What are some unusual ways you’ve applied the math you learned in high school to your life?

My answer to a question on Quora: What are some unusual ways you’ve applied the math you learned in high school to your life?

I once was asked to act as a reviewer of a paper submitted for publication in an academic journal on mathematics education. It was a double blind review: the draft paper sent to me contained no names of authors or their affiliation.

The paper described how the authors set up a website and run online questionnaire among staff at mathematics departments of two British universities on the following issue: what kind of examinations, closed book, or open book, better discriminates between different levels of students’ attainment, and what kind is preferred by the respondents? Three pieces of data were given by the authors:

  • Closed book examinations were selected as the most discriminating or second most discriminating of the assessment methods by 79% of the participants.
  • Closed book examination was selected by 86% of the respondents as their most preferred of the assessment methods.
  • The response rate of the questionnaire was 15%,

What surprised me is that the total number of responses to the on-line questionnaire has not been given in the paper, although omitting the size of the sample from statistical data was unacceptable in published academic research.

However, I calculated the number of responses, and explained in my report to the editors how I did that essentially by mental arithmetic.

This is a cute arithmetic problem; one more general piece of information is needed for solution, but it is something commonsense. Try to think for yourself, it is easy. A solution is given below these warning signs:

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Solution. Indeed, 79% and 86% rates of positive answers to particular questions suggest that 86% – 79% = 7% corresponded to an integer number of people (those who answered positively to one question but not to the other). If 7% consists of 1 person, the number of respondents is 14 or 15. If 7% consist of 2 persons, then the number of respondents is between 28 and 30, but in this case, since the response rate was 15%, the two departments have about 200 mathematics lecturers, which was unlikely in UK universities (here the common sense is used). Hence there were 14 or 15 respondents.

Very conveniently, 11/14 rounds up to 0.79 and 12/14 to 0.86 (here I used a calculator – previous steps had been done by mental arithmetic) 15 respondents would produce not so good rounding of percentages.

I recommended to reject the paper — in my opinion, the paper contained no representative data; a chat in a staff lounge during coffee break, or, even better, on in a pub after a seminar was likely to yield a more representative sample. However, the editors accepted the paper for publication, but asked the authors to reveal the number of respondents – indeed, it was 14.