Three-fifths of the men in chemistry class have beards and two-thirds of the women have long hair. If there are 120 men in the class and 46 are not in the above groups, how many men and how many women are there in the class?

The question doesn't tell the amount of the women so I think the "120 men" mean that there are "120 people"(men+women) in the class. 
That mean 46 people which are not "above group" consist of 2/5 men(without beards) and 1/3 women(without long hair).
You can get this equation:

46= 2/5men + 1/3women
74= 3/5men + 2/3women
120= men + women

You can change the first equation into this
46= 2/5men + 1/3women
1/3women= 46- 2/5 men
women= 138- 6/5 men

Add the first equation into the second
74= 3/5men + 2/3women
74= 3/5men + 2/3(138- 6/5 men)
74= 3/5men + 92- 4/5men
74= -1/5 men + 92
1/5 men= 92-74
men= 18 *5= 90

Using the 3rd equation
120= men + women
women= 120-90= 30

The answer would be: 90 men, 30 women