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Director  Joined: 03 Sep 2006
Posts: 640
If the population of city A is increased by A% in year 2006  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 74% (01:58) correct 26% (02:42) wrong based on 446 sessions

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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%
Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: If the population of city A is increased by A% in year 2006  [#permalink]

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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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Math Expert V
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Re: If the population of city A is increased by A% in year 2006  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If the population of city A is increased by A% in year 2006  [#permalink]

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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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GRE 1: Q169 V154 Re: If the population of city A is increased by A% in year 2006  [#permalink]

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Here is what I did on this one ->
I did not solve it to maximise but rather used hit and trial.
Let the population in 2005 be 100.
Since in most GMAT questions symmetry works perfectly what if a=b=10

Then 100 => 110 => 121

Hence 21% gain is possible.

Scanning the options => 21 is the highest possible value.

Hence E.

But I like what Bunuel did here.

To maximise ab given the sum a+b we must make a=b.

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Re: If the population of city A is increased by A% in year 2006  [#permalink]

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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] 05 Oct 2019, 20:51
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