Working together A and B complete a task in (7)1/2 days B alone takes 8 days less than A alone to complete the task if C alone takes 14 days to complete the same work what is difference between number of days taken by B alone and C alone to complete the task

Options

A. 5
B. 1
C. 4
D. 2

Answer

To solve the problem, we need to find the difference between the number of days taken by B alone and C alone to complete the task. Let's break it down step by step.

### Step 1: Understand the given information
- A and B together complete a task in \(7 \frac{1}{2}\) days.
- B alone takes 8 days less than A alone to complete the task.
- C alone takes 14 days to complete the same work.

### Step 2: Convert mixed fraction to improper fraction
Convert \(7 \frac{1}{2}\) days into an improper fraction:
\[ 7 \frac{1}{2} = \frac{15}{2} \text{ days} \]

### Step 3: Define variables for A and B
Let \(X\) be the number of days A takes to complete the task alone. Then, B takes:
\[ X - 8 \text{ days} \]

### Step 4: Calculate the work done in one day
The work done by A in one day is:
\[ \frac{1}{X} \text{ (work per day)} \]
The work done by B in one day is:
\[ \frac{1}{X - 8} \text{ (work per day)} \]
The combined work done by A and B in one day is:
\[ \frac{1}{X} + \frac{1}{X - 8} \]

### Step 5: Set up the equation for combined work
Since A and B together complete the work in \( \frac{15}{2} \) days, their combined work per day is:
\[ \frac{1}{\frac{15}{2}} = \frac{2}{15} \]
Thus, we can set up the equation:
\[ \frac{1}{X} + \frac{1}{X - 8} = \frac{2}{15} \]

### Step 6: Solve the equation
To solve the equation, find a common denominator:
\[ \frac{(X - 8) + X}{X(X - 8)} = \frac{2}{15} \]
This simplifies to:
\[ \frac{2X - 8}{X(X - 8)} = \frac{2}{15} \]
Cross-multiplying gives:
\[ 15(2X - 8) = 2X^2 - 16X \]
Expanding and rearranging:
\[ 30X - 120 = 2X^2 - 16X \]
\[ 2X^2 - 46X + 120 = 0 \]
Dividing the entire equation by 2:
\[ X^2 - 23X + 60 = 0 \]

### Step 7: Factor the quadratic equation
To factor:
\[ (X - 20)(X - 3) = 0 \]
Thus, \(X = 20\) or \(X = 3\). Since A cannot complete the work in 3 days (as they work together for longer), we take:
\[ X = 20 \text{ days for A} \]

### Step 8: Calculate days taken by B
Now, substituting back to find B's time:
\[ B = X - 8 = 20 - 8 = 12 \text{ days} \]

### Step 9: Days taken by C
We know C takes:
\[ C = 14 \text{ days} \]

### Step 10: Find the difference between B and C
The difference in days taken by B and C is:
\[ |C - B| = |14 - 12| = 2 \text{ days} \]

### Final Answer
The difference between the number of days taken by B alone and C alone to complete the task is:
\[ \boxed{2} \]