Turin, there is an active noise reduction installed indeed. It has microphones scattered all through the cabin (usually located behind those small brass rings) that pick up the noise at their location, then a vibration of the same frequency, albeit with negative amplitude, is issued. This does not make the cabin very quiet, especially around the propellers, but it does reduce the overall noise level noticeably. If for some reason the system is set to "pause" in flight, the propeller hum is still there, but more so than before. It does not work equally well on all aircraft; among our fleet, some are much quieter in the cabin than others.
I do agree with what has been written several times already: if at all possible, take a seat in the last few rows. The propellers are still audible back there, as is a bit of exhaust airflow, but overall, those are the quietest seats in the aircraft by far. The reason why the business class is not back there is simply the passengers having become accustomed to finding the expensive seats in the front of the aircraft. There even was a survey some years ago to find out the preferences of the passengers: Business Class in the front, between the propellers and far from the galley, or rather in the back, close to the galley and much quieter than in the front? Somewhat surprisingly, a majority of passengers preferred the former, valuing seat 1A and a good rattling of their dentures higher than a calm and serene flight on 21F and boarding via the servants entry in the back. So be it then, said the company, and left everything as it is now.
Some numbers about the DH8-400. It can be operated nearly like a jet with a TAS of around 360kts at FL200 (a bit depending on temperature), burning slightly more than a ton/hour. Or it can be flown like an ATR or DH8-300 at much reduced power settings, resulting in around 300kts TAS and a hourly burn of a bit less than 900kg. If You will, the DH8D does the job of a Fokker 70 with half the fuel and lower ATC costs.