Question: In a group of 25 people, only three languages are spoken- English, Spanish and German. If there is atleast one person who speaks all the three languages, how many people can interact with each other in English and German?

1. 4 people speak two languages but do not speak Spanish
2. One fifth of the group speaks more than one language

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Approach Solution (1):
There are 25 people and atleast one of them speaks all three languages
We are trying to find the people who can interact with each other in English and German. That means these people can either speak English and German or English, German and Spanish.
So we need to find the number of people that fall under these two categories:
Let’s look at the remaining information:

1. 4 people speak two languages but not Spanish- this is enough for the first category (i.e. the people that speak English and German only), but we do not know how many people speak all three languages (the question stem says atleast one person)
   Insufficient
2. One fifth of the group speaks more than one language
   Let’s break this down
   One fifth of the group is 25/5 = 5 people. 5 people speak more than one language. We do not know how many of these people speak only English and German or all three languages.
   Insufficient

Now let’s look at both information together:
We know that 4 people speak English and German, and only 5 people speak more than one language (i.e. 2 languages or 3 languages). The information in question stem indicated that there is atleast one person who speaks all three languages, but with these information we can definitely say that there is exactly one person who speaks all three languages. Thus determining that 5 people can interact with each other in English and German.
Correct option: C

Approach Solution (2):
The question asks for the number of people who are able to interact with each other in English and German.
S1 tells us 4 people speak exactly two languages (English and German). However, this figure does not include the individuals that can speak all three languages. An individual that can speak all three languages can certainly
interact in English and German
Insufficient
S2 tells us that 5 individuals speak more than one language.
Clearly, insufficient
Combined, we know 5 individuals speak more than one language. We also know that 4 people speak exactly to languages (English and German). Add the person that can speak all three languages, and we have 5 people who can interact with each other in English and German.
Correct option: C

Approach Solution (3):
According to the question:
Total people = 25

We need to find d + e
According to the question:
Atleast 1 person speaks all 3 languages
\(∴e\geq1\)
S1: 4 people speaks 2 languages but not Spanish
Therefore, d + b + f = 4
(People speaking Spanish = 0)
b + f = 0
Therefore, d = 4
Insufficient
S2: One fifth of the group speaks more than one language
Exactly 2 languages
d + b + f = 25/5 = 5
This equation will give us nothing
All 3 languages + Exactly 2 languages = 5
(We could have found the value of “d + e” if “b + f” was known)
e + d + b + f = 5
\(e\geq1\)
\(∴d+e=4+1=5\)
Solving for option C, d = 4 and b + f = 0
Insufficient
Correct option: C

“In a group of 25 people, only three languages are spoken- English, Spanish and German. If there is atleast one person who speaks all the three languages, how many people can interact with each other in English and German?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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