Honestly, I've never used P1V1/n1T1 = P2V2/n2V2.
I think PV = nRT will solve all of them. If all except one variable are given out of P1, P2, V1, V2, T1 and T2, then I use P1V1/T1 = P2V2/T2 (because it's a little faster) but I could use PV = nRT twice; once to solve P1V1/R*T1 for n, then use PV = nRT again with that n and P2, V2, for example, and solve for T2. So I don't know that it matters which you use. Check my thinking.
I was thinking that if, for example, two different pressures were given, two different volumes were given, two different temperatures were given and only one mole, then you use P1V1/n1T1...etc to find the missing mole.
But, if only one pressure, one volume, and one temperature were given and you needed to find moles, then you use pv=nrt.
That's how I tried figuring out the difference, but I was afraid there was some exception to throw me off (which usually always happens on tests...)
I think your thinking is OK. I tried making up a P1, V1, T1 and calculating n1. Then I made up a P2, V2, T2 and calculated n2, both calculations from PV = nRT.
Then I used the old P1, V1, T1, and the calculated n1 and the old P2, V2, and calculated n2 and calculated T2 from the P1V1/n1T1 = P2V2/n2T2 equation and got the same answer. In fact I tried it with two or three sets of numbers and I always came out with the same answer no matter which way I solved it. Of course, if only ONE P, V, and T are given, you have no choice but to use PV = nRT.
Okay, good. I think I'm all set then. Thank you!