MGMAT Geo series - Need expert help with my calculation

[voodoo_child]

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

[aneesh.kg]

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.

[voodoo_child]

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.....

[cypherskull]

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) "

[voodoo_child]

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....

[Stuart@KaplanGMAT]

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?

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