Solving Work and Time Problems

This problem involves understanding the concept of work rate and how it changes when the number of workers changes or when different types of workers (men and boys) work together. The key is to find the amount of work each individual (man or boy) can do in one day, and then combine their efforts.

Calculating Individual Work Rates

First, let's determine the work rate of one man and one boy.

• We are told that 6 men can complete the job in 2 days.
• This means the total work done by 6 men in 1 day is $1/2$ of the job.
• So, the work done by 1 man in 1 day is $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$ of the job.

Similarly, for the boys:

• We are told that 4 boys can complete the same job in 8 days.
• This means the total work done by 4 boys in 1 day is $1/8$ of the job.
• So, the work done by 1 boy in 1 day is $\frac{1}{4} \times \frac{1}{8} = \frac{1}{32}$ of the job.

We can summarize the individual daily work rates:

Calculating Combined Work Rate

Now, we need to find out how much work 3 men and 6 boys can do together in one day. We add their individual work rates.

• Work done by 3 men in 1 day = $3 \times (\text{Work done by 1 man in 1 day}) = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$ of the job.
• Work done by 6 boys in 1 day = $6 \times (\text{Work done by 1 boy in 1 day}) = 6 \times \frac{1}{32} = \frac{6}{32} = \frac{3}{16}$ of the job.

The total work done by 3 men and 6 boys together in 1 day is the sum of their individual contributions:

Combined work rate = (Work done by 3 men in 1 day) + (Work done by 6 boys in 1 day)

Combined work rate = $\frac{1}{4} + \frac{3}{16}$

To add these fractions, we find a common denominator, which is 16.

Combined work rate = $\frac{1 \times 4}{4 \times 4} + \frac{3}{16} = \frac{4}{16} + \frac{3}{16} = \frac{4+3}{16} = \frac{7}{16}$ of the job per day.

Finding the Time Taken

If 3 men and 6 boys complete $\frac{7}{16}$ of the job in one day, the total number of days required to complete the entire job (which is 1 whole job) is given by:

Time taken = $\frac{\text{Total Work}}{\text{Combined Work Rate}}$

Time taken = $\frac{1}{\frac{7}{16}}$ days

To divide by a fraction, we multiply by its reciprocal:

Time taken = $1 \times \frac{16}{7} = \frac{16}{7}$ days.

Conclusion

Working together, three men and six boys would be able to complete the job in $\frac{16}{7}$ days.