If $\{ x \} = \frac{2} {3}$ and $[ x +\{ x+ [ x+ \{ x +\cdots 100 \text{ times} \} ] \} ] = 5$ then find $x$

Question

If $\{ x \} = \frac{2} {3}$ and $[ x +\{ x+ [ x+ \{ x +\cdots 100 \text{ times} \} ] \} ] = 5$ then find $x$

Where $\{\cdot\}$ and $[\cdot]$ denote the fractional part function and greatest integer functions respectively.

When I tried the question I got $[x] = 5$ and it's given $\{x\} = 2/3$ so $x = [x]+\{x\} = 2/3+5 = 17/3$ but in the answer it is given than $x = 14/3$ and $[x] = 4$. What am I doing wrong?

Answer 1

You are looking for $[x+\{x+[\,\text{stuff}\,]\}]=5$. We don't care what stuff is because when we apply the square brackets it becomes an integer. When we add that to $x$ we get another integer plus $\frac 23$, so the fractional part is $\frac 23$. That gives us $[x+\frac 23]=5$ so we know $x \in [4\frac 13,5\frac 13)$ and knowing that the fractional part of $x$ is $\frac 23$ we know $x=4\frac 23$

Answer 2

If you think about it, this is actually trivial.

$[ x +\color{blue}{\{ x+ \color{red}{[ x+ \{ x +\cdots 100 \text{ times} \} ]} \}}] = 5$

The text in $\color{red}{\text{red}}$ is and integer so:

$\color{blue}{\{ x+ \color{red}{[ x+ \{ x +\cdots 100 \text{ times} \} ]} \}}=\color{blue}{\{x +\color{red}{integer}\}}=\{x\} = \frac 23$

And $x = [x] + \frac 23$

So $[ x +\color{blue}{\{ x+ \color{red}{[ x+ \{ x +\cdots 100 \text{ times} \} ]} \}}] =[([x] + \frac 23) + \frac 23] = [[x]+ \frac 23 + \frac 23] = [[x] + 1\frac 13]= [x] + 1 = 5$.

So $[x] = 4$ and $\{x\} = \frac 23$ and $x = 4 \frac 23$.