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[Smart Math] Percentages Problem 2

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?

1. 25
2. 21
3. 20
4. 23
5. None of these

The Usual Method

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Annual rate of growth = Geometric Mean (GM) as shown below:

$\sqrt[3]{1.25\times 1.30\times 1.10}$ = $\sqrt[3]{1.7875}$ = 1.21

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. $\frac{25+30+10}{3}$ =  $\frac{65}{3}$ = 21

$\frac{2}{3}$ or 21.67%

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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