Translation:In my city there are three private schools of architecture.
Duolingo's translation is a bit better. Here's why.
We have this group of words: "escuelas de arquitectura privadas". The words "escuelas" and "privadas" belong together, you can see that by the fact that "escuelas" and "privadas" have the same ending; they both end in "-as". Linguists call that "grammatical agreement".
In the English language, the endings of the adjectives don't change depending on the noun. The only way to show that a noun and adjective belong together, is by putting them right next to each other.
You wrote: "private architecture schools". In your translation, you put the word "architecture" between the words "private" and "schools". Because of that, in your translation you lose a bit of information that was there in the original Spanish text, namely that "private" modifies the word "schools".
"private architecture schools", does that mean: "private schools of architecture" or "schools of private architecture". From your translation, you can't tell.
Now, this is advanced linguistics, that we´re getting into. It´s the sort of stuff you need to know if you want to become a professional translator. Personally I find it fascinating stuff, but those are very subtle details, it´s not all that important.
I think he, as probably anyone who's made it to this point, is well aware of the fact that the adjective must match the gender and plurality of the noun it's modifying.
The order of the words here is irrelevant, and that answer should be accepted because it's understood in English what is meant regardless of how you place the words. Have you ever seen anything at all relating to private architecture? Much less a school dedicated thereto? Didn't think so.
In the phrase "private architecture schools," I'd contend that the understood meaning is the same for an English speaker as "escuelas de arquitectura privadas" is for a Spanish speaker (though personally, I'd say "escuelas privadas de arquitectura"...).
In the Spanish phrase, the matching inflections indicate that "privadas" modifies "escuelas". English does not have this feature, so you have to indicate that relation through the word order, You have to put the words "private" and "school" right next to each other, to indicate that they belong together.
The fact that you understand what is meant does not make it a correct translation.
Might be a little late on the adjectives here, but jjkemp's question wasn't answered.
If privadas modifies escuelas and not arquitectura, how do you use an adjective so that it modifies arquitectura. In other words, how do you say:
There is a school of good architecture.
And how would it differ from:
There is a good school of architecture.
"Hay" is both singular and plural and can be translated as both "there is" and "there are"; there's no distinction like there is in English. "Han" is a present tense conjugation that has to be paired with a participle (e.g. "han comido"), and doesn't have the same meaning as "hay".
"In my city there are three private architectural schools." I see no reason to declare this sentence wrong, and I've already read the discussion below. One must really twist the mind into the shape of a pretzel to believe "private" is modifying architectural rather than schools.