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# The members of a club were asked whether they speak Cantonese, Mandari

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Math Expert
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The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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01 Jul 2018, 20:50
00:00

Difficulty:

95% (hard)

Question Stats:

37% (02:20) correct 63% (02:19) wrong based on 70 sessions

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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.

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Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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01 Jul 2018, 21:30
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.

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The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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01 Jul 2018, 21:42
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.

$$C = 100 =C_o+CM+CJ+CMJ$$
$$M = 150 =M_o+MJ+CM+CMJ$$
$$J = 200 =J_o+CJ+MJ+CMJ$$..
$$CM+MJ+CJ=120$$
where $$C_o$$ = speaking only Cantonese
$$CM$$,$$MJ$$,and $$CJ$$ = speaking two languages
$$CMJ$$= speaking all languages

total members = $$C_o+M_o+J_o+CM+MJ+CJ+CMJ+None=C+M+J-(CM+MJ+CJ)-2(CMJ)+None=100+150+200-120-2(CMJ)+None=320-2(CMJ)+None$$ ....
so it will depend on number of people speaking all three and number of people not speaking at all

(1) There are twice as many members who speak none of the languages as there are who speak all three languages.
At the first look one may feel nothing, BUT it talks of two things - all 3 and None- and we are looking for these two only
so None $$= 2*CMJ$$
Total = $$320-2(CMJ)+None=320-2(CMJ)+2(CMJ)=320$$
sufficient

(2) Half of the members who speak Japanese and Cantonese also speak Mandarin.
Not what we are looking for
insuff

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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01 Jul 2018, 22:30
After sorting all the data, what we need is CMJ and none.

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

I think twice as many members who speak none of the languages as there are who speak all three languages means 2(none)=CMJ

= -2CMJ+none
=-2CMJ+(CMJ/2)....again we need CMJ value...which we don't have....

so Insufficient

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

we don't have JC value so we don't get any thing.

so Insufficient.

Math Expert
Joined: 02 Aug 2009
Posts: 7202
Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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01 Jul 2018, 22:36
1
mahi816 wrote:
After sorting all the data, what we need is CMJ and none.

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

I think twice as many members who speak none of the languages as there are who speak all three languages means 2(none)=CMJ

= -2CMJ+none
=-2CMJ+(CMJ/2)....again we need CMJ value...which we don't have....

so Insufficient

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

we don't have JC value so we don't get any thing.

so Insufficient.

hi
your inference of coloured portion is not correct..
it is easy to get confused in these wordings but ask yourself WHAT IS MORE?
Quote:
twice as many members who speak none of the languages as there are who speak all three languages means 2(none)=CMJ

this means None is more..
twice as many members X as twice as members Y..
so X is greater ... so it cannot be y=2x, here y becomes greater
so x=2y
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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02 Jul 2018, 10:29
Going with A.

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

Total Members = A+B+C - (Sum of Exactly 2 Languages) - 2* All Languages + Neither of the languages.
Since Neither of the languages = 2* All Languages we have no unknowns.
Sufficent

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

All Languages & Neither of the languages are unknowns.
Insufficent
Intern
Joined: 15 May 2017
Posts: 29
Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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02 Jul 2018, 13:12
Total = J+C+M - 2 * All Languages - (Sum of Exactly 2 Languages) + Neither of the languages.
We know J, C M and people who spoke exactly 2 languages

1. Neither = 2* All Languages
Sufficient

2. we don't know the count of people who Speak Japanese and Cantonese.
Insuff

A
Intern
Joined: 22 Jan 2018
Posts: 17
Re: The members of a club were asked whether they speak Cantonese, Mandari  [#permalink]

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06 Jul 2018, 22:55
(A)

i---->exactly one, ii--->exactly two, iii--->exactly three, n--->none

Given, ii=120

So,we have, i+2ii+3iii=450 ----> eq.1
i+ii+iii+n=Total----> eq.2

From (1)-->n=2iii
Putting the value of n in eq.2 and equating with eq.1 will give Total = 330 ----->Sufficient

From (2)--->Many unknown entities, yet -------->Insufficient
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Re: The members of a club were asked whether they speak Cantonese, Mandari &nbs [#permalink] 06 Jul 2018, 22:55
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