A certain number of persons can complete a piece of work in 55 days. If there were 6 persons more, the work could be finished in 11 days less. How many persons were originally there?

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

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

### Step 2: Calculate the total work done
The total work can be calculated as the product of the number of persons and the number of days they work. If \( x \) persons can complete the work in 55 days, then the total work \( W \) is:
\[ W = 55 \times x \]

### Step 3: Set up the equation for the new scenario
If there are 6 more persons, the total number of persons becomes \( x + 6 \). The work can now be completed in 11 days less, which means it takes \( 55 - 11 = 44 \) days. Therefore, the total work can also be expressed as:
\[ W = 44 \times (x + 6) \]

### 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:
\[ 55x = 44(x + 6) \]

### Step 5: Expand and simplify the equation
Expanding the right side:
\[ 55x = 44x + 264 \]
Now, subtract \( 44x \) from both sides:
\[ 55x - 44x = 264 \]
This simplifies to:
\[ 11x = 264 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 11 to find \( x \):
\[ x = \frac{264}{11} = 24 \]

### Step 7: Conclusion
The original number of persons was \( 24 \).