TITLE: Why in some problems we round up and in some problems we round down?
QUESTION [28 upvotes]: The following was written in the book I am reading.

Suppose you’re organizing an outing for $100$ people, and you're renting minibuses that can hold $15$ people each. How many minibuses do you need? Basically you need to calculate$$100 \,÷\,15 \approx 6.7$$But then you have to take the context into account: you can’t book $0.7$ of a minibus, so you have to round up to $7$ minibuses.
Now consider a different context. You want to send a friend some chocolates in the mail, and a first-class stamp is valid for up to $100$ g. The chocolates weigh $15$ g each, so how many chocolates can you send? You still need to start with the same calculation$$100 \,÷\,15 \approx 6.7$$But this time the context gives a different answer: since you can’t send $0.7$ of a chocolate, you'll need to round down to $6$ chocolates.

I can't help but sense that there's a deeper underlying phenomenon going on here. So I have a few questions.

What is actually the deeper thing going on here?
What might be some exercises worth thinking about (i.e. struggling with) to better grasp what is actually going on here, for someone who has virtually zero proof experience in math?

REPLY [4 votes]: As others have mentioned, you're dealing with linear inequalities. Specifically, you're seeking an integer solution in each circumstance. That is, the answer must be a whole number. There is no rounding that's occurring, but rather, you are selecting the optimal integer solution from the set of all possible real solutions. 
You can find more related material at Khan Academy, under Algebra I: Inequalities. Specifically, this video is similar to your exercises, as are the ones before and after it in the video sequence.
A little more deeply, if you noticed, I used the words "optimal integer solution" above. Because you're really selecting the maximum (or minimum) from some set to find the optimal solution..
Consider then, that the problem you're working on can be classified as a basic example of Optimization. Because you're working with integers, you can refer to it as Discrete Optimization. You may also be interested in the terms Linear Programming (or Optimization) and Integer Linear Programming. Those probably narrow down on these problems best.