How many values of r are there such that \(\lfloor r \rfloor + r = 15.5?\)

I would **NOT** like a direct answer, but I would like some steps and an explanation on how to do these types of problems.

InterestingUsername Oct 3, 2020

#2**0 **

I would welcome further discussion on this, because unlike answer #1,

I think that the equation has no solutions rather than an infinite number of solutions.

Suppose that the integer part of r is n, and the decimal part of r is d.

That is,

\(\displaystyle r = n+d \quad \text{where}\quad n \quad \text{is an integer and}\quad 0 \leq d \leq 1. \)

Then,

\(\displaystyle \lfloor r \rfloor = n\)

so

\(\displaystyle \lfloor r \rfloor +r = 2n+d.\)

If this is to equal 15.5 then does it not imply that 2n = 15, and d = 0.5 ?

If that's the case then n = 7.5 which is a contradiction since n is an integer, so, no solution.

If the integer part of the rhs were even though, there would be a (unique) solution.

If, for example

\(\displaystyle \lfloor r \rfloor +r = 16.5, \quad \text{then} \quad r=8.5.\)

Input from someone who's used to dealing with this sort of stuff would be useful.

Guest Oct 4, 2020