Fifteen men can complete a piece of work in 10 days, working 8 hours per day. How many persons are required to complete double the work in 25 days, working 6 hours per day ?

Answer

To solve the problem step by step, we will use the formula related to work, which states:

\[ M_1 \times N_1 \times D_1 \times H_1 = M_2 \times N_2 \times D_2 \times H_2 \]

Where:
- \(M\) = Number of men
- \(N\) = Number of days
- \(D\) = Number of hours per day
- \(H\) = Work done

### Step 1: Determine the total work done by the initial group of men.
Given:
- 15 men can complete the work in 10 days, working 8 hours per day.

Total work \(W_1\) can be calculated as:
\[ W_1 = M_1 \times N_1 \times D_1 \times H_1 = 15 \times 10 \times 8 \]

### Step 2: Calculate the total work \(W_1\).
\[ W_1 = 15 \times 10 \times 8 = 1200 \text{ man-hours} \]

### Step 3: Since we need to complete double the work, calculate \(W_2\).
\[ W_2 = 2 \times W_1 = 2 \times 1200 = 2400 \text{ man-hours} \]

### Step 4: Set up the equation for the second scenario.
We need to find the number of men \(M_2\) required to complete \(W_2\) in 25 days working 6 hours per day.
Using the formula:
\[ M_1 \times N_1 \times D_1 \times H_1 = M_2 \times N_2 \times D_2 \times H_2 \]
Substituting the known values:
\[ 15 \times 10 \times 8 \times 2 = M_2 \times 25 \times 6 \]

### Step 5: Substitute \(W_2\) into the equation.
\[ 2400 = M_2 \times 25 \times 6 \]

### Step 6: Solve for \(M_2\).
\[ M_2 = \frac{2400}{25 \times 6} \]

### Step 7: Calculate \(M_2\).
\[ M_2 = \frac{2400}{150} = 16 \]

### Conclusion:
Thus, **16 persons** are required to complete double the work in 25 days, working 6 hours per day.

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