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There are $50$ bacteria at 10:00 and each divides simultaneously into 2 every 20 minutes. The process of dividing into two is known as binary fission. Let $n$ be the number of fissions that have occurred since 10:00. At what time does the tenth fission occur?

Expected answer is as follows.

The tenth fission occur $20 \times 10=200$ minutes after 10:00. As 200 minutes equal to 3 hours and 20 minutes, the tenth fission occurs at 13:20.

Unexpected answer is as follows.

As there are 50 bacteria that simultaneously divide themselves so all the first 50 fissions occur at 10:10. Thus the tenth fission, of course, occurs at 10:10 because all the 50 bacteria divide into two simultaneously at 10:10.

Question

How should I reword the question to avoid the second answer above?

(I am not a native English speaker.)

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    $\begingroup$ You mean 10: 20 in the 'unexpected' answer, right? $\endgroup$ – AakashM Feb 19 at 8:55
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    $\begingroup$ What's the point of defining n? You never refer to it later (unless there are additional questions that refer back to this). $\endgroup$ – Barmar Feb 19 at 15:14
  • $\begingroup$ @AakashM: NO. I will update the question to make it much clearer. $\endgroup$ – Money Sets You Free Feb 19 at 16:51
  • $\begingroup$ @Barmar There are other questions referring to this n. But I did not show here the questions for the sake of simplicity. $\endgroup$ – Money Sets You Free Feb 19 at 16:54
  • $\begingroup$ You could use it here: "At what time does n = 10?" But that doesn't address the earlier ambiguity. $\endgroup$ – Barmar Feb 19 at 17:03
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The ambiguous answer is relatively correct (actually you need to know how old they are at the start of the problem). But each fission is an individual splitting, not a generation.

Perhaps this:

A certain species of bacteria splits into two cells ( a process called fission) every 20 minutes (the "generation" length), assuming proper growth conditions. Assume we have a sample of 50 new bacteria at 10 AM at the experiment start. The cells are in a container with proper growth conditions throughout the experiment.

A. Ten generations later, from the experiment start, what time is it?

B. How many bacteria are there in the container?

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  • $\begingroup$ I would rearrange the question to : "Assuming the proper growth conditions, a certain species of bacteria splits into two cells (a process called fission) every 20 minutes (the length of a "generation"). With the proper growth conditions, we start an experiment at 10 AM with 50 new bacteria." I would also change the wording of the questions to "What would the time be after 10 generations from experiment start?" and "What is the number of bacteria after 10 generations from experiment start?". $\endgroup$ – hythis Feb 19 at 21:18
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Use the word "generation" instead of "fission" as also suggested in @guest answer above. Also I couldn't understand the use of this sentence fragment, "Let n be the number of fissions that have occurred since 10:00."

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    $\begingroup$ I originally thought that this was the ambiguity! It seems this is just asking for an off-by-one error: does the first fission occur at 10:00? $\endgroup$ – Carser Feb 19 at 12:29
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Neither of your sample answers are correct.

You ask :"At what time does the tenth fission occur?" The problem is that "Fission" is the incorrect word, as that would relate to individual fissions of individual bacteria.
If 5 bacteria reproduce at the same instant, then in that instant 5 fissions occurred.

I believe you intend to ask when the tenth generation has occurred. Or possibly the more precise but verbose "At what time will a given cell in the sample undergo the tenth consecutive fission?"

Secondly:
As there is no information given on the age distribution of the individual cells in the sample,
the first fission could occur at 10:00 + 0 seconds(very unlikely!). In which case that bacteria's descendants will undergo their 10th generation of fission at 13:00. The first fission could also occur at 10:00 + 20 minutes(also very unlikely!). In which case that bacteria's descendants will undergo their 10th generation of fission at 13:20.

The correct answer is a statistical distribution between 13:00 and 13:20

You need to add information that the bacteria are at the beginning of their lifecycle.

Also: Why bother complicating the question by starting with a count of 50? It has no relevance to this question, because the answer you want is the same whether the starting colony is 1 or 50 or 1 billion.

The students are likely to stumble on this question because they know the chance of 50 bacteria having the exact same age, deciding to fission in perfect synchrony, is zero. Making this part of the lesson about a single starting cell removes this invisible and unneeded stumbling block.

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