A group of persons can do a work in 10 days. If 5 persons remains absent from first day, then the work is completed in 12 days by remaining persons. Tell how many persons were in the group initially?

Options

A. 1. 20
B. 2. 25
C. 3. 45
D. 4. 30

Answer

To solve the problem step-by-step, we will use the relationship between the number of workers, the number of days they work, and the total work done.

### Step 1: Define Variables
Let the initial number of persons in the group be \( X \).

### Step 2: Calculate Total Work
The total work can be expressed in terms of the number of persons and the number of days. If \( X \) persons can complete the work in 10 days, then the total work \( W \) can be calculated as:
\[ W = X \times 10 \]

### Step 3: Work Done with Absentees
If 5 persons are absent, then the number of persons working becomes \( X - 5 \). According to the problem, the remaining persons complete the work in 12 days. Therefore, the total work can also be expressed as:
\[ W = (X - 5) \times 12 \]

### Step 4: Set the Equations Equal
Since both expressions represent the same total work \( W \), we can set them equal to each other:
\[ X \times 10 = (X - 5) \times 12 \]

### Step 5: Expand and Rearrange the Equation
Expanding the right side gives:
\[ 10X = 12X - 60 \]
Now, rearranging the equation to isolate \( X \):
\[ 10X - 12X = -60 \]
\[ -2X = -60 \]
\[ 2X = 60 \]

### Step 6: Solve for \( X \)
Dividing both sides by 2 gives:
\[ X = 30 \]

### Conclusion
Thus, the initial number of persons in the group is \( \boxed{30} \).