Bunuel wrote:

The members of a club were asked whether they speak Cantonese, Mandarin and Japanese. 100 said that they spoke Cantonese, 150 said that they spoke Mandarin and 200 said that they spoke Japanese. 120 said that they spoke exactly two of the three languages. How many members does the club have?

(1) There are twice as many members who speak none of the languages as there are who speak all three languages.

(2) Half of the members who speak Japanese and Cantonese also speak Mandarin.

Since we have many move variables than equations, we'll trying testing easy numbers

This is an Alternative approach.

We'll first start by assuming that the 120 people who spoke 2 languages are split evenly between the pairs. That is 40 spoke Mandarin and Chinese, 40 spoke Mandarin and Japanese and 40 spoke Japanese and Cantonese.

(1) say 0 members spoke all 3 languages. Then 100 - 80 = 20 spoke only Cantonese, 150 - 80 = 70 only Mandarin and 200 - 80 = 120 only Japanese and we can finish calculating to get (only one language) + (exactly two languages) = (20+70+120) + (120) total number of people.

Say 10 members spoke all 3. Then 20 spoke none. so 100 - 80 - 10 = 10 spoke only Cantonese, 150 - 80 - 10 spoke only Mandarin and 200 - 80 - 10 spoke only Japanese.

If we're careful, we'll notice that we've added and subtracted the same number of people: we've subtracted 30 people from the (only one language) group, added 10 to the (all three languages) group and added 20 to the (no languages group). So our total hasn't changed.

Moreover, this always happens - if x is the number who speak all three than we subtract x from each of the single languages, meaning we subtract 3x total but then add back x for all 3 languages and 2x for no languages.

Sufficient.

(2) Based on (1), we know it is critical to know how many people speak 0 languages. Since we currently have no information on this at all, (2) cannot be sufficient.

Insufficient.

(A) is our answer.

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