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.