To save some people time because I know how busy some people can be. The answer to this question is Yes.
For those after a bit more insight into the question of whether an enjoyment of Radio 4 can be shared with the angry rants of a rap song, please read on kind people.
I have recently discovered a real interest in talk radio. This has led me to listening to a lot of Radio 4, 5live and talkSport. I am not just talking about listening to football chatter, I am also mean other stuff like politics and other human interest debates.
One radio listening session which did catch my attention was a live debate in the houses of parliament where Cameron and Milliband were taking it in turns to accuse each other of failing stuff. (and this is going to solve the countries problems).
This is a rough transcript of what they said

Milliband (M): You said you would make sure the country would not be downgraded in its credit rating. Does this mean you have failed?
Cameron (C): We must work to gain this credit rating again
M: So you failed?
C: We must improve the country?
M: So you failed?
And so on… Pointless but amusing. He might as well as said ‘Cameron, you are a dick’ and then Cameron could come back and say ‘Milliband, you are a dick’ which then could see the two of them going ‘no, you are’… ‘no YOU ARE’. One of them could then storm out and slam the door, but it won’t because its one of those silly safety fire doors that closes slowly on its on (never good when you want a dramatic door slam exit).
Anyway back to the original point. So I love listening to this stuff now. But to totally turn it on its head, on my way to presenting my hospital radio show Saturday I listening to a bit of Kanye Wests latest album Cruel Summer which features the song Mercy where the lyrics ‘So much hate, I need an AK’ are used. Pictures it if you will, a white 27 year old man, wearing glasses, driving a Peugeot 307 at 29mph along Shoreham High Street singing along to ‘Your chick, she’s so thirsty, I am in the yellow Lambo, your chick, she’s trying to jerk me’.

If Kanye West is reading this, I would like to know what he means by doing Suicides on the Tour Bus and in the Private Jet.
Anyway, I know I may have personality defects, but at the same time I feel many others do and its ok to like Radio 4 and Rap Music, just never expect the two to cross paths.

A has assumed B wants to know how he is so has just jumped in there and told him without being asked! This awkward moment is usually ignored by both parties, so from that point of view it’s not too much of an issue.