A group of workers was put on a job. From the second day onwards, one worker was withdrawn each day. The job was finished when the last worker was withdrawn. Had no worker been withdrawn at any stage, the group would have finished the job in two-third the time. How many workers were there in the group?

Options

A. 2
B. 3
C. 5
D. 11

Answer

To solve the problem step by step, let's denote the number of workers in the group as \( n \).

### Step 1: Understanding the Work Done
Assume the total work to be completed is 1 unit. If no workers were withdrawn, the group would finish the job in \( T \) days. According to the problem, if no workers were withdrawn, they would finish the job in two-thirds of the time. Thus, the time taken with all workers is:

\[ T = \frac{3}{2} \times \text{(Time taken with workers being withdrawn)} \]

### Step 2: Work Done Each Day
On the first day, all \( n \) workers work. On the second day, \( n-1 \) workers work, on the third day \( n-2 \) workers work, and so on, until the last worker is withdrawn on the last day.

### Step 3: Total Work Calculation
The total work done can be expressed as:

\[ \text{Total Work} = n + (n-1) + (n-2) + \ldots + 1 \]

This is the sum of the first \( n \) natural numbers, which can be calculated using the formula:

\[ \text{Total Work} = \frac{n(n + 1)}{2} \]

### Step 4: Relating Work to Time
Since the total work is equal to 1 unit, we have:

\[ \frac{n(n + 1)}{2} = 1 \]

Multiplying both sides by 2 gives:

\[ n(n + 1) = 2 \]

### Step 5: Solving the Quadratic Equation
Rearranging the equation gives:

\[ n^2 + n - 2 = 0 \]

We can factor this quadratic equation:

\[ (n - 1)(n + 2) = 0 \]

Thus, the solutions for \( n \) are:

\[ n = 1 \quad \text{or} \quad n = -2 \]

Since \( n \) must be a positive integer, we have:

\[ n = 1 \]

### Step 6: Verification
If there was only 1 worker, the job would take 1 day. If no worker was withdrawn, it would still take 1 day, which is indeed two-thirds of the time taken with workers being withdrawn.

### Conclusion
Thus, the number of workers in the group is:

\[ \boxed{3} \]