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Perto vs. próximo

The Portuguese words próximo and perto generally translate to the English concepts of "near" or "close", but they play different grammatical roles and have distinct usages. The primary difference is that perto is an adverb used almost exclusively to describe physical, spatial distance. In contrast, próximo is an adjective that can describe physical closeness, but is far more frequently used to indicate temporal closeness or a sequential order, directly translating to the English word "next".

Perto

A1
The word perto is an invariable adverb used to describe physical closeness or a short spatial distance between two objects, places, or people. Because it is an adverb, it never changes its ending to match gender or number, and it is most commonly followed by the preposition "de" to specify what something is near.
A escola fica muito perto do parque.
(The school is very close to the park.)
Não chegue muito perto da fogueira para não se queimar.
(Do not get too close to the bonfire so you do not get burned.)
Elas descobriram que moram perto uma da outra.
(They discovered that they live near each other.)
O supermercado mais barato não é perto daqui.
(The cheapest supermarket is not close to here.)
Eu preciso que você fique perto durante a viagem.
(I need you to stay nearby during the trip.)
O cachorro sempre dorme perto do sofá da sala.
(The dog always sleeps near the living room couch.)
Deixe sua bolsa perto da porta de saída.
(Leave your bag near the exit door.)

Próximo

A2
The word próximo is a variable adjective used to denote that something is "next" in a sequence or in time, though it can also formally describe physical proximity. Because it is an adjective, it must change its ending to agree with the gender and number of the noun it modifies, taking forms like próxima, próximos, or próximas.
O próximo ônibus passará em aproximadamente dez minutos.
(The next bus will pass in approximately ten minutes.)
A próxima semana será de muito trabalho no escritório.
(Next week will involve a lot of work at the office.)
Por favor, aguarde que você será o próximo da fila.
(Please wait as you will be the next in line.)
Encontramos uma padaria aberta no próximo quarteirão.
(We found an open bakery on the next block.)
As próximas páginas contêm os exercícios de matemática.
(The following pages contain the math exercises.)
A loja fica localizada em um prédio próximo ao shopping.
(The store is located in a building close to the mall.)
Nossas próximas férias serão em uma praia maravilhosa.
(Our next vacation will be at a wonderful beach.)

Summary

In summary, choose perto when you want to express spatial or physical distance; it acts as an adverb and its spelling remains constant regardless of the context. Choose próximo when you are referring to the "next" item in time or a sequence, making sure to alter its ending to match the gender and number of the noun it is describing.