1. What do the following compound words have in common?

FAIRGROUND
DOWNTIME
ROOMMATE
DOUGHBOY
DATEBOOK

2. What do the following words have in common?

RAT
DINE
STUD
CHART
DRAMA

HINTS:
1. Both halves of each compound word have the same word in common.
2. The same two letters will change each word into a new one.

Answers:

1. Both halves of each compound word can be preceded by PLAY:

PLAY FAIR - PLAYGROUND
PLAY DOWN - PLAYTIME
PLAYROOM - PLAYMATE
PLAY DOUGH - PLAYBOY
PLAY DATE - PLAYBOOK

2. Adding or inserting the letters IO will change each word into a new one:

RATIO
IODINE
STUDIO
CHARIOT
DIORAMA

Credits:
Both: NY Times Crossword
#1: December 18, 2012
      Adam G. Perl
#2: December 30, 2012
       Steve Savoy
       PLUS TEN