Other or Another

Other and another seem like they would mean the same thing but their difference is easy to distinguish. Other refers to something different and can be used with singular or plural nouns. Another refers to something additional and is used with singular nouns.

Is there any other way to solve the problem?

Carla has other friends that are more fun.

May I have another olive in my martini?

I added another new song to my iPod.

Give me another expample if you can :)