To solve this problem, we can use the concept of work done and proportionality. Here's the step-by-step process:

Step 1: Calculate the work done by 1 man in 1 day

• Given: 5 men can build a 45-meter long wall in 9 days.
• Total work: $45$ meters.

The work done by 5 men in 9 days: $\text{Work done by 5 men in 9 days} = 45 \text{ meters}$

The work done by 1 man in 1 day: $\text{Work done by 1 man in 1 day} = \frac{45 \text{ meters}}{5 \times 9 \text{ days}} = \frac{45}{45} = 1 \text{ meter per day}$

Step 2: Calculate the total work required for the new task

• New task: Build a 30-meter long wall in 5 days.
• Total work required: 30 meters.

Step 3: Calculate the number of men required

Let $x$ be the number of men required to build a 30-meter long wall in 5 days.

The total work done by $x$ men in 5 days: $\text{Work done by } x \text{ men in 5 days} = x \times 1 \times 5 = 5x \text{ meters}$

Set this equal to the total work required (30 meters): $5x = 30$

Step 4: Solve for $x$

$x = \frac{30}{5} = 6$

Conclusion:

To build a 30-meter long wall in 5 days, 6 men are required.

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Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How many men are needed to build a 60-meter long wall in 10 days?
2. If the wall length is doubled, how many additional men are required?
3. What if 8 men were available—how many days would they take to build the 45-meter long wall?
4. How would the time change if only 4 men were available to build the 30-meter wall?
5. What is the relationship between the number of men and the days required for a fixed length?

Tip: When dealing with work problems, it's helpful to first calculate the work rate (e.g., work per person per day) and then use proportional reasoning to find the unknown.