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MGMAT Geo series - Need expert help with my calculation

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MGMAT Geo series - Need expert help with my calculation

by voodoo_child » Mon May 07, 2012 7:22 am
The rate of bacterial population growth is 'x' every 'y' minutes. How much time would it take for the population to be 10^4 times the original population, given that x^ (1/y) = 10?

For geo seires, Population as a function of y: P(y) = a (x^y) ; a = initial population.

a*10^4 = a (x^y) => 10^4 = (x^y) ...(i)

Now if X^ (1/y) = 10 => X = 10^y ... (ii)

Now, substitute (ii) in (i)

(10^4) = (10^y)^y

Therefore, 10000 = 10^ (y^2) => 4 = y^2 => y=2. However, OA is 4. Where am I going wrong?


Please help me....

Thanks
Voodoo
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by aneesh.kg » Mon May 07, 2012 7:44 am
I don't see any mistake with what you've done.

Let's go backwards and check our calculations.

We got: y = 2 minutes
For y = 2
(x)^1/2 = 10
x = 100

No mistake so far for sure.

The language of the problem isn't great. If, by 'growth is 'x' ', the problem means that population becomes x times then
If the initial population is 'a', it will become P(y = 2 minutes) = A*100*100 = 10^4 times of A.

which is what was required.

So, No mistake.
IMO, OA is wrong.
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by voodoo_child » Mon May 07, 2012 8:35 am
Nope. OA is correct. Let's use another method.

Let's say that Population P = M (x)^ (t/I) --- This is another way to calculate bacterial population questions.

x= multiplication factor (or rate)
t = time
I = interval
M= initial population.
Therefore,
10^4 M = M *x ^ (t/y)

Now we know that x^(1/y) =10 => Therefore,
(10^4) = (10^t) => t= 4!

Even though the above formula is dervied from Geo Series, I am not sure why 'geo series' is not yielding the correct answer.....
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by cypherskull » Mon May 07, 2012 10:30 am
The geo series is one of weaker areas [amongst many others! :(]

Would you please explain how did u get to the following:-


"For geo seires, Population as a function of y: P(y) = a (x^y) ; a = initial population.

a*10^4 = a (x^y) => 10^4 = (x^y) ...(i) "
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by voodoo_child » Fri May 11, 2012 7:48 am
cypherskull wrote:The geo series is one of weaker areas [amongst many others! :(]

Would you please explain how did u get to the following:-


"For geo seires, Population as a function of y: P(y) = a (x^y) ; a = initial population.

a*10^4 = a (x^y) => 10^4 = (x^y) ...(i) "
That's the concept of geometric series. Please see this wiki page for geo series:https://en.wikipedia.org/wiki/Exponential_growth


Essentially, in a series, the number gets multiplied at a rate r; the series becomes a ar ar^2 ar^3....

nth term = a * r (n-1); a = first term

That's what I did above. I have just substituted n-1=m because n starts from 1,2,3....however, real life data, including time 't', starts from t=0, 1, 2, ....

Therefore, t-th term = a*r^t;

I hope that helps....
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by Stuart@KaplanGMAT » Fri May 11, 2012 9:12 am
I have to admit that I'm not an expert on geometric series - I'd always attack this kind of question by picking numbers, which to me is much simpler. However, I think the problem is that you're using "y" to mean different things in different places.

You say that:
For geo seires, Population as a function of y: P(y) = a (x^y) ; a = initial population.
In that formula, isn't y the number of population increases and x the increase per period? (In other words, it's basically the compound interest formula: x=(1+r) and y=t.)

However, in your next equation you're using the y from the question stem, which is the duration of each period, not the number of periods. Since y in your first equation is a completely different variable from y in your second equation, you can't set them equal to each other.

In other words, the first equation is really:

a*10^4 = a*(x^t)

and the second equation is:

x = 10^y

and since t doesn't equal y, there's no way you can say that 10^4 = (10^y)^y.

Let's solve by picking numbers:

We know that x = 10^y, so let's pick x=100 and y=2. Let's pick an original population of 1. We also have to assume (which you'll never have to do on a real GMAT question - a big strike against this one) that "growth of x" means that we multiply the population by x every period.

start: 1... goal: 10^4*1 = 10000
after 2 minutes: 100
after 4 minutes: 10000

So, it took us 4 minutes... choose 4!

Wasn't that way easier than using fancy formulas?


voodoo_child wrote:The rate of bacterial population growth is 'x' every 'y' minutes. How much time would it take for the population to be 10^4 times the original population, given that x^ (1/y) = 10?

For geo seires, Population as a function of y: P(y) = a (x^y) ; a = initial population.

a*10^4 = a (x^y) => 10^4 = (x^y) ...(i)

Now if X^ (1/y) = 10 => X = 10^y ... (ii)

Now, substitute (ii) in (i)

(10^4) = (10^y)^y

Therefore, 10000 = 10^ (y^2) => 4 = y^2 => y=2. However, OA is 4. Where am I going wrong?


Please help me....

Thanks
Voodoo
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