A man, a boy and a woman can finish a work in 10 days, 15 days and 30 days, respectively. In how many days can the work be finished by a man, a woman and a boy when all of them work together?

To determine how long it takes for a man, a woman, and a boy to complete the work together, we first need to find their individual work rates.

– The man’s work rate is ( frac{1}{10} ) of the work per day.

– The boy’s work rate is ( frac{1}{15} ) of the work per day.

– The woman’s work rate is ( frac{1}{30} ) of the work per day.

Now, we add their work rates together:

[

text{Total work rate} = frac{1}{10} + frac{1}{15} + frac{1}{30}

]

To add these fractions, we need a common denominator, which in this case is 30.

[

frac{1}{10} = frac{3}{30}, quad frac{1}{15} = frac{2}{30}, quad frac{1}{30} = frac{1}{30}

]

Now, we can add them up:

[

text{Total work rate} = frac{3}{30} + frac{2}{30} + frac{1}{30} = frac{6}{30} = frac{1}{5}

]

This means that together, they complete ( frac{1}{5} ) of the work in one day. Therefore,