This problem involves understanding the concept of work rate and how it applies to multiple individuals working together. We are given information about the time it takes for individuals and groups to complete a certain amount of work, and we need to find the time it takes for a single man to complete the same work.

Understanding Work Rate

Work rate is the amount of work done by a person or group in a unit of time, usually one day. If a person takes $D$ days to complete a work, their daily work rate is $\frac{1}{D}$ of the total work.

• Work Rate = $\frac{\text{Amount of Work}}{\text{Time Taken}}$
• If the total work is considered as 1 unit, then Work Rate = $\frac{1}{\text{Time Taken}}$
• Total Work = Work Rate $\times$ Time Taken

Calculating Individual and Group Work Rates

We are given the following information:

• A single woman takes 150 days to complete the work.
• 20 women and 15 men together complete the work in 6 days.

Let's define the work rates:

• Let the daily work rate of a single woman be $W$.
• Let the daily work rate of a single man be $M$.

From the first piece of information:

The daily work rate of a single woman ($W$) is $\frac{1}{150}$ of the work per day.

So, $W = \frac{1}{150}$.

Now, consider the group working together:

• The combined daily work rate of 20 women is $20 \times W = 20 \times \frac{1}{150}$.
• The combined daily work rate of 15 men is $15 \times M$.

When they work together, their total combined daily work rate is the sum of their individual combined rates:

Total combined daily work rate = $20W + 15M$

We know that this group completes the work in 6 days. This means their total combined daily work rate is $\frac{1}{6}$ of the total work per day.

So, we can set up the equation:

$20W + 15M = \frac{1}{6}$

Solving for the Man's Work Rate

Now, substitute the value of $W$ into the equation:

$20 \left(\frac{1}{150}\right) + 15M = \frac{1}{6}$

Simplify the term for women's work:

$\frac{20}{150} + 15M = \frac{1}{6}$

Reduce the fraction $\frac{20}{150}$:

$\frac{20}{150} = \frac{2}{15}$

The equation becomes:

$\frac{2}{15} + 15M = \frac{1}{6}$

To find $15M$, subtract $\frac{2}{15}$ from both sides of the equation:

$15M = \frac{1}{6} - \frac{2}{15}$

To subtract the fractions on the right side, find a common denominator for 6 and 15. The least common multiple (LCM) of 6 and 15 is 30.

• $\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}$
• $\frac{2}{15} = \frac{2 \times 2}{15 \times 2} = \frac{4}{30}$

Now perform the subtraction:

$15M = \frac{5}{30} - \frac{4}{30}$

$15M = \frac{1}{30}$

To find $M$ (the daily work rate of a single man), divide both sides by 15:

$M = \frac{1}{30 \times 15}$

$M = \frac{1}{450}$

Determining the Time for a Single Man

The daily work rate of a single man is $M = \frac{1}{450}$. This means a single man completes $\frac{1}{450}$ of the total work in one day.

The time taken for a single man to complete the entire work is the reciprocal of his daily work rate:

Time Taken for Single Man = $\frac{1}{M} = \frac{1}{1/450} = 450$ days.

Therefore, a single man can complete the work in 450 days.

Summary of Steps

1. Identify the daily work rate of a single woman based on the time given.
2. Set up an equation for the combined daily work rate of the group (20 women and 15 men) based on the total time they take.
3. Substitute the woman's work rate into the equation.
4. Solve the equation to find the daily work rate of a single man.
5. Calculate the time taken for a single man by taking the reciprocal of his daily work rate.

The calculation shows that a single man takes 450 days to complete the work.

Revision Table: Work and Time Concepts

Understanding the relationship between work, rate, and time is key to solving these types of problems.

Additional Information: Solving Work Problems

Work and time problems often involve multiple people or machines working at different rates. Here are some common strategies:

• Assume Total Work as 1: It's standard to consider the total work to be completed as '1 unit'. This simplifies calculations as rates become fractions like 1/Time.
• Calculate Daily Rates: Always convert the given time into a daily rate (or rate per hour, etc., depending on the time unit).
• Combine Rates: If people work together, add their individual rates to find their combined rate. This is only true if they work simultaneously on the same task and their efficiency doesn't change.
• Use Equations: Set up equations based on the fact that the sum of the work done by each person or group equals the total work (usually 1) when they work for a certain time.
• Solve for Unknown Rate/Time: Use algebraic methods to solve the equation for the unknown work rate or the unknown time taken.

These problems are a common application of inverse proportionality, as rate and time are inversely proportional (higher rate means less time to complete the same work).