I need to figure out an annual lease payment... in the question I have been given both the carrying value of $162,500, and the fair market value of $225,000. The rate is 10% and the lease term is 5 years. I am unsure of which amount to use to determine the lease payment... do I use the carrying value or the fair market value?

Use the FMV. A lease is really just another way of selling an asset, or at least selling a portion of an asset. And of course, whenever you sell an asset you'll sell it at its fair value, regardless of its carrying amount on the books.