Which is correct:
- One fifth of the population is children.
- One fifth of the population are children.
Which is correct:
- One fifth of the population is children.
- One fifth of the population are children.
One fifth of the population is / are children.
The simple answer is that both verb-forms are correct.
Fractions are noun phrases taking an of complement. They belong with the number-transparent nouns (like "lot" and "plenty") where the number of the complement of the preposition of determines the number of the whole noun phrase. Thus plural noun complements require plural agreement and singular noun complements singular agreement. (cf. One fifth of the students take drugs ~ One fifth of the cheese was contaminated)
The complement of of in your example is the singular population, but since this is a collective noun, singular agreement can if wished be overridden and the plural used.
For the general case, most native speakers would base their "plurality" choice on the preceding (singular) fraction (a half, in this case)...
But circumstances alter cases, and when the following referent is so obviously plural (children, in this case), nobody likes the mismatch (there aren't enough written singular instances to make the chart)...
I'm fairly sure this is an example of a construction where AmE is more conservative / "rule-bound" than BrE (so they stick more doggedly to the principle a half = singular and ignore the resulting semantic clash with children = plural).
I'm no expert in using mathematical expressions in NGrams, but I'm pretty sure this one is confirming that Brits are more likely to adapt their choice according the semantics, rather than slavishly adhering to inappropriate "contextless" rules of logic/grammar...
TL;DR: Idiomatically, most people (particularly Brits) would go for OP's second choice (plural). But mostly it's a matter of personal choice and/or exact context, not grammatical rules as such.
Both the sentences presented are grammatical.
Population is a collective noun; it can take either a singular verb or a plural verb. However, the use of a singular verb is more common.
"Is" is correct because the verb "is" modifies the subject (one fifth).
Let's perform a litmus test of this against similar constructs:
My favorite animal is turtles. (subject is animal, even though turtles are plural).
One way to reach Jerusalem is by the many roads leading to it. (subject is way, even though roads are plural).
"Are" would be the correct usage, because "children" are plural and one fifth of something is not (necessarily) singular.
It really comes down to whether you consider "one fifth of the population" to be a singular group or a plurality of objects (people, or groups of people). My reading is that it is plural, and "are" sounds more natural to me.
In order to avoid this problem, you could instead use "one fifth of the population consists of children" or "one fifth of the population is made up of children".