Lately, I was reading the applications of Hermite's Identity from this site; the second example was quite unconvincing to me though I could conceive how it was solved.
In the problem $$\left\lfloor r+\frac{19}{100}\right\rfloor+\left\lfloor r+\frac{20}{100}\right\rfloor+\left\lfloor r+\frac{21}{100}\right\rfloor +\cdots+\left\lfloor r+\frac{99}{100}\right\rfloor= 546$$ was given & we had to find $\lfloor 100r\rfloor.$
It was solved taking into consideration that each of the terms on L.H.S. would either take the value $\lfloor r\rfloor$ or $\lfloor r\rfloor+1$. It was found out that the first $i$ terms have the value $7$ each & the remaining ones $8$; $i$ was found to be equal to $38$; so the first $57$ terms were equal to $7$ & the remaining ones equal to $8.$
But one thing I couldn't comprehend what $r$ actually is; it was $7$ for the first $57$ terms & $8$ for the next terms. But how could it be? It's a variable & take any value but if I take it $7$, it should be both $7$ at L.H.S. & R.H.S. How can it take two values at the same time? Somehow I'm confused. Could anyone please help me shun the confusion?
$\lfloor 5.8+0.1\rfloor=\lfloor 5.8\rfloor$ but $\lfloor 5.8+0.2\rfloor=\lfloor 5.8\rfloor+1$. Think about this carefully and you'll understand.