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They shared the care of the children.
This is a straightforward sentence telling us that "they looked after the children".
The study showed a deep fear among the elderly of being abandoned to the care of strangers.
Why does this sentence mean that "Strangers look after the elderly",not "the elderly looks after strangers"?
You're right: you have identified an ambiguity in English.
When a noun is formed from a verb, like "care", it is sometimes ambiguous whether an "of" complement is semantically the subject or the object of the verb.
Other examples:
The experience of poverty ("poverty" is the object of the verb "experience")
The experience of people in the past ("people" is the subject of "experience")
The choice of colours is soothing ("colours" is the object of "choose")
The choice of the majority of visitors was not to take a tour ("the majority of visitors" is the subject of "choose").
In this case only real-world experience lets us conclude that "the children" is the object of "care", but "strangers" is the subject.
You are not seeing the difference between "care" as a noun, and "care" as a verb.
noun: care
the provision of what is necessary for the health, welfare, maintenance, and protection of someone or something.
verb: care
look after and provide for the needs of.
They shared the care of the children.
This is care as a noun. The parents look after their children and that provision is called "care".
The study showed a deep fear among the elderly of being abandoned to the care of strangers.
This is care as a verb - the action of strangers looking after the elderly.
