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Rimanere vs. restare vs. avanzare

In Italian, the verbs rimanere, restare, and avanzare can all express the idea of something being left or still available, but they have distinct nuances. While rimanere and restare are broad, often interchangeable synonyms used to describe what is generally left of a quantity, time, or abstract concept, avanzare specifically describes a surplus or something left over after a process of consumption, use, or distribution.

Rimanere

The verb rimanere is a highly common and versatile word used to indicate that a physical quantity, a certain amount of time, or an abstract concept still exists or is still available.
Ci rimangono solo due biglietti per il concerto di stasera.
(We only have two tickets left for the concert tonight.)
Non mi rimane molto tempo per finire questo progetto importante.
(I do not have much time left to finish this important project.)
Sono rimasti alcuni biscotti al cioccolato nella scatola in cucina.
(There are a few chocolate cookies left in the box in the kitchen.)
Di quella magnifica casa antica ormai rimangono solo le rovine.
(Of that magnificent ancient house, only ruins remain now.)
Rimane una sola fetta di torta nel frigorifero.
(Only one slice of cake remains in the fridge.)

Restare

The verb restare is practically synonymous with rimanere in the context of availability, and it is frequently used interchangeably to state that a resource, time, or option is left.
Quanti soldi ti restano nel portafoglio dopo aver fatto la spesa?
(How much money do you have left in your wallet after doing the groceries?)
Mi resta ancora un po' di vernice per completare il lavoro sulla porta.
(I still have a little paint left to complete the work on the door.)
Restano solo tre giorni prima della nostra partenza per le vacanze.
(Only three days remain before our departure for the holidays.)
Non resta niente da dire su questo argomento difficile.
(There is nothing left to say on this difficult topic.)
Ci restano pochissime opzioni a questo punto della negoziazione.
(We have very few options left at this point in the negotiation.)

Avanzare

The verb avanzare specifically means to be left over, emphasizing a surplus, excess, or remainder after a process of using, eating, or spending has taken place.
È avanzata molta pasta dal pranzo che ho preparato oggi.
(A lot of pasta was left over from the lunch I prepared today.)
Se ti avanza del nastro adesivo dai pacchi, me lo puoi prestare?
(If you have some packing tape left over, can you lend it to me?)
Ho comprato tutto quello che serviva e mi sono anche avanzati dieci euro.
(I bought everything needed and I even had ten euros left over.)
Non voglio che avanzi troppo cibo dopo la festa di compleanno.
(I do not want too much food to be left over after the birthday party.)
Con questa stoffa che avanza possiamo fare un cuscino per il divano.
(With this leftover fabric we can make a pillow for the sofa.)

Summary

To summarize, you can use rimanere and restare interchangeably when you simply want to state that a quantity, amount of time, or option is still available or currently exists. You should choose avanzare when you want to emphasize that something is a leftover or a surplus generated after an activity, such as uneaten food from a meal or unspent money from a budget.