Let the number be x and y

x+y=24,or y=24-x..(1)

Xy=120,or y=120/x..(2) equate (1) and (2)

Equate (1) and (2)

24-x=120/x,or

x^2-24x+120= 0

x1=[24+9.7]/2

=16.8 and y2=7.1

x2=[24-(576-480)^0.5]/2

=7.1 and y2= 16.8

The sum of the reciprocals=1/16.8+1/7.1

=1/5 or 0.5