8 children and 12 men complete a certain piece of work in 9 days. Each child takes twice the time taken by a man to finish the work. In how many days will 12 men finish the same work?

Options

A. 9 days
B. 18 days
C. 12 days
D. 15 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Work Done by Children and Men
We know that 8 children and 12 men complete a piece of work in 9 days. We also know that each child takes twice the time taken by a man to finish the work.

### Step 2: Establish the Work Rate of Men and Children
Let the work done by one man in one day be \( M \) and the work done by one child in one day be \( C \). Since each child takes twice the time of a man, we can express their work rates as:
\[ C = \frac{M}{2} \]

### Step 3: Calculate the Total Work Done
The total work done by 8 children and 12 men in one day can be expressed as:
\[ \text{Total work in one day} = 8C + 12M \]
Substituting \( C \) in terms of \( M \):
\[ = 8 \left(\frac{M}{2}\right) + 12M = 4M + 12M = 16M \]

### Step 4: Calculate Total Work in 9 Days
Since they complete the work in 9 days, the total work \( W \) can be calculated as:
\[ W = \text{Total work in one day} \times \text{Number of days} = 16M \times 9 = 144M \]

### Step 5: Calculate Work Done by 12 Men
Now, we need to find out how many days it will take for 12 men to complete the same work. The work done by 12 men in one day is:
\[ \text{Work done by 12 men in one day} = 12M \]

### Step 6: Calculate the Number of Days for 12 Men to Complete the Work
Let \( D \) be the number of days required for 12 men to finish the work. The total work can also be expressed as:
\[ W = \text{Work done by 12 men in one day} \times D = 12M \times D \]
Setting this equal to the total work calculated earlier:
\[ 12M \times D = 144M \]
Dividing both sides by \( 12M \):
\[ D = \frac{144M}{12M} = 12 \]

### Conclusion
Thus, 12 men will finish the same work in **12 days**.

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