12 men or 15 women or 30 boys can do a work in 45 days. Then in how many days will 12 men, 15 women and 30 boys do the work?

Answer

To solve the problem, we need to determine how many days 12 men, 15 women, and 30 boys will take to complete the work together.

### Step-by-step Solution:

1. **Calculate the Work Done by Each Group Individually:**
- We know that:
- 12 men can complete the work in 45 days.
- 15 women can complete the same work in 45 days.
- 30 boys can also complete the work in 45 days.

2. **Determine the Work Rate of Each Group:**
- The total work can be represented in terms of "man-days."
- For 12 men working for 45 days:
\[ \text{Total Work} = 12 \text{ men} \times 45 \text{ days} = 540 \text{ man-days} \]
- For 15 women working for 45 days:
\[ \text{Total Work} = 15 \text{ women} \times 45 \text{ days} = 675 \text{ woman-days} \]
- For 30 boys working for 45 days:
\[ \text{Total Work} = 30 \text{ boys} \times 45 \text{ days} = 1350 \text{ boy-days} \]

3. **Convert Work Rates to a Common Unit:**
- To find a common work rate, we can express the work done by each group in terms of a single unit (let's use "man-days"):
- Let the work done by 1 man in 1 day = 1 unit of work.
- Assume the efficiency of women and boys relative to men:
- If 12 men = 15 women, then 1 man = 1.25 women.
- If 12 men = 30 boys, then 1 man = 2.5 boys.
- Thus, we can express:
- 1 woman = 0.8 men (since 1.25 women = 1 man).
- 1 boy = 0.4 men (since 2.5 boys = 1 man).

4. **Calculate the Total Work Rate of 12 Men, 15 Women, and 30 Boys:**
- The work rate of 12 men:
\[ 12 \text{ men} = 12 \text{ units of work per day} \]
- The work rate of 15 women:
\[ 15 \text{ women} = 15 \times 0.8 = 12 \text{ units of work per day} \]
- The work rate of 30 boys:
\[ 30 \text{ boys} = 30 \times 0.4 = 12 \text{ units of work per day} \]

5. **Combine the Work Rates:**
- Total work rate of 12 men, 15 women, and 30 boys:
\[ \text{Total Work Rate} = 12 + 12 + 12 = 36 \text{ units of work per day} \]

6. **Calculate the Total Days to Complete the Work Together:**
- Since the total work is 540 man-days, we can find the number of days required:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Total Work Rate}} = \frac{540 \text{ man-days}}{36 \text{ units of work per day}} = 15 \text{ days} \]

### Final Answer:
12 men, 15 women, and 30 boys will complete the work in **15 days**.