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Ranho vs. muco vs. catarro

While ranho, muco, and catarro all relate to nasal secretions or bodily fluids, the choice depends heavily on social context (formal vs. informal) and the consistency of the fluid. Muco is scientific, catarro usually refers to sickness with congestion, and ranho is the colloquial, often gross, term for snot.

Ranho

B2
This is the informal, everyday word for snot. It usually refers to the runny liquid that comes out of the nose. It can be considered slightly vulgar or simply colloquial depending on the company. In Portugal, it is also strongly associated with acting like a brat; a ranhoso is a grumpy or badly behaved person.
O menino está com o nariz cheio de ranho.
(The boy's nose is full of snot.)
Limpa o ranho do nariz com um lenço.
(Clean the snot from your nose with a tissue.)
Que nojo, aquilo é ranho na tua manga?
(How gross, is that snot on your sleeve?)

Muco

B2
This is the formal, technical, and medical term for mucus. It is used by doctors or in biology contexts. Unlike the other two words, muco can refer to protective fluids produced anywhere in the body, such as the stomach or lungs, not just the nose. It is used in exactly the same way in both Brazil and Portugal.
O médico explicou que o excesso de muco é normal.
(The doctor explained that excess mucus is normal.)
O muco serve para proteger as vias respiratórias.
(Mucus serves to protect the airways.)
A análise do muco revelou uma infecção.
(The analysis of the mucus revealed an infection.)

Catarro

B1
This word translates best as phlegm. It usually refers to the thick, viscous mucus associated with a cold, flu, or chest infection. While ranho represents a runny nose, catarro implies congestion in the throat or lungs that needs to be coughed up. It is common in both Brazil and Portugal, though Brazil also uses words like pigarro for throat clearing.
Ele está com uma tosse cheia de catarro.
(He has a cough full of phlegm.)
Não consigo respirar bem por causa do catarro.
(I cannot breathe well because of the phlegm.)
Beber muita água ajuda a soltar o catarro.
(Drinking lots of water helps loosen the phlegm.)

Summary

Use muco in a hospital or science class. Use catarro when referencing a cold with thick phlegm or chest congestion. Use ranho in informal casual settings to describe a runny nose or snot, but be aware that it creates a slightly gross mental image.