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# Help is appreciated.

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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.

Oct 3, 2020

#1
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I hope the following helps: To be clear, when I wrote "There are an infinite number of possible values..."  I meant "There are an infinite number of possible values for r ..."

Oct 3, 2020
edited by Alan  Oct 3, 2020
#2
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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.

Oct 4, 2020
#3
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Correction to last answer, should be

d < 1 rather than d <= 1.

Guest Oct 4, 2020