I have a quad-core 64bit CPU and have Windows 7 32bits installed (don't ask why.)
Will a 32bit OS run slower if installed on a 64bit CPU as compared to a 32bit CPU ?
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No, and it will run faster than an Windows x64 bit OS.
Mostly all CPUs released in the past few years are 64 bit, but have the ability to run 32 bit code.
During testing when I performed many benchmarks a few years ago on Windows Vista, it was clear that running x86/32-bit operating systems on modern 64 bit CPUs was faster than running native 64 bit operating systems. However this was just starting Windows, when actually in and started, the x64 bit processors showed that they used slightly more memory on applications - however you can also have a lot more, and actual speed for starting up programs was pretty much the same give or take 5% on each platform.
Is a 32bits OS will run slower if installed on a 64bits CPU computer rather than on a 32bits CPU computer?
Depends how fast the CPU is in 32 bit mode. Most modern CPUs are fast enough (as in: no significant speed impact) if I remember correctly. This is why it is the A64 architecture - this is thanks to AMD which realized being ast in 32 bit mode is important.
From two or three years ago almost all CPU's, with the exception of some Atom chips, are dual 32/64 bits CPU's (either using IA64 or Intel64), but most computers were shipped with 32 bit OS because of the compatibility issues of 64 bits Windows (since the vast majority of applications currently available are still 32 bits).
They shouldn't be slower in 32 bit mode because they natively support the x86 instruction set, but would be faster in 64 bits because of the advantages of that mode (more CPU registers, 64bit operations, etc.)
If you want to run Windows in 32 bit CPU you would need to use a Pentium 4 or a netbook Atom, and definitely it will run slower than using a Core Duo or an Athlon.
You should find your answer here
The 64-bit version of Windows handles large amounts of random access memory (RAM) more effectively than a 32-bit system
Older processors where better on 32bit than 64. Ive witnessed this on old Q6600, core 2 duos etc... Older cpus emulated the 64bit instructions. I remember windows 10 older revisions not supporting the older quads because of this. Microsoft had to fix that due to older hardware still being good enough and not everyone can afford new pcs.