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Retired Moderator B
Joined: 27 Aug 2012
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If a certain culture of bacteria increases by a constant  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 53% (01:51) correct 47% (01:51) wrong based on 282 sessions

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If a certain culture of bacteria increases by a constant factor of x every y minutes, how long will it take for the culture to increase to ten-thousand times its original size?

1. $$x=10^y$$
2.In two minutes, the culture will increase to one hundred times its original size.

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Posts: 60647
Re: If a certain culture of bacteria increases by a constant  [#permalink]

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Bunuel wrote:
ankitaprsd wrote:
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

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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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

Maybe this will help

from 1 we are given x=10^y, let initial quantity be m

So after y minutes the quantity => m*x =>m*10^y , now we need the new value to be 10,000 times the original => m*10^y=m*1000 => y=4 minutes
hence statement 1 is sufficient.

from statement 2

In 2 minutes the culture will increase to 100 times this can be interpreted as follows
1) every 1 minute the quantity is multiplied by 10
or
2) every 2 minutes the quantity is multiplied by 100

Either ways after 4 minutes the quantity will be multiplied by 10,000 hence y=4 minutes

Hence statement 2 is sufficient.
Hope the solution is clear.
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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1
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
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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

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If a certain culture of bacteria  [#permalink]

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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
Director  Joined: 25 Apr 2012
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Location: India
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Re: If a certain culture of bacteria  [#permalink]

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ankitaprsd wrote:
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.
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Joined: 03 Jun 2017
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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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

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Joined: 05 Apr 2017
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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JJSHHShank wrote:
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

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 Intern  B
Joined: 03 Jun 2017
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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thanks smitsharma...
overlooked that... always of the notion that the factor doubles and it remains in the exponential form...
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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bagdbmba wrote:
If a certain culture of bacteria increases by a constant factor of x every y minutes, how long will it take for the culture to increase to ten-thousand times its original size?

1. $$x=10^y$$
2.In two minutes, the culture will increase to one hundred times its original size.

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
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If a certain culture of bacteria increases by a constant  [#permalink]

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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.
OE
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Re: If a certain culture of bacteria increases by a constant  [#permalink]

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bagdbmba wrote:
If a certain culture of bacteria increases by a constant factor of x every y minutes, how long will it take for the culture to increase to ten-thousand times its original size?

1. $$x=10^y$$
2.In two minutes, the culture will increase to one hundred times its original size.

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) Re: If a certain culture of bacteria increases by a constant   [#permalink] 06 Apr 2019, 19:07
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