Question

Total number of positive real values of x satisfying $2\left[x\right]=x+\left\{x\right\}$, where $[.]$ and {.} denote the greatest integer function and fractional part respectively, is equal to

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

The correct option is A $2$

$2\left[x\right]=x+\left\{x\right\}\Rightarrow 2\left[x\right]=\left[x\right]+2\left\{x\right\}\Rightarrow \left\{x\right\}=\frac{\left[x\right]}{2}\Rightarrow 0\le \frac{\left[x\right]}{2}<1\Rightarrow 0\le \left[x\right]<2\Rightarrow \left[x\right]=0,1$

For, $\left[x\right]=0$ we get $\left\{x\right\}=0$

$\Rightarrow x=0$

For, $\left[x\right]=1$ we get $\left\{x\right\}=\frac{1}{2}$

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

Therefore, number of positive values is $2$.