Here is what I remember of the book:

It is about a girl named Patty or Patti whose parents were just divorced. She is living with her mother and because of the divorce, money is tight and her mom cannot afford a lot of basic things for the girl. The girl wears old clothes/an old bathing suit that is falling apart and gets teased by other girls who call her “Safety Pins” because she wears a bathing suit or clothes held up with safety pins. She volunteers or goes to some stables to help take care of horses, and cares for them, but cannot afford to ride them. There is another girl there named Stacey or Stacy who is rich, spoiled, and mean to Patty, and also treats her own horse badly. Stacey throws a tantrum because her horse doesn’t get a fancy twisted bit for her bridle, but a plain one, and screams at Patty for this oversight. I believe (this may not be correct) that Patty earns her lessons to ride a horse because she works hard at the stables and takes good care of the horses there.

The other thing I remember about this short paperback is that it had very simple black and white line drawings in it–maybe one in every other chapter or so.

It was written for kids in elementary school, I believe.