AMD A10-7850k versus CPU + GPU for a budget PC

At $500 the apu is almost a given if you need the OS in the budget. Just get 2133+ mhz memory to speed the onboard video up and at least 8 gigs

http://m.newegg.com/Product/index?itemnumber=20-231-667 the ones you mentioned will be fine these have the exact same specs, just a few bucks cheaper at least on new egg. Anything 2400 mhz with cl of 11 or under is pretty solid stuff and will make a good bit of difference on the onboard video

In short yes the faster the system ram it will help the Igp solution of an FM2 based cpu/Igp.
Because the texture memory for the Igpu is used from your main system memory of the pc.
So the faster the better along with the capacity of system ram you have fitted to your motherboard.

DDR3 2400Mhz will be faster.

If I was opting for a A 10 cpu/igpu I would opt for as much system ram as I could afford, making sure the graphics solution had a decent amount of ram reserved out of the system ram, but enough was left over for windows to use as to not run low on system memory also as it would cripple the system with some game requirements of available graphics card and system ram to run smoothly without any lag or glitches.

The Igpu will take at least 2Gb of the system ram, or more and reserve it for igpu usage only for storage space of game textures ect.

http://www.tomshardware.com/answers/id-2832528/kaveri-apu-compatible-a88x-chipset.html#xtor=EPR-8809 Didn't we already go over this? The $500 build I put together for you with the 860k and 750ti is MUCH better than the 7850k build you have going for the same price.

The APU is a nice idea , but the price makes it a bad call for a gamer in this range

Use the 860K and a 750ti . It will game better . Lots better

PCPartPicker part list: http://pcpartpicker.com/p/NJLvJx
Price breakdown by merchant: http://pcpartpicker.com/p/NJLvJx/by_merchant/

CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($69.99 @ Newegg)
Motherboard: Asus A88XM-A Micro ATX FM2+ Motherboard ($52.99 @ SuperBiiz)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Newegg)
Storage: A-Data Premier Pro SP600 128GB 2.5" Solid State Drive ($44.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($45.89 @ OutletPC)
Video Card: PNY GeForce GTX 750 Ti 2GB Enthusiast Edition Video Card ($93.99 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($41.99 @ Newegg)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive ($13.89 @ OutletPC)
Operating System: Microsoft Windows 8.1 OEM (64-bit) ($87.98 @ OutletPC)
Total: $511.69
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 02:01 EDT-0400

or
PCPartPicker part list: http://pcpartpicker.com/p/PDYyCJ
Price breakdown by merchant: http://pcpartpicker.com/p/PDYyCJ/by_merchant/

CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($69.99 @ Newegg)
Motherboard: Asus A88XM-A Micro ATX FM2+ Motherboard ($52.99 @ SuperBiiz)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($45.89 @ OutletPC)
Video Card: EVGA GeForce GTX 950 2GB SSC ACX 2.0 Video Card ($129.99 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($41.99 @ Newegg)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive ($13.89 @ OutletPC)
Operating System: Microsoft Windows 8.1 OEM (64-bit) ($87.98 @ OutletPC)
Total: $502.70
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 02:04 EDT-0400

or

PCPartPicker part list: http://pcpartpicker.com/p/g69hHx
Price breakdown by merchant: http://pcpartpicker.com/p/g69hHx/by_merchant/

CPU: Intel Core i3-4170 3.7GHz Dual-Core Processor ($126.98 @ Newegg)
Motherboard: Gigabyte GA-B85M-DS3H-A Micro ATX LGA1150 Motherboard ($41.98 @ Newegg)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($45.89 @ OutletPC)
Video Card: PNY GeForce GTX 750 Ti 2GB Enthusiast Edition Video Card ($93.99 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($41.99 @ Newegg)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive ($13.89 @ OutletPC)
Operating System: Microsoft Windows 8.1 OEM (64-bit) ($87.98 @ OutletPC)
Total: $497.68
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 02:09 EDT-0400

Are all better options than yours:

PCPartPicker part list: http://pcpartpicker.com/p/zZsdRB
Price breakdown by merchant: http://pcpartpicker.com/p/zZsdRB/by_merchant/

