Let me proceed in steps. s1) The sentence is of the kind: /A/ e` molto piu` conveniente che /B/ s2) The terms of comparison are actions, both expressed with a verb and an object. s3) when you compare actions we don't use the usual /di/ but /che/. s4) /A/ is an impersonal sentence, expressed with /infinitive verb/ + /object/, implying that it holds for each subject; A : comprare una casa s5) /B/ is also an impersonal sentence, expressed with /infinitive verb/ + /object/ + /ne/, B : affittarne una extended form: affittare una /delle cose riferite precedentemente/ /-ne/ is what we call /pronome partitivo/, that has two meanings: a) it refers to something from the previous simple sentence; b) it only refers to /a part/ of the whole To better understand /-ne/ here is a typical example: Queste castagne sono molto buone, ne vuoi mangiare ? /ne/ refers to /castagne/ (chestnuts) and means: some chestnuts, some of them.

The only different I can see among the two sentences is the use of "affittarNe" instead of "affittare". It is correct to say "affittarne" because it stands of "rent ONE of the houses". It needs an object and with "affittarNe" the object result implied. You have to say "affittare something"("rent something" in English). If you don't want to use an implied object you can say "Scambiare una casa è molto più conveniente di affittare una casa" --> However, in this way there is a repetition of "casa", and so we(Italian) tend to say "affittarNe" so that it's shorter and we don't need to repeat "una casa" at the end of the sentence.

Hello. I guess the mistake they made is in the noun "Scambiare una casA è molto più conveniente che affittarne una." Other possible solutions are : Scambiare casa è molto più conveniente che affittarNe una/Exchanging house is just cheaper than renting one of those. Scambiare una casa è molto più conveniente che affittarla/Exchanging an house is just cheaper than renting it. Scambiare case è molto più conveniente che affittarle/Exchanging houses is just cheaper than renting those. unA casA is right "unA casE" is obviously, like you found, wrong because the indefinitive article is singular while the noun is plural. affittarne, affittare+NE, of that/those/there, is necessary otherwise it would be needed to write "affitare una casa" Is it clear ? :-)