# The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours?

The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours?

I figured out this bacteria $=500(3)^{96/24}$

but then my mate told me it's bacteria $=500(1+3/1)^{96/24 }$

what do I do what is the next step.

• Your interpretation is the normal one. – André Nicolas May 18 '13 at 18:00
• Why is there a bounty on this question? The OP already has the answer, and the answers below debate the fine points of the problem statement (to a pretty strong conclusion). What is the problem here? – Mario Carneiro May 27 '13 at 21:54

It depends on the intended meaning of multiply threefold. You’re interpreting it as meaning increase by a factor of $3$, which is how I would interpret it. Your mate is interpreting it to mean that the amount by which the population increases is $3$ times the present population; I would call that a fourfold increase. However, I can imagine someone using the term multiply threefold in that way; the only way to be certain what’s intended is to ask the person who set the problem.

• Well that's the question... write from the book :( and I can't check my solutions... it sucks because I have to finish this course in 2 weeks and im stupid at it :( – user73122 May 18 '13 at 17:41
• @user73122: I’d go with your solution, with the population multiplying by $3$ every $24$ hours. I think that it’s more likely to be the right interpretation. Do you get to explain your answer, or do you just submit the number? – Brian M. Scott May 18 '13 at 17:42
• I need to explain it Brian do you have skype? – user73122 May 18 '13 at 19:20
• @user73122: I’m afraid not. But it’s good that you’re supposed to explain it: that means that even if your interpretation of the wording is wrong, the grader will see that the reasoning made sense. The explanation is no real problem: on your interpretation the population triples every $24$ hours. after one day it’s $500\cdot3$; after two days it’s $(500\cdot3)\cdot3=500\cdot3^2$; after three days its $(500\cdot3^2)\cdot3=500\cdot3^3$; and in general after $n$ days its $500\cdot3^n$. You know how many days there are in $96$ hours, so you know the right value of $n$ to use. – Brian M. Scott May 18 '13 at 19:26

Your mate is simply wrong. I see no ambiguity here at all.

• It’s not nearly so common as the ambiguity in three times more, which for some of us natively means four times as much, not three times as much, but I’m pretty sure that I have encountered it in the wild. (Whether the OP’s mate actually has this usage or is simply applying the wrong formula is another question.) – Brian M. Scott May 18 '13 at 19:39
• I agree with TonyK that "multiply threefold" is not ambiguous. "Increase threefold" is ambiguous, and it seems whoever posed the question was careful to avoid that. If I were to criticize the formulation, it would be for "can multiply", which is intended to mean that the population not only can but in fact does multiply. – Andreas Blass May 21 '13 at 17:25

Regardless of the ambiguity, that solution doesn't look right because that doesn't seem like the right rate constant. In order to solve an exponential growth problem we normally assume a growth given by $$P(t) = P(0) e ^{k t}$$ We can find k by using the given information (there is where the ambiguity lies, I agree that threefold prabably means 3 times the present value like the rest). If we then use that $$P(24) = 3 P(0) = P(0) e ^{k 24}$$ we can then find k and the problem will be a little easier, since once we have k we can then find $P(t)$ for any future value of $t$.

• at least when I do bacterial growth I normally use exponential models, just my 2 cents – Triatticus May 26 '13 at 11:08