How does processing speed affect algorithm performance?
Let's say that the execution time of an algorithm is $\Theta(f(n))$. That means the execution time is between c·f(n) and C·f(n) for all n ≥ N, for some N, c and C. Because of the constant, you can't say anything definite about the maximum problem size that can be solved in time T on a slow and a fast computer. 

You can assume that the actual execution time is c·f(n) for a c that you know; that assumption is neither completely unreasonable nor completely justified. Then define for example n' as the value where c·f(n') = k · c·f(n), to get something meaningful for a machine that is k times faster. 

Calculating n' depends on the nature of f(n). If f(n) is linear, then n' = k·n. If f(n) is cubic, then n' = $k^{1/3}·n$. If f(n) = c·n·ln n then n' is a bit less than k·n, the exact result is difficult. If f(n)=$2^n$, then n' = $n + \log k$. The last shows why efficient algorithms don't become less important with faster computers in practice: Because the less efficient the algorithm, the less a faster computer can improve the size of a problem that can be solved in fixed time.