5 men working 8 hours a day can completely build a wall of length 20 metres, breadth `1/4` metre and height 6 metres in 3 days. How many days will 8 men working 6 hours a day require to build a wall of length 120 meters, breadth `1/2` metre and height 4 metres.

Answer

To solve the problem, we will use the concept of work done, which can be expressed in terms of men, days, and hours. We will denote the work done in the first scenario and use it to find the time required in the second scenario.

### Step-by-Step Solution:

1. **Calculate the Work Done in the First Scenario:**
- The dimensions of the wall are:
- Length (L1) = 20 meters
- Breadth (B1) = 1/4 meters
- Height (H1) = 6 meters
- The volume of the wall (which represents the work done) can be calculated as:
\[ W_1 = L_1 \times B_1 \times H_1 = 20 \times \frac{1}{4} \times 6 = 30 \text{ cubic meters} \]

2. **Determine the Total Man-Hours in the First Scenario:**
- Number of men (M1) = 5
- Number of days (D1) = 3
- Number of hours worked per day (H1) = 8
- Total man-hours (M1 × D1 × H1):
\[ \text{Total Man-Hours} = M_1 \times D_1 \times H_1 = 5 \times 3 \times 8 = 120 \text{ man-hours} \]

3. **Calculate Work Done Per Man-Hour:**
- Work done per man-hour can be calculated as:
\[ \text{Work per man-hour} = \frac{W_1}{\text{Total Man-Hours}} = \frac{30}{120} = \frac{1}{4} \text{ cubic meters per man-hour} \]

4. **Calculate the Work Required in the Second Scenario:**
- The dimensions of the new wall are:
- Length (L2) = 120 meters
- Breadth (B2) = 1/2 meters
- Height (H2) = 4 meters
- The volume of the new wall (W2):
\[ W_2 = L_2 \times B_2 \times H_2 = 120 \times \frac{1}{2} \times 4 = 240 \text{ cubic meters} \]

5. **Determine the Total Man-Hours Required for the Second Scenario:**
- Number of men (M2) = 8
- Number of hours worked per day (H2) = 6
- Let D2 be the number of days required.
- The total man-hours for the second scenario can be expressed as:
\[ \text{Total Man-Hours} = M_2 \times D_2 \times H_2 = 8 \times D_2 \times 6 \]

6. **Set Up the Equation:**
- The total work done in man-hours for the second scenario must equal the work required:
\[ \frac{240}{\frac{1}{4}} = 8 \times D_2 \times 6 \]
- Simplifying the left side:
\[ 240 \times 4 = 960 \]
- Thus, we have:
\[ 960 = 48 D_2 \]

7. **Solve for D2:**
- Rearranging gives:
\[ D_2 = \frac{960}{48} = 20 \]

### Final Answer:
The number of days required for 8 men working 6 hours a day to build the wall is **20 days**.