450 man-days of work can be completed by a certain number of men in some days. If the number of people (men) are increased by 27, then the. number of day required to complete the same work is decreased- by 15. The number of days are required to complete the" three times work (than the previous/actual work) by 27 men?

Options

A. 50
B. 60
C. 54
D. 45

Answer

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

### Step 1: Define Variables
Let:
- \( N \) = initial number of men
- \( D \) = initial number of days required to complete the work

### Step 2: Set Up the Equation for Work
The total work can be expressed in terms of man-days:
\[ \text{Total Work} = N \times D = 450 \text{ man-days} \]
From this, we can express \( D \) in terms of \( N \):
\[ D = \frac{450}{N} \]

### Step 3: Set Up the New Scenario
When the number of men is increased by 27, the new number of men is \( N + 27 \) and the new number of days required is \( D - 15 \). The total work remains the same:
\[ (N + 27)(D - 15) = 450 \]

### Step 4: Substitute \( D \) in the New Equation
Substituting \( D = \frac{450}{N} \) into the new equation:
\[ (N + 27)\left(\frac{450}{N} - 15\right) = 450 \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ (N + 27)\left(\frac{450 - 15N}{N}\right) = 450 \]
Multiplying both sides by \( N \):
\[ (N + 27)(450 - 15N) = 450N \]
Expanding this gives:
\[ 450N + 12150 - 15N^2 - 405N = 450N \]
This simplifies to:
\[ 12150 - 15N^2 - 405N = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ 15N^2 + 405N - 12150 = 0 \]
Dividing the entire equation by 15:
\[ N^2 + 27N - 810 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 27, c = -810 \):
\[ N = \frac{-27 \pm \sqrt{27^2 - 4 \times 1 \times (-810)}}{2 \times 1} \]
Calculating the discriminant:
\[ 27^2 + 3240 = 729 + 3240 = 3969 \]
Taking the square root:
\[ \sqrt{3969} = 63 \]
Now substituting back:
\[ N = \frac{-27 \pm 63}{2} \]
Calculating the two possible values for \( N \):
1. \( N = \frac{36}{2} = 18 \)
2. \( N = \frac{-90}{2} = -45 \) (not valid since number of men cannot be negative)

Thus, \( N = 18 \).

### Step 8: Find Initial Days \( D \)
Now substituting \( N \) back to find \( D \):
\[ D = \frac{450}{N} = \frac{450}{18} = 25 \text{ days} \]

### Step 9: Calculate Days for Three Times the Work
The total work for three times the original work is:
\[ 3 \times 450 = 1350 \text{ man-days} \]
Now, we need to find the number of days required by 27 men to complete this work:
\[ \text{Days} = \frac{1350}{27} = 50 \text{ days} \]

### Final Answer
The number of days required to complete three times the work by 27 men is **50 days**.

---