33 men can do a job in 30 days. If 44 men started the job together and after every day of the work, one person leaves. What is the minimum number of days required to complete the whole work?

Options

A. 21
B. 42
C. 45
D. none of these

Answer

To solve the problem step by step, we need to determine how many days it will take for 44 men to complete a job if one man leaves at the end of each day.

### Step 1: Calculate the total work
We know that 33 men can complete the work in 30 days. Therefore, the total work can be calculated as follows:

\[ \text{Total Work} = \text{Number of Men} \times \text{Number of Days} = 33 \times 30 = 990 \text{ man-days} \]

### Step 2: Determine the daily work done by 44 men
On the first day, 44 men will work together. Thus, the work done on the first day is:

\[ \text{Work done on Day 1} = 44 \text{ units} \]

### Step 3: Calculate the work done on subsequent days
Since one man leaves at the end of each day, the number of men working each day decreases by 1. Therefore, the work done on each subsequent day will be as follows:

- Day 1: 44 men → 44 units
- Day 2: 43 men → 43 units
- Day 3: 42 men → 42 units
- ...
- Day N: \( (44 - (N - 1)) \) men → \( (44 - (N - 1)) \) units

### Step 4: Set up the equation for total work
The total work done over N days can be expressed as the sum of an arithmetic series:

\[ \text{Total Work} = 44 + 43 + 42 + \ldots + (44 - (N - 1)) \]

This can be simplified using the formula for the sum of an arithmetic series:

\[ \text{Sum} = \frac{N}{2} \times (\text{First Term} + \text{Last Term}) \]

Where:
- First Term = 44
- Last Term = \( 44 - (N - 1) = 45 - N \)

Thus, the equation becomes:

\[ \text{Total Work} = \frac{N}{2} \times (44 + (45 - N)) = \frac{N}{2} \times (89 - N) \]

### Step 5: Set the equation equal to total work and solve for N
We know the total work is 990 units:

\[ \frac{N}{2} \times (89 - N) = 990 \]

Multiplying both sides by 2 to eliminate the fraction:

\[ N(89 - N) = 1980 \]

Rearranging gives us a quadratic equation:

\[ N^2 - 89N + 1980 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = -89, c = 1980 \):

\[ N = \frac{89 \pm \sqrt{(-89)^2 - 4 \times 1 \times 1980}}{2 \times 1} \]

Calculating the discriminant:

\[ (-89)^2 = 7921 \]
\[ 4 \times 1 \times 1980 = 7920 \]
\[ \sqrt{7921 - 7920} = \sqrt{1} = 1 \]

Thus, substituting back into the formula:

\[ N = \frac{89 \pm 1}{2} \]

Calculating the two possible values for N:

1. \( N = \frac{90}{2} = 45 \)
2. \( N = \frac{88}{2} = 44 \)

Since we need the minimum number of days, we take \( N = 45 \).

### Conclusion
The minimum number of days required to complete the whole work is **45 days**.