Cum pater e tablino exit, puella ei librum amissum ostendit.

Because cum can be two different kinds of word:

• as a preposition with the ablative, it means with
• as a conjunction introducing a clause, it can mean when, since, or although

Here it is followed by a whole clause, pater e tablino exit, so it is a conjunction. In this sentence it is temporal, so when is the right sense.

Latin uses cum with different moods depending on the meaning.

Here the sentence is simply saying when father comes out..., with a straightforward time relationship. For that, Latin commonly uses cum + indicative.

A cum clause with the subjunctive is more common when the clause is circumstantial, causal, or concessive, especially in more literary past narrative.

So:

• cum ... exit = when ... comes out
• not a more interpretive idea like since or although in this context

E or ex means out of, from, and it takes the ablative case.

So:

• dictionary form: tablinum
• ablative singular: tablino

That is why Latin says e tablino = out of the tablinum/study

As for e vs ex:

• e is commonly used before consonants
• ex is commonly used before vowels or h

Since tablino begins with t, e tablino is perfectly normal.

Mainly from case and the structure of the sentence.

• pater is the subject of exit
• puella is the subject of ostendit

The sentence has two clauses:

1. Cum pater e tablino exit
2. puella ei librum amissum ostendit

In each clause, the nominative noun is the subject:

• pater = nominative singular
• puella = nominative singular

So the father is the one who comes out, and the girl is the one who shows.

Ei is the dative singular form of is, ea, id and here means to him.

It is used because with ostendere the person something is shown to is commonly put in the dative.

So the pattern is:

• ei = to him
• librum = the book as the direct object

Thus:

• puella ei librum ostendit = the girl shows him the book

In context, him refers to pater.

Yes, the form ei can be ambiguous by itself.

Depending on context, it can be:

• to him
• to her
• to it

It can also be a different form entirely in other contexts, such as nominative plural masculine. But here the sentence structure makes it clearly a dative singular, and since it refers back to pater, it means to him.

Librum is the direct object of ostendit, so it is in the accusative singular.

The girl shows the book, so the book is the thing directly affected by the verb.

Amissum modifies librum, so it must agree with it in:

• gender: masculine
• number: singular
• case: accusative

So:

• librum = accusative singular masculine
• amissum = accusative singular masculine

That agreement is why both forms look the way they do.

Amissum is the perfect passive participle of amittere, meaning lost.

Literally, it has the sense of having been lost. In this sentence it is being used like an adjective to describe librum.

So:

• librum amissum = the lost book

Because it is a passive participle, the book is understood as the thing that has been lost, not the thing doing the losing.

Because Latin word order is much more flexible than English word order.

English depends heavily on order:

• The girl shows him the lost book

Latin depends more on endings:

• puella shows who the subject is
• ei shows the indirect object
• librum shows the direct object
• amissum agrees with librum
• ostendit is the verb

That means Latin can move words around more freely. In this sentence:

• putting ostendit at the end is very normal
• putting ei before librum is also normal
• placing amissum after librum is a common adjective position

So the order is not random, but it does not need to match English order.

This is a very common question, because ostendit can be either:

• he/she shows — present
• he/she showed / has shown — perfect

The form is identical, so you have to decide from context.

Here the surrounding sentence strongly suggests the present:

• Cum pater e tablino exit = When father comes out
• so puella ... ostendit is naturally the girl shows

So in this sentence, ostendit is best understood as present tense.

Because Latin has no articles.

A Latin noun like pater can mean:

• father
• the father
• a father

The same is true for puella and librum. English translations add the or a based on context and what sounds natural.

So Latin leaves that unstated unless some other word makes it more specific.