If where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then is equal to

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Solution

Q.
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 ________ [2024]

Ans.
(18)

$We have, S = {a \in ℝ : | 2 a - 1 | = 3 [a] + 2 {a}}$

$Since for x \in ℝ, x = [x] + {x}$

$∴ 3 [a] + 2 {a} = 2 a + [a]$

$∴ | 2 a - 1 | = 2 a + [a]$

$For 2 a - 1 < 0, we have$

$- 2 a + 1 = 2 a + [a]$

$\Rightarrow 4 a = 1 - [a] \Rightarrow 4 a + [a] - 1 = 0$

$\Rightarrow a = \frac{1}{4}$

$Now, if 2 a - 1 > 0 \Rightarrow a > \frac{1}{2}$

$\Rightarrow 2 a - 1 = 2 a + [a]$

$\Rightarrow [a] = - 1 which is not possible as a > \frac{1}{2}$

$S o, a = \frac{1}{4} \in S$

$∴ 72 \sum_{a \in S} a = 72 \times \frac{1}{4} = 18$

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