# Average Percentages over 3 years

## Main Question or Discussion Point

I am just having trouble with percentages.

Lets say a town has a population of

1,000,000 in 2006
1,200,000 in 2007
1,800,000 in 2008

The percentage increase from
2006-2007 is 20%
2007-2008 is 50%

So, what I want to work out is the AVERAGE percentage per year.

The increase from 2006 to 2008 is 80%. So would it be 80%/2=40% per year?

OR

Would it be (20%+50%)/2 =35%

Neither of these actually give the right answer if you do 1,000,000*1.4^2, so was just wondering if this was even possible?

Since you know the the average squared must equal the growth over the two years, taking he square root of the growth will give you the average. Now you just need to observe that the two year growth is the product of the two yearly growths: $1.2 \cdot 1.5 = 1.8$, so $avg = \sqrt {1.2 \cdot 1.5} = 1.34$. So, 34% average growth.

This type of average is called geometric, as opposed to the more familiar arithmetic average.

Hi,

(1+x)^2=1,8
x=1,8^(0,5)-1=0,3416
34,16% of average augmentation each years