A certain number of men can do a work in 15 days working 8 hours a days.If the number of men is decreased `(1)/(3)`, then in how many days can twice the previous work be completed by the remaining men working 5 hours per day?

Answer

To solve the problem step by step, we will use the concept of work done, which can be expressed using the formula:

\[ \text{Men} \times \text{Days} \times \text{Hours} = \text{Work} \]

### Step 1: Determine the total work done by the original number of men.
Let the number of men be \( X \). The work done in the original scenario can be calculated as follows:

- Number of days = 15
- Number of hours per day = 8

So, the total work \( W_1 \) can be expressed as:
\[ W_1 = X \times 15 \times 8 \]

### Step 2: Calculate the new number of men after the decrease.
The number of men is decreased by \( \frac{1}{3} \). Therefore, the new number of men \( M_2 \) is:
\[ M_2 = X - \frac{1}{3}X = \frac{2}{3}X \]

### Step 3: Determine the total work to be done in the new scenario.
We need to complete twice the previous work, so:
\[ W_2 = 2W_1 = 2(X \times 15 \times 8) \]

### Step 4: Set up the equation for the new scenario.
Now, we need to find out how many days \( D_2 \) it will take for the remaining men to complete this work while working 5 hours a day. The equation can be set up as follows:
\[ M_2 \times D_2 \times H_2 = W_2 \]
Substituting the known values:
\[ \left(\frac{2}{3}X\right) \times D_2 \times 5 = 2(X \times 15 \times 8) \]

### Step 5: Simplify the equation.
Now, we can simplify the equation:
\[ \frac{2}{3}X \times D_2 \times 5 = 2 \times 15 \times 8 \times X \]
Dividing both sides by \( X \) (assuming \( X \neq 0 \)):
\[ \frac{2}{3}D_2 \times 5 = 2 \times 15 \times 8 \]
Now, we can simplify further:
\[ \frac{10}{3}D_2 = 2 \times 15 \times 8 \]
Calculating the right side:
\[ \frac{10}{3}D_2 = 240 \]

### Step 6: Solve for \( D_2 \).
Now, multiply both sides by \( \frac{3}{10} \):
\[ D_2 = 240 \times \frac{3}{10} = 72 \]

### Conclusion:
The remaining men can complete twice the previous work in **72 days**.

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