A takes 27 days more than (A + B) together to complete a work. B takes 3 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, let's denote the time taken by (A + B) together to complete the work as \( x \) days.

### Step 1: Define the time taken by A and B
According to the problem:
- A takes 27 days more than (A + B) together, so:
\[ \text{Time taken by A} = x + 27 \]
- B takes 3 days more than (A + B) together, so:
\[ \text{Time taken by B} = x + 3 \]

### Step 2: Use the formula for work
The work done by A in one day is:
\[ \frac{1}{x + 27} \]
The work done by B in one day is:
\[ \frac{1}{x + 3} \]
The work done by (A + B) in one day is:
\[ \frac{1}{x} \]

### Step 3: Set up the equation
Since the work done by A and B together is equal to the sum of the work done by A and the work done by B, we can write:
\[ \frac{1}{x} = \frac{1}{x + 27} + \frac{1}{x + 3} \]

### Step 4: Find a common denominator and solve the equation
The common denominator of the right-hand side is \((x + 27)(x + 3)\):
\[ \frac{1}{x} = \frac{(x + 3) + (x + 27)}{(x + 27)(x + 3)} \]
This simplifies to:
\[ \frac{1}{x} = \frac{2x + 30}{(x + 27)(x + 3)} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ (x + 27)(x + 3) = x(2x + 30) \]

### Step 6: Expand both sides
Expanding the left side:
\[ x^2 + 30x + 81 = 2x^2 + 30x \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ x^2 + 81 = 0 \]

### Step 8: Solve for x
This implies:
\[ x^2 = 81 \]
Taking the square root of both sides gives:
\[ x = 9 \]

### Conclusion
Thus, (A + B) will complete the work in **9 days**.

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