If they have 24 hours in a day, and use 1/6 of those 24 hours on news, they use 24*(1/6) or 4 hours on news. Assuming that they use the leftover time for just music and entertainment, we have 20h - 4h, so thats 20h. You were right. Not sure on their reasoning either.
For the smaller triangle ;
15^2 + b^2 = 17^2
b = 8
To check the work
15^2 + 8^2 = 289
17^2 = 289
(This is if the bottom number on the smaller triangle says 15, its a bit hard to read..)
(9.is 12 and 2)cghcjgcmhgcghccgckghc