At a certain university, there are s students, \(w\) of whom are female and m of whom are male. The number of female phy

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At a certain university, there are s students, \(w\) of whom are female and m of whom are male. The number of female physics students is denoted by \(p_w\) and is exactly \(12\%\) of the female enrollment. Similarly, the number of male physics students is denoted by \(p_m\) and is exactly \(25\%\) of the male enrollment. If the overall number of physics students is denoted by \(p,\) which of the following must be true?

A. \(p_w<p_m\)

B. \(w>0.06s\)

C. \(w<p<m\)

D. \(p_m=0.125s\)

E. \(0.12s<p<0.25s\)

Answer: E

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It's just a weighted average or mixtures situation. We know 12% of one group (women) and 25% of another group (men) are in physics. So overall, when we look at men and women together, somewhere between 12% and 25% are in physics. So the number of physics students p is somewhere between 12% of all students (which is 0.12s) and 25% of all students (which is 0.25s), and E is the correct answer.

The other answers can all be false. Answer A is false if we have far more female students than male students overall. Answer B is false if only a tiny fraction of the student population is female. Answer C can clearly be false, just ignoring the middle of the inequality, because w > m can be true. Answer D will only be true in the one case where men make up exactly half of the entire student body.
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