Here’s and example of a SMART MATH problem for PERCENTAGES.
Problem
What is the annual rate of growth of population of bacteria which has increased by 25% in the first hour 30% in the second hour and 10% in the third hour?
- 25
- 21
- 20
- 23
- None of these
The Usual Method
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Annual rate of growth = Geometric Mean (GM) as shown below:
![\sqrt[3]{1.25\times 1.30\times 1.10}](http://s0.wp.com/latex.php?latex=%5Csqrt%5B3%5D%7B1.25%5Ctimes+1.30%5Ctimes+1.10%7D&bg=ffffff&fg=000&s=0 "\sqrt[3]{1.25\times 1.30\times 1.10}")
![\sqrt[3]{1.7875}](http://s0.wp.com/latex.php?latex=%5Csqrt%5B3%5D%7B1.7875%7D&bg=ffffff&fg=000&s=0 "\sqrt[3]{1.7875}")
i.e. 21 %
(Ans: 2)
Estimated Time to arrive at the answer = 120 seconds as this method requires finding a cube root which can best be done in an iterative manner without the use of a calculator
Using Technique
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Simply find the Arithmetic Mean (AM) of the three percentage values
i.e. 
We already know that AM > GM and that the difference is small when the individual values are ‘near’ each other. Thus the actual answer has to be less than 21.67% but close to it. Hence, the answer must be 21% or option ‘2’.
(Ans: 2)
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