Work Calculation: 5 Women vs 5 Men

The problem involves calculating the time required for a group of men to complete a work, given information about women completing the same work and the relative work capacities of men and women.

Total Work Assessment

First, determine the total amount of work. The total work is the product of the number of workers, their individual rate, and the time taken.

Given: 5 women can do the work in 36 days.

Let $R_w$ represent the work rate of one woman.

Total Work = (Number of Women) $\times$ (Days) $\times$ (Rate of one Woman)

Total Work $= 5 \times 36 \times R_w = 180 R_w$ units of work.

Man's Work Rate Determination

The ratio between the capacity (work rate) of a man ($R_m$) and a woman ($R_w$) is given as 3:1.

This means $\frac{R_m}{R_w} = \frac{3}{1}$, which implies $R_m = 3 R_w$. A man works three times as fast as a woman.

Time Calculation for 5 Men

Now, calculate the time it takes for 5 men to complete the same total work.

Let $D$ be the number of days it takes for 5 men to complete the work.

Total Work = (Number of Men) $\times$ (Days) $\times$ (Rate of one Man)

Using the total work calculated earlier:

$180 R_w = 5 \times D \times R_m$

Substitute the relationship $R_m = 3 R_w$ into the equation:

$180 R_w = 5 \times D \times (3 R_w)$

$180 R_w = 15 D R_w$

To find $D$, divide both sides by $15 R_w$ (assuming $R_w \neq 0$):

$D = \frac{180}{15}$

$D = 12$

Therefore, it will take 5 men 12 days to complete the same work.