Question

In `2006` the deficit was `$248bn`. By `2013` the deficit was `$680bn`. What is the average annual rate of change?

Answer 1

If the period is `7` years then the average rate of change is:

`($680bn-$248bn)/7 ~= $61.71bn`

This is the average annual increase in the deficit over that `7` year period.

Explanation:

If these deficit figures are from the same time of the year, then the period is `7` years and the average rate of change is `$61.71bn`.

We are told that at some time in `2006` the deficit was `$248bn` and that by `2013` it was `$680bn`. Assuming that that last clause means that the deficit was `$680bn` on 1st January 2013 then the period could be anywhere between just over `6` years (from 31st December 2006) up to `7` years (from 1st January 2006).

`($680bn-$248bn)/6 ~= $72bn`

So the average rate of change is somewhere between `$61.71bn` and `$72bn`.

In reality, it is more likely that the curve is more exponential, reflecting a certain annual percentage change. To calculate that, first divide:

`($680bn)/($248bn) ~= 2.7419`

This is a `174%` increase over `6` or `7` years.

If the period is `7` years, then the average annual percentage increase is `(2.7419^(1/7) - 1) * 100% ~= 15.5%`

If the period is `6` years, then the average annual percentage increase is `(2.7419^(1/6)-1)*100% ~= 18.3%`