Are we nearly there yet equation

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In mathematics, the are we nearly there yet equation is a humorous equation formulated by University of Warwick mathematician Dwight Barkley to describe the time at which a child on a car journey will first ask the seemingly inevitable question, "Are we nearly there yet?" It is an example of mathematical modelling of a common process in everyday life.

On his website, Professor Barkley says that he agreed to help Skoda Auto researcher Rebecca Wallace by formulating the equation "as a fun exercise for people to think about as they set off on holiday with their children."

[edit] The equation

The time $T$ at which a child first asks, "Are we nearly there yet?" is given by

$T = t_{0} + \frac{1 + \beta A}{\alpha C^{2}},$

where

$t 0$ is the time at which the journey began;
$A$ is the number of diversionary activities provided to keep the children amused;
$C$ is the number of children in the car; and
$α,β$ are parameters.

The equation was formulated as a fun exercise to illustrate an application of mathematics, accompanied by simple arguments justifying the overall "shape" of the equation.

As a simplified mathematical model, the equation considers two effects on the duration of time $T - t 0$ between setting off and being asked "Are we nearly there yet?": the diversionary activities provided and the number of children in the car. Each distraction is assumed to keep the children amused (and therefore uninterested in the time of arrival at the destination) for a fixed amount of time, $β$ . On the other hand, the more children there are in the car, the more likely it is that one of them will ask the question within a given period of time. Moreover, not only could one of the $C$ children ask one of the adults driving the car, but he/she could also ask one of the other $(C - 1)$ children: therefore, the probability that the question is asked in a given period of time might reasonably be expected to grow like $C (C - 1) \approx C^{2}$ , up to a constant of proportionality $α$ . Finally, the " $1 +$ " reflects the fact that even the most impatient child is unlikely to ask the question before the journey has even got underway.