10 men working 6 hours a day can complete a work in 18 days. How many hours a day must 15 men work to complete the same work in 12 days.

Answer

To solve the problem, we will first calculate the total amount of work done in terms of man-hours, and then determine how many hours per day the 15 men need to work to complete the same work in 12 days.

### Step 1: Calculate the total work done by 10 men in 18 days.
- The formula for total work in man-hours is:
\[ \text{Total Work} = \text{Number of Men} \times \text{Hours per Day} \times \text{Number of Days} \]
- Here, we have:
- Number of Men = 10
- Hours per Day = 6
- Number of Days = 18

Now, substituting the values:
\[ \text{Total Work} = 10 \times 6 \times 18 \]
\[ \text{Total Work} = 1080 \text{ man-hours} \]

### Step 2: Set up the equation for 15 men working for a certain number of hours per day to complete the same work in 12 days.
- Let \( x \) be the number of hours per day that 15 men need to work.
- The total work done by 15 men in 12 days can be expressed as:
\[ \text{Total Work} = 15 \times x \times 12 \]

### Step 3: Set the two expressions for total work equal to each other.
Since both expressions represent the same total work, we can set them equal:
\[ 15 \times x \times 12 = 1080 \]

### Step 4: Solve for \( x \).
- First, simplify the left side:
\[ 180x = 1080 \]
- Now, divide both sides by 180 to isolate \( x \):
\[ x = \frac{1080}{180} \]
\[ x = 6 \]

### Conclusion:
Thus, 15 men must work **6 hours a day** to complete the same work in 12 days.

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