A takes 9 days more than B to do a certain piece of work. Together they can do the work in 6 days. How many days will A alone take to do the work?

Answer

To solve the problem, we need to find out how many days A alone will take to complete the work. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( y \) = number of days B takes to complete the work.
- \( x \) = number of days A takes to complete the work.

From the problem, we know that:
\[ x = y + 9 \]
(This means A takes 9 days more than B.)

### Step 2: Set Up the Equation for Combined Work
When A and B work together, they can complete the work in 6 days. Therefore, their combined work rate can be expressed as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{6} \]

### Step 3: Substitute for x
Substituting \( x \) from Step 1 into the equation:
\[ \frac{1}{y + 9} + \frac{1}{y} = \frac{1}{6} \]

### Step 4: Find a Common Denominator
To combine the fractions on the left-hand side, we need a common denominator:
\[ \frac{y + y + 9}{y(y + 9)} = \frac{1}{6} \]
This simplifies to:
\[ \frac{2y + 9}{y(y + 9)} = \frac{1}{6} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ 6(2y + 9) = y(y + 9) \]

### Step 6: Expand and Rearrange
Expanding both sides:
\[ 12y + 54 = y^2 + 9y \]

Rearranging the equation:
\[ y^2 + 9y - 12y - 54 = 0 \]
This simplifies to:
\[ y^2 - 3y - 54 = 0 \]

### Step 7: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ (y - 9)(y + 6) = 0 \]

### Step 8: Solve for y
Setting each factor to zero gives:
1. \( y - 9 = 0 \) → \( y = 9 \)
2. \( y + 6 = 0 \) → \( y = -6 \) (not valid since days cannot be negative)

Thus, \( y = 9 \).

### Step 9: Find x
Now, substituting \( y \) back into the equation for \( x \):
\[ x = y + 9 = 9 + 9 = 18 \]

### Conclusion
A alone will take **18 days** to complete the work.

---