X number of men can do a work in (X – 28) days, while (X + 12) number of men can do the same work in (X – 32) days. Find$\frac{3}{4}X$ women together can do the same work in how many days.

The correct option is $\frac{80}{3}$ days.

Step 1: Understand the work relationship
The total work done is equal to the product of the number of men and the number of days taken to complete the work.
$\text{Total Work}=\text{Number of Men}\times \text{Number of Days}$

Step 2: Equate total work to solve for X
According to the problem statement:
Case 1: $X$ men can complete the work in $(X-28)$ days.
Case 2: $(X+12)$ men can complete the same work in $(X-32)$ days.

Since the total work is constant in both cases, we can write:

$X(X-28)=(X+12)(X-32)$

Expanding both sides:

$X^2-28X=X^2-32X+12X-384$

Simplify the right side:

$X^2-28X=X^2-20X-384$

Subtracting $X^2$ from both sides:

$-28X=-20X-384$

Rearranging the terms to isolate $X$:

$384=28X-20X$

$8X=384$

$X=\frac{384}{8}=48$

Step 3: Calculate Total Work
Substitute $X=48$ into the work expression:

$\text{Total Work}=48\times (48-28)=48\times 20=960\text{man-days}$

Step 4: Determine the number of days required by the given group
Assuming standard efficiency where 1 woman's rate is equal to 1 man's rate (or considering group work capacity directly):
We need to find the number of days taken by $\frac{3}{4}X$ individuals.

Calculate the number of individuals in the group:

$\frac{3}{4}X=\frac{3}{4}\times 48=36$

Now, calculate the number of days taken by 36 individuals to do 960 units of work:

$\text{Number of days}=\frac{\text{Total Work}}{\text{Number of workers}}=\frac{960}{36}$

Simplify the fraction by dividing both numerator and denominator by 12:

$\text{Number of days}=\frac{80}{3}\text{days}$

Thus, $\frac{3}{4}X$ women can complete the same work in $\frac{80}{3}$ days.