Learn Portuguese (Portugal): Não estudou; por conseguinte, chumbou no exame. - He didn't study; consequently, he failed the exam.

It is quite formal and usually reserved for writing or formal speech, meaning "consequently" or "therefore". In everyday spoken Portuguese, you are much more likely to hear por isso or portanto.

"Chumbar" literally means to plug with lead. Is this the normal word for failing an exam?

Yes! While its literal meaning is to seal or weight something with lead, chumbar is the standard, everyday term in European Portuguese for failing a test, a class, or a school year. If you want a more formal equivalent, you can use the verb reprovar.

Why do we say "chumbou no exame" instead of just "chumbou o exame"?

When meaning "to fail an assessment", the verb chumbar requires the preposition em. Since exame is a masculine noun, em combines with the definite article o to become no. You literally say that someone failed "in the" exam.

Since the subject is dropped, how do we know who failed the exam?

The verbs estudou and chumbou are in the third-person singular of the past tense (pretérito perfeito). This means the dropped subject must be ele (he), ela (she), or você (you, formal). The exact person would be easily understood from the surrounding conversation.