Can carpet on stairs be cleaned?
Absolutely, there is no reason carpet on stairs cannot be cleaned. If you have your carpet cleaned by a truck mounted carpet cleaner, which you should be, they will probably have special tools just to clean stairs with. Some carpet cleaners, choose to use the same tool on stairs that they do on all of the carpet, but many use stair cleaning tools. The benefit of stair cleaning tools is that it can get into tighter spaces and is much more maneuverable. Hopefully, the carpet cleaner you choose will use one of these specialty tools.
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As far as pricing goes, most carpet cleaners will charge by the stair or will count the staircase itself as a room. You might think that this is excessive because the amount of carpet on a stairway is often less than what is in a room but there is a reason for this pricing. Stairs take just as long to clean as a full room of carpet and are much more work in general.
The only thing that you have to keep in mind when cleaning stairs is that they are subjected to much more wear then the rest of your carpet. What this means is that because of the concentrated traffic down the middle of your stairway, your carpet is probably permanently worn. Your carpet cleaner can greatly improve the appearance of your stairs but you should not expect them to look brand-new.