$25.$ In the current year, Penny exchanges an investment property in Santa Barbara with a mountain view for a lot with an ocean view in a qualifying like$-$kind exchange. Penny's basis in the land given up is $$100,000,$ and the property has a fair market value of $$250,000.$ In exchange for her property, Penny receives land with a fair market value of $$200,000$ and cash of $$20,000.$ In addition, the other party to the exchange assumes a mortgage loan on Penny's property of $$30,000.$

a$.$ Calculate Penny's realized gain, if any, on the exchange.

b$.$ Calculate Penny's recognized gain, if any, on the exchange.

c$.$ Calculate Penny's basis in the property received.