Find the solutions of $4\left\{x\right\}=x+\left[x\right]$ where $\left\{.\right\},\left[.\right]$ represents fractional part and greatest integer function.
• A
$0$
• B
$\dfrac{1}{3}$
• C
$\dfrac{5}{3}$
• D
$1$

#### Solution As we know, to find number of solutions of two curves we should find the point of intersection of two curves.
$\therefore 4\left\{x\right\}=x+\left[x\right]$
$\Rightarrow 4\left(x-\left[x\right]\right)=x+\left[x\right]$
$\because x=\left[x\right]+\left\{x\right\}$
$\Rightarrow 4x-x=4\left[x\right]+\left[x\right]$
$\Rightarrow 3x=5\left[x\right]$
$\Rightarrow x=\dfrac{5}{3}\left[x\right]$                    .................$\left(1\right)$
$\therefore$ To plot the graph of both
$y=\left[x\right]$ and $y=\dfrac{3}{5}x$
Clearly, the two graphs intersects when
$\left[x\right]=0$ and $\left[x\right]=1$                  .................$\left(2\right)$
$\therefore x=\dfrac{5}{3}\left[x\right]$                   (from eqns$\left(1\right)$ and $\left(2\right)$)
$\therefore x=\dfrac{5}{3},0$ and $x=\dfrac{5}{3}\left(1\right)$
$\therefore x=0$ and $x=\dfrac{5}{3}$ are the only two solutions.