CPU: AMD A10-7850K 3.7GHz Quad-Core Processor ($119.99 @ NCIX US)
CPU Cooler: Cooler Master Hyper T4 70.0 CFM Rifle Bearing CPU Cooler ($19.99 @ NCIX US)
Motherboard: Asus A88X-PLUS ATX FM2+ Motherboard ($63.89 @ OutletPC)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-2400 Memory ($46.99 @ Newegg)
Storage: Kingston SSDNow V300 Series 120GB 2.5" Solid State Drive ($47.99 @ Amazon)
Storage: Western Digital Caviar Blue 500GB 3.5" 7200RPM Internal Hard Drive ($35.95 @ Amazon)
Case: DIYPC Zondda-S ATX Mid Tower Case ($39.59 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($17.89 @ OutletPC)
Operating System: Microsoft Windows 10 Home OEM (64-bit) ($97.89 @ OutletPC)
Total: $490.17
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 02:10 EDT-0400

^ 860k + gtx 950 is a great option out of those build you've done mate - well worth sactificing the ssd for.

It's hard to say, I almost want to root for the i3/750ti combo just because it has more upgrade overhead. The GTX 950 has been doing pretty well for it's price at 1080p though. I think if he could move his budget up $100 he could be in a better position for the long run:

PCPartPicker part list: http://pcpartpicker.com/p/NnbNCJ
Price breakdown by merchant: http://pcpartpicker.com/p/NnbNCJ/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($174.89 @ OutletPC)
Motherboard: Gigabyte GA-B85M-DS3H-A Micro ATX LGA1150 Motherboard ($41.98 @ Newegg)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Newegg)
Storage: A-Data Premier Pro SP600 128GB 2.5" Solid State Drive ($44.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($45.89 @ OutletPC)
Video Card: PNY GeForce GTX 750 Ti 2GB Enthusiast Edition Video Card ($93.99 @ Newegg)
Case: Xion XON-560 MicroATX Mini Tower Case ($29.99 @ Newegg)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($41.99 @ Newegg)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive ($13.89 @ OutletPC)
Operating System: Microsoft Windows 8.1 OEM (64-bit) ($87.98 @ OutletPC)
Total: $612.58
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 02:52 EDT-0400

Or just drop the SSD for the GTX950 again.

It's not that your build is bad so don't worry about it, there is just more performance per dollar to be had. A year ago, you would have been stuck with an APU with a $500 budget including the OS. Fortunately, that's not the case right now. The FX8300 is actually $95 including a cooler on tiger direct if you pay with visa debit/credit. Also they have a 32bit windows 7(might be 8.1) for $69. I wouldn't install the 32 bit, just download the 64 bit ISO and install that, and use the key from the 32 bit during activation. So, there is about $50 off of that build to put towards a SSD or a GTX 950 if you wanted. Or get it all:

PCPartPicker part list: http://pcpartpicker.com/p/Dspx3C
Price breakdown by merchant: http://pcpartpicker.com/p/Dspx3C/by_merchant/

CPU: AMD FX-8300 3.3GHz 8-Core OEM/Tray Processor ($95.00 @ TigerDirect) <-with visa payment
CPU Cooler: Arctic Cooling ACALP64-GT 25.6 CFM CPU Cooler ($0.00 @ TigerDirect) <-included with fx8300
Motherboard: ASRock 970M PRO3 Micro ATX AM3+/AM3 Motherboard ($41.98 @ Newegg)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Newegg)
Storage: A-Data Premier Pro SP600 128GB 2.5" Solid State Drive ($44.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($45.89 @ OutletPC)
Video Card: EVGA GeForce GTX 950 2GB SSC ACX 2.0 Video Card ($129.99 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: SeaSonic S12II 520W 80+ Bronze Certified ATX Power Supply ($41.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 OEM (32-bit) ($69.00 @ TigerDirect) <-Do not install, DL 64bit ISO use key
Total: $528.82
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-21 17:15 EDT-0400

Please skip the apu it's for the best, It's just a waste of money if you are building a gaming pc.

