In a class, there are 170 students, each plays at least one out of 3

Question

In a class, there are 170 students, each plays at least one out of 3 outdoor sports: Hockey, Tennis or Cricket. Of these students, 55 play exactly one sport. Is there any student in the class who plays Tennis and Cricket but not Hockey?

(1) 25 students play all the three types of sports.
(2) 35 students play Hockey and exactly one of the other two sports.

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georgec · Answer

This is an overlapping sets question of three groups. Hockey (set A), Tennis (set B), Cricket (set C)
We are given that 170 is the total number of participants. Each plays at least one sport, so 0 play no sport. First equation is a+b+c+d+e+f+k=170.
We are also given that 55 play exactly one sport. Second equation is a+b+c=55.
We can combine our equations into 55+d+e+f+k=170 -> d+e+f+k=115
Question is asking about the group that plays Tennis and Cricket, but not Hockey - which is e.

1) 25 play all 3 sports - which is k.
we replace k with 25 to get d+e+f+25=115; so d+e+f=90
This is clearly NOT SUFFICIENT to find e

2) 35 play Hockey and exactly 1 of other 2 sports - which are d and f. Not k because k is a group that plays all 3 sports.
we replace d+f with 35 to get 35+e+k=115; so e+k=80; 
This is clearly NOT SUFFICIENT to find e

1&2 combined) k=25, d+f=35, 
so d+e+f+k=115=e+35+25
e=55 which answers the question.

Answer is C

ManifestDreamMBA · Answer

Statement 1
Total = Exactly 1 + Exactly 2 + All 3
170 = 55 + Exactly 2 + 25
Exactly 2 = 170-80=90
We don't know how many of these 90 play Tennis and Cricket but not Hockey
Insufficient

Statement 2
Exactly 2 = HT + HC + TC
170 = Exactly 1 + HT + HC + TC + All 3
170 = 55 + 35 +TC + All 3
We don't know All 3, to identify TC, i.e. students who play Tennis and Cricket but not Hockey
Insufficient

Combined, we have All 3 = 25, which we can use in the above eqn to find TC
Sufficient

HarshavardhanR · Answer

Exactly 1 -> I
Exactly 2 -> II
Exactly 3 -> III
None -> N

I + II + III + N = 170.

Here, 
N = 0; So,

I + II + III = 170

I = 55. Given.

Question:  Is (T and C only) > 0?

(1) III = 25

I + II + III = 170 
 
55 + II + 25 = 170
=> II = 100

i.e., (T and C only) + (H and C only) + (T and H only) = 100.

It is possible that (T and C only) = 0. Or it could be some non-zero number.

S1 - Insufficient.

(2) (H and C only) + (T and H only) = 35

Just from this, we cannot find (T and C only)

S2 - Insufficient.

S1 and S2 together
  
 (T and C only) + (H and C only) + (T and H only) = 100 
 (H and C only) + (T and H only) = 35 
=> (T and C only) = 65. So, there are students who play (T and C only).
 
S1 and S2 together - Sufficient. Choice C.

---
Harsha

nammolestias · Answer

https://gmatclub.com/forum/download/file.php?id=89068 combining both the statement we get the ans therefore ans is option c GMAT-Club-Forum-gc7jilru.png

In a class, there are 170 students, each plays at least one out of 3

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GMAT Data Sufficiency (DS) Questions

In a class, there are 170 students, each plays at least one out of 3

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