What processor do you have? Athlon XP? Sempron? Athlon 64? There are quite a difference in these processors, and upgrading might be rather pointless(or require more than just a processor
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As with any duel core, you have two cores, so basically 2 processors. So yes, you have two 2.4GHz cores in some processors. This does not equal 4.8GHz in any way. Under many circumstances, the a duel core is no faster, perhaps even slower than a single core at equal speeds. Your only real advantages are in multitasking and games that can utilize both cores.