Question

Find the solutions of $4\left\{x\right\}=x+\left[x\right]$ where $\{.\},[.]$ represents fractional part and greatest integer function.

• A. $0$
• B. $\frac{1}{3}$
• C. $\frac{5}{3}$
• D. $1$

Answer 1

Answer: (A), (C)

The correct options are
A $0$
C $\frac{5}{3}$

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