Discipuli gaudent, cum finis horae venit et initium prandii iam prope est.

Discipuli is nominative plural. It is the subject of gaudent, so it has to be in the nominative case. The verb is plural, so the subject is plural too.

Because its subject, discipuli, is plural.
Gaudent means they rejoice / they are glad.

Grammatically, gaudent is:

• 3rd person plural
• present tense
• indicative
• from gaudēre

So Latin is matching the verb to the subject just as English does in the students rejoice.

No. Here cum is a conjunction, not a preposition.

It means when in this sentence, introducing a time clause:

• cum finis horae venit
• when the end of the hour comes

Latin cum can mean different things depending on how it is used:

• cum
  • ablative noun = with
• cum introducing a clause = often when, sometimes since or although

Here it clearly introduces a clause, so it means when.

Because this is a straightforward temporal use of cum: it simply tells when something happens.

The verb in the cum clause is venit, which is indicative. That is normal when cum means a simple when.

Learners often notice that cum sometimes takes the subjunctive. That also happens, but usually in other kinds of clauses, especially:

• cum circumstantial
• cum causal
• cum concessive

Here the idea is just: the students are glad when the hour ends. That simple time relationship calls for the indicative.

Because Latin often expresses of with the genitive case.

So:

• finis = end
• horae = of the hour

Together, finis horae literally means end of the hour.

This is a very common Latin pattern:

• initium belli = the beginning of the war
• porta urbis = the gate of the city
• finis viae = the end of the road

In form, yes: horae can be either:

• genitive singular = of the hour
• nominative plural = hours

But context shows that it must be genitive singular here.

Why? Because finis horae makes excellent sense as the end of the hour, while the end hours would not fit the sentence at all.

This is very common in Latin: a form may be ambiguous by itself, but the surrounding words make the meaning clear.

Because finis is the subject of venit.

In the clause cum finis horae venit, the thing that comes is the end of the hour. So finis has to be nominative.

You can think of the structure like this:

• finis = subject
• horae = genitive dependent on finis
• venit = verb

So literally: when the end of the hour comes.

Because prandii is genitive singular, meaning of lunch.

So:

• initium = beginning
• prandii = of lunch

Together: the beginning of lunch

If you used prandium, that would be nominative or accusative singular, and it would not express the of relationship that Latin needs here.

This is exactly parallel to finis horae:

• finis horae = end of the hour
• initium prandii = beginning of lunch

Here prope is being used as an adverb, meaning near or close by.

So:

• initium prandii iam prope est
• the beginning of lunch is already near

Because prope is functioning adverbially here, it does not change form to agree with initium.

This can feel unusual to an English speaker because English also uses near in a similar flexible way. In Latin, prope can be:

• an adverb: near
• a preposition with the accusative: near something

Here it is adverbial, with est.

Iam means already, now, or by this point, depending on context.

In this sentence it adds the idea that lunch is not just near in general, but already near. It gives a sense of anticipation.

So:

• prope est = is near
• iam prope est = is already near

Because Latin is describing the action as a present, regular, or immediate event: the end of the hour comes.

The Latin present can often be translated in more than one natural English way depending on context:

• comes
• is coming
• sometimes even a more general present such as arrives

Here the present tense fits well because the sentence describes what happens at that moment or what regularly happens.

Because Classical Latin has no articles. There is no separate word for the or a/an.

So Latin simply says:

• discipuli = students or the students
• finis horae = end of the hour or the end of the hour
• initium prandii = beginning of lunch or the beginning of lunch

English has to choose whether to use the, a, or nothing at all, but Latin usually leaves that to context.

Latin word order is much more flexible than English word order because the endings show the grammatical roles.

This sentence uses a very natural order:

• main clause first: Discipuli gaudent
• then the cum clause: cum finis horae venit
• then the second coordinated part: et initium prandii iam prope est

But Latin could rearrange parts for emphasis without changing the basic meaning. For example, putting iam or prope in a different place could shift emphasis slightly.

So the order matters mainly for style and emphasis, not for basic grammar as much as it does in English.