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

Question

At a company, 124 employees completed at least one of the following three training modules: Safety, Data Privacy, and First Aid. Of these employees, 40 completed exactly one module. Is there at least one employee who completed both Data Privacy and First Aid, but not Safety?

(1) 16 employees completed all three training modules.

(2) The combined number of employees who completed Safety and Data Privacy only and those who completed Safety and First Aid only is 27.

Bunuel · Accepted Answer

GMAT Club Official Solution:

At a company, 124 employees completed at least one of the following three training modules: Safety, Data Privacy, and First Aid. Of these employees, 40 completed exactly one module. Is there at least one employee who completed both Data Privacy and First Aid, but not Safety?

Let:
x = number who completed Safety and Data Privacy only
y = number who completed Safety and First Aid only
z = number who completed Data Privacy and First Aid only
t = number who completed all three

Since 40 employees completed exactly one module, the remaining 124 - 40 = 84 employees completed either exactly two modules or all three. So:

x + y + z + t = 84

(1) 16 employees completed all three training modules.

This gives t = 16. Therefore:

x + y + z = 68

But this does not tell us whether z > 0.

Not sufficient.

(2) The combined number of employees who completed Safety and Data Privacy only and those who completed Safety and First Aid only is 27.

This tells us that:

x + y = 27

So:

z + t = 84 - 27 = 57

But this still does not tell us whether z > 0, because t could be 57 and z could be 0.

Not sufficient.

(1)+(2):

From statement (1), t = 16.
From statement (2), x + y = 27.

So:

z = 84 - 16 - 27 = 41

Thus z > 0, so there is at least one employee who completed Data Privacy and First Aid but not Safety.

Sufficient.

Answer: C.

Adit_ · Answer

Statement 1:
Clearly INSUFFICIENT.
We only know all 3 and we dont know about no.of people pursuing two thing at a time.

Statement 2:
INSUFFICIENT.
Again we dont know how many did all 3 here.

Combining the two:
We can include all 3 things at a time = 16,
We can also include a+b=27  
Using the venn diagram equation we can clearly find value for c.
SUFFICIENT.

Answer:  Option C

BongBideshini · Answer

Total = (Exactly 1) + (Exactly 2) + (Exactly 3)
Given:
Total = 124
Exactly 1 = 40
Exactly 2 + Exactly 3 = 124 - 40 = 84
Let the exactly 2 regions be:
a = Safety & Data Privacy only
b = Safety & First Aid only
c = Data Privacy & First Aid only
x = Exactly 3
So,
a + b + c + x = 84
We need to find the value of c

Statement (1): x = 16
a + b + c + 16 = 84
a + b + c = 68
Without knowing a or b, we cannot determine c.
 Insufficient

Statement (2): a + b = 27
a + b + c + x = 84
27 + c + x = 84
c + x = 57
Without knowing x, we cannot determine c.
 Insufficient

Combined (1)+(2)
27 + c + 16 = 84
43 + c = 84
c = 41
 Sufficient 
 Correct Answer: C

Edskore · Answer

Answer is C.

This is a three-overlapping-sets Data Sufficiency problem. The real question being asked: is {Data Privacy & First Aid only} > 0?

Set up the basics first:
Total = Exactly-1 + Exactly-2 + Exactly-3
124 = 40 + Exactly-2 + Exactly-3
So Exactly-2 + Exactly-3 = 84

The Exactly-2 bucket splits into three regions:
a = Safety & Data Privacy only
b = Safety & First Aid only
c = Data Privacy & First Aid only (this is what we need)

So a + b + c + Exactly-3 = 84.

Statement (1) alone: Exactly-3 = 16, so a + b + c = 68. But we have no idea how that 68 splits between a, b, and c. c could be anywhere from 0 to 68. INSUFFICIENT.

Statement (2) alone: a + b = 27. So c + Exactly-3 = 57. But without knowing Exactly-3, c is still free to vary. INSUFFICIENT.

Together: From (1), Exactly-3 = 16 and a + b + c = 68. From (2), a + b = 27. So c = 68 - 27 = 41. Since 41 > 0, yes, at least one employee completed Data Privacy and First Aid but not Safety. SUFFICIENT.

Answer: C

The trap is thinking Statement (1) is enough. It pins down the all-three count, but the "exactly two" group still has three unknown pieces. You need (2) to split that bucket further.

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