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.
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).