The number of the students who failed the chemistry examination has increased to fifteen.
翻译：化学考试不及格的人数已达十五名.
上句中主语的是The number数字,作主语,所以后面的谓语动词用has.
of the students who failed the chemistry examination 是介词短语,修饰the number.
who failed the chemistry examination 是定语从句,修饰the students.
我们将介词短语省略后就成了： The number has increased to fifteen.显而易见,要用has,而不能用have.