Let's solve the problem step-by-step.

Step 1: Calculate Total Work in Man-Hours

The total work is to be completed in 56 days by 104 men, each working 8 hours per day.

Total work (in man-hours) can be calculated as:

$\text{Total work} = \text{Number of men} \times \text{Number of days} \times \text{Number of hours per day}$

$\text{Total work} = 104 \times 56 \times 8$

Let's compute this:

$\text{Total work} = 46592 \text{ man-hours}$

Step 2: Calculate Work Completed in 30 Days

It is given that after 30 days, $\frac{2}{5}$ of the work is completed.

Work completed in 30 days:

$\text{Work completed} = \frac{2}{5} \times \text{Total work}$

$\text{Work completed} = \frac{2}{5} \times 46592 = 18636.8 \text{ man-hours}$

Step 3: Calculate Remaining Work

Remaining work to be done:

$\text{Remaining work} = \text{Total work} - \text{Work completed}$

$\text{Remaining work} = 46592 - 18636.8 = 27955.2 \text{ man-hours}$

Step 4: Calculate Remaining Days and Hours per Day

The work must be completed in 56 days. Since 30 days have already passed, the remaining days are:

$\text{Remaining days} = 56 - 30 = 26 \text{ days}$

Now, each man is working 9 hours per day instead of 8 hours.

Step 5: Calculate Total Number of Man-Hours Required Per Day

Let $x$ be the number of additional men required.

The total number of men working now will be:

$\text{Total men} = 104 + x$

Total number of man-hours required per day to complete the remaining work:

$\text{Man-hours per day} = (\text{Number of men}) \times (\text{Hours per day})$

$\text{Man-hours per day} = (104 + x) \times 9$

Step 6: Set up the Equation for Remaining Work

The total man-hours required to complete the remaining work in 26 days is:

$26 \times (104 + x) \times 9 = 27955.2$

Simplify this equation to find $x$:

$(104 + x) \times 9 \times 26 = 27955.2$

$(104 + x) \times 234 = 27955.2$

$104 + x = \frac{27955.2}{234}$

$104 + x = 119.4$

$x = 119.4 - 104$

$x = 15.4$

Since the number of men cannot be a fraction, we round up to the nearest whole number:

$x = 16$

Conclusion

16 additional men should be employed to complete the work on time.

Would you like more details or have any questions?

Related Questions:

1. How would the solution change if the initial number of men was different?
2. What happens if the number of working hours per day is increased further?
3. How much work would each man need to do if no additional workers were hired?
4. What if the work completed in the first 30 days was more than $\frac{2}{5}$?
5. How would the calculations change if the total duration was less than 56 days?

Tip: Always consider the impact of changes in work hours or number of workers on the overall timeline for project completion.