It depends on its context or how you look at it. In sentence No. 1,"a quarter" refers to "a quarter of people", so its plural after which "were" is used. While in sentence No.2 ,"a quarter" might be used as an integral whole, so it is reckoned sigular.

"Quarter" can be singular or plural depending on the situation. So to make it clear insert what quarter refers to in the sentence. Examples: - A quarter (of the apple PIE) IS yours. - A quarter of the basketball GAME IS over. - A quarter of my INCOME WAS cut. often two different subject can talk about the same thing: - A quarter of the world (POPULATION) IS malnourished. [even though we are talking about people, population is singular.] - A quart of the world's PEOPLE ARE malnourished. [same meaning but plural] or - A quarter of the STUDENTS ARE absent. - A quarter of the CLASS IS absent. [same meaning] As for your example, we need to see the whole context to know for sure what the quarter refers to. - Its research found that a quarter (of the PEOPLE) WERE aged under 18. [this is probably the best] but these might also be right: - Its research found that a quarter (of the POPULATION) WAS aged under 18. - Its research found that a quarter (of the RESEARCH GROUP) WAS aged under 18. Fractions and decimals get a little tricky. I think this is right but I don't know grammatical rules very well, so maybe someone can confirm or deny this answer. Good Question, keep 'em coming

It all depends on "a quarter of SOMETHING"... if SOMETHING is singular, then use singuar, else it is plural. A quarter of an apple was eaten... 1/4 of one apple A quarter of the apples were eaten 1/4 of (maybe referring to some apples in a basket)