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In developing countries, where three quarters of the world's population lives, women produce more than half of the food.

Shouldn't it be live in agreement with three quarters?

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Where collective nouns (like population) are concerned, the only rule is to be consistent. You will find numerous instances of the couple is and the couple are. Similarly, the army is/are, the team is/are, the enemy is/are.....and so on.

It depends partly on whether you are considering the collective noun as a unit comprising various elements or as the elements themselves - and this is really an individual decision. If you work in the media, your style guide will express a preference.

When it comes to fractions, it depends on the item concerned.

We say:

Half the food has been spilled on the floor

and

Half the beans have been spilled on the floor.

In your example, the question is whether you consider the population to be a single group or numerous individuals. So you may talk about the population is or the population are. The same is true for the tribe, the group, the family and other collective nouns. Your choice will generally depend on what follows the collective noun as well as your personal preference.

https://www.gingersoftware.com/content/grammar-rules/nouns/collective-nouns/

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