It’s driver’s licence renewal time in our household, and the State Government of South Australia helpfully allows her subjects to choose renewal periods ranging from one year for $42 up to ten years for $285. The renewal periods and their costs are summarised below:

**Table 1: SA licence renewal options, total cost & annualised
**

Renewal period(years) |
Licence fee(total period) |
Licence fee(annualised) |

1 | $42 | $42.00 |

2 | $69 | $34.50 |

3 | $96 | $32.00 |

4 | $123 | $30.75 |

5 | $150 | $30.00 |

6 | $177 | $29.50 |

7 | $204 | $29.14 |

8 | $231 | $28.88 |

9 | $258 | $28.67 |

10 |
$285 | $28.50 |

It’s nice that my government offers so much flexibility in the way she collects her taxes, but I almost feel paralysed by choice here. Which option is best? Obviously renewing once a year, every single year, is a bad strategy (and in my view unfairly penalises people on low incomes). On the other hand, $285 for ten years, which does offer the best *per annum* rate over time, is quite a lot to fork over all at once. That money could potentially be better spent elsewhere. And by “better spent”, I mean on shiny toys like new iPhones obviously.

Luckily mathematics is there to sort it all out.

The annualised cost per year for a driver’s licence renewal is described by the formula:

*annualised cost* = $15/*year* + $27

which graphically looks a bit like this:

The turning point of the above function that I’m looking for occurs where the tangent (given by the first derivative) is equal to -1 (i.e. a negative slope on a 45 degree angle).

That is -15/*year*^{2} = -1

re-arranging yields when *year* = sqrt[15] = 3.9

So calculus tells us that renewing for four years strikes a good balance between savings realised from an extended renewal period and minimising capital outlay.

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Filed under: research, statistics | Tagged: calculus, driver's licence | 7 Comments »