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If the population of city A is increased by A% in year 2006

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If the population of city A is increased by A% in year 2006 [#permalink] New post 04 Apr 2012, 07:50
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If the population of city A is increased by A% in year 2006 from year 2005 and b% in 2007 from year 2006. If a+b=20, then what is the maximum percentage increase in population of city A from 2005 to year 2007?

A. 20.19%
B. 20.36%
C. 20.51%
D. 20.84%
E. 21%
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Re: If the population of city A is increased by A% in year 2006 [#permalink] New post 04 Apr 2012, 08:17
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LM wrote:
If the population of city A is increased by A% in year 2006 from year 2005 and b% in 2007 from year 2006. If a+b=20, then what is the maximum percentage increase in population of city A from 2005 to year 2007?

A. 20.19%
B. 20.36%
C. 20.51%
D. 20.84%
E. 21%


Say the population of the city was 100 in 2005, then in 2007 it would be 100(1+\frac{a}{100})(1+\frac{b}{100})=\frac{ab}{100}+a+b+100=\frac{ab}{100}+120 (since a+b=20).

So, in order to maximize this expression we need to maximize the value of ab, when given that a+b=20. Useful property: for given sum of two numbers, their product is maximized when they are equal. Hence, ab will be maximized for a=b=10 --> 120+\frac{ab}{100}=120+1=121.

Answer: E.
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Re: If the population of city A is increased by A% in year 2006 [#permalink] New post 14 Jun 2013, 04:27
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Re: If the population of city A is increased by A% in year 2006 [#permalink] New post 29 Aug 2014, 04:12
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Re: If the population of city A is increased by A% in year 2006 [#permalink] New post 30 Aug 2014, 01:58
2005 ------2006-------------2007
x --------x(1+\frac{a}{100})----------x(1+\frac{a}{100})(1+\frac{b}{100})

% increase=\frac{x(1+\frac{a}{100})(1+\frac{b}{100})-x}{x}*100

=\frac{(100+a)(100+b)-100^2}{100^2}*100

=\frac{a+b+ab}{100}
as given a+b=20
=20+\frac{ab}{100}

To maximize \frac{ab}{100} we must keep a=b=10.

i.e. %increase = 20 + 1=21 (E)
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Re: If the population of city A is increased by A% in year 2006   [#permalink] 30 Aug 2014, 01:58
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