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
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