I would never present a problem like this without brackets. As you have aptly demonstrated, PEMDAS can be quite confusing. Deciding to do the first operation of the DA pair is an artificial choice. What this means is that, if I were to write this equation in a slightly wrong order, and to decide to solve it sometime later, I would end up with the wrong answer. Far smarter to use brackets, eh. PEMDAS should then be expressed as: PE(MD)(AS), to clarify that the MD and the AS pairs can go backward or forward. But that wouldn't be as cool or as easy to remember. If I ever taught math, I would certainly teach PEMDAS, but I would sternly caution my students to use brackets when writing their own equations. I would also mark down any equation that could use brackets, but didn't have them.
This is the best way? I would teach to clear each fraction one term at a time. Multiply through by 3, then by 5, then by 2 and then by 3, No worrying about LCD (which you seem to be fixated on). everyone can, or rather it's easier to, multiply by these numbers than 60.
