If $S=\{a \in R:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal

Question

If $S=\{a \in R:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to $\_\_\_\_$ .\

Professor PC Thomas Classes · Answer

(18) : We have, $$ S=\{a \in \mathbb{R}:|2 a-1|=3[a]+2\{a\}\} $$ Since for $x \in \mathbb{R}, x=[x]+\{x\}$\ $	herefore \quad 3[a]+2\{a\}=2 a+[a]$\ $	herefore \quad|2 a-1|=2 a+[a]$\ For $2 a-1<0$, we have $$ -2 a+1=2 a+[a] $$ $\Rightarrow \quad 4 a=1-[a] \Rightarrow 4 a+[a]-1=0$\ $\Rightarrow a=\frac{1}{4}$\ Now, if $2 a-1>0 \Rightarrow a>\frac{1}{2}$\ $\Rightarrow \quad 2 a-1=2 a+[a]$\ $\Rightarrow \quad[a]=-1$ which is not possible as $a>\frac{1}{2}$\ So, $a=\frac{1}{4} \in S$\ $	herefore \quad 72 \sum_{a \in S} a=72 	imes \frac{1}{4}=18$\