A & B together can complete a piece of work in 9 days. Time taken by A alone to complete the same work is 7.5 days less than time taken by B alone to complete the same work. In how many days B alone will complete `(2)/(9)` of the work?

Options

A. 8 days
B. 6 days
C. 7 days
D. 5 days

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define Variables
Let the time taken by A alone to complete the work be \( x \) days. Then, the time taken by B alone to complete the work will be \( x + 7.5 \) days.

### Step 2: Write the Work Rates
The work done by A in one day is \( \frac{1}{x} \) and the work done by B in one day is \( \frac{1}{x + 7.5} \).

### Step 3: Write the Combined Work Rate
Together, A and B can complete the work in 9 days, so their combined work rate is:
\[ \frac{1}{x} + \frac{1}{x + 7.5} = \frac{1}{9} \]

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply through by \( 9x(x + 7.5) \):
\[ 9(x + 7.5) + 9x = x(x + 7.5) \]

### Step 5: Simplify the Equation
Expanding both sides:
\[ 9x + 67.5 + 9x = x^2 + 7.5x \]
This simplifies to:
\[ 18x + 67.5 = x^2 + 7.5x \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ x^2 - 10.5x - 67.5 = 0 \]

### Step 7: Solve the Quadratic Equation
To solve the quadratic equation \( x^2 - 10.5x - 67.5 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -10.5, c = -67.5 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-10.5)^2 - 4 \cdot 1 \cdot (-67.5) = 110.25 + 270 = 380.25 \]
Now substituting into the quadratic formula:
\[ x = \frac{10.5 \pm \sqrt{380.25}}{2} \]
Calculating \( \sqrt{380.25} \approx 19.5 \):
\[ x = \frac{10.5 \pm 19.5}{2} \]
This gives us two possible solutions:
1. \( x = \frac{30}{2} = 15 \)
2. \( x = \frac{-9}{2} \) (not valid since time cannot be negative)

Thus, \( x = 15 \) days (time taken by A).

### Step 8: Find Time Taken by B
Now, time taken by B alone is:
\[ x + 7.5 = 15 + 7.5 = 22.5 \text{ days} \]

### Step 9: Calculate Time for B to Complete \( \frac{2}{9} \) of the Work
The work done by B in one day is \( \frac{1}{22.5} \). To find how many days B will take to complete \( \frac{2}{9} \) of the work:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{\frac{2}{9}}{\frac{1}{22.5}} = \frac{2}{9} \times 22.5 = \frac{2 \times 22.5}{9} = 5 \text{ days} \]

### Final Answer
B alone will complete \( \frac{2}{9} \) of the work in **5 days**.

---