# Problem about resulting ratio after dividing a rope in half, repeatedly.

A student was given a piece of rope and told to cut it into two equal pieces, keep one piece, and pass the other piece to the next student. Each student was to repeat this process until every student in the class had exactly one piece of rope. Which of the following could be the fraction of the original rope that one of the students had?

(A) $\displaystyle\frac{1}{4}$

(B) $\displaystyle\frac{1}{15}$

(C) $\displaystyle\frac{1}{16}$

(D) $\displaystyle\frac{1}{17}$

My answer sheet tells me the answer, which is (C). But why? This question is basically asking how many kids are in the classroom. There could be a random amount of students as far as the question goes, so why does it have to (C) that's correct?

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If I interprete the problem correctly, (C) is also possible. – Hagen von Eitzen Nov 7 '12 at 19:27
This problem has absolutely nothing to do with probability. – rschwieb Nov 7 '12 at 19:28
@rschwieb Sorry I confused this with another question I have. – David Nov 7 '12 at 19:29
@HagenvonEitzen Sorry, it is C which is the correct answer. But my question still stands. – David Nov 7 '12 at 19:31
student $n$ will have $(1/2^n)$ of the length of the rope. So (A) and (C) are possible. – jay-sun Nov 7 '12 at 19:35

If there are $n$ srtudents, then the first will keep $\frac12$, the second $\frac14$ and so on until the $(n-1)$st and the $n$th who both keep $\frac1{2^{n-1}}$ (because the last student will not cut his piece). Thus as soon as $n>2$, at least one student will have $\frac14$ and as soon as $n>4$, at least one student will have $\frac1{16}$ of the rope. Therefore, both (A) and (C) are possible answers (with (A) occuring even for smaller groups of students).
Why do we go with the larger amount of students (i.e $\frac{1}{16}$)? – David Nov 7 '12 at 19:43