Many years ago there was a grammar rule that said that sentences should not be ended with a preposition. The sentence would have been written as follows: ".... and no water 'with which' to wash or cook." But in the past few years this rule has been relaxed and now it is not considered to be a mistake to end a sentence with a preposition. ******************************************************************

You're basically asking why English sentences can end in a preposition. There is a really simple answer to this. English hates repetition, so if the object of the preposition has already been mentioned, we do not say it again. In your example, we "wash with water" and we "cook with water", but "no water" is at the beginning of the phrase. Our English brains see no reason at all to repeat the word! By the way, they are NOT phrasal verbs. This is just natural word order in an English sentence.

"cook with" and "wash with" are phrasal verbs so "with" stays after the verb and should not be separated. In the first example, "with no drinking water", "with" is a preposition

Imagine a world with no drinking water, and no WATER to wash or cook WITH. > no water to cook with .> no water to do your cooking when its needed to prepare food Imagine a world with no drinking water, and NO WATER to wash or TO COOK > no water to cook . Do you cook water ?? Boil water , right?