kevincanspain wrote:If 20 of his friends speak at least two languages, does Seth have more than 100 friends?

(1) 8% of his friends speak exactly two languages
(2) 60 of his friends speak English and 50 of his friends speak Japanese

Hi Kevin:

Let the total number of friends be x. We need to find if x>100. We are also told that number of people speaking two or more languages is 20. With that in mind, let's look at the statements:

(1) 8% of his friends speak exactly two languages
We know that 8% of x = friends who speak exactly two languages.
Now since 8% of x is a positive integer, 8% =8/100=2/25. Hence x is a multiple of 25.
For x = 25, 2 people speak exactly two languages. Hence 18 people speak more than two, and 5 speak exactly one.
For x=200, 16 speak exactly two languages, 4 speak more than two, and 180 speak exactly one language.
So we can't be sure whether x > 100. Insufficient.


(2) 60 of his friends speak English and 50 of his friends speak Japanese[/quote]
Here we are told that 60 are English speakers and 50 are Japanese speakers.
Best case scenario: 20 people speak both English and Japanese and these are the only two languages spoken by Seth's friends. Then:
Min. no. of people who speak only English=60-20=40
Min. no. of people who speak only Japanese=50-20=30
So, Total number of people = 40+30+20 = 90. So there must be at least 90 people present, of which at least 70 speak exactly one language (either English or Japanese).

But we can also have as one of the possibilities, 60 people speaking only English, 50 people
speaking only Japanese, and 20 more people who speak French and Spanish. Then total = 60+ 50+20 = 130
So we can't be sure whether x > 100. Insufficient.

Let's consider them together. From 1, we know that x is a multiple of 25. From 2, we know that x is no less than 90. So we need to only check, x=100 (25*4) case to determine whether it fulfills all our conditions. If it does, answer is E. If it doesn't, answer is C.

When x=100, 8%*100=8 people speak exactly two languages and 20-8=12 people speak more than two. So, 80 people speak exactly one language. Since we already established that at least 70 people speak only English or only Japanese, let's test that case. For simplicity, assume that there is only one more language being spoken, and that language is Spanish. Hence, no. of people speaking only Spanish = 80-70 = 10. This satisfies all our conditions. Hence Seth can have 100 friends. So we can't be sure that x>100. Insufficient again.

So the final answer is E.

PS: One can check the 100 case as follows:

100 = (exactly one) + (exactly 2) + (exactly 3)
100 = (40+30+10) + (8) + (12).
English speakers = 40+8+12=60
Japanese speakers = 30+8+12=50
Spanish speakers = 10+0+12=22
People speaking at least two = 8+12=20
8% of people speaking exactly two languages = 8