Tired workers

Let $x$ be the number of days in which the 150 workers finish the work.

According to the given information, $\large 150x = 150 + 146 + 142 + ........ (x + 8) terms$ .

The sequence is in A.P. where the first term $a = 150$ , common difference $d = -4$ and number of terms are $(x + 8)$

We know that, $\large S_n = \frac{n}{2} [2a + (n - 1)d]$

$\large \implies 150x = \frac{(x + 8)}{2} [2(150) + (x + 8 - 1)(-4)]$

$\large \implies 150x = (x + 8)[(150) + (x + 7)(-2)]$

$\large \implies 150x = (x + 8)(136 - 2x)$

$\large \implies 75x = (x + 8)(68 - x)$

$\large \implies x^2 + 15x - 544 = 0$

$\large \implies (x-17) (x+32) = 0$

$\large \implies x = 17 or x = -32$

Negative is rejected. $\because$ Time cannot be in negative.

$\large \therefore x = 17$

$\therefore$ Originally, the number of days in which work was completed is 17.

Thus, the required number of days $\large = (17 + 8) = \boxed{25} Days$