The maths question below is asking me to find the number of people. I have found the answer to be $57553.0834$, but obviously there can't be $0.0834$ of a human.
The mark scheme states the answer is $57553$ as it's been rounded to the nearest integer. However is this not wrong as you're essentially excluding the $0.0834$ of a human? Should all numbers of humans which has a decimal be rounded up?
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There is a dichotomy between $P_t$ representing the number of people, which as you say must be an integer, and the recursion which involves the non-integral multiplier of 1.03. This means that most values of $t$, $P_t$ will not be an integer and, thus, cannot be an exact representation. Instead, it is at best approximate. Whether that means in any specific case rounding up, down, to the nearest integer, or even making a larger adjustment, is not clear. In many cases, the assumption will be to either round down or to the nearest integer, but always rounding up is also a fairly reasonable thing to assume. The question should, ideally, state how to handle non-integral values.
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It doesn't matter in the slightest. In particular, all of that is way beyond the accuracy limit of your model. I'd even object to the answer scheme, since I can't see any basis for going much past, say, 2 significant figures.
On
The end result is that it depends on the question. There are certainly scenarios where you would want to round up, and other scenarios where you would want to round down, and other scenarios still where you might want to leave it as a decimal and not round at all.
Consider the following three questions:
We have ten people in a class. We want to form a group to work on a project and at least a third of the class should be in the group. What is the smallest number of people allowed in the group then? --- Ans: $\lceil\frac{10}{3}\rceil = \lceil 3.\overline{3}\rceil = 4$
We have ten people in a class. We want to form a group to work on a project and no more than a third of the class should be in the group. What is the largest number of people allowed in the group then? --- Ans: $\lfloor \frac{10}{3}\rfloor = \lfloor 3.\overline{3}\rfloor = 3$
We have ten people in a class. They each roll a fair six-sided die. What is the expected number of students who rolled a $5$ or $6$? --- Ans: $10\times \frac{1}{3}=\frac{10}{3}=3.\overline{3}$
For your specific problem, it sounds to me like the problem is improperly worded, but as soon as it were to be properly worded it would be phrased in terms of expectations in which case it would fall into the third category I describe above where rounding should not occur at all. "We expect there to be approximately 57553.0834 people after 3 years."
One convention is to round to the nearest integer. (Various approaches exist for tie-breaking with $n+\frac12$.) In this case you're calculating a mean value of something stochastic, so rounding to an integer might not be expected. In fact, for exams it's worth referring to the preferred number of significant figures.