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Silly problem about multiple downloads

I was using the downTHEMall! Firefox plugin to download the pdfs for an online book when I thought of a simple question. To phrase it mathematically:

If you are downloading n files, each of size x_k, how many times does the overall percentage downloaded, OP, equal the percentage of the current file being downloaded, CP_k? And what are these percentages?

I was pretty certain that the answer to the first question is n, but had to work it out for certainty, and an answer to the second question. Assume OP = CP_k for some k, then

\displaystyle OP = \frac{\sum_{j=1}^{k-1} x_j + CP_k x_k}{ \sum_1^n x_j} = CP_k \Rightarrow CP_k = \frac{\sum_{j=1}^{k-1} x_j}{\sum_1^n x_j - x_k}.

Assuming n>0, this is satisfied exactly once for each k, so OP=CP_k n times.

Silly problem, but nice to know.

This entry was posted on Saturday, September 24th, 2005 at 5:09 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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