In 1 hour, Jack can paint 1/10 of the room.
In 1 hour, Kate can paint 1/14 of the room.
Together, in hour, they can paint (1/10) + (1/14) = 6 / 35 of the room.
Hence, time taken for them to paint whole room
= 1 / (6 / 35) hours
= 35 / 6 hours
= 5 and 5/6 hours
= 5 hours and 50 minutes