Of the 600 students in a class, each French speaker also speaks German

Of the 600 students in a class, each French speaker also speaks German, and 140 of students only speak English. If no member can speak all 3 languages, how many of the members speak 2 of the 3 languages?

(1) 120 members speak only German
(2) 40 students do not speak any of the 3 languages

Please see attached file for solution.

IMO C

Statement 1 is not sufficient since there is no information about the number who don’t speak any language.

Statement 2 is also insufficient since no information is available about the number who speak only German.

1+2 however provides sufficient information to get the number who speak only two languages as follows:

600-140-120-40=300

The answer therefore is C

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We need the shaded part.
1 & 2 Independently insufficient, but 1+2 gives the required answer

From the stem itself we know that the overall number of students is \(600\). Since \(140\) students speak only English, the max possible number of students who speak \(2\) languages is \(600-140=460\). If each French speaker also speaks German, then no student speaks only French. French speakers for sure speak two languages. However, German speakers may speak only German. We know nothing about them from the stem. Additionally, no one can speak all three languages.

ST1. \(120\) members speak only German. Now the max possible number of students who speak \(2\) languages is \(460-120=340.\) However, we still don’t know about all students. There may be those who don’t speak all languages. In such problems we usually have such groups and thus should be alert. In overlapping sets problems it would be wise to take a look at both ST1 and ST2 before we make conclusion.

ST2. \(40\) students do not speak any of the 3 languages. Bingo! That was something we were curious about while reading ST1. But ST2 itself is not enough too.

ST1+ST2. Finally we have all what we need: \(340-40=300\).

Hence C

Total = 600
English only = 140
French only = 0 (every French speaker also speaks German)
German only = G

English+French+German = 0 (no member can speak all 3 languages)
English+French = 0 (every French speaker also speaks German and no member can speak all 3 languages)
English+German = y
French+German = x
No language = z

QUESTION: How many of the members speak 2 of the 3 languages? --> x + y ?

(1) 120 members speak only German
G = 120 --> 600 = (140+0+120) + (x+y+0+0) + z --> 340 = (x+y)+z
(1) is NOT SUFFICIENT, because z is not known.


(2) 40 students do not speak any of the 3 languages
z = 40 --> 600 = (140+0+G) + (x+y+0+0) + 40 --> 420 = (x+y)+G
(2) is NOT SUFFICIENT, because G is not known.


Using both: G=120, z=40
600 = (140+0+120) + (x+y+0+0) + 40
300 = (x+y)
Combination of (1) and (2) is SUFFICIENT


Correct answer is (C)

Total = Germany only + French only + English only + G∩E only + G∩F only + F∩E only + G∩F∩E + Neither

translating the given:
Of the 600 students in a class --> Total = 600
each French speaker also speaks German --> French only = 0, and F∩E only = 0
140 of students only speak English --> English only = 140
no member can speak all 3 languages --> G∩F∩E = 0

formula after substituting known values:
600 = Germany only + 0 + 140 + G∩E only + G∩F only + 0 + 0 + Neither

question : what is the value of (G∩E only + G∩F only)

statement (1): Germany only = 120 --> insufficient because we don't know the Neither value
statement (2): Neither = 40 --> insufficient because we don't know the Germany Only value

Combining (1) and (2):
600 = 120 + 0 + 140 + G∩E only + G∩F only + 0 + 0 + 40
G∩E only + G∩F only = 420 --> sufficient

C

Let students that can speak french= a
Let students that can speak only German= b
Let students that can speak only English= c
Let students that can speak both German and English=d
and students who can speak none = e

Given- a+b+c+d+e=600, and c=140
members that can speak 2 of the 3 languages= a+d
Hence if we have value of b and e, we can find out a+d

Statement 1
b=120. Still e is unknown
Insufficient

Statement 2
e= 40, b is unknown.
Insufficient

Combining both statements
We have value of b and e, hence we can find out a+d
Sufficient

IMO C

Of the 600 students in a class, each French speaker also speaks German, and 140 of students only speak English. If no member can speak all 3 languages, how many of the members speak 2 of the 3 languages?

