Answer: A

Solution:

Since our room has $\dfrac 25$ of the people wearing gloves, the number of people must be a multiple of $5.$ Since our room has $\dfrac 34$ of the people wearing hats, the number of people must be a multiple of $4.$ Therefore, the people in the room must be a multiple of $20.$

Now, we can also use the following formula by the principle of inclusion exclusion: Fraction of people wearing both = Fraction of people wearing gloves + Fraction of people wearing hats - Fraction of people wearing either.

This makes our desired fraction equal to $\dfrac{2}{5} + \dfrac 34 -$ Fraction of people who wear either. If we wish to minimize the number who wear both, we maximize the fraction of people who wear either, up to $1.$ Therefore, the fraction of people that wear both is $\dfrac{2}{5} + \dfrac 34- 1 = \dfrac 3{20}.$

Since our number is a (positive) multiple of $20,$ we have the number of people wearing both as $3$ if we choose to have just $20$ people.

Therefore, A is the correct answer.