I've been struggling around this problem for 5 hours straight (yes, straight), and I've been getting different answers from different people and sources, so I wrote everything on a paper and tried to get a logical thinking from both results.
Let's forget about time at a moment, let talk about speed.
Faster than implies how much more speed you has, while slower means how much less speed you has.
Now, talking about time, time is inversely proportional to speed, and when we talk about faster than, we are talking about speed, not time (but they are related).
So, if X is faster than Y, X must have more speed than Y, so its percentage must be greater than Y (in regards to faster than percentage), otherwise it means it is slower.
So the formula of @turkeyhundt is correct:
$$
\frac{5.106 - 0.851}{0.851} = 5.0 = {500}\%
$$
If you got the greater value at the divider:
$$
\frac{5.106 - 0.851}{5.106} = 0.833... = {83.333...}\%
$$
And now the most important part of the question, if I say that B is X% faster than A, I must be able to resolve the A value without knowing it, just from the B and the percentage:
B (0.851) is 500% faster than A:
$$
0.851 + (0.851 * 5.0) = A
$$
$$
5.106 = A
$$
The same as if I said that A (5.106) is 83.333% slower than B:
$$
5.106 - (5.106 * 0.8333333333333334) = B
$$
$$
0.851 = B
$$
The decision between +
or -
is just if the result should be greater or lesser than, by evaluating the faster (+) or slower (-) assumption.
I was just thinking as if this problem were a math test, I would not have A or B when questioning about faster or slower than, so that was my conclusion:
If the percentage of A speed is greater than B, A is faster, otherwise, A is slower.
So it does mean that, if you've got something like B is 83% faster than A and A is 500% slower than B, that is probably incorrect, it is the other way around, the greater percentage means faster (more speed), the lesser percentage means slower.
We also have some semantics problem here, faster than is talking about speed, but we also interpret it as time, and they are inversely proportional.
If the question was what percent of time A takes more than B, then you could say A takes 500% more time than B (A = B + 500%
), and B takes 83% less time than A (B = A - 83%
), which is the thing that @ashleedawg has stated as faster than, but is in fact, how much more or less time one takes. And here you can see, how time and speed are inversely proportional.
Just as an advisory, I'm a software engineer, but not that good at mathematics, just sharing the conclusion that I had after those 5 hours.
Edit:
I've made a mistake by assuming that it should always be greater than 100% because I've not taken into account slight fractional differences, like A taking 1.4s
and B taking 1.1s
, that would've resulted in B being 27% faster than A, and A 21% slower, so I've fixed the answer.
Also, this does not applies to money and costs, since they are not inversely proportional: more money means more expensive, less money means cheaper. While time and speed are inversely proportional, more speed is equals less time, and less speed is equals to more time.
Just one more thing to keep in mind, and I think that is the reason we confuse this: the less time it takes, faster it is, and because time is decreasing, we calculate as if it were proportional, by having always the greater percentage meaning more time and lesser percentage meaning less time, however, faster than or slower than is not about time, but speed, and again, time and speed are inversely proportional.