The work done by a woman in 8 hours is equal to the work done by a man in 6 hours and by a boy in 12 hours. If working 6 hours per day 9 men can complete a work in 6 days, then in how many days can 12 men, 12 women and 12 boys together finish the same work working 8 hours per day?

Options

A. `1(1)/(3)` days
B. `3(2)/(3)` days
C. 3 days
D. `1(1)/(2)` days

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the work done by each individual (man, woman, boy)
Given:
- Work done by a woman in 8 hours = Work done by a man in 6 hours = Work done by a boy in 12 hours.

Let’s denote the work done by a woman in 1 hour as W, by a man as M, and by a boy as B.

From the information given:
- 1 woman does W work in 8 hours, so her rate is \( \frac{W}{8} \) per hour.
- 1 man does W work in 6 hours, so his rate is \( \frac{W}{6} \) per hour.
- 1 boy does W work in 12 hours, so his rate is \( \frac{W}{12} \) per hour.

To find the ratio of their efficiencies, we can take the LCM of the denominators (8, 6, and 12), which is 24.

- Woman's efficiency: \( \frac{24}{8} = 3 \)
- Man's efficiency: \( \frac{24}{6} = 4 \)
- Boy's efficiency: \( \frac{24}{12} = 2 \)

Thus, the ratio of efficiencies is:
- Woman : Man : Boy = 3 : 4 : 2

### Step 2: Calculate the total work done by 9 men in 6 days
Given that 9 men can complete the work in 6 days working 6 hours per day.

The efficiency of 1 man is 4, so the total efficiency of 9 men is:
\[ 9 \times 4 = 36 \text{ units of work per hour} \]

The total work done in 6 days working 6 hours per day is:
\[ \text{Total Work} = \text{Efficiency} \times \text{Hours per day} \times \text{Days} \]
\[ \text{Total Work} = 36 \times 6 \times 6 = 1296 \text{ units of work} \]

### Step 3: Calculate the total efficiency of 12 men, 12 women, and 12 boys
Now we need to find the total efficiency of 12 men, 12 women, and 12 boys.

- Efficiency of 12 men = \( 12 \times 4 = 48 \)
- Efficiency of 12 women = \( 12 \times 3 = 36 \)
- Efficiency of 12 boys = \( 12 \times 2 = 24 \)

Total efficiency of 12 men, 12 women, and 12 boys:
\[ \text{Total Efficiency} = 48 + 36 + 24 = 108 \text{ units of work per hour} \]

### Step 4: Calculate the number of days to complete the work
Now, we need to find out how many days it will take for 12 men, 12 women, and 12 boys to complete the same work of 1296 units, working 8 hours per day.

Let \( D \) be the number of days required.
The total work done per day by 12 men, 12 women, and 12 boys is:
\[ \text{Work per day} = \text{Total Efficiency} \times \text{Hours per day} = 108 \times 8 = 864 \text{ units of work per day} \]

Now, we can set up the equation:
\[ \text{Total Work} = \text{Work per day} \times D \]
\[ 1296 = 864 \times D \]

Solving for \( D \):
\[ D = \frac{1296}{864} = \frac{3}{2} \text{ days} \]

### Final Answer
Thus, the number of days required for 12 men, 12 women, and 12 boys to finish the work is \( 1.5 \) days or \( 1 \frac{1}{2} \) days.

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