Let's break down the problem step by step to find the solution.

Step 1: Define the variables

Let:

• $B$ = work done by one boy in one day
• $M$ = work done by one man in one day

Step 2: Set up the equations

From the first part of the problem:

• Three boys and two men can complete the work in 10 days.
• The total work done by three boys and two men in one day is $\frac{1}{10}$.

So, the equation for their daily work is: $3B + 2M = \frac{1}{10}$

From the second part of the problem:

• Two boys and three men can complete the work in 8 days.
• The total work done by two boys and three men in one day is $\frac{1}{8}$.

So, the equation for their daily work is: $2B + 3M = \frac{1}{8}$

Step 3: Solve the system of equations

We have the following system of equations: $3B + 2M = \frac{1}{10}$ $2B + 3M = \frac{1}{8}$

Let's solve this system using the substitution or elimination method.

First, multiply the first equation by 3 and the second equation by 2 to eliminate one of the variables: $9B + 6M = \frac{3}{10}$ $4B + 6M = \frac{2}{8} = \frac{1}{4}$

Now subtract the second equation from the first: $(9B + 6M) - (4B + 6M) = \frac{3}{10} - \frac{1}{4}$ $5B = \frac{3}{10} - \frac{1}{4}$

Now, find a common denominator and solve for $B$: $5B = \frac{12 - 10}{40} = \frac{2}{40} = \frac{1}{20}$ $B = \frac{1}{100}$

So, one boy can do $\frac{1}{100}$ of the work in one day.

Step 4: Substitute $B$ back to find $M$

Using $3B + 2M = \frac{1}{10}$ and substituting $B = \frac{1}{100}$: $3 \times \frac{1}{100} + 2M = \frac{1}{10}$ $\frac{3}{100} + 2M = \frac{1}{10}$ $2M = \frac{1}{10} - \frac{3}{100} = \frac{10 - 3}{100} = \frac{7}{100}$ $M = \frac{7}{200}$

So, one man can do $\frac{7}{200}$ of the work in one day.

Step 5: Calculate the time for one boy and two men to complete the work

Now, let's calculate the work done by one boy and two men in one day: $B + 2M = \frac{1}{100} + 2 \times \frac{7}{200} = \frac{1}{100} + \frac{14}{200} = \frac{1}{100} + \frac{7}{100} = \frac{8}{100} = \frac{2}{25}$

So, the time taken by one boy and two men to complete the work is the reciprocal of their combined work rate: $\text{Time} = \frac{1}{\frac{2}{25}} = \frac{25}{2} = 12.5 \text{ days}$

Final Answer:

The work can be done by one boy and two men jointly in 12.5 days.

Do you want any further details or have any questions? Here are 5 related questions to consider:

1. How would the time change if one more man joined the group?
2. What if one of the boys was absent? How would that affect the time taken?
3. Can you verify the results using another method, like algebraic substitution?
4. What if the work rate of the men and boys was given differently? How would the solution adapt?
5. How would the solution change if they were working different hours each day?

Tip: Always check your final results by substituting the values back into the original equations to ensure accuracy.