Riya remembers that the birthday is after 18^th June but before 21st June, while Annu remembers that birthday is before 24^th June but after 19^th June . On which date of June is the birthday?

20

According to question , 

According to Riya  birthday is after 18^th June but before 21st June

Possible date of birthday= 19 , 20   ------ ( 1 ) 

According to Annu  birthday is before 24^th June but after 19^th june

Possible date of birthday = 20 , 21 , 22 , 23      ----- (2 ) 

from ( 1 ) & ( 2) 

Common satisfying date is 20 june 

So birthday is on 20th june .