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avatar+187 

Let t be a real number such that

\(\begin{aligned} \lfloor t \rfloor &= 4, \\ \lfloor t + \{t\} \rfloor &= 4, \\ \lfloor t + 2 \{ t \} \rfloor & = 5, \end{aligned}\)

where \(\{t\} = t - \lfloor t \rfloor\) Find all possible values for t 

 Aug 9, 2019
 #1
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\(\text{let $t=4+f,~f\in [0,1)$}\\~\\ 4 \leq (4+f)+f < 5\\ 0 \leq 2f < 1\\ 0\leq f < \dfrac 1 2\\~\\ 5 \leq 4+f + 2f < 6\\ 1 \leq 3f < 2\\ \dfrac 1 3 \leq f < \dfrac 2 3\\~\\ \text{both of these inequalities must be met}\\ \dfrac 1 3 \leq f < \dfrac 1 2\\ \dfrac{13}{3} \leq t < \dfrac 9 2\)

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 Aug 9, 2019

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