A group of laboures promises to do a plece of work in 10 days, but five of them become absent. If the remaining labourers complete the work in 12 days. Find their original number in the group.

Answer

To solve the problem step by step, we will use the relationship between work, number of laborers, and time.

### Step 1: Define the variables
Let \( x \) be the original number of laborers in the group.

### Step 2: Set up the equation for the promised work
The group of laborers promises to complete the work in 10 days. Therefore, the total work can be expressed as:
\[ \text{Work} = \text{Number of laborers} \times \text{Time} = x \times 10 \]
So, the total work is \( 10x \).

### Step 3: Set up the equation for the actual work done
When 5 laborers are absent, the number of laborers becomes \( x - 5 \). They complete the work in 12 days, so the work can also be expressed as:
\[ \text{Work} = \text{Number of laborers} \times \text{Time} = (x - 5) \times 12 \]
So, the total work is \( 12(x - 5) \).

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

### Step 5: Simplify the equation
Distributing the right side:
\[ 10x = 12x - 60 \]

### Step 6: Rearrange the equation
Now, we will move all terms involving \( x \) to one side and constants to the other:
\[ 10x - 12x = -60 \]
This simplifies to:
\[ -2x = -60 \]

### Step 7: Solve for \( x \)
Dividing both sides by -2 gives:
\[ x = 30 \]

### Conclusion
The original number of laborers in the group is \( 30 \).