8 men and 3 women can do a piece of work in 7 days, whereas 5 men and 1 woman can do it in 12 days. How many men should assist 8 women so that the same work is completed in \(4\frac{1}{5}\) days ?

Given:

8 men and 3 women can complete a work in 7 days.

5 men and 1 woman can complete the same work in 12 days.

We need to find how many men should assist 8 women to complete the work in 4 and 1/5 days (which is 21/5 days).

Formula used:

Work = (Number of Men × Days × Hours) / Work Done (assuming efficiency of all workers is constant and work done is 1 unit).

In cases involving different types of workers (men and women), we first find the relationship between their efficiencies.

Total Work = (M1 × D1) = (M2 × D2) (where M represents man-days or man-equivalent-days)

Calculation:

Let 'm' be the efficiency of 1 man (work done by 1 man in 1 day).

Let 'w' be the efficiency of 1 woman (work done by 1 woman in 1 day).

From the first condition:

(8m + 3w) × 7 = Total Work ......(Equation 1)

From the second condition:

(5m + 1w) × 12 = Total Work ......(Equation 2)

Equating Equation 1 and Equation 2 to find the relationship between 'm' and 'w':

7(8m + 3w) = 12(5m + w)

⇒ 56m + 21w = 60m + 12w

⇒ 21w - 12w = 60m - 56m

⇒ 9w = 4m

⇒ m/w = 9/4

This means 1 man's efficiency is equivalent to 9/4 times a woman's efficiency, or 4 men do the same work as 9 women.

Now, let's find the total work using either equation. Using Equation 2:

Total Work = (5m + w) × 12

Substitute m = (9/4)w into the Total Work equation:

Total Work = (5 × (9/4)w + w) × 12

⇒ Total Work = 49w × 3 = 147w (units of work)

Now, let 'x' be the number of men required to assist 8 women to complete the work in 4 and 1/5 days (21/5 days).

(x men + 8 women) × (21/5 days) = Total Work

(xm + 8w) × (21/5) = 147w

Substitute m = (9/4)w:

(x × (9/4)w + 8w) × (21/5) = 147w

(9x/4 + 8) × (21/5) = 147

9x/4 + 8 = 147 × (5/21)

⇒ 9x/4 + 8 = 7 × 5

⇒ 9x/4 = 35 - 8

⇒ 9x/4 = 27

⇒ x = 12

∴ The correct answer is option 3.