Should I allow tenant to pay for a partial month?

I rent the rooms in the basement of my primary residence. I have one person moving out on the 18th, he wants me to breakdown the the per day rate and pay me only that. Is that something I should do? Is that normal in a rental situation?
How would you guys handle this situation

Thanks in advance!

P.S.
Also, his room now has a bad musty smell because he never cleans or washes his clothes. There is literally a cloud of funk that comes out when he opens his door. Should he have to pay for having the carpet cleaned and the smell removed? Other than that the room seems in decent shape. I have a security deposit from him.

Do you have an actual lease for any of these or are you just letting people in with money and a handshake? Yes, pro-rating rent is normal although most people are on a lease until the end of a month and that’s when they move.
If you have no lease, your options are limited.

We don’t have a lease. I only require 30 days notice when you leave, which he did give me.

And what about the smell in the room?

I’m sorry, but no lease is not good. I mean, there should be a lease which of course the tenant will have a copy and yourself. This is where you and your tenant had agreed upon prior to accepting him and let him live in your property. Justin is right, if you don’t have a lease then your options are limited. With the smell and some related issues, read this.

Repairs

The landlord will normally be responsible for carrying out repairs to your home. See ‘Disrepair in Privately Rented Accommodation’. If there are any items of disrepair then tenant should tell the landlord straightaway. However, if tenant or tenant’s visitors break or damage something in the property, it will be tenant’s responsibility to get it repaired or replaced. If not, the landlord could be entitled to some or all of tenant’s deposit to pay for the works himself. Minor repairs such as changing a light bulb, internal decoration and keeping the property clean are the responsibility of the tenant.

Writing a lease agreement and having the tenant sign it doesn’t take a lot of time or effort.
You could and should simply write down what is expected from the tenant and what you will provide. Also list the consequences for not doing what is required.
That is where you should start the if the room is left damaged then the tenant will be required to pay or lose their deposit.