses the problem. This is exactly that type of question. If you tried to draw Venn diagrams or track every language separately, it would become messy. The question is really checking whether you can identify the right way to represent the information.

Remember not to get stressed over any one question. Learn what you can from it and revisit it a few days later. The main takeaway from this question is to try to limit the number of variables you use. A good setup often turns a difficult-looking problem into a straightforward one, while too many variables can create unnecessary work and clutter your thinking.

At a conference, there are 250 attendees: 90 speak French, 8

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