One post > Two blogs

I have two blogs. One is my academic site and the other is a personal site. On the personal site, I have a category titled “Teaching and Learning”. I’d like to figure out a way for stuff I post to that category to automatically update the academic blog as well. Basically, write once, post twice. I know I can import the entire personal blog or include an RSS widget to pull the entries, but it pulls ALL the entries. How to I get just a category on one blog to update another blog, if that is even possible?

Thanks,

Debby