A survey was conducted to find out the number of languages spoken by

Question

A survey was conducted to find out the number of languages spoken by the 210 employees of a company. It was found that 60 employees did not speak English, 150 employees did not speak German and 170 employees did not speak French. If there were 20 employees who did not speak German or English and 20 employees who did not speak French or English, what was the maximum number of employees who spoke only English? Assume that each of the employees spoke at least one language and no employee spoke any language other than English, French and German.

A. 110
B. 120
C. 130
D. 140
E. 140

A. 110
B. 120
C. 130
D. 140
E. 140

Kritisood · Accepted Answer

Number of employees = 210
Number of employees who did not speak English = 60
So, the number of employees who spoke English = 210 – 60 = 150

Number of employees who did not speak German = 150
So, number of employees who spoke German = 210 – 150 = 60

Number of employees who did not speak French = 170
So, number of employees who spoke French = 210 – 170 = 40

Number of employees who did not speak German or English = 20
So, these employees must speak only French (because it’s given that each employee speaks at least one language)

Number of employees who did not speak French and English = 20
So, these employees must speak only German (because it’s given that each employee speaks at least one language)

To Find: Maximum number of employees who spoke only English

To max the no. who only spoke English we need to reduce as little as possible from 150 (no. who spoke English)
We know French in total is 40, which means that max overlap between French/German/English can be 20. That leaves 20 of german out of the overlapping circles. This 20 needs to get subtracted from 150 of English. Hence max English will be  150-20=130

The explanation will help greatly if you try to understand it with a diagram.

shashankism · Answer

According to the data given and venn diagram
c+f+g = 60 .......................1
a+d+g = 150.......................2
a+b+c = 170........................3
g=20
c=20

From 1, f = 20
From 2, a+d = 130
From 3, a+b = 150

Also, a+b+c+d+e+f+g = 210 
150 + 20+d+e+20+20 = 210
d=-e
that means both d=e = 0

so, a+d = 130 means a = 130
and b = 20

The no. of people who spoke only english = a = 130
 Answer C

aayush.works · Answer

the only tricky part in this question is to understand the meaning of the term "only english speaking employee" and what segment it represents in a venn diagram consisting of data like not E, not G, not F etc.. Its actually very simple once we draw the venn diagram (draw it using the data below).

Given:-
Not E = 60 (not English speaking employees)
Not G = 150
Not F = 170
p = not E,G
q = not G,F
r = not E,F
k = not E,G,F
p+k = 20
k+r = 20
N=0

To find :-
q (since q is outside Not E and inside Not G and Not F it is only English speaking employees, 
trust me once you draw the venn its very intuitive!)

hard work done! plug in all values and arrive at your answer:-
T-N = not E + not G + not F - (p+q+r) - 2k
210-0 = 60+150+170-(p+k) -(k+r) -q {p+k = 20, k+r = 20, substituting the values in this equantion}
q = 130

RanonBanerjee · Answer

Can anyone clarify if my concept is right here?

So from the info given we can say that 60-20= 40 people speaks English and french and 40-20=20 people speaks English and German. So from 150 (the number of people who speaks English) we will reduce either 20 or 40 to get maximum and minimum number of English speaking people respectively.

Min number of English speaking people 110 and max number of English speaking people will be 130.

Posted from my mobile device

kabirgandhi · Answer

Hello, could anyone explain this question through a venn diagram/ using formulae?

egmat · Answer

kabirgandhi   Great question!  Set problems with multiple overlapping groups are perfect for Venn diagrams. Let me walk you through this systematically.

Setting Up the Information

First, let's translate the "did not speak" information into "spoke" information:
 
  210  total employees 
  60  did NOT speak English →  150  speak English 
  150  did NOT speak German →  60  speak German  
  170  did NOT speak French →  40  speak French 
  20  did NOT speak German or English →  20  speak  only French  
  20  did NOT speak French or English →  20  speak  only German

Venn Diagram Regions

Let me label the  7  regions in our three-circle Venn diagram:
 
  a  = only English 
  b  = English and German only (not French) 
  c  = English and French only (not German) 
  d  = all three languages 
 Only French =  20  (given) 
 Only German =  20  (given) 
  x  = German and French only (not English)

Using the Formulas

From the total: 
 a + b + c + d + 20 + 20 + x = 210 
 a + b + c + d + x = 170  ... (equation 1)

From English speakers: 
 a + b + c + d = 150  ... (equation 2)

From German speakers: 
 20 + b + d + x = 60 
 b + d + x = 40  ... (equation 3)

From French speakers: 
 20 + c + d + x = 40 
 c + d + x = 20  ... (equation 4)

Solving the System

Substituting equation 2 into equation 1:
 150 + x = 170 
 x = 20

Now we know  German and French only  =  20

Substituting  x = 20  into equation 3:
 b + d = 20

Substituting  x = 20  into equation 4:
 c + d = 0

Since  c  and  d  cannot be negative:  c = 0  and  d = 0

Therefore:  b = 20

Finally, from equation 2:
 a + 20 + 0 + 0 = 150 
 a = 130

Answer: C. 130

Key Strategic Insight

The phrase "did not speak X or Y" is crucial - it means the person speaks  only  the remaining language(s). This immediately gives you concrete regions in your Venn diagram. Always start by filling in what you know for certain, then use the set formulas to find the unknowns. To maximize "only English," minimize all regions that include other languages - which the constraints naturally force you to do.

Hope this helps!

ledzep10 · Answer

I solved this by an approach of working with not speaking the languages.

If we have Total = 210
We can take a 3 set Venn Diagaram. 
Set 1: Not speaking English = 60
Set 2: Not speaking French = 170
Set 3: Not speaking German = 150
We also know,
The overlap between No English and No French = 20
The overlap between No English and No German = 20
 They have mentioned everyone speaks a language and they only speak English, German and French => From this we can gather that no English and no French and no German = 0
 Therefore the overlaps that we have found become an exactly 2 overlap:
i.e no English and no French means that speaking  only German  = 20
and no English and no German means that speaking  only French  = 20
and no French and no German means that speaking  only English  =  x,  which we have to find.

From the overlapping sets equation : 
Total = no English + no French + no German - ((no english and no French)+ (no English and no German) + (no French and no German)) + None
Total = no English + no French + no German - ((only German)+ (only French) + (only English)) + 0
210 = 60 + 170 + 150 - ((20)+ (20) + (x))
On Solving,  x = 130

Tanpok · Answer

It should be 60 + 170 +150  -  (20+20+x)

ledzep10 · Answer

Thanks for pointing it out. I have made the necessary changes to the original post.

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