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I want to build a budget PC for old games and emulation. An APU will likely be used. Can something low as a A6-5400k work while giving consistent 30+ fps? If not, what is the cheapest ＡＰＵ that will?

That cpu will probably be okay (depending on the game of course) but it's a good idea to check out this list

http://forums.pcsx2.net/Thread-CPU-Bench...rk_results

No CPU/GPU will guarantee 100% speed 100% of the time. The better the CPU (and GPU, depending on the resolution used), a higher percentage of the time will be at 100% speed. Above some HW threshold, many games will run at 100% most of the time. If you use a little slower CPU/GPU, then a bit less games will run at 100% speed, and at a bit less of the time.

There's just no absolute threshold beyond which a CPU/GPU combo could be declared 100% perfect for PCSX2. It also depends on what people declare as "playable". Some could live with 90% of the speed much of the time and won't mind occasional slowdown to 30-50% speed, while others would declare a game unplayable if it ever get as slow as 95% of full speed.

In general, the more money you could spend on hardware, you'll get higher percentages of playing at 100% speed. But you won't ever reach full 100% all the time, regardless of CPU/GPU.

If you want to use emulators like PCSX2 and Dolphin you should probably go for something better, like an A8 (5500 if you want low power consumption, 5600k if you plan to overclock). That A6 it's a single module (dual core) APU, and due to it's particular architecture even with softwares that use just two cores it will be slower than a dual module (quad core) APU.

For emulation of older machines and old PC games that A6 should be fine.

A Trinity or Richland A8 or A10 would work fine. They're quad-cores and can overclock pretty easily past the 4.0ghz range where they will emulate most games at fullspeed with ease. I would recommend the Richland variants since they're still quite cheap and can overclock a lot better than the Trinity versions could. If you get one, you're gonna have to pair it up with some good dual-channel RAM to get the most out of the on-die GPU. Recommended speeds of 1866mhz between 2400mhz DDR3 RAM.

FYI less than 2 months away from Kaveri. I'd wait.