If $S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a \}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_\limits{a \in S} a$ is equal to _________.
Answer (integer)
18
Solution
<p>$$\begin{aligned}
& S:\{a \in R:|2 a-1|=3[a]+2\{a\}\} \\
& |2 a-1|=3[a]+2(a-[a]) \\
& |2 a-1|=[a]+2 a
\end{aligned}$$</p>
<p>Case I: If $0 < a < \frac{1}{2}$</p>
<p>$$\begin{aligned}
& 1-2 a=0+2 a \\
& \Rightarrow a=\frac{1}{4}
\end{aligned}$$</p>
<p>Case II: If $\frac{1}{2} < a < 1$</p>
<p>$2 a-1=0+2 a$</p>
<p>No solution</p>
<p>Case III: If $1 \leq a<2$</p>
<p>$2 a-1=1+2 a$</p>
<p>$\Rightarrow$ No solution</p>
<p>$\therefore$ only solution is $a=\frac{1}{4}$</p>
<p>$72 \sum_\limits{a \in S} a=72 \times \frac{1}{4}=18$</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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