VJesus12 wrote: ↑

Wed Nov 17, 2021 12:55 pm

At an engineering firm, 80 employees have graduate degrees (PhDs or Masters degrees). How many employees have PhDs?

(1) One-third of the employees with Masters degrees also have PhDs.

(2) One-half of the employees with PhDs also have Masters degrees.

Aswer: C

Source: Veritas Prep

Given; At an engineering firm, 80 employees have graduate degrees (PhDs or Masters degrees).

Asked: How many employees have PhDs?

Let employees with PhD be P alone, employees with Masters be M alone and employees with PhD and Masters be PM.

\(P + M + PM = 80\)
\(Q. P + PM = ?\)

1) One third of the employees with Masters degrees also have PhDs.
\(1/3 (M+PM) = PM\)
\(PM = M 1/2\)
Since \(P\) is unknown. Not Sufficient \(\Large{\color{red}\chi}\)

2) One half of the employees with PhDs also have Masters degrees.
\(1/2(P+PM) = PM\)
\(P = PM\)
Since \(M\) is unknown. Not Sufficient \(\Large{\color{red}\chi}\)

Combining \(1 \& 2\)

1) One third of the employees with Masters degrees also have PhDs.
\(1/3 (M+PM) = PM\)
\(PM = M 1/2; M = 2PM\)

2) One half of the employees with PhDs also have Masters degrees.
\(1/2(P+PM) = PM\)
\(P = PM\)
\(P + M + PM = 80\)
\(P + 2P + P = 80\)
\(P = 20 = PM\)
\(P + PM = 40\) Sufficient \(\Large{\color{green}\checkmark}\)