Correct Option (3)

Let M₁ represent the initial number of men and D₁ represent the initial number of days, which is 6k. Let M₂ represent the new number of men and D₂ represent the new number of days, which is 5k.

Assuming the total work (W) remains constant, the relationship between men and days is given by the formula: M₁ × D₁ = M₂ × D₂.

Substituting the given values:

M₁ × 6k = M₂ × 5k

To determine the new number of men (M₂):

M₂ = $\frac{M_{1} \times 6 k}{5 k}$

M₂ = $\frac{6 M_{1}}{5}$

The increase in the number of men is calculated as M₂ - M₁:

Increase = $\frac{6 M_{1}}{5} - M_{1} = \frac{M_{1}}{5}$

The percentage increase is determined by dividing the increase in men by the initial number of men and multiplying by 100:

Percentage Increase = $\frac{\frac{M_{1}}{5}}{M_{1}} \times 100 = \frac{1}{5} \times 100 = 20 %$

Therefore, the number of men should be increased by 20% to complete the work in 5k days.

Incorrect Options:

Options 1 (10%), 2 ( $\frac{50}{3}$ %), and 4 (25%) are incorrect. These values do not correspond to the calculated percentage increase derived from the inverse proportionality between the number of men and the time required to complete a fixed amount of work.