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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 $72 \sum_{a \in S} a$ is equal to ______
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18
$\begin{aligned} & |2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\} \\ & |2 \mathrm{a}-1|=[\mathrm{a}]+2 \mathrm{a}\end{aligned}$
$\begin{aligned} & \text { Case-1 }: \mathrm{a}>\frac{1}{2} \\ & 2 a-1=[a]+2 a \\ & {[a]=-1 \quad \therefore a \in[-1,0) \text { Reject }} \\ & \text { Case-2 }: \mathrm{a} < \frac{1}{2} \\ & -2 \mathrm{a}+1=[\mathrm{a}]+2 \mathrm{a} \\ & \mathrm{a}=\mathrm{I}+\mathrm{f} \\ & -2(\mathrm{I}+\mathrm{f})+1=\mathrm{I}+2 \mathrm{I}+2 \mathrm{f} \\ & \mathrm{I}=0, \mathrm{f}=\frac{1}{4} \quad \therefore \mathrm{a}=\frac{1}{4} \\ & \text { Hence } \mathrm{a}=\frac{1}{4} \\ & 72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}=72 \times \frac{1}{4}=18\end{aligned}$
$\begin{aligned} & \text { Case-1 }: \mathrm{a}>\frac{1}{2} \\ & 2 a-1=[a]+2 a \\ & {[a]=-1 \quad \therefore a \in[-1,0) \text { Reject }} \\ & \text { Case-2 }: \mathrm{a} < \frac{1}{2} \\ & -2 \mathrm{a}+1=[\mathrm{a}]+2 \mathrm{a} \\ & \mathrm{a}=\mathrm{I}+\mathrm{f} \\ & -2(\mathrm{I}+\mathrm{f})+1=\mathrm{I}+2 \mathrm{I}+2 \mathrm{f} \\ & \mathrm{I}=0, \mathrm{f}=\frac{1}{4} \quad \therefore \mathrm{a}=\frac{1}{4} \\ & \text { Hence } \mathrm{a}=\frac{1}{4} \\ & 72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}=72 \times \frac{1}{4}=18\end{aligned}$
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