(1) 120 members speak only German
(2) 40 students do not speak any of the 3 languages

The scenario will be as shown in the figure

Now given
we have to find f+x
as per statement 1
f+120+x+140+ y = 600
thus f+x+y = 340
we need y also
Hence not sufficient

As per statement 2

f+g+ x+ 140+40 = 600
again we have an extra g that we should know
Hence no suffciient
Combine 1 & 2

f+120+x+140+40 = 600
f+x = 300
this is what was the answer needed
Thus C is

The 2 formulas for 3 intersecting sets are,

1. TOTAL = A + B + C - (sum of 2 set overlaps) + (3 set overlap) + Neither
2. TOTAL = A + B + C - (sum of exactly 2 set overlaps) - 2 * (3 set overlap) + Neither

In our question,

Knowns are,

1. Total students in class = 600
2. Each French speaker also speaks German.
3. No members can speak all 3 languages.
=> 4. From 2 and 3 No French speaker also speaks English.

Unknowns are,

1. Number of students who speak both English and German.
2. Number of students who speak only German.
3. Number of students who speak both German and French.
4. Number of students who do not speak any of the languages.

What we need to find?

Number of students who speak 2 of the 3 languages.
=> Number of students who speak either
English and German OR
German and French OR
French and English

Now from our knowns we know,
French and English = 0

Options 1: 120 members speak only GERMAN.

As per all the above information, this is not enough as we still do not know the number of students who speak all the 3 languages.

Option 1 alone insufficient.

Option 2: 40 students do not speak any of the 3 languages.

As per all the above information, this is not enough as we still do not know how many students speak only GERMAN.

Option 2 alone insufficient.

However if we use the information available in both the options above together we get,

2 out of our 4 unknowns.
The other 2 unknowns are what we need to find out cumulatively, since we deduced that the number of students who speak only English and French is 0.

Hence, both statements together are sufficient.

Answer: C

2 categories = 0 + Y + Z = Y + Z

Formula: Total(T) = Neither(N) + 140 + 0 + C + (0 + Y + Z) + 0
--> 600 = N + 140 + C + (0 + Y + Z)
--> Y + Z = 460 - (N + C)

(1) 120 members speak only German
--> C = 120
--> Y + Z = 460 - (N + C)
--> Y + Z = 460 - (N + 120)
--> Y + Z = 340 - N

Insufficient

(2) 40 students do not speak any of the 3 languages
--> N = 40
--> Y + Z = 460 - (N + C)
--> Y + Z = 460 - (C + 40)
--> Y + Z = 420 - C

Insufficient

Combining (1) & (2)
C = 120 & N = 40
--> Y + Z = 460 - (N + C)
--> Y + Z = 460 - (120 + 40)
--> Y + Z = 300

Sufficient

IMO Option C

Pls Hit Kudos if you like the solution

As shown in the image,

(a+b+c) + (p+q+r) + s = 600 is given.
c=140 is given
Each French speaker also speaks German = a & q both are zero.
This I think is the most important inference.

Now, we need to find (p+r) and we have (p+r) + (b+s) = 460
So, we need to find values of b & s.

(1) 120 members speak only German

b=120 but we still don't know s. Insufficient.

(2) 40 students do not speak any of the 3 languages

s = 40 but we don't know b. Insufficient.

If we combine two, we get an answer. So, Ans should be (C)

----each French speaker also speaks German (given)
----students only speak English = 140 (given)
combining statement (1)&(2) we get the answer
C is the answer

Simple one. Drawing a Venn diagram will be much helpful while solving this Q.

Quickly looking at the information provided in the question stem, each French speaker also speaks German means People who ONLY speak French = 0, and 140 of students only speak English means People who ONLY speak English = 140

Given that a + b + c + d + e + f + g + n = 600; where a = ONLY French, b = ONLY German, c = ONLY English, d = BOTH French and German, e = BOTH German and English, f = BOTH English and French, g = ALL THREE, n = NONE OF THE THREE

We already have a = 0, c = 140, g = 0, and we need to find d + e + f

Looking at Statement (1):

We have b = 120

Putting in the values, we have 0 + 120 + 140 + d + e + f + 0 + n = 600

d + e + f + n = 600 - (120 + 140)

d + e + f + n = 340, since we don't know the value of n we can't compute the value of d + e + f

STATEMENT (1), hence, is NOT SUFFICIENT

Looking at Statement (2):

We have n = 40

Since here, we don't know the value of b, we again have insufficient information in this Statement, and hence, STATEMENT (2) is NOT SUFFICIENT


Combining both (1) and (2), we now have values of both n and b, putting in which we get 0 + 120 + 140 + d + e + f + 0 + 40 = 600

d + e + f = 300

And hence, (C) is the correct answer here

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