What is the subject in the sentence? Why? One-third of the people are suffering.

Good question.

Single one word ( compound word too) is always a subject and it will never be after preposition like here people!

"one-third" is subject of the verb

"one third of people" subject of the sentence--determiner is only one-third, no people.

Like, The cost of new cars is high.

Cost is subject no CARS, clear?