Den tredje bog er lettere end den første.

Both dens function as the.

• den tredje bog = the third book
• den første = the first one / the first book

In the second phrase, the noun bog is left out because it is already understood from the context. So the second den is not extra; it is just part of another definite phrase.

Also, here den is not the number one. It is the definite article used with a common-gender noun phrase.

Because Danish normally uses a different pattern from English when a definite noun has an adjective or ordinal before it.

With an adjective or ordinal, Danish usually uses:

den/det/de + adjective/ordinal + noun in its basic form

So:

• den tredje bog = correct
• den tredje bogen = not standard here

Compare:

• bogen = the book
• den røde bog = the red book
• den tredje bog = the third book

So once den is there before the adjective/ordinal, the noun itself usually does not take the definite ending.

Because bog is a common-gender noun, also called an en-word.

• en bog = a book
• therefore den bog / den tredje bog

If the noun were neuter, an et-word, you would use det instead:

• et hus = a house
• det tredje hus = the third house

So the choice between den and det depends on the gender of the noun.

They are ordinal numbers: words like first, second, third.

• første = first
• tredje = third

They are different from cardinal numbers such as:

• en = one
• tre = three

So this sentence is talking about order or position, not quantity.

That is the normal form of these ordinals in this kind of phrase.

In Danish, ordinals used before a noun are commonly seen in forms like:

• første
• anden
• tredje
• fjerde

So in:

• den første bog
• den tredje bog

the -e form is exactly what you expect.

Because Danish can leave out a noun when it is obvious from the context.

Here, once bog has already been mentioned, den første naturally means:

• the first one
• or more specifically, the first book

English does this too:

• The third book is lighter than the first.

You do not need to repeat book in English, and Danish works the same way here.

lettere is the comparative form of let.

That means it works like English lighter or easier, depending on context.

A very common Danish pattern is:

adjective + -ere for the comparative

For example:

• stor → større = bigger
• hurtig → hurtigere = faster
• let → lettere

So er lettere means is lighter / is easier.

end means than in comparisons.

So the pattern is:

X er [comparative] end Y
= X is [comparative] than Y

In this sentence:

• Den tredje bog er lettere end den første.
• The third book is lighter/easier than the first.

More examples:

• Hun er ældre end mig. = She is older than me.
• Denne bil er dyrere end den anden. = This car is more expensive than the other one.

Yes. In this sentence, the structure is very close to English:

• Den tredje bog = subject
• er = verb
• lettere end den første = complement/comparison

So it follows the same basic order as:

The third book / is / lighter than the first.

That makes this sentence fairly straightforward for an English speaker, even though some details, like den tredje bog, follow specifically Danish grammar.