[StephenBatey]

As a non-mathematical answer to your final question (and assuming that we're referring to lenses) the simple answer is that you can believe the packaging. But only when you use the product under the conditions to which the figures apply. Here's an illustration that most photographers should be familiar with, based on the maximum aperture.

To keep things very simple, we won't even consider zoom lenses which may have a variable (and specified) maximum aperture, and just look at prime lenses. Your lens is marked to be f/2. You take a meter reading, and see that at f/2 the exposure time should be (say) 1/1000th second. You make the exposure, and we'll assume that a) it's on film and b) you have a densitometer available. You measure the effect of 1/1000th at f/2 and find that the exposure is perfect. Your f/2 lens really is f/2.

Or is it? Refocus on a really close object, so close that it's reproduced at 1:1 on the film. The light is exactly the same, so the exposure should still be 1/1000th at f/2, shouldn't it? Take the photo, and you'll find that it's not the same result. It will (probably) be two stops underexposed. The shutter speed hasn't changed, so the lens isn't f/2 any more, but f/4. I'm assuming that you're sufficiently up on photography to know what's happened. As you focus closer, the effective aperture of the lens changes. The value marked on the box (and the barrel) is correct, but only within the parameters under which it is defined. You'll also find (if you care to try it) that the f/2 value won't apply if you reverse the lens, as some do when using a lens close up.

Similarly, if you reverse the lens, the 2 stop exposure increase needed when working at 1:1 won't apply in most cases. The amount depends on the lens design, and the discrepancy can actually be very large.From data published by Zeiss on two of its lenses, our 4 times increase could be less than that (3.78) or nearly 3 times that figure (11.06).

The focal length works in the same way. The marked value will be accurate (to within about 1% usually) when it is measured in the way that the focal length is defined. Modern lens design (with both zoom lenses and internal focusing ones) means that the focal length will only be accurate at infinity focusing.

Since I mentioned the 1% in an earlier post, I'll just emphasise that I mean 1%, so 1mm in 100mm, 10mm in 1000mm, and not 1mm. A 1mm difference in focal length can make a whopping diffference when the focal length is very small (as with a compact digital camera, where 1mm is about 15% of the focal length).