The work done by (x - 2) men in (4x + 1) days and the work done by (4x + 1) men in (2x - 3) days are in the ratio 3 : 8. Find the value of x.

Amount of work done by (x - 2) men in (4x + 1) days = (x - 2)(4x + 1),

Amount of work done by (4x + 1) men in (2x - 3) days = (4x + 1)(2x - 3)

According to question,

$\Rightarrow \dfrac{(x - 2)(4x + 1)}{(4x + 1)(2x - 3)} = \dfrac{3}{8} \\[1em] \Rightarrow \dfrac{4x^2 + x - 8x - 2}{8x^2 - 12x + 2x - 3} = \dfrac{3}{8} \\[1em] \Rightarrow \dfrac{4x^2 - 7x - 2}{8x^2 - 10x - 3} = \dfrac{3}{8} \\[1em] \Rightarrow 8(4x^2 - 7x - 2) = 3(8x^2 - 10x - 3) \\[1em] \Rightarrow 32x^2 - 56x - 16 = 24x^2 - 30x - 9 \\[1em] \Rightarrow 32x^2 - 24x^2 - 56x + 30x - 16 + 9 = 0 \\[1em] \Rightarrow 8x^2 - 26x - 7 = 0 \\[1em] \Rightarrow 8x^2 - 28x + 2x - 7 = 0 \\[1em] \Rightarrow 4x(2x - 7) + 1(2x - 7) = 0 \\[1em] \Rightarrow (4x + 1)(2x - 7) = 0 \\[1em] \Rightarrow 4x + 1 = 0 \text{ or } 2x - 7 = 0 \\[1em] \Rightarrow x = -\dfrac{1}{4} \text{ or } x = \dfrac{7}{2}.$

x ≠ $-\dfrac{1}{4}$ as that will make number of men negative which is not possible.

Hence, x = $\dfrac{7}{2}$ = 3.5.