Jumătatea aceasta este mare.

Because Romanian usually puts the definite article at the end of the noun.

• jumătate = a half
• jumătatea = the half

In this sentence, the noun appears in the definite form because it is followed by the demonstrative aceasta. That is a very common Romanian pattern.

When Romanian uses this/that after the noun, the noun normally takes the definite article.

So the pattern is:

• jumătatea aceasta = this half
• cartea aceasta = this book
• fata aceasta = this girl

So jumătate aceasta would sound wrong in standard Romanian.

Yes, it can also go before the noun.

Both of these are correct:

• jumătatea aceasta
• această jumătate

They both mean this half.

A useful rule is:

• before the noun: use această jumătate
• after the noun: use jumătatea aceasta

The version with the demonstrative after the noun is extremely common in Romanian.

They are related forms of the same demonstrative, but they are used in different positions.

• această is used before a feminine singular noun:
  această jumătate
• aceasta is used after the noun, or by itself as a pronoun:
  jumătatea aceasta
  Aceasta este mare. = This one is big.

So here aceasta is correct because it comes after jumătatea.

It is feminine singular.

You can tell because:

• jumătate is a feminine noun
• aceasta is the feminine singular form of this

In this sentence, the whole phrase is also the subject, so it is in the nominative case.

Este is the 3rd person singular present of a fi, which means to be.

So:

• eu sunt = I am
• tu ești = you are
• el/ea este = he/she is

In everyday speech, Romanians very often shorten este to e:

• Jumătatea aceasta e mare.

That sounds natural and common in conversation.

Because many Romanian adjectives have the same singular form for masculine and feminine.

For mare:

• masculine singular: mare
• feminine singular: mare
• plural: mari

So:

• băiatul este mare = the boy is big
• fata este mare = the girl is big
• copiii sunt mari = the children are big

The adjective is agreeing correctly here; it just happens that the feminine singular form looks the same as the masculine singular form.

It can mean either big or large, depending on context.

Romanian mare is a very common general word for size, and English can translate it in different ways:

• big
• large
• sometimes even great, depending on context

In this sentence, big and large are both natural translations.

They are very close in meaning, but not in style.

• aceasta is more standard or careful
• asta is more colloquial and common in everyday speech

So you may hear:

• Jumătatea aceasta este mare.
• Jumătatea asta e mare.

Both are natural, but the second one sounds more conversational.

The Romanian ă is a short central vowel, similar to the a in English about.

So in jumătatea, the ă sounds are not like the a in cat or father. They are softer and more neutral.

This vowel is very common in Romanian, so it is worth practicing early.