Fractions- differences in interpretation - part of a whole

Question

Imagine a fraction of the form $x/y$, where $y>x.$ It seems to me that this can be interpreted in two ways. The first way is to split one unit (say an object), into $y$ parts, and populate $x$ of those. For instnace, splitting one apple four ways can be represented as $1/4.$ At the same time, both $x$ and~ $y$ can represent whole numbers/units themselves. For instance, only $1$ out of $4$ people passed the exam, or $1/4.$ Is this a distinction without a difference? They seem to be different applications: in the former, we have a literal whole which we are splitting up, but in the latter, the whole is considered as the collection of all (already whole) units. Is making a distinction like this reasonable?

Answer 1

Yes, it is reasonable to distinguish between these two interpretations of $\frac14:$ divvying up $4$ apples among $4$ persons, versus slicing an apple into $4$ quarters.

But it is frequently useful to reason abstractly with fractions, such that even $\frac28$—which is differently-specified but denotes the same rational number—is treated as equivalent to (belonging to the same equivalence class as) the previous fraction.

Or even to treat $\frac14$ and $\frac28$ as equal solutions to $12x-3=0.$

P.S. This question feels like the flip-side of numeral vs. number.