A certain number of men can complete a piece of work in 6k days, where k is a natural number. By what percent should the number of men be increased…

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 6k}{5k}$5kM1​×6k​

M₂ = $\frac{6M_1}{5}$56M1​​

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

Increase = $\frac{6M_1}{5} - M_1 = \frac{M_1}{5}$56M1​​−M1​=5M1​​

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

Percentage Increase = $\frac{\frac{M_1}{5}}{M_1} \times 100 = \frac{1}{5} \times 100 = 20\%$M1​5M1​​​×100=51​×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}$350​%), 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.