Let's denote the work rates of the man, woman, and boy as follows:

• Let $M$ be the work rate of the man.
• Let $W$ be the work rate of the woman.
• Let $B$ be the work rate of the boy.

Given:

• The man can complete the job in 3 days, so his work rate $M$ is: $M = \frac{1}{3}$
• The woman can complete the job in 4 days, so her work rate $W$ is: $W = \frac{1}{4}$
• The boy can complete the job in 12 days, so his work rate $B$ is: $B = \frac{1}{12}$

We need to find how many boys, $n$, must assist one man and one woman to complete the job in $\frac{1}{4}$ of a day. The combined work rate of one man, one woman, and $n$ boys must be equal to the work rate required to complete the job in $\frac{1}{4}$ of a day.

The combined work rate of one man, one woman, and $n$ boys is: $M + W + nB$

The work rate required to complete the job in $\frac{1}{4}$ of a day is: $\frac{1}{\frac{1}{4}} = 4$

Therefore, we have the equation: $M + W + nB = 4$

Substitute the values of $M$, $W$, and $B$ into the equation: $\frac{1}{3} + \frac{1}{4} + n \cdot \frac{1}{12} = 4$

First, find a common denominator for the fractions: $\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{12} = \frac{1}{12}$

So the equation becomes: $\frac{4}{12} + \frac{3}{12} + \frac{n}{12} = 4$

Combine the fractions: $\frac{4 + 3 + n}{12} = 4$

Simplify the equation: $\frac{7 + n}{12} = 4$

Multiply both sides by 12 to solve for $n$: $7 + n = 48$

Subtract 7 from both sides: $n = 41$

Therefore, 41 boys must assist one man and one woman to complete the job in $\frac{1}{4}$ of a day.

Would you like any details or have any questions about this solution?

Here are some related questions for further exploration:

1. How long would it take for 2 men and 3 women to complete the same job?
2. If a boy works half as fast as a man, how many boys are needed to complete the job in 2 days?
3. What is the combined work rate of 2 men and 2 boys?
4. If only boys are available, how many are needed to complete the job in 1 day?
5. How much of the job can a woman complete in 6 hours?
6. What fraction of the job is done by one man in half a day?
7. If a man and a woman work together, how long will it take them to complete half of the job?
8. How many days will it take for one boy and one man working together to complete the job?

Tip: When solving work rate problems, converting individual work rates to a common unit and summing them up can simplify finding the combined work rate.