9 children can complete a plece of work in 360 days. 18 men can complete the same plece of work in 72 days and 12 women can complete the piece of work in 162 days. In how many days can 4 men, 12 women and 10 children together complete the piece of work?

Options

A. 124
B. 81
C. 68
D. 96

Answer

To solve the problem step by step, we will first determine the work done by each group (children, men, and women) in a day and then find out how long it will take for 4 men, 12 women, and 10 children to complete the work together.

### Step 1: Calculate the work done by 9 children in one day
9 children can complete the work in 360 days. Therefore, the work done by 9 children in one day is:
\[ \text{Work done by 9 children in one day} = \frac{1}{360} \text{ of the work} \]
So, the work done by 1 child in one day is:
\[ \text{Work done by 1 child in one day} = \frac{1}{360} \div 9 = \frac{1}{3240} \text{ of the work} \]

**Hint**: To find the work done by one child, divide the total work done by the group by the number of children.

### Step 2: Calculate the work done by 18 men in one day
18 men can complete the work in 72 days. Therefore, the work done by 18 men in one day is:
\[ \text{Work done by 18 men in one day} = \frac{1}{72} \text{ of the work} \]
So, the work done by 1 man in one day is:
\[ \text{Work done by 1 man in one day} = \frac{1}{72} \div 18 = \frac{1}{1296} \text{ of the work} \]

**Hint**: Similar to the previous step, divide the total work done by the group by the number of men to find the work done by one man.

### Step 3: Calculate the work done by 12 women in one day
12 women can complete the work in 162 days. Therefore, the work done by 12 women in one day is:
\[ \text{Work done by 12 women in one day} = \frac{1}{162} \text{ of the work} \]
So, the work done by 1 woman in one day is:
\[ \text{Work done by 1 woman in one day} = \frac{1}{162} \div 12 = \frac{1}{1944} \text{ of the work} \]

**Hint**: Again, divide the total work done by the group by the number of women to find the work done by one woman.

### Step 4: Calculate the total work done by 4 men, 12 women, and 10 children in one day
Now, we can calculate the total work done by 4 men, 12 women, and 10 children in one day:
\[ \text{Total work done in one day} = 4 \times \text{(work done by 1 man)} + 12 \times \text{(work done by 1 woman)} + 10 \times \text{(work done by 1 child)} \]
Substituting the values:
\[ = 4 \times \frac{1}{1296} + 12 \times \frac{1}{1944} + 10 \times \frac{1}{3240} \]

### Step 5: Calculate each term
Calculating each term:
1. Work done by 4 men:
\[ 4 \times \frac{1}{1296} = \frac{4}{1296} = \frac{1}{324} \]
2. Work done by 12 women:
\[ 12 \times \frac{1}{1944} = \frac{12}{1944} = \frac{1}{162} \]
3. Work done by 10 children:
\[ 10 \times \frac{1}{3240} = \frac{10}{3240} = \frac{1}{324} \]

### Step 6: Combine the work done
Now, we add these fractions:
\[ \text{Total work done in one day} = \frac{1}{324} + \frac{1}{162} + \frac{1}{324} \]
Finding a common denominator (which is 648):
\[ = \frac{2}{648} + \frac{4}{648} + \frac{2}{648} = \frac{8}{648} = \frac{1}{81} \]

### Step 7: Calculate the total days to complete the work
Since the total work done in one day is \(\frac{1}{81}\), the total number of days \(T\) to complete the work is:
\[ T = \frac{1}{\text{Total work done in one day}} = 81 \text{ days} \]

### Final Answer
Thus, 4 men, 12 women, and 10 children together can complete the piece of work in **81 days**.

---