Average of percentage

Jan 2017
Dear friends,
I know this question is very simple but I will ask it hope not to misunderstand, recently I involved in a test which one of the questions like below:
a company revenue growth percentage were 30%, 10% and 90% within 3 years continuously what is the percentage of average yearly growth.
I know average is adding all numbers and divide by the number of years, but in the test there were not answer options like that.
the answer options were 25, 20, 35, 40. I don't remember exactly.


Math Team
Jul 2011
I'd say the correct choice is about 40%. Coming to a solution is not as easy as one would think ...

Let $R$ represent the initial revenue ...

Year 1 growth of 30% yields a new revenue of $1.3 \cdot R$

Year 2 growth of 10% yields a new revenue of $1.1 \cdot (1.3 \cdot R) = 1.43 \cdot R$

Year 3 growth of 90% (wow!) yields a new revenue of $1.9 \cdot (1.43 \cdot R) = 2.717 \cdot R$

So, for the 3 year period, $R$ increased by a factor of $2.717$

Let $a$ be the average yearly growth factor ...

$a^3 = 2.717 \implies a \approx 1.3954 \implies$ average yearly growth percentage is about 39.54%.

To check, say $R=\$100$

growth rate of revenue about 39.54% per year ...

$\$100 \cdot 1.3954^3 = \$271.70$