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two thirds of the world population ............?

a) is

b) are

a third of the world population ...........?

a) is

b) are

Thanks for your help in advance.
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Fractions and percentages are "number transparent". That is, when they are used as the subject of a sentence, the noun in the following "of" phrase determines the agreement with the verb.

A third of the tables are unoccupied.
Two thirds of the tables are unoccupied.
A third of the sugar is missing.
Two thirds of the sugar is missing.

Fifty percent of the managers were fired within one year.
Fifty percent of the money was destroyed in the fire.


If there is no "of" phrase, something else in the context should be used to make the determination.

If speaking of people (plural):

[One fifth / Twenty percent] have been notified. (of the people implied. The agreement is with people, not with either one (fifth) or twenty (percent)).

CJ
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Here are the rules for fractions

Fractions

Fractions modifying singular nouns take singular verbs:

Two thirds of the book is outdated.

Fractions modifying plural nouns take plural verbs:

Two thirds of the books have already arrived.

Fractions modifying a collective noun can take either singular or plural:

Two thirds of the population of Lebanon is Muslim.

Two thirds of the population of Lebanon are Muslims.
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Comments  
It would be 'is' for both.
I might be full of hokum, but I would use [two thirds] are and [one third] is, respectively.

For me, it's all about the ear.
 CalifJim's reply was promoted to an answer.
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 Likeguslee's reply was promoted to an answer.
'is' in both cases

are

Students: We have free audio pronunciation exercises.
Kilimanjarotwo thirds of the world population ......are......?
a) is
b) are
a third of the world population ..........is.?
a) is
b) are
Thanks for your help in advance.