A group of men decided to do a work in 10 days, but 5 of them become absent. if the rest of the group did the work in 12 days, find the original number of men?

Options

A. 20
B. 25
C. 30
D. 35

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the original number of men be \( x \).

### Step 2: Set Up the Work Equation
According to the problem, the group of men decided to complete the work in 10 days. Therefore, the total work done can be represented as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} = x \times 10 = 10x \]

### Step 3: Adjust for Absentees
When 5 men become absent, the number of men remaining is \( x - 5 \). The remaining men complete the work in 12 days. Thus, the total work done by the remaining men can be represented as:
\[ \text{Total Work} = \text{Number of Remaining Men} \times \text{Days} = (x - 5) \times 12 \]

### Step 4: Set the Two Work Equations Equal
Since the total work remains the same, we can set the two equations equal to each other:
\[ 10x = (x - 5) \times 12 \]

### Step 5: Expand and Simplify the Equation
Expanding the right side gives:
\[ 10x = 12x - 60 \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate \( x \):
\[ 10x - 12x = -60 \]
\[ -2x = -60 \]
\[ 2x = 60 \]
\[ x = 30 \]

### Step 7: Conclusion
The original number of men is \( x = 30 \).

Thus, the answer is **30**.

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