Get the athlon 860k or i3 even the am3+ is a better solution

For sure.

This is assuming you can't use the RAM from your old desktop, but possibly the case. Or you have about $30 left to find one:

PCPartPicker part list: http://pcpartpicker.com/p/LncCxr
Price breakdown by merchant: http://pcpartpicker.com/p/LncCxr/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ TigerDirect) <-have to use visa promo
Motherboard: Gigabyte GA-B85M-DS3H-A Micro ATX LGA1150 Motherboard ($41.98 @ Newegg)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1866 Memory ($36.99 @ Amazon)
Video Card: XFX Radeon R9 380 2GB Double Dissipation Video Card ($163.98 @ Newegg)
Power Supply: SeaSonic S12II 620W 80+ Bronze Certified ATX Power Supply ($64.99 @ SuperBiiz)
Total: $467.93
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-22 18:20 EDT-0400

This is assuming you can use your old ram, and that it's 8gb ddr3. if you have 2 or 4, I recommend just buying a new set instead of adding to those:

PCPartPicker part list: http://pcpartpicker.com/p/zXTwBm
Price breakdown by merchant: http://pcpartpicker.com/p/zXTwBm/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ TigerDirect)
Motherboard: ASRock H97 PRO4 ATX LGA1150 Motherboard ($83.89 @ OutletPC)
Video Card: XFX Radeon R9 280X 3GB Double Dissipation Video Card ($193.98 @ Newegg)
Power Supply: SeaSonic S12II 620W 80+ Bronze Certified ATX Power Supply ($64.99 @ SuperBiiz)
Total: $502.85
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-22 18:27 EDT-0400

Here's another route:

PCPartPicker part list: http://pcpartpicker.com/p/R7frkL
Price breakdown by merchant: http://pcpartpicker.com/p/R7frkL/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ TigerDirect))
Motherboard: Gigabyte GA-B85M-DS3H-A Micro ATX LGA1150 Motherboard ($41.98 @ Newegg)
Video Card: Sapphire Radeon R9 290 4GB Video Card ($242.98 @ Newegg)
Power Supply: SeaSonic S12II 620W 80+ Bronze Certified ATX Power Supply ($64.99 @ SuperBiiz)
Total: $509.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-22 18:31 EDT-0400

You need to realize that computer component prices change daily. I would really be asking that question on the day you plan to buy everything. If you look at that build with the RAM included, it's price is now $490 today. Things will have to be shuffled from one day to the next to sit within your budget. In fact, when that Tiger Direct promotion expires tomorrow that i5 will likely have to be turned into an i3 to stay in your budget. Unfortunately, I can't just say you need a xxxx motherboard, xxxx gpu, xxxx psu, etc... mainly because the parts I will recommend to someone are going to be quality components within their tier. Not just any power supply will do. A certain GPU manufacture may make a good gtx 960, but a horrible r9 280, etc... Motherboards are that way too. RAM generally has a little more flexibility, but things like speed, cas, and timing ratio per dollar are still important. If buying today:

PCPartPicker part list: http://pcpartpicker.com/p/jXbX6h
Price breakdown by merchant: http://pcpartpicker.com/p/jXbX6h/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ TigerDirect)
Motherboard: Gigabyte GA-B85M-DS3H Micro ATX LGA1150 Motherboard ($57.89 @ OutletPC)
Memory: Crucial Ballistix Sport XT 8GB (2 x 4GB) DDR3-1600 Memory ($34.99 @ Amazon)
Video Card: XFX Radeon R9 380 2GB Double Dissipation Video Card ($163.98 @ Newegg)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($26.99 @ Newegg)
Total: $443.84
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-23 17:33 EDT-0400

The builds I have put together for you are based off of your want to have future upgrade ability, therefore at the moment I cannot recommend an AMD cpu/motherboard, otherwise the price would go down enough to get you into a higher tier GPU, but you would likely want to replace them both during any upgrade (so pay for good stuff now, or pay later). I have tried to put together a build that you would only need to upgrade the GPU when the time comes or add more RAM if it becomes necessary.