To solve this problem, let's first define the variables and break down the work completed by the men and women:

• Let the work done by 1 man in one day be $M$.
• Let the work done by 1 woman in one day be $W$.

Step-by-Step Solution

1. Establish Equations from Given Information:

   • Equation 1: Two men and five women complete the work in 4 days.

   The amount of work done by 2 men and 5 women in one day can be expressed as: $2M + 5W$

   Since they complete the whole work in 4 days: $4 \times (2M + 5W) = 1 \quad \text{(1 full work)}$ This simplifies to: $2M + 5W = \frac{1}{4} \quad \text{(Equation 1)}$

   • Equation 2: One man and one woman complete the work in 12 days.

   The amount of work done by 1 man and 1 woman in one day is: $M + W$

   Since they complete the whole work in 12 days: $12 \times (M + W) = 1 \quad \text{(1 full work)}$ This simplifies to: $M + W = \frac{1}{12} \quad \text{(Equation 2)}$

2. Solve the System of Equations:

   From Equation 2, we can express $W$ in terms of $M$: $W = \frac{1}{12} - M$

   Substitute this into Equation 1: $2M + 5 \left( \frac{1}{12} - M \right) = \frac{1}{4}$

   Expand and simplify: $2M + \frac{5}{12} - 5M = \frac{1}{4}$

   Combine like terms: $-3M + \frac{5}{12} = \frac{1}{4}$

   Isolate $M$: $-3M = \frac{1}{4} - \frac{5}{12}$

   Find a common denominator (12) for the fractions: $-3M = \frac{3}{12} - \frac{5}{12} = -\frac{2}{12} = -\frac{1}{6}$

   Solve for $M$: $M = \frac{1}{18}$

3. Interpret the Result:

   $M = \frac{1}{18}$ means that one man can do $\frac{1}{18}$ of the work in one day.

   Therefore, the time taken by 1 man to complete the entire work is the reciprocal of $M$: $\text{Time} = \frac{1}{M} = 18 \, \text{days}$

Conclusion

A single man would take 18 days to complete the work alone.

Would you like more details or have any other questions?

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Here are 5 related questions to consider:

1. How much work can 1 woman do in one day?
2. How long would it take for 5 men to complete the same work?
3. If 3 men and 2 women work together, how long will they take to complete the work?
4. How would the time change if the number of men and women were doubled?
5. How does changing the number of days affect the amount of work done per day?

Tip: Understanding how to set up equations based on the rate of work done is crucial for solving work-time problems effectively.