Question

Let $x$ and $\left[x\right]$ denote the fractional and integral parts of a real number $x$ respectively. Then a possible solution of the equation is $4\left\{x\right\}=x+\left[x\right]$

• A. $x=0$
• B. $x=\frac{5}{3}$
• C. $x=5$
• D. $x=3$

Answer 1

Answer: (A), (B)

$x=0$
$x=\frac{5}{3}$

$4\left\{x\right\}=x+\left[x\right]$

When $x$ is an integer the equation becomes,

$2x=0$

$\Rightarrow x=0$

For non-integral values,

$3\left\{x\right\}=x-\left\{x\right\}+\left[x\right]$

$3\left\{x\right\}=2\left[x\right]$

$\left\{x\right\}=\frac{2\left[x\right]}{3}$

$\therefore 0<\frac{2\left[x\right]}{3}<1$

$\therefore 0<\left[x\right]<1.5$

$\Rightarrow \left[x\right]=1$

$\Rightarrow \left\{x\right\}=\frac{2}{3}$

$\therefore x=\left[x\right]+\left\{x\right\}$

$x=\frac{5}{3}$