When working out the pattern for children/rats, I thought it would be a good idea to start with either everything as children or everything as rats. I decided to start with everyting as children, which would be 300 children and 0 rats, since 600 halved is 300.
Then, I knew if I wanted to get all possible solutions, I'd need to come up with a pattern. My pattern was adding some of the childrens legs, to the rats each time. But of course, since rats have twice as many legs as children, that wouldn't work, so I took away two children for every rat I added, as shown in the pattern below.
Children Rats Legs
300 0 600 + 0
298 1 596 + 4
296 2 592 + 8
294 3 588 + 12
I did that on and on, until I was certain, that every time, the legs would total up to 600. By looking at the number of children, and noticing that it started at 300, and got 2 less each time, I divided 300 by 2, and then added on the 1 possibilty, with 0 children, to work out there would be 151 possible solutions for this.