Question

Let the solution of the equation $5\left\{x\right\}=2\left[x\right]+x$ be $a/4.$ Then the value of $a$ is where $\{.\}$ and $[.]$ represents fractional part function and greatest integer function respectively.

Answer 1

$5\left\{x\right\}=2\left[x\right]+x$
$\Rightarrow 5\left\{x\right\}=2\left[x\right]+\left[x\right]+\left\{x\right\}$
$\Rightarrow 4\left\{x\right\}=3\left[x\right]...\left(1\right)$
We know,
$0\le \left\{x\right\}<1$
$\Rightarrow 0\le 4\left\{x\right\}<4$
$\Rightarrow 0\le 3\left[x\right]<4$
$\Rightarrow 0\le \left[x\right]<4/3$
Now, for equation (1)
$4\left\{x\right\}=3\left[x\right]$
$\Rightarrow 4(x-[x\left]\right)=3\left[x\right]$
$\Rightarrow 4x-4\left[x\right]=3\left[x\right]$
$\Rightarrow x=7/4\left[x\right]$
$\Rightarrow x=0,7/4$
Sum $=0+7/4=7/4$
$\therefore \frac{a}{4}=\frac{7}{4}$
$\Rightarrow a=7$