In a club, there are 15 people, 1/3 can speak English, 2/5 can speak French and 2/3 can speak German. There is only 1 who speaks all three languages.

1. How many people exactly can speak two languages?

2. How many people can speak at least two languages?

2. How many people can speak only one language?

## Languages

##### This topic has expert replies

- gaggleofgirls
- Master | Next Rank: 500 Posts
**Posts:**138**Joined:**15 Jan 2009**Location:**Steamboat Springs, CO**Thanked**: 15 times

When you have three groups (as in this case with Eng, French and German), then the formula is:

total - GroupA + GroupsB + GroupC - AB - BC - AC - ABC - ABC + None.

You have to subtract those that are in all three groups twice since they have been counted 3 times and we need them only counted once.

Now, with the percentages, we know that:

5 speak English

10 speak German

6 speak French

1 speaks E, G, & F

In this case, we don't care which two languages they speak, so we can group together all the "speak 2 lang" together (AB, BC and AC) and get:

15 = 5 + 6 + 10 -2lang -1 -1 + None

We are missing a bit of information to really be able to solve this. Without knowing that everyone speaks at least one language, there are multiple solutions.

Solution if everyone speaks at least 1 language:

15 = 5 + 6 + 10 -2lang -1 -1 +0

15 = 21 - 2 - 2lang

15 = 19 - 2lang

4 people speak 2 languages

So the answers to your questions are:

a) exactly 2 lang = 4

b) at least 2 lang = 5

c) only one lang = 10

However, the % still work fine if 5 people speak no languages.

Then you get:

15 = 5 + 6 + 10 - 2lang - 1 -1 +5

15 = 26 - 2 - 2lang

15 = 24 - 2lang

9 = 2lang

So the answers to your questions are:

a) exactly 2 lang = 9

b) at least 2 lang = 10

c) only one lang = 0

Another way to look at this is that there are 21 languages spoken by 15 people.

1 person counts for 3 languages

If the other 14 all speak at least 1 lang, then you have 21 - 14 + 3 = 21-17 = 4 who have to speak 2 languages.

OR

21 languages spoken by 15 people.

1 person counts for 3 languages

9 people count for 2 languages

= 3 + (9*2) = 3 + 18 = 21, so there are 15-(9+1) = 15-10 = 5 people who don't speak any languages.

With the none not = to 0, I am probably looking too hard to be tricked, but since this isn't really in the form of a GMAT question (one question with 5 choices), then it lends itself to this openness.

-Carrie