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MGMAT - Formulas & Functions Advanced Problem

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by champ0007 » Mon Jun 13, 2011 12:46 pm
A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to 40 million in one hour, by what factor does the population increase every 10 minutes?

I did not understand the answer explanation, I tried to use Geometric Progression, but that is not working for me :(

Thanks
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by GMATGuruNY » Mon Jun 13, 2011 12:59 pm
champ0007 wrote:A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to 40 million in one hour, by what factor does the population increase every 10 minutes?

I did not understand the answer explanation, I tried to use Geometric Progression, but that is not working for me :(

Thanks
Use the formula for exponential growth:

Final Amount = Original Amount * (Multiplier)^(Number of changes)

In the problem above:
Final Amount = 40,000,000
Original Amount = 5,000,000
Multiplier = x
Number of changes = 6 (We're looking for the change over 10 minutes, and this change will occur 6 times over the course of 60 minutes.)

Plugging these values into the formula:
40,000,000 = 5,000,000 * x^6
8 = x^6
x = 8^(1/6) = √2.

For many test-takers, the easiest approach would be to plug in the answer choices, which an actual GMAT question would include.
The answer choices would represent the factor by which the population increases every 10 minutes.
Over 60 minutes, the population would be multiplied by the given factor 60/10 = 6 times.

Answer choice: √2.
5,000,000 * (√2)^6 = 5,000,000 * 8 = 40,000,000.

The correct answer is [spoiler]√2[/spoiler].
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by champ0007 » Mon Jun 13, 2011 1:13 pm
Thanks GMATGuruNY,
I could understand your solution, but how do I know when to use exponential growth or Geometric Progression?

using Exponential Growth -> 40 million = 5 million * x^6

using GP -> 40 million = 5 million * x^5 [i.e. a(r^(n-1)) ]

it sounded like a GP case to me ... but having x^6 made the whole difference...

-----------------

Thought over it again, to start with the initial population is 5 million, and at the end of the first minute it does get multiplied
So my first term should be a * r

Nth term = a * r^n

Makes sense...
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by newgmattest » Mon Jun 13, 2011 7:47 pm
Even GP is fine here.

Let's take classic GP : a, ar, ar^2......ar^(n-1) with n terms.

Now as per question every minute population multiply by constant ratio, which is "r" in our case. So at first it was a = 5 million. After 1 minute, it will be ar and so on.

Now after 60 minutes the term will ar^60 i.e. (5 million).r^60, which is equal to 40 million.

solving this we get r^60 = 8.

Now question is r^10 = ?

so r^10 = (8)^(10/60) = 8^(1/6) = sqrt(2).

Hence GP works as well.
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