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 togethe

To find out how many days it will take for the man, woman, and boy to complete the work together, we can use the concept of work rates.

Let's define the work done by each individual per day:

- The man can finish the work in 10 days, so his work rate is \( \frac{1}{10} \) of the work per day.
- The boy can finish the work in 15 days, so his work rate is \( \frac{1}{15} \) of the work per day.
- The woman can finish the work in 30 days, so her work rate is \( \frac{1}{30} \) of the work per day.

When they all work together, their combined work rate is the sum of their individual work rates:

\[
\text{Combined work rate} = \frac{1}{10} + \frac{1}{15} + \frac{1}{30}
\]

To add these fractions, we first find a common denominator. The least common multiple (LCM) of 10, 15, and 30 is 30.

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

Adding these up:

\[
\text{Combined work rate} = \frac{3}{30} + \frac{2}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}
\]

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

Finally, the number of days required to complete the work when all three work together is the reciprocal of their combined work rate:

\[
\text{Number of days} = \frac{1}{\frac{1}{5}} = 5 \text{ days}
\]

So, the work can be finished in **5 days** when the man, woman, and boy work together.

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

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Here are 5 related questions you might find interesting:
1. What if only the man and boy work together? How many days will it take?
2. How would the time change if a second woman, with the same work rate, joined the group?
3. If the woman can work only half as effectively due to fatigue, how many days would it take then?
4. What if the boy worked twice as fast as usual? How many days would it take for the three to finish the work?
5. How would the scenario change if each of them took a day off during the process?

**Tip:** When solving work problems, always remember that adding work rates together gives you the combined rate, which is key to finding how long it takes for multiple workers to complete a task together.

Discover how to calculate the combined work rates of a man, woman, and boy to determine how quickly they can finish a task together. Understand the concept of work rates, fraction addition, and reciprocal calculations in this practical example. Suitable for high school students.

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