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A simple question

If two people complete the same task, and Person A completes the task in 10 minutes, Person B in 8 minutes, what figures can/should be quoted in terms of how much quicker A is than B?

  1. Person B completed the task in 80% of the time it took Person A to complete the task. Therefore Person B is 20% faster?

  2. Person A took 125% of the time Person B took, so Person B is 25% faster?

So Person A completes a task in 100 minutes.

Using method 1, Person B will complete in 80% * 100 minutes = 80 minutes Using method 2, Person B will complete in 100 / 125% = 80 minutes

Obviously no difference, but what should be quoted? When manufacturers of various devices quote, this device is 20% faster than our rivals, what are they usually referring to?

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  • $\begingroup$ In (1), you should say that B took 20% less time than A. In (2), you should say that A took 25% more time than B. It might be okay to say that B is 20% faster than A, or that A is 25% slower than B. But you can't just switch the adjectives and keep the numbers, that's misleading. $\endgroup$ – MPW Jun 22 '14 at 12:45
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While there is certainly an English aspect to this question, it is also a Math question. It was directly addressed by Hennessy and Patterson in their seminal 1990 1st edition book "Computer Architecture - A Quantitative Approach." They explain why B is 25% faster than A (in this example) and recommend against using language like "slower than" because of the confusion noted above.

While they go through a long proof I can't reproduce here, one way to explain their logic is to convert the raw completion time into a rate. Let's say this task is to run one mile. If A runs a mile in 10 minutes, then he is running at a rate of 6 mph, right? And likewise B is running at 7.5 mph. B's velocity is 1.5 mph faster than A, who is running at 6 mph. And 1.5 is 25% of 6. So thought of this way, people are more willing to agree that B is 25% faster. See the book for a more rigorous proof.

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This isn't a mathematics question, but rather an English question. In English, when you say "A than X" where "A" can be any comparative adjective, the 'starting point' for the comparison is "X". This applies to any comparative adjective including "better", "faster", "higher", "bigger", "stronger". As an aside we also have things like "twice as fast" where the 'starting point' is again the entity after "as". The funny ambiguous business comes when people use phrases like "10 times faster than X"; did they mean "11 times the speed of X"???

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