If a certain culture of bacteria increases by a constant

My take:

We need to find y, when xy=10^4.

1) When x=10^y, y*(10^y)=10^4
or could also be written as y = 10^(4-y)
By using substitution, we could say y could be between 3 and 4 (~=3.4). Hence st1 is Sufficient
( And Options B,C,&E could be eliminated)

2) (2)*x=100 i.e. x=50. By substituting value of x in xy=10^4, we could find the time taken. Hence st2 is Sufficient

Hence the solution is
/SM

Hi Bunuel,
It'd be really helpful if you could provide an explanation for this problem to help me understand it.

Thanks in advance.

If a certain culture of bacteria increases by a factor of x every y minutes, how long will it take for the culture to increase to 10000 times its original amount ?
1) (x)^1/y = 10
2) In 2 minutes the culture will increase to 100 times its original amount.

Please could you explain an how to solve such questions which involves population increasing by certain factors

Stem 1 is interesting..

Consider x=10 and y=1...so Initial bacteria (K) will increase by a factor of 10 in 1 minute...so after 1 minute there will be K+10K=11K...so we can find the time it will take for culture to increase to 10000 times

So after 1 minute, we have 11K
After 2 minutes, 11K+110K=121K
After 3 minutes= 121K+1210K=1331K
After 4 minutes...

Consider \(x=\sqrt{10}\), and y =1/2 or 30 seconds, So after 30 seconds, no. of bacteria will increase by \((10^{1/2})^{2}\) or by a factor of 10...in 1 minute it will increase by factor \(\sqrt{10}\)
At 0 seconds=K
After 30 seconds= K+10K=11K
After 1 minute, by a factor\(\sqrt{10}\) so after 1 minute you will have \(11K+\sqrt{10}*11K\)-----> Now here the problem is the increase every y minute is not the same so this case will not be considered...



So St1 is sufficient

St2 is straight forward...

Can you post the Official Explanation to question.

Hi... Pls throw more light about the 1st statement...
i solved it like this...
Initial at time=0, let it be P
after y min, xP
after 2y min, x^2P
after ny min, x^nP

from question x^nP = 10000P
x^n=10^4
from statement 1, x=10^y
10^(yn)=10^4
yn=10
2 unknown and 1 equation, not sufficient

Thanks in advance

keyword is "by a factor of ", is used commonly to mean the same as "multiplied by" or "divided by." If x is INCREASED by a factor of 4, it becomes 4x. If x is DECREASED by a factor of 4, it becomes x/4. The key word is the direction of change i.e. (increased / decreased) by a factor of. so here from both the points, we can derive that every y minute(s) (the number increases/multiplies by a factor of 10) or in case of (2) we straightaway know that at 2 minutes we have 100, so 10^2 = 100 means that the number became 10 at 1 minute and 100 at 2 minutes. Hope it helps

thanks smitsharma...
overlooked that... always of the notion that the factor doubles and it remains in the exponential form...

The question asks "how long it will take for the culture to INCREASE to 10.000x its original size". We need to find "y" minutes.
1. x=10ˆy | If 10.000 = 10ˆ4; When x = 10ˆ4; y=4 minutes; Sufficient
2. 2min: x=100 ; 4 minutes: x=10000

To understand what the question stem is telling us, let's pick some numbers: the bacteria
culture begins with an initial quantity of I = 100 and increases by a factor of x = 2 every y = 3
minutes.

In 3 minutes bacteria population = 2*100
In 6 minutes bacteria population = 2^2*100
.
.
.
.
In 3n (t) minutes bacteria population = 2^n*100

n represents the number of growth periods, and n = t/y where t is time in minutes. For example, the
4th growth period in our chart above ended at 12 minutes, and 4 = 12 minutes/3 minutes.
From this example, we can generalize to a formula for the quantity of bacteria, F:
F = I(x)^t/y
This question asks us how long it will take for the bacteria to grow to 10,000 times their original
amount. In other words, “What is t when F = 10,000 I ?”
F = 10,000 I = I(x)^t/y
10,000 = (x)^t/y
Thus, our final rephrased question is “What is t when 10,000 = (x)^t/y ?”
(1) SUFFICIENT: Note that the yth root of x is equivalent to x to the 1/y power. This statement tells
us that x^1/y = 10. If we plug this value into the equation we can solve for t.
10,000 = (x)^t/y
10,000 = [(x)^1/y]^t
10,000 = (10)^t
10^4 = 10^t
t = 4
(2) SUFFICIENT: The culture grows one-hundredfold in 2 minutes. In other words, the sample
grows by a factor of 10^2. Since exponential growth is characterized by a constant factor of growth
(i.e. by a factor of x every y minutes), in another 2 minutes, the culture will grow by another factor of
102. Therefore, after a total of 4 minutes, the culture will have grown by a factor of 10^2 × 10^2 = 10^4,
or 10,000.
The correct answer is D.
OE

Simply explained:

Let the number of bacteria initially be N
Given N increases by factor of x every y minutes :
Y minutes -> Nx
2Y minutes-> NX^2
3Y Minutes -> NX^3


Statement 1 -> X=10^Y thus:

Y Minutes -> N 10^Y

so for N 10^4 -> Y=4 minutes (Sufficient)


Statement 2 -> When Y=2 , X= 100

2 minutes -> N 10^2
4 minutes -> N 10^4 (Sufficient)

Hence D is the answer

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