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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]  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%
[Reveal] Spoiler: OA
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Re: If the population of city A is increased by A% in year 2006 [#permalink]  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 $$a=b=10$$ --> $$120+\frac{ab}{100}=120+1=121$$.

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

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

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

$$=\frac{a+b+ab}{100}$$
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]  24 Oct 2015, 15:40
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If the population of city A is increased by A% in year 2006   [#permalink] 24 Oct 2015, 15:40
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