When A alone does a piece of work , he takes 16 days more than the time taken by (A+B) to complete the work, while B alone takes 9 days more than the time taken by (A + B) to finish the work. What time,AandBtogether will take to finish this work?

Answer

To solve the problem step by step, we will denote the time taken by A and B together to complete the work as \( x \) days.

### Step 1: Define the time taken by A and B together
Let \( x \) be the time taken by A and B together to complete the work.

### Step 2: Express the time taken by A and B alone
According to the problem:
- A takes \( x + 16 \) days to complete the work alone.
- B takes \( x + 9 \) days to complete the work alone.

### Step 3: Set up the work rates
The work done by A in one day is \( \frac{1}{x + 16} \) and the work done by B in one day is \( \frac{1}{x + 9} \). The work done by A and B together in one day is \( \frac{1}{x} \).

### Step 4: Write the equation for combined work
Since A and B together can complete the work in \( x \) days, we can write the equation:
\[ \frac{1}{x + 16} + \frac{1}{x + 9} = \frac{1}{x} \]

### Step 5: Find a common denominator and simplify
To solve the equation, we will find a common denominator:
\[ \frac{(x)(x + 9) + (x)(x + 16)}{(x + 16)(x + 9)} = \frac{(x + 16)(x + 9)}{x(x + 16)(x + 9)} \]
This simplifies to:
\[ x(x + 9) + x(x + 16) = (x + 16)(x + 9) \]
Expanding both sides:
\[ x^2 + 9x + x^2 + 16x = x^2 + 25x + 144 \]
Combining like terms gives:
\[ 2x^2 + 25x = x^2 + 25x + 144 \]
Subtracting \( x^2 + 25x \) from both sides results in:
\[ x^2 = 144 \]

### Step 6: Solve for x
Taking the square root of both sides gives:
\[ x = 12 \]

### Conclusion
Thus, A and B together will take **12 days** to finish the work.

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