To solve this problem, we need to apply the concept of work and how it relates to the number of men and the number of days. We will use the idea that the total amount of work is constant. This means that if you change the number of men and/or the number of days, the product of these two quantities must remain the same.
Let's go step-by-step:
1. Determine the initial amount of work:
The total work done can be expressed as the product of the number of men and the number of days they work. Initially, 36 men working for 20 days:
[tex]\[
\text{Work Done} = 36 \text{ men} \times 20 \text{ days}
\][/tex]
2. Calculate the reduced number of days:
We want to complete the work in [tex]\(\frac{4}{5}\)[/tex] of the original time:
[tex]\[
\text{Reduced Days} = 20 \text{ days} \times \frac{4}{5} = 16 \text{ days}
\][/tex]
3. Relate the new amount of work to the initial amount of work:
Since the total work done is constant:
[tex]\[
36 \text{ men} \times 20 \text{ days} = \text{New Number of Men} \times 16 \text{ days}
\][/tex]
4. Solve for the new number of men:
To find the new number of men needed, we rearrange the equation:
[tex]\[
\text{New Number of Men} = \frac{36 \text{ men} \times 20 \text{ days}}{16 \text{ days}}
\][/tex]
[tex]\[
\text{New Number of Men} = \frac{720}{16} = 45 \text{ men}
\][/tex]
5. Calculate how many men should be added:
To find out how many men need to be added to the original 36 men:
[tex]\[
\text{Additional Men} = 45 \text{ men} - 36 \text{ men} = 9 \text{ men}
\][/tex]
Therefore, 9 men should be added to the initial 36 men to complete the work in [tex]\( \frac{4}{5} \)[/tex] of the time.
Thus, the correct answer is:
b. 9
