Can anyone name this picture book?

A fellow librarian sent this to me:

"Here is another one of those 'tracking down a book title' questions.
A patron is looking for a children's book (likely a picture book) she remembers about twin girls who alternate days going to school. Classmates wonder why the girl in their class acts differently on different days. The reader learns there are really twin girls, and the parents are sending them on alternate days because they have only one coat between them. We are drawing a blank. We have tried various searches in HIP with no luck. Does this ring a bell with anyone?"

I haven't a clue? Can anyone help?

I know this book! The girl's name is Sarabeth (twins Sara and Beth). I'm pretty sure "Sarabeth" is in the title. Can't search right now, but somebody ought to be able to find it.

Interesting and I love the story line...I searched based on MerryMary's lead and found "Mystery of Sara Beth" by Polly Putnam.

Thanks for your help, MerryMary and Esterbwolf. I sent this info on to my friend and it is The Mystery of Sara Beth.