A takes `4(1)/(2)` days more than (A + B) together to complete a work. B takes 8 days more than (A + B) together to complete a work. In how many days will (A + B) complete this work ?

Answer

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

### Step 1: Define the time taken by A and B
Let:
- A takes \( X + 4.5 \) days to complete the work alone.
- B takes \( X + 8 \) days to complete the work alone.

### Step 2: Calculate the work done by A and B
The work done by A in one day is \( \frac{1}{X + 4.5} \) and the work done by B in one day is \( \frac{1}{X + 8} \).

The work done by A and B together in one day is:
\[ \frac{1}{X + 4.5} + \frac{1}{X + 8} = \frac{(X + 8) + (X + 4.5)}{(X + 4.5)(X + 8)} \]
This simplifies to:
\[ \frac{2X + 12.5}{(X + 4.5)(X + 8)} \]

### Step 3: Set up the equation
Since A and B together can complete the work in \( X \) days, the total work can also be expressed as:
\[ \frac{1}{X} \text{ (work done by A and B together in one day)} \]

Thus, we can set up the equation:
\[ \frac{2X + 12.5}{(X + 4.5)(X + 8)} = \frac{1}{X} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ X(2X + 12.5) = (X + 4.5)(X + 8) \]

### Step 5: Expand both sides
Expanding both sides:
- Left side: \( 2X^2 + 12.5X \)
- Right side: \( X^2 + 8X + 4.5X + 36 = X^2 + 12.5X + 36 \)

### Step 6: Rearrange the equation
Now, rearranging the equation gives:
\[ 2X^2 + 12.5X = X^2 + 12.5X + 36 \]
Subtract \( X^2 + 12.5X \) from both sides:
\[ X^2 = 36 \]

### Step 7: Solve for X
Taking the square root of both sides gives:
\[ X = 6 \]

### Conclusion
Thus, A and B together will complete the work in **6 days**.

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