School A is 40% girls and school B is 60% girls. The ratio

You don't need to take numbers that are multiples of 7. 7 is the total number of girls in both schools together on the ratio scale, but it is not one of our parameters in the question. We just want that number of girls in each school should be multiples of 4 and 3 respectively. So stick to easier numbers such as 200, 100 and then number of girls as 80 and 60 respectively.

Note that when you assume numbers, they should fall in place such that after you transfer the 20 boys, their ratio must be 5:3 (as given). You get a different ratio with your numbers. It means your assumed numbers are not correct. You cannot know whether your assumed numbers are correct without calculating the ratio. Normally, when you have a value increase or decrease and not percentage (i.e. 20 boys transferred), assuming numbers could be tricky. You might need to adjust them around. If it seems cumbersome, algebra would be the way to go.

Often on GMAT, the numbers are quite simple. I took the numbers I mentioned while reading the question and figuring out the logic involved. Luckily, they worked. Else, I would have had to tweak them around depending on whether I was getting a lower or a higher ratio than expected.