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I recently ran into the following question for high school students, which confused me a little. I am curious what you have to say about it.

A temperature $T$ is measured (in Celcius) and rounded to the nearest whole number. The rounded temperature is 38°C. Which of the following statements about $T$ is true (multiple statements might be true).

a) $T \geq 37.5$,

b) $37.5 < T < 38.5$,

c) $37.5 \leq T \leq 38.5$,

d) $37.5 \leq T < 38.5$,

e) $37.5 < T \leq 38.5$.

Note: although not mentioned in the problem formulation, I think it can be assumed that the "usual" way of rounding numbers is applied: so if the number ends with a $5, 6, 7, 8$, or $9$ the number is rounded up, otherwise rounded down.

I find the problem confusing, because in the original version it says that exactly two of the statements are correct, which is not what I think :) I would say that a), c) and d) are true since any number in $[37.5; 38.5)$ is rounded to 38. Thank you in advance.

Cettt
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    The correct ones are (a) and (d). – quasi Mar 06 '17 at 07:47
  • @quasi (a) cannot be true since if $;T=45;$, for example, it then isn't rounded to $;38;$ ... – DonAntonio Mar 06 '17 at 07:48
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    @DonAntonio: No, the question is not asking for an equivalence. It's asking which ones must be true (i.e., which are implied by the rounded result). – quasi Mar 06 '17 at 07:49
  • @quasi I think you might be right. – DonAntonio Mar 06 '17 at 07:56
  • I edited the question: I think that a), c) and d) are true. Also it would be true to say that the true temperature $T$ satisfies $T>4$. This does not mean that every number $T>4$ is rounded to 38, but that every number $T$ which can be rounded to 38 is greater than 4. – Cettt Mar 06 '17 at 07:59
  • rounded to the next whole number Are you sure you transcribed that right? As written, that amounts to $38 = \lceil ,T, \rceil$ which is not equivalent to any of the options, but rather $37 \lt T \le 38,$. – dxiv Mar 06 '17 at 08:09
  • Surely c) can't be true since if you take $T = 38.5$ it doesn't work – mrnovice Mar 06 '17 at 08:09
  • @mrnovice: yeah but the true temperature $T$ for sure satisfies $T \leq 38.5$ as well as $T<100$. Not every number smaller than 100 is rounded to 38, but every number that is rounded to 38 is smaller than 100. – Cettt Mar 06 '17 at 08:11
  • Rounded to the next whole number means rounded up. Presumably the question was written as rounded to the nearest whole number. – quasi Mar 06 '17 at 08:15
  • @Cettt: I agree, it's (a),(c),(d). The only choices that have the potential to be false are (b) and (e). – quasi Mar 06 '17 at 08:16
  • @Cettt After more thought, I agree with your conclusion now – mrnovice Mar 06 '17 at 08:16
  • @quasi, thanks, I edited the question to the nearest whole number. It was a translation error on my side (the original problem is in german). – Cettt Mar 06 '17 at 08:17
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    @Cettt: Good catch. It shows that many of us (myself included) need to be more careful in reading questions. – quasi Mar 06 '17 at 08:21

1 Answers1

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Let's evaluate the statements one-by-one. It's helpful to phrase them clearly as if-then statements.

(a) If $\lceil T \rceil = 38$, then $T \geq 37.5$. This is true.

(b) If $\lceil T \rceil = 38$, then $37.5 < T < 38.5$. This is false. It is possible to have $T = 37.5$, in which case $\lceil T \rceil = 38$.

(c) If $\lceil T \rceil = 38$, then $37.5 \leq T \leq 38.5$. This is true. Note that $T \neq 38.5$, but this doesn't matter: we could just as well write If $\lceil T \rceil = 38$, then $0 \leq T \leq 100$ and we'd still be perfectly correct.

d) If $\lceil T \rceil = 38$, then $37.5 \leq T < 38.5$. No surprises here, this is true. All the possible values of $T$ that let $\lceil T \rceil = 38$ are in this interval, by what you know about rounding.

(e) If $\lceil T \rceil = 38$, then $37.5 < T \leq 38.5$. This is false by the reasoning in (b).

Newb
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