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 find how long it takes for a man, a boy, and a woman to finish the work together, we need to calculate their combined work rate and then determine the total time based on that rate.

### Step-by-step solution:

1. **Work Rate of Each Individual:**
   - A **man** can finish the work in 10 days. So, his work rate is \( \frac{1}{10} \) of the work per day.
   - A **boy** can finish the work in 15 days. So, his work rate is \( \frac{1}{15} \) of the work per day.
   - A **woman** can finish the work in 30 days. So, her work rate is \( \frac{1}{30} \) of the work per day.

2. **Combined Work Rate:**
   The combined work rate when all of them work together is the sum of their individual work rates:
   \[
   \text{Combined rate} = \frac{1}{10} + \frac{1}{15} + \frac{1}{30}
   \]

   Let's find a common denominator:
   \[
   \text{LCM of } 10, 15, \text{ and } 30 = 30
   \]
   Now, express each fraction with the common denominator of 30:
   \[
   \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{30} = \frac{1}{30}
   \]
   Adding them together:
   \[
   \frac{3}{30} + \frac{2}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}
   \]

3. **Total Time to Finish the Work:**
   The combined rate is \( \frac{1}{5} \) of the work per day. Therefore, the time it takes for all three of them to finish the work together is:
   \[
   \text{Time} = \frac{1}{\text{Work rate}} = \frac{1}{\frac{1}{5}} = 5 \text{ days}
   \]

Thus, all three of them working together will finish the work in **5 days**.

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Would you like any further details on this, or do you have any questions?

Here are 5 related questions for further practice:
1. If a man and a boy work together, how many days would it take to finish the work?
2. What would be the combined work rate if a woman worked with two boys instead of one?
3. How long would it take for two men and a boy to finish the same work?
4. If the man worked alone for 3 days and then was joined by the boy, how many more days would it take to finish the work?
5. If a woman could work faster and finish the work in 20 days, how long would it take for all three to complete the work together?

**Tip:** When solving combined work problems, remember to always sum the work rates of each individual to find the overall work rate.

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?

Explore how to solve the combined work problem where a man, a boy, and a woman work together. Each has different rates of finishing the task individually, and this solution explains how to calculate the time for completing the work when all three work simultaneously. Suitable for students in Grades 6-8.

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