Understanding the Work Rates

This problem involves calculating the combined time taken by individuals with different work efficiencies. The core idea is to find the individual rate of work for each person (man, woman, boy) and then sum them up to find their combined rate. The total time taken is the inverse of this combined rate.

• The question states that 3 men can complete a work in 1 day. This means the rate of work for 1 man is $\frac{1}{3}$ of the work per day.
• Similarly, 4 women can complete the same work in 1 day. So, the rate of work for 1 woman is $\frac{1}{4}$ of the work per day.
• And, 12 boys can complete the work in 1 day. Therefore, the rate of work for 1 boy is $\frac{1}{12}$ of the work per day.

Calculating Combined Work Rate

We need to find the time taken when 1 man, 1 woman, and 5 boys work together. First, let's calculate their combined rate of work per day:

Rate of 1 man = $\frac{1}{3}$ per day

Rate of 1 woman = $\frac{1}{4}$ per day

Rate of 5 boys = $5 \times (\text{Rate of 1 boy}) = 5 \times \frac{1}{12} = \frac{5}{12}$ per day

The combined rate is the sum of their individual rates:

Combined Rate = (Rate of 1 man) + (Rate of 1 woman) + (Rate of 5 boys)

Combined Rate = $\frac{1}{3} + \frac{1}{4} + \frac{5}{12}$

To add these fractions, we find a common denominator, which is 12:

Combined Rate = $\frac{1 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} + \frac{5}{12}$

Combined Rate = $\frac{4}{12} + \frac{3}{12} + \frac{5}{12}$

Combined Rate = $\frac{4 + 3 + 5}{12} = \frac{12}{12}$

Combined Rate = $1$ per day

Determining Total Time Taken

The time taken to complete the work is the inverse of the combined rate. If the combined rate is 1 unit of work per day, it means the entire work is completed in 1 day.

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

Assuming the total work is 1 unit:

Time = $\frac{1}{1} = 1$ day

Therefore, 1 man, 1 woman, and 5 boys working together can complete the work in 1 day.