I have troubles trying to figure out why 的 should be present (and its position) in the following sentence:
As far as I know, 的 is used to modify nouns. What is that particle modifying?
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I'll proceed to dissect the clause 坐四十五分钟的二百七路 using systemic grammar:
坐 四十五分钟 的 二百七路 1. \-----------Predicate----------/ 2. \P/ \---------Object---------/ P: Predicator 3. \Modifier/ 的 \Head/
The clause can be analyzed using 3 levels.
As you can see, the object in this clause if constructed using the Modifier-的-Head pattern.
The predicator is 坐, meaning ride.
The modified noun (head) is 二百七路, which literally translate into "207 Line", presumably a bus route.
The modifier here acts as an adjective, describing the "207 Line", with the quality of 四十五分钟, which is "45 minutes".
See, if we take out the modifier (四十五分钟的), the clause is simplified into:
The meaning would simply be: Take (ride) the 207 Line.
This sentence makes sense, and this is a hint that 的 was used as a particle for the Modifier-Head construct and not as a possessive particle.
二百七路 sounds like a bus route number. That would make it a noun
的 is acting as the preposition 'of' in this context. It connects the quantity complement "四十五分钟" to the noun 二百七路 (45 minutes of route 207)
[坐] [四十五分钟] [的] [二百七路]
[ride] [45 minutes] [of] [route 207 (bus)]
Example of quantity complement complements verb:
坐 (v) 四十五分钟 (quantity complement)
坐 (v) 二百七路 (n) 四十五分钟 (quantity complement)
坐四十五分钟的公共汽車 can be reduced to 坐四十五分钟公共汽車 by omitting 的. Which mean 坐四十五分钟的二百七路 can also be reduced to 坐四十五分钟二百七路
As explained in other answers, 的 in 坐四十五分钟的二百七路[sic, 1] is customary and frequently used. However, as a native speaker, I don't find it "mandatory" as you said. The following sounds fine to me:
1: the notation of "二百七路" feels rather unorthodox to me. At least in where I live, if you mean 207, it is usually called 二零七(路), or rarely, 二百零七, but never 二百七. And if you mean 270, although it is called 二百七 sometimes, it's mostly in prices or amount of cash. Otherwise, as a bus route, you would call it 二七零, or rarely, 二百七十.