In the speedometer's case, here's my guess of how it probably works. With the manufacturer's recommended tire size, the speedometer calculates the number of revolutions per minute (RPM), then multiplies it by the circumference of the tire (=pi*diameter), which gives you the distance the car has moved in one minute, which can be easily converted to a unit of speed (e.g., miles per hour or kilometers per hour).

However, this tire diameter must be a kind of system parameter that the speedometer knows and assumes this value never changes. In this case, when you replace the tires with larger sizes (meaning larger diameter now), the speedometer doesn't know and still uses the old diameter value. Since for a given speed, a larger tire takes longer to complete one revolution,, the number of RPMs is less than that with the smaller tire. As a result, the speedometer does the computations in the last paragraph incorrectly and displays a speed that's slower than the actual speed. So the driver may be tempted to step on the gas until the speedometer needle hits the desired speed, which means the actual speed is faster than that indicated.

If my math is right, and the above reasoning is correct (i.e., assuming the tire diameter parameter is a fixed value accepted by the speedometer), then the ratio of the larger tire diameter to the smaller tire diameter is equal to the ratio of the speed run with the larger tire to that run with the smaller one. So knowing the diameters of the two different tires might give you some idea of how fast you're actually driving, assuming you can drive and compute at the same time. :-)

I wonder if any new car now has a speedometer that can automatically sense the size of a new set of tires and therefore uses the new diameter value in its speed computations... Or would the GPS navigation system give a better, more accurate estimate of the speed than the on-board speedometer?