There are an increasing number of people who prefer to go by bike.这句话有语法错误吗?还有increasing前是不是得用are而不是is?

没错，关键在与the number of 和a number of的区别
number of +名词复数
=a lot of =los of+名词复数
the number of+名词复数，但动词用第三人称单数
例句：The number of visitors to Frinley has now increased.（has就是第三人称单数）
Mike is a number of my friends ,he likes eating food
(在这里of后面就要跟名词复数)
1，a namber of 许多，很多，相当于a lot of
There are a namber of ants on the ground.地上有很多蚂蚁
2，the number of ……的数量
The number of the apples is 40 苹果的数量是40
a number of是大量的 等于a lot of一类 后面接名词复数
the number of是......的数量 后面接名词复数
例如:A number of apples are red
许多苹果是红色的
The number of students is 2000
学生的数量是2000人
a number of.. 许多。。很多。。谓语动词要用复数
e.g.A number of students are playing.很多学生在玩。
the number of.. 。。的数量 谓语动词要用单数
e.g.The number of the students is 50.学生的数量是50。

没错吧
这句话缩写一下是：There are people who prefer to go by bike.这个are是针对people的,people这里是复数,所以用are 没错!