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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.
(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.
03 Aug 2026, 08:59
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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
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
