Question

Which of the following is/are correct regarding $4\left\{x\right\}=x+\left[x\right]$?
(where {.} represents fractional part function and [.] represents greatest integer function)

• A. There exist $3$ solutions.
• B. There exist $2$ solutions.
• C. The sum of solutions is $\frac{5}{3}$
• D. The sum of solutions is $\frac{4}{3}$

Answer 1

Answer: (B), (C)

The sum of solutions is $\frac{5}{3}$

Here $4\left\{x\right\}=x+\left[x\right]$
$4(x-[x\left]\right)=x+\left[x\right]$
$4x=x+5\left[x\right]$
$3x=5\left[x\right]$
$\left[x\right]=\frac{3}{5}x$

Plotting the graph of $y=\left[x\right]$ and $y=\frac{3}{5}x$


The two graphs intersect at $\left[x\right]=0,1$
$\therefore x=0,\frac{5}{3}$
Hence, sum of the solutions is $\frac{5}{3}.$